Subgroups — C₂₃×#29158 #19176784

Order 5888.

784 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2851	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2097	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2026	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1918	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1892	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1892	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1808	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1807	C₂	1	1	yes	maximal, characteristic
15	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
30	C₂₃×#391	C₂²	1	1	yes	characteristic
31	C₂₃×#389	C₂²	1	1	yes	characteristic
32	C₂₃×#389	C₂²	1	1	yes	characteristic
33	C₂₃×#389	C₂²	1	1	yes	characteristic
34	C₂₃×#387	—	2	1	no	—
35	C₂₃×#387	—	2	1	no	—
36	C₂₃×#387	—	2	1	no	—
37	C₂₃×#385	—	2	1	no	—
38	C₂₃×#385	—	2	1	no	—
39	C₂₃×#385	—	2	1	no	—
40	C₂₃×#382	C₂²	1	1	yes	characteristic
41	C₂₃×#380	C₂²	1	1	yes	characteristic
42	C₂₃×#380	C₂²	1	1	yes	characteristic
43	C₂₃×#380	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
47	C₂₃×#340	C₂²	1	1	yes	characteristic
48	C₂₃×#340	C₂²	1	1	yes	characteristic
49	C₂₃×#340	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#336	C₂²	1	1	yes	characteristic
56	C₂₃×#336	C₂²	1	1	yes	characteristic
57	C₂₃×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	—	2	2	no	—
64	C₂²×C₈×C₂₃	—	2	2	no	—
65	C₂²×C₈×C₂₃	—	2	2	no	—
66	C₂²×C₈×C₂₃	—	2	2	no	—
67	C₂²×C₈×C₂₃	—	4	1	no	—
68	C₂²×C₈×C₂₃	—	4	1	no	—
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂₃×#125	—	4	1	no	—
74	C₂₃×#125	—	4	1	no	—
75	C₂₃×C₂²⋊Q₈	—	2	2	no	—
76	C₂₃×C₂²⋊Q₈	Q₈	1	4	yes	—
77	C₄×Q₈×C₂₃	—	4	1	no	—
78	C₄×Q₈×C₂₃	—	4	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	—	2	1	no	—
98	C₂×C₄²×C₂₃	—	2	1	no	—
99	C₂×C₄²×C₂₃	—	4	1	no	—
100	C₂×C₄²×C₂₃	—	4	1	no	—
101	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×#107	—	2	2	no	—
106	C₂₃×#107	—	4	1	no	—
107	C₂₃×#106	—	2	2	no	—
108	C₂₃×#106	—	4	1	no	—
109	C₂₃×#103	—	2	2	no	—
110	C₂₃×#103	—	2	2	no	—
111	C₂₃×#103	—	4	1	no	—
112	C₂₃×#103	—	4	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	—	2	1	no	—
117	C₂₃×#95	—	2	1	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	1	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Q₈×C₂₃	—	2	1	no	—
126	C₂×Q₈×C₂₃	—	2	1	no	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	—	2	1	no	—
129	C₂×Q₈×C₂₃	—	2	1	no	—
130	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	4	1	no	—
147	C₂²×C₄×C₂₃	—	4	1	no	—
148	C₂²×C₄×C₂₃	—	4	1	no	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
155	C₂²×C₄×C₂₃	#41	1	2	yes	—
156	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	4	4	no	—
168	C₂×C₈×C₂₃	—	4	4	no	—
169	C₂×C₈×C₂₃	#41	1	2	yes	—
170	C₂×C₈×C₂₃	#41	1	2	yes	—
171	C₄⋊C₄×C₂₃	—	2	4	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	4	2	no	—
186	C₄⋊C₄×C₂₃	—	4	2	no	—
187	C₄⋊C₄×C₂₃	—	4	2	no	—
188	C₄⋊C₄×C₂₃	—	4	2	no	—
189	C₄⋊C₄×C₂₃	—	4	2	no	—
190	C₄⋊C₄×C₂₃	—	4	2	no	—
191	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
192	C₂²⋊C₄×C₂₃	—	2	4	no	—
193	C₂²⋊C₄×C₂₃	—	4	2	no	—
194	C₂²⋊C₄×C₂₃	—	4	2	no	—
195	C₂²⋊C₄×C₂₃	—	4	2	no	—
196	C₄²×C₂₃	—	2	1	no	—
197	C₄²×C₂₃	—	2	1	no	—
198	C₄²×C₂₃	—	2	1	no	—
199	C₄²×C₂₃	—	2	1	no	—
200	C₄²×C₂₃	—	4	1	no	—
201	C₄²×C₂₃	—	4	1	no	—
202	C₄²×C₂₃	—	4	2	no	—
203	C₄²×C₂₃	—	4	2	no	—
204	#29158	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₂₃	—	2	1	no	—
206	C₂³×C₂₃	—	2	1	no	—
207	C₂³×C₂₃	—	2	1	no	—
208	C₂³×C₂₃	—	2	1	no	—
209	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
210	C₂³×C₂₃	#135	1	2	yes	—
211	C₂³×C₂₃	#137	1	2	yes	—
212	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₂₃	—	2	2	no	—
214	Q₈×C₂₃	—	2	2	no	—
215	Q₈×C₂₃	—	4	1	no	—
216	Q₈×C₂₃	—	4	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	4	4	no	—
237	C₂×C₄×C₂₃	—	4	4	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
