Subgroups — C₂₃×#29204 #19176830

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2328	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2067	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2065	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1961	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1957	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1864	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1862	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1847	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#107	C₂²	1	2	yes	—
16	C₂×C₂₃×#106	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×#102	C₂²	1	2	yes	—
19	C₂×C₂₃×#102	C₂²	1	2	yes	—
20	C₂×C₂₃×#102	C₂²	1	2	yes	—
21	C₂×C₂₃×#102	C₂²	1	2	yes	—
22	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#400	—	2	2	no	—
25	C₂₃×#398	—	2	2	no	—
26	C₂₃×#390	C₂²	1	1	yes	characteristic
27	C₂₃×#390	C₂²	1	1	yes	characteristic
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#389	C₂²	1	2	yes	—
30	C₂₃×#386	—	2	2	no	—
31	C₂₃×#386	—	2	2	no	—
32	C₂₃×#384	C₂²	1	1	yes	characteristic
33	C₂₃×#382	C₂²	1	1	yes	characteristic
34	C₂₃×#382	C₂²	1	1	yes	characteristic
35	C₂₃×#376	C₂²	1	1	yes	characteristic
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂²×D₄×C₂₃	—	2	2	no	—
40	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₂₃	—	2	1	no	—
43	C₂³×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂₃×#127	—	2	2	no	—
47	C₂₃×#127	Q₈	1	4	yes	—
48	C₄×D₄×C₂₃	—	4	2	no	—
49	C₄×D₄×C₂₃	—	4	2	no	—
50	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#107	—	4	2	no	—
65	C₂₃×#106	—	4	2	no	—
66	C₂₃×C₄⋊C₈	—	2	4	no	—
67	C₂₃×#102	—	2	4	no	—
68	C₂₃×#102	—	2	4	no	—
69	C₂₃×#102	—	4	2	no	—
70	C₂₃×#102	—	4	2	no	—
71	C₂₃×#97	—	2	2	no	—
72	C₄×C₈×C₂₃	—	2	2	no	—
73	C₂₃×#95	—	2	2	no	—
74	C₂₃×#95	—	2	2	no	—
75	C₂₃×#95	—	2	2	no	—
76	C₂₃×#95	—	2	2	no	—
77	C₂₃×#95	D₄	1	2	yes	—
78	C₂₃×#95	D₄	1	2	yes	—
79	C₂⁴×C₂₃	—	2	1	no	—
80	C₂⁴×C₂₃	—	2	1	no	—
81	C₂×D₄×C₂₃	—	2	8	no	—
82	C₂×D₄×C₂₃	—	2	2	no	—
83	C₂×D₄×C₂₃	—	2	2	no	—
84	C₂×D₄×C₂₃	—	2	1	no	—
85	C₂×D₄×C₂₃	—	2	1	no	—
86	C₂×D₄×C₂₃	—	2	2	no	—
87	C₂×D₄×C₂₃	—	2	2	no	—
88	C₂×D₄×C₂₃	—	8	2	no	—
89	C₂×D₄×C₂₃	—	8	2	no	—
90	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
91	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₂₃	—	2	1	no	—
93	C₂²×C₄×C₂₃	—	2	1	no	—
94	C₂²×C₄×C₂₃	—	2	2	no	—
95	C₂²×C₄×C₂₃	—	2	2	no	—
96	C₂²×C₄×C₂₃	—	4	2	no	—
97	C₂²×C₄×C₂₃	—	4	2	no	—
98	C₂²×C₄×C₂₃	—	8	2	no	—
99	C₂²×C₄×C₂₃	—	8	2	no	—
100	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₂₃	—	2	2	no	—
106	C₂×C₈×C₂₃	—	2	2	no	—
107	C₂×C₈×C₂₃	—	2	2	no	—
108	C₂×C₈×C₂₃	—	2	2	no	—
109	C₂×C₈×C₂₃	—	2	2	no	—
110	C₂×C₈×C₂₃	#41	1	4	yes	—
111	C₄⋊C₄×C₂₃	—	2	2	no	—
112	C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₄⋊C₄×C₂₃	—	2	2	no	—
114	C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	8	2	no	—
118	C₄⋊C₄×C₂₃	—	8	2	no	—
119	C₄⋊C₄×C₂₃	—	8	2	no	—
120	C₂²⋊C₄×C₂₃	—	4	2	no	—
121	C₂²⋊C₄×C₂₃	—	4	2	no	—
122	C₂²⋊C₄×C₂₃	—	8	2	no	—
123	C₂²⋊C₄×C₂₃	—	8	2	no	—
124	C₄²×C₂₃	—	2	2	no	—
125	C₄²×C₂₃	—	2	2	no	—
126	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
127	C₄²×C₂₃	#41	1	2	yes	—
128	#29204	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₂₃	—	2	2	no	—
130	C₂³×C₂₃	—	2	2	no	—
131	C₂³×C₂₃	—	4	1	no	—
132	C₂³×C₂₃	—	4	1	no	—
133	C₂³×C₂₃	—	4	1	no	—
134	C₂³×C₂₃	—	4	1	no	—
135	C₂³×C₂₃	—	4	2	no	—
136	C₂³×C₂₃	—	4	2	no	—
137	C₂³×C₂₃	—	4	2	no	—
138	C₂³×C₂₃	—	4	2	no	—
