Subgroups — C₂₃×#26347 #19173973

Order 5888.

858 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2441	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2363	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2065	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1921	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1893	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#1875	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1854	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1850	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂₃×#478	—	2	4	no	—
23	C₂₃×#427	—	2	4	no	—
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₃×#399	—	2	1	no	—
38	C₂₃×#396	—	2	1	no	—
39	C₂₃×#390	—	2	1	no	—
40	C₂₃×#390	C₂²	1	1	yes	characteristic
41	C₂₃×#389	—	2	1	no	—
42	C₂₃×#389	C₂²	1	1	yes	characteristic
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#388	—	2	1	no	—
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#386	—	2	1	no	—
48	C₂₃×#384	C₂²	1	1	yes	characteristic
49	C₂₃×#380	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	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×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	—	2	1	no	—
62	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	—	2	2	no	—
66	C₂²×C₈×C₂₃	—	2	2	no	—
67	C₂²×C₈×C₂₃	—	2	2	no	—
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
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	C₂₃×#125	—	2	2	no	—
73	C₂₃×#125	—	4	1	no	—
74	C₂₃×#121	—	2	2	no	—
75	C₂₃×#121	Q₈	1	4	yes	—
76	C₄×D₄×C₂₃	—	2	2	no	—
77	C₄×D₄×C₂₃	—	4	1	no	—
78	C₂₃×#117	—	2	2	no	—
79	C₂₃×#117	—	2	2	no	—
80	C₂₃×#117	—	4	1	no	—
81	C₂₃×#117	—	4	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	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
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×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	—	2	1	no	—
106	C₂×C₄²×C₂₃	—	2	1	no	—
107	C₂×C₄²×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂₃×#107	—	2	2	no	—
111	C₂₃×#107	—	4	1	no	—
112	C₂₃×#107	D₄	1	4	yes	complemented
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×#106	—	2	2	no	—
115	C₂₃×#106	—	2	2	no	—
116	C₂₃×C₄⋊C₈	—	2	2	no	—
117	C₂₃×C₄⋊C₈	—	2	2	no	—
118	C₂₃×#102	—	2	2	no	—
119	C₂₃×#102	—	2	2	no	—
120	C₂₃×#102	—	2	2	no	—
121	C₂₃×#102	—	4	1	no	—
122	C₂₃×C₂²⋊C₈	—	2	2	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	D₄	1	1	yes	characteristic
129	C₂₃×#95	D₄	1	1	yes	characteristic
130	C₂⁴×C₂₃	—	2	1	no	—
131	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂×D₄×C₂₃	—	2	4	no	—
133	C₂×D₄×C₂₃	—	2	1	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	8	1	no	—
139	C₂×D₄×C₂₃	—	8	1	no	—
140	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	4	2	no	—
153	C₂²×C₄×C₂₃	—	4	2	no	—
154	C₂²×C₄×C₂₃	—	8	1	no	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
161	C₂²×C₄×C₂₃	#41	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
163	C₂×C₈×C₂₃	—	2	8	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	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
175	C₂×C₈×C₂₃	#41	1	2	yes	—
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	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	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	—	8	1	no	—
192	C₄⋊C₄×C₂₃	—	8	1	no	—
193	C₄⋊C₄×C₂₃	—	8	1	no	—
194	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
197	C₂²⋊C₄×C₂₃	—	2	4	no	—
198	C₂²⋊C₄×C₂₃	—	2	2	no	—
199	C₂²⋊C₄×C₂₃	—	4	1	no	—
200	C₂²⋊C₄×C₂₃	—	4	2	no	—
201	C₂²⋊C₄×C₂₃	—	4	2	no	—
202	C₂²⋊C₄×C₂₃	—	4	2	no	—
203	C₂²⋊C₄×C₂₃	—	4	2	no	—
204	C₂²⋊C₄×C₂₃	—	8	1	no	—
205	C₂²⋊C₄×C₂₃	—	8	1	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	4	2	no	—
211	C₄²×C₂₃	—	4	2	no	—
212	C₄²×C₂₃	—	8	1	no	—
213	#26347	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂³×C₂₃	—	2	1	no	—
215	C₂³×C₂₃	—	2	1	no	—
216	C₂³×C₂₃	—	2	1	no	—
217	C₂³×C₂₃	—	2	1	no	—
218	C₂³×C₂₃	—	2	1	no	—
219	C₂³×C₂₃	—	2	1	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	C₂³×C₂₃	—	4	1	no	—
223	C₂³×C₂₃	—	4	1	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	—	4	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
228	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
229	D₄×C₂₃	—	2	2	no	—
230	D₄×C₂₃	—	2	2	no	—
231	D₄×C₂₃	—	4	1	no	—
232	D₄×C₂₃	—	4	1	no	—