261	C₂×C₄×C₂₃	#135	1	2	yes	—
262	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
267	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
268	C₈×C₂₃	—	2	2	no	—
269	C₈×C₂₃	—	2	2	no	—
270	C₈×C₂₃	—	4	1	no	—
271	C₈×C₂₃	—	4	1	no	—
272	#2851	C₂×C₂₃	1	1	yes	characteristic
273	C₄×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
274	#2101	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#2097	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#2026	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1918	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1892	C₂×C₂₃	1	1	yes	characteristic
281	#1892	C₂×C₂₃	1	1	yes	characteristic
282	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1808	C₂×C₂₃	1	1	yes	characteristic
285	#1807	C₂×C₂₃	1	1	yes	characteristic
286	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
287	C₂²×C₂₃	—	2	2	no	—
288	C₂²×C₂₃	—	2	2	no	—
289	C₂²×C₂₃	—	2	2	no	—
290	C₂²×C₂₃	—	2	2	no	—
291	C₂²×C₂₃	—	2	2	no	—
292	C₂²×C₂₃	—	2	2	no	—
293	C₂²×C₂₃	#479	1	4	yes	—
294	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
295	C₂²×C₂₃	#532	1	1	yes	characteristic
296	C₂²×C₂₃	#534	1	1	yes	characteristic
297	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
298	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
299	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
300	C₂²×C₂₃	#575	1	1	yes	characteristic
301	C₄×C₂₃	—	2	2	no	—
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	4	2	no	—
304	C₄×C₂₃	—	4	2	no	—
305	C₄×C₂₃	—	4	1	no	—
306	C₄×C₂₃	—	4	1	no	—
307	C₄×C₂₃	—	4	1	no	—
308	C₄×C₂₃	—	8	1	no	—
309	C₄×C₂₃	—	8	1	no	—
310	C₄×C₂₃	—	8	1	no	—
311	C₄×C₂₃	—	8	1	no	—
312	C₄×C₂₃	—	8	1	no	—
313	C₄×C₂₃	—	8	1	no	—
314	C₄×C₂₃	#551	1	1	yes	characteristic
315	C₄×C₂₃	#551	1	1	yes	characteristic
316	C₄×C₂₃	#554	1	1	yes	characteristic
317	C₄×C₂₃	#554	1	1	yes	characteristic
318	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
319	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
332	#391	C₂²×C₂₃	1	1	yes	characteristic
333	#389	C₂²×C₂₃	1	1	yes	characteristic
334	#389	C₂²×C₂₃	1	1	yes	characteristic
335	#389	C₂²×C₂₃	1	1	yes	characteristic
336	#387	—	2	1	no	—
337	#387	—	2	1	no	—
338	#387	—	2	1	no	—
339	#385	—	2	1	no	—
340	#385	—	2	1	no	—
341	#385	—	2	1	no	—
342	#382	C₂²×C₂₃	1	1	yes	characteristic
343	#380	C₂²×C₂₃	1	1	yes	characteristic
344	#380	C₂²×C₂₃	1	1	yes	characteristic
345	#380	C₂²×C₂₃	1	1	yes	characteristic
346	C₄×C₄⋊C₄	—	2	1	no	—
347	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
348	C₄×C₂²⋊C₄	—	2	1	no	—
349	#340	C₂²×C₂₃	1	1	yes	characteristic
350	#340	C₂²×C₂₃	1	1	yes	characteristic
351	#340	C₂²×C₂₃	1	1	yes	characteristic
352	#340	C₂²×C₂₃	1	1	yes	characteristic
353	#340	C₂²×C₂₃	1	1	yes	characteristic
354	#340	C₂²×C₂₃	1	1	yes	characteristic
355	#340	C₂²×C₂₃	1	1	yes	characteristic
356	#340	C₂²×C₂₃	1	1	yes	characteristic
357	#336	C₂²×C₂₃	1	1	yes	characteristic
358	#336	C₂²×C₂₃	1	1	yes	characteristic
359	#336	C₂²×C₂₃	1	1	yes	characteristic
360	#336	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₂₃	—	2	4	no	—
362	C₂×C₂₃	#2564	1	1	yes	characteristic
363	C₂×C₂₃	C₂×#479	1	1	yes	characteristic
364	C₂×C₂₃	#3076	1	1	yes	characteristic
365	C₂×C₂₃	#3168	1	1	yes	characteristic
366	C₂×C₂₃	#3181	1	1	yes	characteristic
367	C₂×C₂₃	#3190	1	1	yes	characteristic
368	C₂×C₂₃	#3191	1	1	yes	characteristic
369	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
370	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
371	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
372	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂²×C₈	—	2	2	no	—
374	C₂²×C₈	—	2	2	no	—
375	C₂²×C₈	—	2	2	no	—
376	C₂²×C₈	—	2	2	no	—
377	C₂²×C₈	—	4	1	no	—
378	C₂²×C₈	—	4	1	no	—
379	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
383	#125	—	4	1	no	—
384	#125	—	4	1	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	Q₈×C₂₃	1	4	yes	—