139	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	D₄×C₂₃	—	2	4	no	—
141	D₄×C₂₃	—	2	4	no	—
142	D₄×C₂₃	—	4	2	no	—
143	D₄×C₂₃	—	4	2	no	—
144	D₄×C₂₃	—	8	8	no	—
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	1	no	—
149	C₂×C₄×C₂₃	—	2	1	no	—
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	4	2	no	—
155	C₂×C₄×C₂₃	—	4	1	no	—
156	C₂×C₄×C₂₃	—	4	1	no	—
157	C₂×C₄×C₂₃	—	4	2	no	—
158	C₂×C₄×C₂₃	—	4	2	no	—
159	C₂×C₄×C₂₃	—	4	2	no	—
160	C₂×C₄×C₂₃	—	4	2	no	—
161	C₂×C₄×C₂₃	—	4	2	no	—
162	C₂×C₄×C₂₃	—	4	2	no	—
163	C₂×C₄×C₂₃	—	4	2	no	—
164	C₂×C₄×C₂₃	—	8	4	no	—
165	C₂×C₄×C₂₃	—	8	4	no	—
166	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
167	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
170	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
171	C₈×C₂₃	—	2	4	no	—
172	C₈×C₂₃	—	4	2	no	—
173	#2647	C₂×C₂₃	1	1	yes	characteristic, complemented
174	#2328	C₂×C₂₃	1	1	yes	characteristic
175	#2067	C₂×C₂₃	1	2	yes	complemented
176	#2065	C₂×C₂₃	1	2	yes	complemented
177	#1961	C₂×C₂₃	1	1	yes	characteristic
178	#1957	C₂×C₂₃	1	1	yes	characteristic
179	#1864	C₂×C₂₃	1	2	yes	complemented
180	#1862	C₂×C₂₃	1	2	yes	complemented
181	#1847	C₂×C₂₃	1	1	yes	characteristic, complemented
182	#1835	C₂×C₂₃	1	1	yes	characteristic, complemented
183	#1768	C₂×C₂₃	1	1	yes	characteristic, complemented
184	C₂²×C₂₃	—	4	2	no	—
185	C₂²×C₂₃	—	4	2	no	—
186	C₂²×C₂₃	—	4	4	no	—
187	C₂²×C₂₃	—	4	4	no	—
188	C₂²×C₂₃	—	8	2	no	—
189	C₂²×C₂₃	—	8	2	no	—
190	C₂²×C₂₃	—	8	1	no	—
191	C₂²×C₂₃	—	8	1	no	—
192	C₂²×C₂₃	—	8	1	no	—
193	C₂²×C₂₃	—	8	1	no	—
194	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
195	C₂²×C₂₃	#532	1	1	yes	characteristic
196	C₂²×C₂₃	#535	1	1	yes	characteristic
197	C₂²×C₂₃	#576	1	2	yes	—
198	C₂²×C₂₃	#577	1	2	yes	—
199	C₄×C₂₃	—	2	4	no	—
200	C₄×C₂₃	—	4	2	no	—
201	C₄×C₂₃	—	4	1	no	—
202	C₄×C₂₃	—	4	1	no	—
203	C₄×C₂₃	—	8	2	no	—
204	C₄×C₂₃	—	8	2	no	—
205	C₄×C₂₃	—	8	1	no	—
206	C₄×C₂₃	—	8	1	no	—
207	C₄×C₂₃	#553	1	2	yes	—
208	C₄×C₂₃	#554	1	2	yes	—
209	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
211	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
212	C₂×#107	C₂²×C₂₃	1	2	yes	—
213	C₂×#106	C₂²×C₂₃	1	2	yes	—
214	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
215	C₂×#102	C₂²×C₂₃	1	2	yes	—
216	C₂×#102	C₂²×C₂₃	1	2	yes	—
217	C₂×#102	C₂²×C₂₃	1	2	yes	—
218	C₂×#102	C₂²×C₂₃	1	2	yes	—
219	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
221	#400	—	2	2	no	—
222	#398	—	2	2	no	—
223	#390	C₂²×C₂₃	1	1	yes	characteristic
224	#390	C₂²×C₂₃	1	1	yes	characteristic
225	#389	C₂²×C₂₃	1	2	yes	—
226	#389	C₂²×C₂₃	1	2	yes	—
227	#386	—	2	2	no	—
228	#386	—	2	2	no	—
229	#384	C₂²×C₂₃	1	1	yes	characteristic
230	#382	C₂²×C₂₃	1	1	yes	characteristic
231	#382	C₂²×C₂₃	1	1	yes	characteristic
232	#376	C₂²×C₂₃	1	1	yes	characteristic
233	#340	C₂²×C₂₃	1	2	yes	—
234	#340	C₂²×C₂₃	1	2	yes	—
235	#336	C₂²×C₂₃	1	2	yes	—
236	C₂×C₂₃	—	8	2	no	—
237	C₂×C₂₃	—	8	2	no	—
238	C₂×C₂₃	#2566	1	1	yes	characteristic
239	C₂×C₂₃	#3081	1	2	yes	—
240	C₂×C₂₃	#3179	1	2	yes	—
241	C₂×C₂₃	#3220	1	1	yes	characteristic
242	C₂×C₂₃	#3222	1	1	yes	characteristic
243	C₂²×D₄	—	2	2	no	—
244	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
245	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
246	C₂³×C₄	—	2	1	no	—
247	C₂³×C₄	—	2	1	no	—
248	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
249	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
250	#127	—	2	2	no	—
251	#127	Q₈×C₂₃	1	4	yes	—
