Subgroups — C₂₃×#29114 #19176740

Order 5888.

866 conjugacy classes of subgroups.

The subgroup lattice has 432 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, complemented
2	C₂₃×#2297	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2074	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2073	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2026	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2025	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1936	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1935	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1825	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1788	C₂	1	2	yes	maximal
11	C₂₃×#1749	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#103	C₂²	1	2	yes	—
17	C₂×C₂₃×#103	C₂²	1	2	yes	—
18	C₂×C₂₃×#102	C₂²	1	2	yes	—
19	C₂×C₂₃×#102	C₂²	1	2	yes	—
20	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂₃×#391	C₂²	1	1	yes	characteristic
23	C₂₃×#391	C₂²	1	1	yes	characteristic
24	C₂₃×#390	C₂²	1	1	yes	characteristic
25	C₂₃×#390	C₂²	1	1	yes	characteristic
26	C₂₃×#389	C₂²	1	2	yes	—
27	C₂₃×#387	—	2	2	no	—
28	C₂₃×#386	—	2	2	no	—
29	C₂₃×#385	—	2	2	no	—
30	C₂₃×#382	C₂²	1	1	yes	characteristic
31	C₂₃×#380	C₂²	1	2	yes	—
32	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₂₃×#340	C₂²	1	2	yes	—
34	C₂₃×#340	C₂²	1	2	yes	—
35	C₂₃×#340	C₂²	1	2	yes	—
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂₃×#336	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	1	yes	characteristic
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×#41×C₂₃	—	2	2	no	—
41	C₂×#41×C₂₃	—	2	4	no	—
42	C₂×#41×C₂₃	—	4	2	no	—
43	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	1	no	—
47	C₂³×C₄×C₂₃	D₄	1	2	yes	—
48	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂₃×#125	—	4	2	no	—
52	C₂₃×#117	—	2	4	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₂×C₄²×C₂₃	—	2	1	no	—
64	C₂×C₄²×C₂₃	—	4	2	no	—
65	C₂×C₄²×C₂₃	D₄	1	2	yes	—
66	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#103	—	2	4	no	—
69	C₂₃×#103	—	4	2	no	—
70	C₂₃×#102	—	2	4	no	—
71	C₂₃×#102	—	4	2	no	—
72	C₂₃×C₂²⋊C₈	—	2	4	no	—
73	C₂₃×#97	—	4	2	no	—
74	C₂₃×#95	—	2	2	no	—
75	C₂₃×#95	—	2	2	no	—
76	C₂₃×#95	—	2	2	no	—
77	C₂₃×#95	—	2	2	no	—
78	C₂₃×#95	—	2	2	no	—
79	C₂⁴×C₂₃	—	2	1	no	—
80	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
81	#41×C₂₃	—	4	8	no	—
82	#41×C₂₃	—	4	8	no	—
83	C₂×Q₈×C₂₃	—	2	1	no	—
84	C₂×Q₈×C₂₃	—	2	2	no	—
85	C₂×Q₈×C₂₃	—	2	2	no	—
86	C₂×Q₈×C₂₃	#41	1	2	yes	—
87	C₂×D₄×C₂₃	—	2	4	no	—
88	C₂×D₄×C₂₃	—	2	2	no	—
89	C₂×D₄×C₂₃	—	2	1	no	—
90	C₂×D₄×C₂₃	—	2	2	no	—
91	C₂×D₄×C₂₃	—	4	2	no	—
92	C₂×D₄×C₂₃	—	4	2	no	—
93	C₂×D₄×C₂₃	#41	1	2	yes	—
94	C₂²×C₄×C₂₃	—	2	1	no	—
95	C₂²×C₄×C₂₃	—	2	1	no	—
96	C₂²×C₄×C₂₃	—	2	2	no	—
97	C₂²×C₄×C₂₃	—	2	4	no	—
98	C₂²×C₄×C₂₃	—	2	1	no	—
99	C₂²×C₄×C₂₃	—	2	1	no	—