233	D₄×C₂₃	—	8	4	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	2	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	2	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	8	4	no	—
273	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄×C₂₃	#135	1	2	yes	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
279	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
281	C₈×C₂₃	—	2	2	no	—
282	C₈×C₂₃	—	2	2	no	—
283	C₈×C₂₃	—	4	1	no	—
284	C₈×C₂₃	—	4	1	no	—
285	C₂×#478	C₂×C₂₃	1	1	yes	characteristic, complemented
286	C₂×#427	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#2441	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#2363	C₂×C₂₃	1	1	yes	characteristic
289	#2093	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#2086	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#2065	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#2059	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1921	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1893	C₂×C₂₃	1	1	yes	characteristic
296	#1875	C₂×C₂₃	1	1	yes	characteristic
297	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1854	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1850	C₂×C₂₃	1	1	yes	characteristic, complemented
300	C₂²×C₂₃	—	2	2	no	—
301	C₂²×C₂₃	—	2	2	no	—
302	C₂²×C₂₃	—	2	2	no	—
303	C₂²×C₂₃	—	4	1	no	—
304	C₂²×C₂₃	—	4	1	no	—
305	C₂²×C₂₃	—	4	1	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	4	2	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	8	1	no	—
311	C₂²×C₂₃	—	8	1	no	—
312	C₂²×C₂₃	—	8	1	no	—
313	C₂²×C₂₃	—	8	1	no	—
314	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
315	C₂²×C₂₃	D₄²	1	1	yes	characteristic
316	C₂²×C₂₃	#547	1	1	yes	characteristic
317	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
318	C₂²×C₂₃	#575	1	1	yes	characteristic
319	C₂²×C₂₃	#576	1	1	yes	characteristic
320	C₂²×C₂₃	#577	1	1	yes	characteristic
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	2	2	no	—
323	C₄×C₂₃	—	4	2	no	—
324	C₄×C₂₃	—	4	1	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	2	no	—
327	C₄×C₂₃	—	4	2	no	—
328	C₄×C₂₃	—	4	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	—	8	1	no	—
332	C₄×C₂₃	—	8	1	no	—
333	C₄×C₂₃	#525	1	1	yes	characteristic
334	C₄×C₂₃	#539	1	1	yes	characteristic
335	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
336	C₄×C₂₃	#554	1	1	yes	characteristic
337	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
343	#478	—	2	4	no	—
344	#427	—	2	4	no	—
345	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
358	#399	—	2	1	no	—
359	#396	—	2	1	no	—
360	#390	—	2	1	no	—
361	#390	C₂²×C₂₃	1	1	yes	characteristic
362	#389	—	2	1	no	—
363	#389	C₂²×C₂₃	1	1	yes	characteristic
364	#389	C₂²×C₂₃	1	1	yes	characteristic
365	#389	C₂²×C₂₃	1	1	yes	characteristic
366	#388	—	2	1	no	—
367	#387	—	2	1	no	—
368	#386	—	2	1	no	—
369	#384	C₂²×C₂₃	1	1	yes	characteristic
370	#380	C₂²×C₂₃	1	1	yes	characteristic
371	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
372	C₄×C₂²⋊C₄	—	2	1	no	—
373	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#340	C₂²×C₂₃	1	1	yes	characteristic
378	#340	C₂²×C₂₃	1	1	yes	characteristic
379	#340	C₂²×C₂₃	1	1	yes	characteristic
380	#336	C₂²×C₂₃	1	1	yes	characteristic
381	#336	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×C₂₃	—	4	2	no	—
383	C₂×C₂₃	—	8	2	no	—
384	C₂×C₂₃	#2449	1	1	yes	characteristic
385	C₂×C₂₃	#3076	1	1	yes	characteristic
386	C₂×C₂₃	#3081	1	1	yes	characteristic
387	C₂×C₂₃	#3267	1	1	yes	characteristic
388	C₂×C₂₃	#3287	1	1	yes	characteristic
389	C₂×C₂₃	#3337	1	1	yes	characteristic
390	C₂×C₂₃	#3347	1	1	yes	characteristic
391	C₂²×D₄	—	2	1	no	—
392	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂³×C₄	—	2	1	no	—
394	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
395	C₂²×C₈	—	2	2	no	—
396	C₂²×C₈	—	2	2	no	—
397	C₂²×C₈	—	2	2	no	—
398	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
402	#125	—	2	2	no	—
403	#125	—	4	1	no	—
404	#121	—	2	2	no	—
405	#121	Q₈×C₂₃	1	4	yes	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	4	1	no	—
408	#117	—	2	2	no	—
409	#117	—	2	2	no	—
410	#117	—	4	1	no	—
411	#117	—	4	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄²	—	2	1	no	—
436	C₂×C₄²	—	2	1	no	—
437	C₂×C₄²	—	2	1	no	—
438	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
440	#107	—	2	2	no	—
441	#107	—	4	1	no	—
442	#107	D₄×C₂₃	1	4	yes	complemented
443	#106	—	2	2	no	—
444	#106	—	2	2	no	—
445	#106	—	2	2	no	—