252	C₄×D₄	—	4	2	no	—
253	C₄×D₄	—	4	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
261	C₂×C₂²⋊C₄	—	2	1	no	—
262	C₂×C₂²⋊C₄	—	2	1	no	—
263	C₂×C₂²⋊C₄	—	2	2	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
268	#107	—	4	2	no	—
269	#106	—	4	2	no	—
270	C₄⋊C₈	—	2	4	no	—
271	#102	—	2	4	no	—
272	#102	—	2	4	no	—
273	#102	—	4	2	no	—
274	#102	—	4	2	no	—
275	#97	—	2	2	no	—
276	C₄×C₈	—	2	2	no	—
277	#95	—	2	2	no	—
278	#95	—	2	2	no	—
279	#95	—	2	2	no	—
280	#95	—	2	2	no	—
281	#95	D₄×C₂₃	1	2	yes	—
282	#95	D₄×C₂₃	1	2	yes	—
283	C₂₃	#29204	1	1	yes	characteristic, direct product factor, Sylow
284	C₂⁴	—	2	1	no	—
285	C₂⁴	—	2	1	no	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	8	2	no	—
294	C₂×D₄	—	8	2	no	—
295	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
296	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	8	2	no	—
304	C₂²×C₄	—	8	2	no	—
305	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
309	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	#41×C₂₃	1	4	yes	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	8	2	no	—
323	C₄⋊C₄	—	8	2	no	—
324	C₄⋊C₄	—	8	2	no	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	8	2	no	—
328	C₂²⋊C₄	—	8	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
332	C₄²	#41×C₂₃	1	2	yes	—
333	C₂³	—	2	2	no	—
334	C₂³	—	2	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	1	no	—
340	C₂³	—	4	1	no	—
341	C₂³	—	4	1	no	—
342	C₂³	—	4	1	no	—
343	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
344	D₄	—	2	4	no	—
345	D₄	—	2	4	no	—
346	D₄	—	4	2	no	—
347	D₄	—	4	2	no	—
348	D₄	—	8	8	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	4	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
371	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
373	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
374	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
375	C₈	—	2	4	no	—
376	C₈	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	4	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	1	no	—
384	C₂²	—	8	1	no	—
385	C₂²	—	8	1	no	—
386	C₂²	—	8	1	no	—
387	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
388	C₂²	C₂₃×#532	1	1	yes	characteristic
389	C₂²	C₂₃×#535	1	1	yes	characteristic
390	C₂²	C₂₃×#576	1	2	yes	—
391	C₂²	C₂₃×#577	1	2	yes	—
392	C₄	—	2	4	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	1	no	—
400	C₄	C₂₃×#553	1	2	yes	—
401	C₄	C₂₃×#554	1	2	yes	—
402	C₂	—	8	2	no	—
403	C₂	—	8	2	no	—
404	C₂	C₂₃×#2566	1	1	yes	characteristic
405	C₂	C₂₃×#3081	1	2	yes	—
406	C₂	C₂₃×#3179	1	2	yes	—
407	C₂	C₂₃×#3220	1	1	yes	characteristic
408	C₂	C₂₃×#3222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2647	C₂	semidirect
3. C₂₃×#2067	C₂	semidirect
4. C₂₃×#2065	C₂	semidirect
7. C₂₃×#1864	C₂	semidirect
8. C₂₃×#1862	C₂	semidirect
9. C₂₃×#1847	C₂	semidirect
10. C₂₃×#1835	C₂	semidirect
11. C₂₃×#1768	C₂	semidirect
128. #29204	C₂₃	direct (N × H)
173. #2647	C₂×C₂₃	semidirect
175. #2067	C₂×C₂₃	semidirect
176. #2065	C₂×C₂₃	semidirect
179. #1864	C₂×C₂₃	semidirect
180. #1862	C₂×C₂₃	semidirect
181. #1847	C₂×C₂₃	semidirect
182. #1835	C₂×C₂₃	semidirect
183. #1768	C₂×C₂₃	semidirect
283. C₂₃	#29204	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×#2647	2
C₂₃×#2328	2
C₂₃×#2067	2
C₂₃×#2065	2
C₂₃×#1961	2
C₂₃×#1957	2
C₂₃×#1864	2
C₂₃×#1862	2
C₂₃×#1847	2
C₂₃×#1835	2
C₂₃×#1768	2
#29204	23

Subgroups — C23×#29204 #19176830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29204 #19176830