100	C₂²×C₄×C₂₃	—	2	1	no	—
101	C₂²×C₄×C₂₃	—	2	2	no	—
102	C₂²×C₄×C₂₃	—	2	2	no	—
103	C₂²×C₄×C₂₃	—	4	2	no	—
104	C₂²×C₄×C₂₃	—	4	2	no	—
105	C₂²×C₄×C₂₃	—	4	2	no	—
106	C₂²×C₄×C₂₃	—	4	2	no	—
107	C₂²×C₄×C₂₃	—	4	2	no	—
108	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	#41	1	4	yes	—
112	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₄⋊C₄×C₂₃	—	2	2	no	—
122	C₄⋊C₄×C₂₃	—	2	2	no	—
123	C₄⋊C₄×C₂₃	—	2	2	no	—
124	C₄⋊C₄×C₂₃	—	2	2	no	—
125	C₄⋊C₄×C₂₃	—	4	4	no	—
126	C₄⋊C₄×C₂₃	—	4	4	no	—
127	C₂²⋊C₄×C₂₃	—	4	4	no	—
128	C₂²⋊C₄×C₂₃	—	4	4	no	—
129	C₂²⋊C₄×C₂₃	—	4	4	no	—
130	C₂²⋊C₄×C₂₃	—	4	2	no	—
131	C₄²×C₂₃	—	2	2	no	—
132	C₄²×C₂₃	—	2	2	no	—
133	C₄²×C₂₃	—	4	2	no	—
134	C₄²×C₂₃	—	4	2	no	—
135	#29114	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂³×C₂₃	—	2	1	no	—
137	C₂³×C₂₃	—	2	1	no	—
138	C₂³×C₂₃	—	2	2	no	—
139	C₂³×C₂₃	—	2	2	no	—
140	C₂³×C₂₃	—	2	2	no	—
141	C₂³×C₂₃	—	4	1	no	—
142	C₂³×C₂₃	—	4	1	no	—
143	C₂³×C₂₃	—	4	2	no	—
144	C₂³×C₂₃	—	4	2	no	—
145	C₂³×C₂₃	#124	1	2	yes	—
146	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₂₃	—	2	4	no	—
148	Q₈×C₂₃	—	4	2	no	—
149	D₄×C₂₃	—	2	4	no	—
150	D₄×C₂₃	—	4	8	no	—
151	D₄×C₂₃	—	4	2	no	—
152	C₂×C₄×C₂₃	—	2	1	no	—
153	C₂×C₄×C₂₃	—	2	1	no	—
154	C₂×C₄×C₂₃	—	2	4	no	—
155	C₂×C₄×C₂₃	—	2	4	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	2	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	2	no	—
161	C₂×C₄×C₂₃	—	2	2	no	—
162	C₂×C₄×C₂₃	—	2	2	no	—
163	C₂×C₄×C₂₃	—	4	8	no	—
164	C₂×C₄×C₂₃	—	4	8	no	—
165	C₂×C₄×C₂₃	—	4	1	no	—
166	C₂×C₄×C₂₃	—	4	1	no	—
167	C₂×C₄×C₂₃	—	4	2	no	—
168	C₂×C₄×C₂₃	—	4	1	no	—
169	C₂×C₄×C₂₃	—	4	1	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	4	2	no	—
172	C₂×C₄×C₂₃	—	4	2	no	—
173	C₂×C₄×C₂₃	—	4	2	no	—
174	C₂×C₄×C₂₃	—	4	2	no	—
175	C₂×C₄×C₂₃	—	4	2	no	—
176	C₂×C₄×C₂₃	—	4	2	no	—
177	C₂×C₄×C₂₃	#124	1	2	yes	—
178	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
181	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
182	C₈×C₂₃	—	4	2	no	—
183	C₈×C₂₃	—	4	2	no	—
184	#2851	C₂×C₂₃	1	1	yes	characteristic, complemented
185	#2297	C₂×C₂₃	1	1	yes	characteristic
186	#2074	C₂×C₂₃	1	1	yes	characteristic, complemented
187	#2073	C₂×C₂₃	1	1	yes	characteristic, complemented
188	#2026	C₂×C₂₃	1	1	yes	characteristic, complemented
189	#2025	C₂×C₂₃	1	1	yes	characteristic, complemented
190	#1936	C₂×C₂₃	1	2	yes	complemented
191	#1935	C₂×C₂₃	1	2	yes	complemented
192	#1825	C₂×C₂₃	1	2	yes	complemented
193	#1788	C₂×C₂₃	1	2	yes	—
194	#1749	C₂×C₂₃	1	1	yes	characteristic, complemented
195	C₂²×C₂₃	—	2	2	no	—
196	C₂²×C₂₃	—	2	2	no	—
197	C₂²×C₂₃	—	2	2	no	—
198	C₂²×C₂₃	—	4	2	no	—
199	C₂²×C₂₃	—	4	2	no	—
200	C₂²×C₂₃	—	4	2	no	—
201	C₂²×C₂₃	—	4	4	no	—
202	C₂²×C₂₃	—	8	1	no	—
203	C₂²×C₂₃	—	8	1	no	—
204	C₂²×C₂₃	—	8	2	no	—
205	C₂²×C₂₃	#525	1	1	yes	characteristic
