Subgroups — C₂₃×#19157 #19166783

Order 5888.

704 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3035	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1702	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1701	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1694	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1693	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1688	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1687	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1540	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#135	—	2	2	no	—
13	C₂₃×#532	C₂²	1	2	yes	—
14	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	1	yes	characteristic
16	C₂₃×#493	C₂²	1	1	yes	characteristic
17	C₂₃×#486	C₂²	1	2	yes	—
18	C₂₃×#466	—	2	1	no	—
19	C₂₃×#466	—	2	1	no	—
20	C₂₃×#464	—	2	2	no	—
21	C₂₃×#463	—	2	2	no	—
22	C₂₃×#462	C₂²	1	2	yes	—
23	C₂₃×#461	C₂²	1	2	yes	—
24	C₂₃×#460	C₂²	1	2	yes	—
25	C₂₃×#459	C₂²	1	2	yes	—
26	C₂₃×#446	C₂²	1	1	yes	characteristic
27	C₂₃×#446	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₂₃	—	2	1	no	—
29	C₄×D₈×C₂₃	—	2	1	no	—
30	C₂₃×#436	C₂²	1	1	yes	characteristic
31	C₂₃×#436	C₂²	1	1	yes	characteristic
32	C₂₃×#435	—	2	1	no	—
33	C₂₃×#435	—	2	1	no	—
34	C₂₃×#431	—	2	1	no	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
37	C₂₃×#416	—	2	2	no	—
38	C₄×M₁₆×C₂₃	—	2	1	no	—
39	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#334	C₂²	1	1	yes	characteristic
41	C₂₃×#334	C₂²	1	1	yes	characteristic
42	C₂₃×#334	C₂²	1	2	yes	—
43	C₂₃×#332	C₂²	1	2	yes	—
44	C₂₃×#332	C₂²	1	2	yes	—
45	C₂₃×#331	C₂²	1	2	yes	—
46	C₂₃×#331	C₂²	1	2	yes	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
49	C₂₃×#135	—	4	8	no	—
50	C₂×Q₁₆×C₂₃	—	2	2	no	—
51	C₂×SD₁₆×C₂₃	—	2	2	no	—
52	C₂×SD₁₆×C₂₃	—	2	2	no	—
53	C₂×D₈×C₂₃	—	2	2	no	—
54	C₂×D₈×C₂₃	—	2	1	no	—
55	C₂×D₈×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	—	2	1	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₂₃×#127	C₂³	1	2	yes	—
63	C₂₃×#124	—	2	2	no	—
64	C₂₃×#121	—	2	2	no	—
65	C₂₃×#121	—	2	2	no	—
66	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
67	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₂₃	—	2	1	no	—
69	C₄×D₄×C₂₃	—	2	1	no	—
70	C₄×D₄×C₂₃	—	2	2	no	—
71	C₄×D₄×C₂₃	—	2	2	no	—
72	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
73	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#117	—	2	1	no	—
75	C₂₃×#117	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₂₃	—	2	1	no	—
77	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂₃×#107	—	2	1	no	—
79	C₂₃×#107	—	2	1	no	—
80	C₂₃×C₄⋊C₈	—	2	1	no	—
81	C₂₃×C₄⋊C₈	—	2	1	no	—
82	C₂₃×C₄⋊C₈	—	2	2	no	—
83	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
86	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
87	C₂₃×#103	—	2	2	no	—
88	C₂₃×#103	—	2	2	no	—
89	C₂₃×#102	—	2	2	no	—
90	C₂₃×#102	—	2	2	no	—
91	C₂₃×#102	—	2	2	no	—
92	C₂₃×#102	—	2	2	no	—
93	C₂₃×C₂²⋊C₈	—	2	1	no	—
94	C₂₃×C₂²⋊C₈	—	2	1	no	—
95	C₂₃×C₂²⋊C₈	—	2	1	no	—
96	C₂₃×#97	—	2	1	no	—
97	C₂₃×#97	—	2	1	no	—
98	C₄×C₈×C₂₃	—	2	1	no	—
99	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
100	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
101	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