387	C₄×Q₈	—	4	1	no	—
388	C₄×Q₈	—	4	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	4	1	no	—
410	C₂×C₄²	—	4	1	no	—
411	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
412	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
415	#107	—	2	2	no	—
416	#107	—	4	1	no	—
417	#106	—	2	2	no	—
418	#106	—	4	1	no	—
419	#103	—	2	2	no	—
420	#103	—	2	2	no	—
421	#103	—	4	1	no	—
422	#103	—	4	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	#95	—	2	1	no	—
431	#95	—	2	1	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	C₂₃	#29158	1	1	yes	characteristic, direct product factor, Sylow
435	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	4	1	no	—
458	C₂²×C₄	—	4	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
465	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
466	C₂²×C₄	#41×C₂₃	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	4	4	no	—
479	C₂×C₈	—	4	4	no	—
480	C₂×C₈	#41×C₂₃	1	2	yes	—
481	C₂×C₈	#41×C₂₃	1	2	yes	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	4	2	no	—
505	C₂²⋊C₄	—	4	2	no	—
506	C₂²⋊C₄	—	4	2	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	4	1	no	—
512	C₄²	—	4	1	no	—
513	C₄²	—	4	2	no	—
514	C₄²	—	4	2	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
520	C₂³	C₂₃×#135	1	2	yes	—
521	C₂³	C₂₃×#137	1	2	yes	—
522	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
523	Q₈	—	2	2	no	—
524	Q₈	—	2	2	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	4	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
571	C₂×C₄	C₂₃×#135	1	2	yes	—
572	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
573	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
574	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
575	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
576	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
577	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
578	C₈	—	2	2	no	—
579	C₈	—	2	2	no	—
580	C₈	—	4	1	no	—
581	C₈	—	4	1	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	C₂₃×#479	1	4	yes	—
589	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
590	C₂²	C₂₃×#532	1	1	yes	characteristic
591	C₂²	C₂₃×#534	1	1	yes	characteristic
592	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
593	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
594	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
595	C₂²	C₂₃×#575	1	1	yes	characteristic
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	8	1	no	—
604	C₄	—	8	1	no	—
605	C₄	—	8	1	no	—
606	C₄	—	8	1	no	—
607	C₄	—	8	1	no	—
608	C₄	—	8	1	no	—
609	C₄	C₂₃×#551	1	1	yes	characteristic
610	C₄	C₂₃×#551	1	1	yes	characteristic
611	C₄	C₂₃×#554	1	1	yes	characteristic
612	C₄	C₂₃×#554	1	1	yes	characteristic
613	C₂	—	2	4	no	—
614	C₂	C₂₃×#2564	1	1	yes	characteristic
615	C₂	C₂×C₂₃×#479	1	1	yes	characteristic
616	C₂	C₂₃×#3076	1	1	yes	characteristic
617	C₂	C₂₃×#3168	1	1	yes	characteristic
618	C₂	C₂₃×#3181	1	1	yes	characteristic
619	C₂	C₂₃×#3190	1	1	yes	characteristic
620	C₂	C₂₃×#3191	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2101	C₂	semidirect
4. C₂₃×#2097	C₂	semidirect
5. C₂₃×#2059	C₂	semidirect
6. C₂₃×#2026	C₂	semidirect
7. C₂₃×#1920	C₂	semidirect
8. C₂₃×#1918	C₂	semidirect
11. C₂₃×#1865	C₂	semidirect
12. C₂₃×#1863	C₂	semidirect
204. #29158	C₂₃	direct (N × H)
274. #2101	C₂×C₂₃	semidirect
275. #2097	C₂×C₂₃	semidirect
276. #2059	C₂×C₂₃	semidirect
277. #2026	C₂×C₂₃	semidirect
278. #1920	C₂×C₂₃	semidirect
279. #1918	C₂×C₂₃	semidirect
282. #1865	C₂×C₂₃	semidirect
283. #1863	C₂×C₂₃	semidirect
434. C₂₃	#29158	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2851	2
C₄×C₂₃×C₂²⋊Q₈	2
C₂₃×#2101	2
C₂₃×#2097	2
C₂₃×#2059	2
C₂₃×#2026	2
C₂₃×#1920	2
C₂₃×#1918	2
C₂₃×#1892	2
C₂₃×#1865	2
C₂₃×#1863	2
C₂₃×#1808	2
C₂₃×#1807	2
C₂×C₂₃×#336	2
#29158	23

Subgroups — C23×#29158 #19176784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29158 #19176784