206	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
207	C₂²×C₂₃	#534	1	1	yes	characteristic
208	C₂²×C₂₃	#575	1	2	yes	—
209	C₂²×C₂₃	#575	1	2	yes	—
210	C₄×C₂₃	—	2	2	no	—
211	C₄×C₂₃	—	4	2	no	—
212	C₄×C₂₃	—	4	2	no	—
213	C₄×C₂₃	—	4	2	no	—
214	C₄×C₂₃	—	8	2	no	—
215	C₄×C₂₃	—	8	1	no	—
216	C₄×C₂₃	—	8	1	no	—
217	C₄×C₂₃	—	8	2	no	—
218	C₄×C₂₃	#551	1	2	yes	—
219	C₄×C₂₃	#551	1	2	yes	—
220	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
221	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
222	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
223	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×#103	C₂²×C₂₃	1	2	yes	—
225	C₂×#103	C₂²×C₂₃	1	2	yes	—
226	C₂×#102	C₂²×C₂₃	1	2	yes	—
227	C₂×#102	C₂²×C₂₃	1	2	yes	—
228	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
229	C₂×#97	C₂²×C₂₃	1	2	yes	—
230	#391	C₂²×C₂₃	1	1	yes	characteristic
231	#391	C₂²×C₂₃	1	1	yes	characteristic
232	#390	C₂²×C₂₃	1	1	yes	characteristic
233	#390	C₂²×C₂₃	1	1	yes	characteristic
234	#389	C₂²×C₂₃	1	2	yes	—
235	#387	—	2	2	no	—
236	#386	—	2	2	no	—
237	#385	—	2	2	no	—
238	#382	C₂²×C₂₃	1	1	yes	characteristic
239	#380	C₂²×C₂₃	1	2	yes	—
240	C₄×C₂²⋊C₄	—	2	2	no	—
241	#340	C₂²×C₂₃	1	2	yes	—
242	#340	C₂²×C₂₃	1	2	yes	—
243	#340	C₂²×C₂₃	1	2	yes	—
244	#340	C₂²×C₂₃	1	2	yes	—
245	#336	C₂²×C₂₃	1	1	yes	characteristic
246	#336	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×C₂₃	—	4	2	no	—
248	C₂×C₂₃	—	8	2	no	—
249	C₂×C₂₃	#2561	1	1	yes	characteristic
250	C₂×C₂₃	#3111	1	2	yes	—
251	C₂×C₂₃	#3135	1	1	yes	characteristic
252	C₂×C₂₃	#3142	1	1	yes	characteristic
253	C₂×C₂₃	#3197	1	2	yes	—
254	C₂×#41	—	2	2	no	—
255	C₂×#41	—	2	2	no	—
256	C₂×#41	—	2	4	no	—
257	C₂×#41	—	4	2	no	—
258	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
259	C₂²×D₄	—	2	1	no	—
260	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
261	C₂³×C₄	—	2	1	no	—
262	C₂³×C₄	D₄×C₂₃	1	2	yes	—
263	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
265	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
266	#125	—	4	2	no	—
267	#117	—	2	4	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
271	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₄²	—	2	1	no	—
279	C₂×C₄²	—	4	2	no	—
280	C₂×C₄²	D₄×C₂₃	1	2	yes	—
281	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
283	#103	—	2	4	no	—
284	#103	—	4	2	no	—
285	#102	—	2	4	no	—
286	#102	—	4	2	no	—
287	C₂²⋊C₈	—	2	4	no	—
288	#97	—	4	2	no	—
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	—	2	2	no	—
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	C₂₃	#29114	1	1	yes	characteristic, direct product factor, Sylow
295	C₂⁴	—	2	1	no	—
296	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
297	#41	—	4	8	no	—
298	#41	—	4	8	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	#41×C₂₃	1	2	yes	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂×D₄	#41×C₂₃	1	2	yes	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	—	4	2	no	—