103	#41×C₂₃	—	2	4	no	—
104	#41×C₂₃	—	4	4	no	—
105	#41×C₂₃	—	4	4	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₂₃	—	2	1	no	—
109	C₂×D₄×C₂₃	—	2	2	no	—
110	C₂×D₄×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	2	2	no	—
112	C₂×D₄×C₂₃	—	4	2	no	—
113	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	—	2	1	no	—
115	C₂²×C₄×C₂₃	—	2	1	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	4	2	no	—
118	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
121	Q₁₆×C₂₃	—	4	4	no	—
122	SD₁₆×C₂₃	—	4	4	no	—
123	SD₁₆×C₂₃	—	4	4	no	—
124	D₈×C₂₃	—	4	2	no	—
125	D₈×C₂₃	—	4	4	no	—
126	D₈×C₂₃	—	4	2	no	—
127	M₁₆×C₂₃	—	4	2	no	—
128	M₁₆×C₂₃	—	4	2	no	—
129	C₂×C₈×C₂₃	—	2	2	no	—
130	C₂×C₈×C₂₃	—	2	2	no	—
131	C₂×C₈×C₂₃	—	2	2	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	—	4	1	no	—
135	C₂×C₈×C₂₃	—	4	4	no	—
136	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
150	C₂²⋊C₄×C₂₃	—	4	2	no	—
151	C₄²×C₂₃	—	2	1	no	—
152	C₄²×C₂₃	—	2	1	no	—
153	C₄²×C₂₃	—	2	1	no	—
154	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₂₃	#41	1	1	yes	characteristic
157	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
158	#19157	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₂₃	—	2	1	no	—
160	C₂³×C₂₃	—	4	2	no	—
161	C₂³×C₂₃	#121	1	1	yes	characteristic
162	Q₈×C₂₃	—	2	2	no	—
163	Q₈×C₂₃	—	4	2	no	—
164	Q₈×C₂₃	—	4	2	no	—
165	D₄×C₂₃	—	2	2	no	—
166	D₄×C₂₃	—	4	2	no	—
167	D₄×C₂₃	—	4	2	no	—
168	D₄×C₂₃	—	4	2	no	—
169	D₄×C₂₃	—	4	2	no	—
170	D₄×C₂₃	—	4	2	no	—
171	D₄×C₂₃	—	8	2	no	—
172	D₄×C₂₃	—	8	4	no	—
173	C₂×C₄×C₂₃	—	2	1	no	—
174	C₂×C₄×C₂₃	—	2	1	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	4	1	no	—
179	C₂×C₄×C₂₃	—	4	2	no	—
180	C₂×C₄×C₂₃	—	4	1	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	8	4	no	—
185	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
186	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
187	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
188	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
189	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
192	C₈×C₂₃	—	2	1	no	—
193	C₈×C₂₃	—	2	1	no	—
194	C₈×C₂₃	—	2	1	no	—
195	C₈×C₂₃	—	4	1	no	—
196	C₈×C₂₃	—	4	2	no	—
197	C₈×C₂₃	—	4	2	no	—
198	C₈×C₂₃	—	4	1	no	—
199	C₈×C₂₃	#135	1	2	yes	complemented
200	#3169	C₂×C₂₃	1	1	yes	characteristic, complemented
201	#3035	C₂×C₂₃	1	2	yes	complemented
202	#2965	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#1702	C₂×C₂₃	1	2	yes	complemented
204	#1701	C₂×C₂₃	1	2	yes	complemented
205	#1694	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#1693	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#1688	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#1687	C₂×C₂₃	1	1	yes	characteristic, complemented
209	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#1540	C₂×C₂₃	1	2	yes	complemented
211	C₂²×C₂₃	—	2	1	no	—
212	C₂²×C₂₃	—	2	1	no	—
213	C₂²×C₂₃	—	4	1	no	—
214	C₂²×C₂₃	—	4	1	no	—
215	C₂²×C₂₃	—	4	1	no	—
216	C₂²×C₂₃	—	8	2	no	—
217	C₂²×C₂₃	—	8	2	no	—
218	C₂²×C₂₃	—	8	2	no	—
219	C₂²×C₂₃	#469	1	2	yes	—
220	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
221	C₄×C₂₃	—	2	1	no	—
222	C₄×C₂₃	—	2	1	no	—
223	C₄×C₂₃	—	2	1	no	—
224	C₄×C₂₃	—	4	1	no	—
225	C₄×C₂₃	—	4	1	no	—
226	C₄×C₂₃	—	4	1	no	—
227	C₄×C₂₃	—	8	2	no	—