446	C₄⋊C₈	—	2	2	no	—
447	C₄⋊C₈	—	2	2	no	—
448	#102	—	2	2	no	—
449	#102	—	2	2	no	—
450	#102	—	2	2	no	—
451	#102	—	4	1	no	—
452	C₂²⋊C₈	—	2	2	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	D₄×C₂₃	1	1	yes	characteristic
459	#95	D₄×C₂₃	1	1	yes	characteristic
460	C₂₃	#26347	1	1	yes	characteristic, direct product factor, Sylow
461	C₂⁴	—	2	1	no	—
462	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	8	1	no	—
470	C₂×D₄	—	8	1	no	—
471	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	8	1	no	—
486	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
492	C₂²×C₄	#41×C₂₃	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
494	C₂×C₈	—	2	8	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
506	C₂×C₈	#41×C₂₃	1	2	yes	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	4	2	no	—
519	C₄⋊C₄	—	4	2	no	—
520	C₄⋊C₄	—	4	2	no	—
521	C₄⋊C₄	—	4	2	no	—
522	C₄⋊C₄	—	8	1	no	—
523	C₄⋊C₄	—	8	1	no	—
524	C₄⋊C₄	—	8	1	no	—
525	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
526	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
527	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	4	1	no	—
531	C₂²⋊C₄	—	4	2	no	—
532	C₂²⋊C₄	—	4	2	no	—
533	C₂²⋊C₄	—	4	2	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₂²⋊C₄	—	8	1	no	—
536	C₂²⋊C₄	—	8	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	4	2	no	—
542	C₄²	—	4	2	no	—
543	C₄²	—	8	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	2	1	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
558	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
559	D₄	—	2	2	no	—
560	D₄	—	2	2	no	—
561	D₄	—	4	1	no	—
562	D₄	—	4	1	no	—
563	D₄	—	8	4	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	8	4	no	—
603	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
604	C₂×C₄	C₂₃×#135	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
606	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
608	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
609	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
610	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
611	C₈	—	2	2	no	—
612	C₈	—	2	2	no	—
613	C₈	—	4	1	no	—
614	C₈	—	4	1	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	8	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
630	C₂²	D₄²×C₂₃	1	1	yes	characteristic
631	C₂²	C₂₃×#547	1	1	yes	characteristic
632	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
633	C₂²	C₂₃×#575	1	1	yes	characteristic
634	C₂²	C₂₃×#576	1	1	yes	characteristic
635	C₂²	C₂₃×#577	1	1	yes	characteristic
636	C₄	—	2	2	no	—
637	C₄	—	2	2	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	2	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	—	8	1	no	—
647	C₄	—	8	1	no	—
648	C₄	C₂₃×#525	1	1	yes	characteristic
649	C₄	C₂₃×#539	1	1	yes	characteristic
650	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
651	C₄	C₂₃×#554	1	1	yes	characteristic
652	C₂	—	4	2	no	—
653	C₂	—	8	2	no	—
654	C₂	C₂₃×#2449	1	1	yes	characteristic
655	C₂	C₂₃×#3076	1	1	yes	characteristic
656	C₂	C₂₃×#3081	1	1	yes	characteristic
657	C₂	C₂₃×#3267	1	1	yes	characteristic
658	C₂	C₂₃×#3287	1	1	yes	characteristic
659	C₂	C₂₃×#3337	1	1	yes	characteristic
660	C₂	C₂₃×#3347	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#478	C₂	semidirect
2. C₂×C₂₃×#427	C₂	semidirect
3. C₂₃×#2441	C₂	semidirect
5. C₂₃×#2093	C₂	semidirect
6. C₂₃×#2086	C₂	semidirect
7. C₂₃×#2065	C₂	semidirect
8. C₂₃×#2059	C₂	semidirect
9. C₂₃×#1948	C₂	semidirect
10. C₂₃×#1921	C₂	semidirect
13. C₂₃×#1864	C₂	semidirect
14. C₂₃×#1854	C₂	semidirect
15. C₂₃×#1850	C₂	semidirect
112. C₂₃×#107	D₄	semidirect
213. #26347	C₂₃	direct (N × H)
285. C₂×#478	C₂×C₂₃	semidirect
286. C₂×#427	C₂×C₂₃	semidirect
287. #2441	C₂×C₂₃	semidirect
289. #2093	C₂×C₂₃	semidirect
290. #2086	C₂×C₂₃	semidirect
291. #2065	C₂×C₂₃	semidirect
292. #2059	C₂×C₂₃	semidirect
293. #1948	C₂×C₂₃	semidirect
294. #1921	C₂×C₂₃	semidirect
297. #1864	C₂×C₂₃	semidirect
298. #1854	C₂×C₂₃	semidirect
299. #1850	C₂×C₂₃	semidirect
442. #107	D₄×C₂₃	semidirect
460. C₂₃	#26347	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×#478	2
C₂×C₂₃×#427	2
C₂₃×#2441	2
C₂₃×#2363	2
C₂₃×#2093	2
C₂₃×#2086	2
C₂₃×#2065	2
C₂₃×#2059	2
C₂₃×#1948	2
C₂₃×#1921	2
C₂₃×#1893	2
C₂₃×#1875	2
C₂₃×#1864	2
C₂₃×#1854	2
C₂₃×#1850	2
#26347	23

Subgroups — C23×#26347 #19173973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26347 #19173973