Subgroups — C₂₃×#25557 #19173183

Order 5888.

1086 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#552	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2415	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2036	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2026	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2001	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1878	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1873	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1873	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1870	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1736	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#552	—	2	4	no	—
19	C₂₃×#552	—	2	4	no	—
20	C₂×C₂₃×#126	—	2	1	no	—
21	C₂×C₂₃×#125	—	2	1	no	—
22	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#123	—	2	1	no	—
24	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#117	—	2	1	no	—
31	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#484	—	2	4	no	—
35	C₂₃×#470	—	2	4	no	—
36	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
41	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	C₂₃×#397	—	2	1	no	—
49	C₂₃×#394	—	2	1	no	—
50	C₂₃×#391	C₂²	1	1	yes	characteristic
51	C₂₃×#389	—	2	1	no	—
52	C₂₃×#389	C₂²	1	1	yes	characteristic
53	C₂₃×#388	—	2	1	no	—
54	C₂₃×#380	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#340	C₂²	1	1	yes	characteristic
60	C₂₃×#340	C₂²	1	1	yes	characteristic
61	C₂₃×#336	C₂²	1	1	yes	characteristic
62	C₂₃×#336	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₂₃	—	2	2	no	—
70	C₂×Q₁₆×C₂₃	—	2	2	no	—
71	C₂×Q₁₆×C₂₃	—	4	1	no	—
72	C₂×M₁₆×C₂₃	—	2	2	no	—
73	C₂×M₁₆×C₂₃	—	2	2	no	—
74	C₂×M₁₆×C₂₃	—	4	1	no	—
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×#126	—	2	8	no	—
80	C₂₃×#125	—	2	8	no	—
81	C₂₃×#125	—	2	2	no	—
82	C₂₃×#125	—	2	2	no	—
83	C₂₃×#123	—	2	8	no	—
84	C₂₃×C₂²⋊Q₈	—	2	2	no	—
85	C₂₃×C₂²⋊Q₈	—	2	2	no	—
86	C₂₃×C₂²⋊Q₈	—	2	2	no	—
87	C₂₃×C₂²⋊Q₈	—	2	2	no	—
88	C₂₃×C₂²⋊Q₈	—	2	2	no	—
89	C₂₃×C₂²⋊Q₈	—	2	2	no	—
90	C₂₃×C₂²⋊Q₈	—	4	1	no	—
91	C₂₃×C₂²⋊Q₈	D₄	1	4	yes	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×D₄×C₂₃	—	2	2	no	—
95	C₄×D₄×C₂₃	—	2	2	no	—
96	C₂₃×#117	—	2	8	no	—
97	C₂₃×#117	—	2	2	no	—
98	C₂₃×#117	—	2	2	no	—
99	C₂₃×#117	—	2	2	no	—
100	C₂₃×#117	—	4	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
127	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	—	2	1	no	—
129	C₂×C₄²×C₂₃	—	2	1	no	—
130	C₂×C₄²×C₂₃	—	2	1	no	—
131	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
132	C₂₃×#107	—	2	2	no	—
133	C₂₃×#107	—	2	2	no	—
134	C₂₃×#106	—	2	2	no	—
135	C₂₃×#103	—	2	2	no	—
136	C₂₃×#103	—	2	2	no	—
137	C₂₃×#103	—	2	2	no	—
138	C₂₃×#103	—	2	2	no	—
139	C₂₃×#103	—	2	2	no	—
140	C₂₃×#103	—	4	1	no	—
141	C₂₃×C₂²⋊C₈	—	2	2	no	—
142	C₂₃×C₂²⋊C₈	—	2	2	no	—
143	C₂₃×C₂²⋊C₈	—	4	1	no	—
144	C₂₃×#95	—	2	1	no	—
145	C₂₃×#95	—	2	1	no	—
146	C₂₃×#95	—	2	1	no	—
147	C₂₃×#95	—	2	1	no	—
148	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	2	1	no	—
153	C₂×Q₈×C₂₃	—	2	1	no	—
154	C₂×Q₈×C₂₃	—	2	1	no	—