228	C₄×C₂₃	#469	1	2	yes	—
229	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
230	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
231	C₄×C₂₃	#575	1	1	yes	characteristic
232	C₂×#135	—	2	2	no	—
233	#532	C₂²×C₂₃	1	2	yes	—
234	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
235	#500	C₂²×C₂₃	1	1	yes	characteristic
236	#493	C₂²×C₂₃	1	1	yes	characteristic
237	#486	C₂²×C₂₃	1	2	yes	—
238	#466	—	2	1	no	—
239	#466	—	2	1	no	—
240	#464	—	2	2	no	—
241	#463	—	2	2	no	—
242	#462	C₂²×C₂₃	1	2	yes	—
243	#461	C₂²×C₂₃	1	2	yes	—
244	#460	C₂²×C₂₃	1	2	yes	—
245	#459	C₂²×C₂₃	1	2	yes	—
246	#446	C₂²×C₂₃	1	1	yes	characteristic
247	#446	C₂²×C₂₃	1	1	yes	characteristic
248	C₄×D₈	—	2	1	no	—
249	C₄×D₈	—	2	1	no	—
250	#436	C₂²×C₂₃	1	1	yes	characteristic
251	#436	C₂²×C₂₃	1	1	yes	characteristic
252	#435	—	2	1	no	—
253	#435	—	2	1	no	—
254	#431	—	2	1	no	—
255	#431	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
257	#416	—	2	2	no	—
258	C₄×M₁₆	—	2	1	no	—
259	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
260	#334	C₂²×C₂₃	1	1	yes	characteristic
261	#334	C₂²×C₂₃	1	1	yes	characteristic
262	#334	C₂²×C₂₃	1	2	yes	—
263	#332	C₂²×C₂₃	1	2	yes	—
264	#332	C₂²×C₂₃	1	2	yes	—
265	#331	C₂²×C₂₃	1	2	yes	—
266	#331	C₂²×C₂₃	1	2	yes	—
267	C₂×C₂₃	—	2	2	no	—
268	C₂×C₂₃	—	8	1	no	—
269	C₂×C₂₃	—	16	2	no	—
270	C₂×C₂₃	#3035	1	1	yes	characteristic
271	C₂×C₂₃	C₂×#469	1	1	yes	characteristic
272	C₂×C₂₃	#3067	1	1	yes	characteristic
273	C₂×#41	—	2	2	no	—
274	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
275	#135	—	4	8	no	—
276	C₂×Q₁₆	—	2	2	no	—
277	C₂×SD₁₆	—	2	2	no	—
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×D₈	—	2	2	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	—	2	1	no	—
282	C₂×M₁₆	—	2	1	no	—
283	C₂×M₁₆	—	2	1	no	—
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
286	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
287	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
288	#127	C₂³×C₂₃	1	2	yes	—
289	#124	—	2	2	no	—
290	#121	—	2	2	no	—
291	#121	—	2	2	no	—
292	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
293	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
294	C₄×D₄	—	2	1	no	—
295	C₄×D₄	—	2	1	no	—
296	C₄×D₄	—	2	2	no	—
297	C₄×D₄	—	2	2	no	—
298	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
299	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
300	#117	—	2	1	no	—
301	#117	D₄×C₂₃	1	1	yes	characteristic
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
304	#107	—	2	1	no	—
305	#107	—	2	1	no	—
306	C₄⋊C₈	—	2	1	no	—
307	C₄⋊C₈	—	2	1	no	—
308	C₄⋊C₈	—	2	2	no	—
309	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
311	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
312	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
313	#103	—	2	2	no	—
314	#103	—	2	2	no	—
315	#102	—	2	2	no	—
316	#102	—	2	2	no	—
317	#102	—	2	2	no	—
318	#102	—	2	2	no	—
319	C₂²⋊C₈	—	2	1	no	—
320	C₂²⋊C₈	—	2	1	no	—
321	C₂²⋊C₈	—	2	1	no	—
322	#97	—	2	1	no	—
323	#97	—	2	1	no	—
324	C₄×C₈	—	2	1	no	—
325	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
326	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
327	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
329	C₂₃	#19157	1	1	yes	characteristic, direct product factor, Sylow