321	C₂²×C₄	—	4	2	no	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
326	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
327	C₂²×C₄	#41×C₂₃	1	4	yes	—
328	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	4	4	no	—
342	C₄⋊C₄	—	4	4	no	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₂²⋊C₄	—	4	4	no	—
345	C₂²⋊C₄	—	4	4	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	4	2	no	—
350	C₄²	—	4	2	no	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	1	no	—
359	C₂³	—	4	1	no	—
360	C₂³	C₂₃×#124	1	2	yes	—
361	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	4	2	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	8	no	—
366	D₄	—	4	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	C₂₃×#124	1	2	yes	—
393	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
396	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
397	C₈	—	4	2	no	—
398	C₈	—	4	2	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	2	2	no	—
401	C₂²	—	2	2	no	—
402	C₂²	—	4	2	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	4	4	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	1	no	—
408	C₂²	—	8	1	no	—
409	C₂²	C₂₃×#525	1	1	yes	characteristic
410	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
411	C₂²	C₂₃×#534	1	1	yes	characteristic
412	C₂²	C₂₃×#575	1	2	yes	—
413	C₂²	C₂₃×#575	1	2	yes	—
414	C₄	—	2	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	8	2	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	1	no	—
421	C₄	—	8	1	no	—
422	C₄	C₂₃×#551	1	2	yes	—
423	C₄	C₂₃×#551	1	2	yes	—
424	C₂	—	4	2	no	—
425	C₂	—	8	2	no	—
426	C₂	C₂₃×#2561	1	1	yes	characteristic
427	C₂	C₂₃×#3111	1	2	yes	—
428	C₂	C₂₃×#3135	1	1	yes	characteristic
429	C₂	C₂₃×#3142	1	1	yes	characteristic
430	C₂	C₂₃×#3197	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	C₂²
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
1. C₂₃×#2851	C₂	semidirect
3. C₂₃×#2074	C₂	semidirect
4. C₂₃×#2073	C₂	semidirect
5. C₂₃×#2026	C₂	semidirect
6. C₂₃×#2025	C₂	semidirect
7. C₂₃×#1936	C₂	semidirect
8. C₂₃×#1935	C₂	semidirect
9. C₂₃×#1825	C₂	semidirect
11. C₂₃×#1749	C₂	semidirect
135. #29114	C₂₃	direct (N × H)
184. #2851	C₂×C₂₃	semidirect
186. #2074	C₂×C₂₃	semidirect
187. #2073	C₂×C₂₃	semidirect
188. #2026	C₂×C₂₃	semidirect
189. #2025	C₂×C₂₃	semidirect
190. #1936	C₂×C₂₃	semidirect
191. #1935	C₂×C₂₃	semidirect
192. #1825	C₂×C₂₃	semidirect
194. #1749	C₂×C₂₃	semidirect
294. C₂₃	#29114	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
C₂₃×#2297	2
C₂₃×#2074	2
C₂₃×#2073	2
C₂₃×#2026	2
C₂₃×#2025	2
C₂₃×#1936	2
C₂₃×#1935	2
C₂₃×#1825	2
C₂₃×#1788	2
C₂₃×#1749	2
#29114	23

Subgroups — C23×#29114 #19176740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29114 #19176740