155	C₂×Q₈×C₂₃	—	2	1	no	—
156	C₂×Q₈×C₂₃	—	2	1	no	—
157	C₂×Q₈×C₂₃	—	2	1	no	—
158	C₂×Q₈×C₂₃	—	2	1	no	—
159	C₂×Q₈×C₂₃	—	2	1	no	—
160	C₂×Q₈×C₂₃	—	2	1	no	—
161	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
162	C₂×D₄×C₂₃	—	2	2	no	—
163	C₂×D₄×C₂₃	—	4	1	no	—
164	C₂×D₄×C₂₃	—	4	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	4	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	2	1	no	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	2	1	no	—
179	C₂²×C₄×C₂₃	—	2	1	no	—
180	C₂²×C₄×C₂₃	—	2	1	no	—
181	C₂²×C₄×C₂₃	—	2	1	no	—
182	C₂²×C₄×C₂₃	—	2	1	no	—
183	C₂²×C₄×C₂₃	—	2	1	no	—
184	C₂²×C₄×C₂₃	—	4	2	no	—
185	C₂²×C₄×C₂₃	—	4	2	no	—
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	Q₁₆×C₂₃	—	4	4	no	—
194	M₁₆×C₂₃	—	4	4	no	—
195	C₂×C₈×C₂₃	—	2	1	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	2	1	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	—	2	1	no	—
206	C₂×C₈×C₂₃	#41	1	2	yes	—
207	C₄⋊C₄×C₂₃	—	2	4	no	—
208	C₄⋊C₄×C₂₃	—	2	4	no	—
209	C₄⋊C₄×C₂₃	—	2	4	no	—
210	C₄⋊C₄×C₂₃	—	2	4	no	—
211	C₄⋊C₄×C₂₃	—	2	4	no	—
212	C₄⋊C₄×C₂₃	—	2	4	no	—
213	C₄⋊C₄×C₂₃	—	2	2	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	2	1	no	—
219	C₄⋊C₄×C₂₃	—	2	1	no	—
220	C₄⋊C₄×C₂₃	—	2	1	no	—
221	C₄⋊C₄×C₂₃	—	2	1	no	—
222	C₄⋊C₄×C₂₃	—	4	2	no	—
223	C₄⋊C₄×C₂₃	—	4	2	no	—
224	C₄⋊C₄×C₂₃	—	4	2	no	—
225	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
226	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
227	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
228	C₄⋊C₄×C₂₃	#41	1	2	yes	—
229	C₂²⋊C₄×C₂₃	—	2	4	no	—
230	C₂²⋊C₄×C₂₃	—	2	4	no	—
231	C₂²⋊C₄×C₂₃	—	2	4	no	—
232	C₂²⋊C₄×C₂₃	—	2	2	no	—
233	C₂²⋊C₄×C₂₃	—	2	2	no	—
234	C₂²⋊C₄×C₂₃	—	4	2	no	—
235	C₂²⋊C₄×C₂₃	—	4	2	no	—
236	C₂²⋊C₄×C₂₃	—	4	2	no	—
237	C₂²⋊C₄×C₂₃	—	4	2	no	—
238	C₄²×C₂₃	—	2	4	no	—
239	C₄²×C₂₃	—	2	4	no	—
240	C₄²×C₂₃	—	2	1	no	—
241	C₄²×C₂₃	—	2	1	no	—
242	C₄²×C₂₃	—	4	2	no	—
243	#25557	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂³×C₂₃	—	2	1	no	—
245	C₂³×C₂₃	—	2	1	no	—
246	C₂³×C₂₃	—	2	1	no	—
247	C₂³×C₂₃	—	2	1	no	—
248	C₂³×C₂₃	—	2	1	no	—
249	C₂³×C₂₃	—	2	1	no	—
250	C₂³×C₂₃	—	2	1	no	—
251	C₂³×C₂₃	—	2	1	no	—
252	C₂³×C₂₃	—	2	1	no	—
253	C₂³×C₂₃	—	2	1	no	—
254	C₂³×C₂₃	—	2	1	no	—
255	C₂³×C₂₃	—	2	1	no	—
256	C₂³×C₂₃	—	2	1	no	—
257	C₂³×C₂₃	#120	1	2	yes	—
258	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₂₃	—	2	2	no	—
260	Q₈×C₂₃	—	2	2	no	—
261	Q₈×C₂₃	—	4	1	no	—
262	Q₈×C₂₃	—	4	1	no	—
263	Q₈×C₂₃	—	4	1	no	—
264	Q₈×C₂₃	—	4	1	no	—
265	Q₈×C₂₃	—	4	1	no	—
266	Q₈×C₂₃	—	4	1	no	—
267	D₄×C₂₃	—	4	4	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	1	no	—
270	C₂×C₄×C₂₃	—	2	2	no	—
271	C₂×C₄×C₂₃	—	2	2	no	—
272	C₂×C₄×C₂₃	—	2	2	no	—
273	C₂×C₄×C₂₃	—	2	2	no	—
274	C₂×C₄×C₂₃	—	2	1	no	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	2	1	no	—
280	C₂×C₄×C₂₃	—	2	2	no	—
281	C₂×C₄×C₂₃	—	2	2	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	1	no	—
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	1	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	4	8	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	—	4	1	no	—