330	#41	—	2	4	no	—
331	#41	—	4	4	no	—
332	#41	—	4	4	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
347	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
348	Q₁₆	—	4	4	no	—
349	SD₁₆	—	4	4	no	—
350	SD₁₆	—	4	4	no	—
351	D₈	—	4	2	no	—
352	D₈	—	4	4	no	—
353	D₈	—	4	2	no	—
354	M₁₆	—	4	2	no	—
355	M₁₆	—	4	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	4	1	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
366	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
373	C₂²⋊C₄	—	2	1	no	—
374	C₂²⋊C₄	—	2	1	no	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
382	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
383	C₄²	#41×C₂₃	1	1	yes	characteristic
384	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
385	C₂³	—	2	1	no	—
386	C₂³	—	4	2	no	—
387	C₂³	C₂₃×#121	1	1	yes	characteristic
388	Q₈	—	2	2	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	4	2	no	—
391	D₄	—	2	2	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	8	2	no	—
398	D₄	—	8	4	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	8	4	no	—
411	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
412	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
413	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
414	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
415	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
418	C₈	—	2	1	no	—
419	C₈	—	2	1	no	—
420	C₈	—	2	1	no	—
421	C₈	—	4	1	no	—
422	C₈	—	4	2	no	—
423	C₈	—	4	2	no	—
424	C₈	—	4	1	no	—
425	C₈	C₂₃×#135	1	2	yes	complemented
426	C₂²	—	2	1	no	—
427	C₂²	—	2	1	no	—
428	C₂²	—	4	1	no	—
429	C₂²	—	4	1	no	—
430	C₂²	—	4	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	C₂₃×#469	1	2	yes	—
435	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	4	1	no	—
442	C₄	—	8	2	no	—
443	C₄	C₂₃×#469	1	2	yes	—
444	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
445	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
446	C₄	C₂₃×#575	1	1	yes	characteristic
447	C₂	—	2	2	no	—
448	C₂	—	8	1	no	—
449	C₂	—	16	2	no	—
450	C₂	C₂₃×#3035	1	1	yes	characteristic
451	C₂	C₂×C₂₃×#469	1	1	yes	characteristic
452	C₂	C₂₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3169	C₂	semidirect
2. C₂₃×#3035	C₂	semidirect
3. C₂₃×#2965	C₂	semidirect
4. C₂₃×#1702	C₂	semidirect
5. C₂₃×#1701	C₂	semidirect
6. C₂₃×#1694	C₂	semidirect
7. C₂₃×#1693	C₂	semidirect
8. C₂₃×#1688	C₂	semidirect
9. C₂₃×#1687	C₂	semidirect
10. C₂×C₂₃×#334	C₂	semidirect
11. C₂₃×#1540	C₂	semidirect
158. #19157	C₂₃	direct (N × H)
199. C₈×C₂₃	#135	semidirect
200. #3169	C₂×C₂₃	semidirect
201. #3035	C₂×C₂₃	semidirect
202. #2965	C₂×C₂₃	semidirect
203. #1702	C₂×C₂₃	semidirect
204. #1701	C₂×C₂₃	semidirect
205. #1694	C₂×C₂₃	semidirect
206. #1693	C₂×C₂₃	semidirect
207. #1688	C₂×C₂₃	semidirect
208. #1687	C₂×C₂₃	semidirect
209. C₂×#334	C₂×C₂₃	semidirect
210. #1540	C₂×C₂₃	semidirect
329. C₂₃	#19157	direct (N × H)
425. C₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3169	2
C₂₃×#3035	2
C₂₃×#2965	2
C₂₃×#1702	2
C₂₃×#1701	2
C₂₃×#1694	2
C₂₃×#1693	2
C₂₃×#1688	2
C₂₃×#1687	2
C₂×C₂₃×#334	2
C₂₃×#1540	2
#19157	23

Subgroups — C23×#19157 #19166783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19157 #19166783