302	C₂×C₄×C₂₃	—	4	1	no	—
303	C₂×C₄×C₂₃	—	4	1	no	—
304	C₂×C₄×C₂₃	—	4	1	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	4	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	4	1	no	—
312	C₂×C₄×C₂₃	#120	1	2	yes	—
313	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
314	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
315	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
316	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
317	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
318	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
319	C₈×C₂₃	—	2	2	no	—
320	C₈×C₂₃	—	4	1	no	—
321	C₈×C₂₃	—	4	1	no	—
322	C₈×C₂₃	—	4	1	no	—
323	C₂×#552	C₂×C₂₃	1	1	yes	characteristic
324	C₂×#484	C₂×C₂₃	1	1	yes	characteristic, complemented
325	C₂×#470	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#2415	C₂×C₂₃	1	1	yes	characteristic
327	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#2036	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#2026	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#2001	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#1878	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#1873	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1873	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1870	C₂×C₂₃	1	1	yes	characteristic
335	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1854	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1736	C₂×C₂₃	1	1	yes	characteristic, complemented
338	C₂²×C₂₃	—	2	2	no	—
339	C₂²×C₂₃	—	2	2	no	—
340	C₂²×C₂₃	—	2	2	no	—
341	C₂²×C₂₃	—	4	1	no	—
342	C₂²×C₂₃	—	4	1	no	—
343	C₂²×C₂₃	—	4	1	no	—
344	C₂²×C₂₃	—	4	1	no	—
345	C₂²×C₂₃	—	4	1	no	—
346	C₂²×C₂₃	—	4	1	no	—
347	C₂²×C₂₃	—	4	1	no	—
348	C₂²×C₂₃	—	4	1	no	—
349	C₂²×C₂₃	—	4	1	no	—
350	C₂²×C₂₃	—	4	1	no	—
351	C₂²×C₂₃	—	4	1	no	—
352	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
353	C₂²×C₂₃	#546	1	1	yes	characteristic
354	C₂²×C₂₃	#547	1	1	yes	characteristic
355	C₂²×C₂₃	#575	1	1	yes	characteristic
356	C₂²×C₂₃	#575	1	1	yes	characteristic
357	C₂²×C₂₃	#578	1	1	yes	characteristic
358	C₂²×C₂₃	#578	1	1	yes	characteristic
359	C₄×C₂₃	—	2	2	no	—
360	C₄×C₂₃	—	4	1	no	—
361	C₄×C₂₃	—	4	2	no	—
362	C₄×C₂₃	—	4	1	no	—
363	C₄×C₂₃	—	4	2	no	—
364	C₄×C₂₃	—	4	2	no	—
365	C₄×C₂₃	—	4	1	no	—
366	C₄×C₂₃	—	4	2	no	—
367	C₄×C₂₃	—	4	2	no	—
368	C₄×C₂₃	—	8	1	no	—
369	C₄×C₂₃	—	8	1	no	—
370	C₄×C₂₃	—	8	1	no	—
371	C₄×C₂₃	#532	1	1	yes	characteristic
372	C₄×C₂₃	#537	1	1	yes	characteristic
373	C₄×C₂₃	#546	1	1	yes	characteristic
374	C₄×C₂₃	#547	1	1	yes	characteristic
375	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
376	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
377	#552	—	2	4	no	—
378	#552	—	2	4	no	—
379	C₂×#126	—	2	1	no	—
380	C₂×#125	—	2	1	no	—
381	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×#123	—	2	1	no	—
383	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
389	C₂×#117	—	2	1	no	—
390	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
392	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
393	#484	—	2	4	no	—
394	#470	—	2	4	no	—
395	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
397	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
399	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
400	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
401	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
402	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
404	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
405	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
407	#397	—	2	1	no	—
408	#394	—	2	1	no	—
409	#391	C₂²×C₂₃	1	1	yes	characteristic
410	#389	—	2	1	no	—
411	#389	C₂²×C₂₃	1	1	yes	characteristic
412	#388	—	2	1	no	—
413	#380	C₂²×C₂₃	1	1	yes	characteristic
414	#340	C₂²×C₂₃	1	1	yes	characteristic
415	#340	C₂²×C₂₃	1	1	yes	characteristic
416	#340	C₂²×C₂₃	1	1	yes	characteristic
417	#340	C₂²×C₂₃	1	1	yes	characteristic
418	#340	C₂²×C₂₃	1	1	yes	characteristic
419	#340	C₂²×C₂₃	1	1	yes	characteristic
420	#336	C₂²×C₂₃	1	1	yes	characteristic
421	#336	C₂²×C₂₃	1	1	yes	characteristic
422	C₂×C₂₃	—	4	2	no	—
423	C₂×C₂₃	—	8	1	no	—
424	C₂×C₂₃	#2430	1	1	yes	characteristic
425	C₂×C₂₃	#3005	1	1	yes	characteristic
426	C₂×C₂₃	#3011	1	1	yes	characteristic
427	C₂×C₂₃	#3315	1	1	yes	characteristic
428	C₂×C₂₃	#3327	1	1	yes	characteristic
429	C₂×C₂₃	#3349	1	1	yes	characteristic
430	C₂×C₂₃	#3352	1	1	yes	characteristic
431	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
434	C₂³×C₄	—	2	1	no	—
435	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
436	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
437	C₂×Q₁₆	—	2	2	no	—
438	C₂×Q₁₆	—	2	2	no	—
439	C₂×Q₁₆	—	4	1	no	—
440	C₂×M₁₆	—	2	2	no	—
441	C₂×M₁₆	—	2	2	no	—
442	C₂×M₁₆	—	4	1	no	—
443	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
444	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
446	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
447	#126	—	2	8	no	—
448	#125	—	2	8	no	—
449	#125	—	2	2	no	—
450	#125	—	2	2	no	—
451	#123	—	2	8	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	2	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	4	1	no	—
459	C₂²⋊Q₈	D₄×C₂₃	1	4	yes	—
460	C₄×Q₈	—	2	2	no	—
461	C₄×Q₈	—	2	2	no	—
462	C₄×D₄	—	2	2	no	—
463	C₄×D₄	—	2	2	no	—
464	#117	—	2	8	no	—
465	#117	—	2	2	no	—
466	#117	—	2	2	no	—
467	#117	—	2	2	no	—
468	#117	—	4	1	no	—
469	C₂×C₄⋊C₄	—	2	1	no	—
470	C₂×C₄⋊C₄	—	2	2	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄⋊C₄	—	2	1	no	—
473	C₂×C₄⋊C₄	—	2	1	no	—
474	C₂×C₄⋊C₄	—	2	1	no	—
475	C₂×C₄⋊C₄	—	2	1	no	—
476	C₂×C₄⋊C₄	—	2	1	no	—
477	C₂×C₄⋊C₄	—	2	1	no	—
478	C₂×C₄⋊C₄	—	2	1	no	—
479	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
481	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
482	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
483	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
484	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
485	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
486	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
487	C₂×C₂²⋊C₄	—	2	1	no	—
488	C₂×C₂²⋊C₄	—	2	1	no	—
489	C₂×C₂²⋊C₄	—	2	1	no	—
490	C₂×C₂²⋊C₄	—	2	1	no	—
491	C₂×C₂²⋊C₄	—	2	1	no	—
492	C₂×C₂²⋊C₄	—	2	1	no	—
493	C₂×C₂²⋊C₄	—	2	1	no	—
494	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
495	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
496	C₂×C₄²	—	2	1	no	—
497	C₂×C₄²	—	2	1	no	—
498	C₂×C₄²	—	2	1	no	—
499	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
500	#107	—	2	2	no	—
501	#107	—	2	2	no	—
502	#106	—	2	2	no	—
503	#103	—	2	2	no	—
504	#103	—	2	2	no	—
505	#103	—	2	2	no	—
506	#103	—	2	2	no	—
507	#103	—	2	2	no	—
508	#103	—	4	1	no	—
509	C₂²⋊C₈	—	2	2	no	—
510	C₂²⋊C₈	—	2	2	no	—
511	C₂²⋊C₈	—	4	1	no	—
512	#95	—	2	1	no	—
513	#95	—	2	1	no	—
514	#95	—	2	1	no	—
515	#95	—	2	1	no	—
516	C₂₃	#25557	1	1	yes	characteristic, direct product factor, Sylow
517	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
518	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
519	C₂×Q₈	—	2	1	no	—
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	1	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
531	C₂×D₄	—	2	2	no	—
532	C₂×D₄	—	4	1	no	—
533	C₂×D₄	—	4	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	1	no	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	1	no	—
545	C₂²×C₄	—	2	1	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	1	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	4	2	no	—
554	C₂²×C₄	—	4	2	no	—
555	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
559	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
560	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
561	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
562	Q₁₆	—	4	4	no	—
563	M₁₆	—	4	4	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	2	1	no	—
570	C₂×C₈	—	2	1	no	—
571	C₂×C₈	—	2	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	#41×C₂₃	1	2	yes	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	4	no	—
578	C₄⋊C₄	—	2	4	no	—
579	C₄⋊C₄	—	2	4	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	2	2	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	1	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	4	2	no	—
592	C₄⋊C₄	—	4	2	no	—
593	C₄⋊C₄	—	4	2	no	—
594	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
595	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
596	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
597	C₄⋊C₄	#41×C₂₃	1	2	yes	—
598	C₂²⋊C₄	—	2	4	no	—
599	C₂²⋊C₄	—	2	4	no	—
600	C₂²⋊C₄	—	2	4	no	—
601	C₂²⋊C₄	—	2	2	no	—
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	4	2	no	—
604	C₂²⋊C₄	—	4	2	no	—
605	C₂²⋊C₄	—	4	2	no	—
606	C₂²⋊C₄	—	4	2	no	—
607	C₄²	—	2	4	no	—
608	C₄²	—	2	4	no	—
609	C₄²	—	2	1	no	—
610	C₄²	—	2	1	no	—
611	C₄²	—	4	2	no	—
612	C₂³	—	2	1	no	—
613	C₂³	—	2	1	no	—
614	C₂³	—	2	1	no	—
615	C₂³	—	2	1	no	—
616	C₂³	—	2	1	no	—
617	C₂³	—	2	1	no	—
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	—	2	1	no	—
623	C₂³	—	2	1	no	—
624	C₂³	—	2	1	no	—
625	C₂³	C₂₃×#120	1	2	yes	—
626	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
627	Q₈	—	2	2	no	—
628	Q₈	—	2	2	no	—
629	Q₈	—	4	1	no	—
630	Q₈	—	4	1	no	—
631	Q₈	—	4	1	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	4	1	no	—
635	D₄	—	4	4	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	4	8	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	4	1	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	C₂₃×#120	1	2	yes	—
681	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
682	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
683	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
684	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
685	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
686	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
687	C₈	—	2	2	no	—
688	C₈	—	4	1	no	—
689	C₈	—	4	1	no	—
690	C₈	—	4	1	no	—
691	C₂²	—	2	2	no	—
692	C₂²	—	2	2	no	—
693	C₂²	—	2	2	no	—
694	C₂²	—	4	1	no	—
695	C₂²	—	4	1	no	—
696	C₂²	—	4	1	no	—
697	C₂²	—	4	1	no	—
698	C₂²	—	4	1	no	—
699	C₂²	—	4	1	no	—
700	C₂²	—	4	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	—	4	1	no	—
704	C₂²	—	4	1	no	—
705	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
706	C₂²	C₂₃×#546	1	1	yes	characteristic
707	C₂²	C₂₃×#547	1	1	yes	characteristic
708	C₂²	C₂₃×#575	1	1	yes	characteristic
709	C₂²	C₂₃×#575	1	1	yes	characteristic
710	C₂²	C₂₃×#578	1	1	yes	characteristic
711	C₂²	C₂₃×#578	1	1	yes	characteristic
712	C₄	—	2	2	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	2	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	2	no	—
720	C₄	—	4	2	no	—
721	C₄	—	8	1	no	—
722	C₄	—	8	1	no	—
723	C₄	—	8	1	no	—
724	C₄	C₂₃×#532	1	1	yes	characteristic
725	C₄	C₂₃×#537	1	1	yes	characteristic
726	C₄	C₂₃×#546	1	1	yes	characteristic
727	C₄	C₂₃×#547	1	1	yes	characteristic
728	C₂	—	4	2	no	—
729	C₂	—	8	1	no	—
730	C₂	C₂₃×#2430	1	1	yes	characteristic
731	C₂	C₂₃×#3005	1	1	yes	characteristic
732	C₂	C₂₃×#3011	1	1	yes	characteristic
733	C₂	C₂₃×#3315	1	1	yes	characteristic
734	C₂	C₂₃×#3327	1	1	yes	characteristic
735	C₂	C₂₃×#3349	1	1	yes	characteristic
736	C₂	C₂₃×#3352	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₂₃×#484	C₂	semidirect
3. C₂×C₂₃×#470	C₂	semidirect
5. C₂₃×#2059	C₂	semidirect
6. C₂₃×#2036	C₂	semidirect
7. C₂₃×#2026	C₂	semidirect
8. C₂₃×#2001	C₂	semidirect
9. C₂₃×#1878	C₂	semidirect
10. C₂₃×#1873	C₂	semidirect
11. C₂₃×#1873	C₂	semidirect
13. C₂₃×#1863	C₂	semidirect
14. C₂₃×#1854	C₂	semidirect
15. C₂₃×#1736	C₂	semidirect
243. #25557	C₂₃	direct (N × H)
324. C₂×#484	C₂×C₂₃	semidirect
325. C₂×#470	C₂×C₂₃	semidirect
327. #2059	C₂×C₂₃	semidirect
328. #2036	C₂×C₂₃	semidirect
329. #2026	C₂×C₂₃	semidirect
330. #2001	C₂×C₂₃	semidirect
331. #1878	C₂×C₂₃	semidirect
332. #1873	C₂×C₂₃	semidirect
333. #1873	C₂×C₂₃	semidirect
335. #1863	C₂×C₂₃	semidirect
336. #1854	C₂×C₂₃	semidirect
337. #1736	C₂×C₂₃	semidirect
516. C₂₃	#25557	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#552	2
C₂×C₂₃×#484	2
C₂×C₂₃×#470	2
C₂₃×#2415	2
C₂₃×#2059	2
C₂₃×#2036	2
C₂₃×#2026	2
C₂₃×#2001	2
C₂₃×#1878	2
C₂₃×#1873	2
C₂₃×#1870	2
C₂₃×#1863	2
C₂₃×#1854	2
C₂₃×#1736	2
#25557	23

Subgroups — C23×#25557 #19173183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25557 #19173183