Subgroups — C₂₃×#19202 #19166828

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1686	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1666	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
10	C₂²×SD₁₆×C₂₃	—	2	1	no	—
11	C₂²×SD₁₆×C₂₃	—	2	1	no	—
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
15	C₂₃×#498	—	2	1	no	—
16	C₂₃×#492	—	2	1	no	—
17	C₂₃×#492	—	2	1	no	—
18	C₂₃×#492	—	2	1	no	—
19	C₂₃×#492	C₂²	1	2	yes	—
20	C₂₃×#492	C₂²	1	2	yes	—
21	C₂₃×#487	—	2	1	no	—
22	C₂₃×#486	—	2	1	no	—
23	C₂₃×#483	—	2	2	no	—
24	C₂₃×#475	C₂²	1	4	yes	complemented
25	C₂₃×#460	C₂²	1	4	yes	—
26	C₂₃×#450	—	2	2	no	—
27	C₂₃×#446	—	2	1	no	—
28	C₂₃×#446	C₂²	1	2	yes	complemented
29	C₈×Q₈×C₂₃	—	2	1	no	—
30	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
31	C₂₃×#431	C₂²	1	1	yes	characteristic
32	C₂₃×#431	C₂²	1	1	yes	characteristic
33	C₂₃×#424	—	2	1	no	—
34	C₂₃×#423	C₂²	1	1	yes	characteristic
35	C₂₃×#405	—	2	1	no	—
36	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#335	C₂²	1	2	yes	complemented
38	C₂₃×#334	C₂²	1	2	yes	complemented
39	C₂₃×#332	C₂²	1	4	yes	complemented
40	C₂₃×#331	C₂²	1	4	yes	—
41	C₂²×Q₈×C₂₃	—	2	1	no	—
42	C₂²×D₄×C₂₃	—	2	1	no	—
43	C₂²×D₄×C₂₃	—	2	1	no	—
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂×SD₁₆×C₂₃	—	2	4	no	—
46	C₂×SD₁₆×C₂₃	—	2	2	no	—
47	C₂×SD₁₆×C₂₃	—	2	2	no	—
48	C₂×SD₁₆×C₂₃	—	2	2	no	—
49	C₂×SD₁₆×C₂₃	—	2	2	no	—
50	C₂×SD₁₆×C₂₃	—	4	4	no	—
51	C₂×SD₁₆×C₂₃	—	4	4	no	—
52	C₂×M₁₆×C₂₃	—	2	1	no	—
53	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
59	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₂₃×#127	—	2	1	no	—
62	C₂₃×#127	—	2	1	no	—
63	C₂₃×#127	—	2	1	no	—
64	C₂₃×#127	C₂³	1	2	yes	—
65	C₂₃×#125	—	2	1	no	—
66	C₂₃×C₂²⋊Q₈	—	2	1	no	—
67	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
68	C₄×Q₈×C₂₃	—	2	1	no	—
69	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
70	C₂₃×#117	—	2	1	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#106	—	2	4	no	—
75	C₂₃×C₄⋊C₈	—	2	2	no	—
76	C₂₃×C₄⋊C₈	—	2	2	no	—
77	C₂₃×C₄⋊C₈	—	2	1	no	—
78	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
79	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
80	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
81	C₂₃×#103	—	2	4	no	—
82	C₂₃×#102	—	2	4	no	—
83	C₂₃×C₂²⋊C₈	—	2	1	no	—
84	C₂₃×C₂²⋊C₈	—	2	1	no	—
85	C₂₃×#97	—	2	2	no	—
86	C₄×C₈×C₂₃	—	2	2	no	—
87	C₄×C₈×C₂₃	—	2	1	no	—
88	C₄×C₈×C₂₃	—	2	1	no	—
89	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
91	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
93	C₂⁴×C₂₃	—	4	1	no	—
94	C₂×Q₈×C₂₃	—	2	1	no	—
95	C₂×Q₈×C₂₃	—	2	2	no	—
96	C₂×Q₈×C₂₃	—	4	1	no	—
97	C₂×Q₈×C₂₃	—	4	2	no	—
98	C₂×Q₈×C₂₃	—	4	2	no	—
99	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
100	C₂×D₄×C₂₃	—	2	2	no	—
101	C₂×D₄×C₂₃	—	2	2	no	—
102	C₂×D₄×C₂₃	—	2	2	no	—
103	C₂×D₄×C₂₃	—	4	1	no	—
104	C₂×D₄×C₂₃	—	4	1	no	—
105	C₂×D₄×C₂₃	—	4	1	no	—
106	C₂×D₄×C₂₃	—	4	2	no	—
107	C₂×D₄×C₂₃	—	4	2	no	—
108	C₂×D₄×C₂₃	—	8	2	no	—
109	C₂×D₄×C₂₃	—	8	2	no	—
110	C₂²×C₄×C₂₃	—	4	1	no	—
111	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
114	SD₁₆×C₂₃	—	4	4	no	—
115	SD₁₆×C₂₃	—	4	4	no	—
116	SD₁₆×C₂₃	—	4	4	no	—
117	SD₁₆×C₂₃	—	4	4	no	—
118	SD₁₆×C₂₃	—	8	4	no	—
119	M₁₆×C₂₃	—	4	2	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	SD₁₆	1	4	yes	complemented
125	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₂₃	—	2	1	no	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	—	4	1	no	—
137	C₄⋊C₄×C₂₃	—	4	1	no	—
138	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₂₃	—	2	2	no	—
140	C₄²×C₂₃	—	2	2	no	—
141	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₂₃	#41	1	1	yes	characteristic
144	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
145	#19202	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₂₃	—	4	2	no	—
147	C₂³×C₂₃	—	8	2	no	—
148	C₂³×C₂₃	—	8	2	no	—
149	C₂³×C₂₃	—	8	2	no	—
150	C₂³×C₂₃	#121	1	1	yes	characteristic
151	Q₈×C₂₃	—	2	4	no	—
152	Q₈×C₂₃	—	4	2	no	—
153	Q₈×C₂₃	—	4	2	no	—
154	Q₈×C₂₃	—	4	2	no	—
155	Q₈×C₂₃	—	8	2	no	—
156	D₄×C₂₃	—	4	2	no	—
157	D₄×C₂₃	—	4	2	no	—
158	D₄×C₂₃	—	4	4	no	—
159	D₄×C₂₃	—	8	2	no	—
160	D₄×C₂₃	—	8	2	no	—
161	D₄×C₂₃	—	8	2	no	—
162	D₄×C₂₃	—	8	2	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	4	2	no	—
168	C₂×C₄×C₂₃	—	8	2	no	—
169	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
170	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
171	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
173	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
177	C₈×C₂₃	—	2	1	no	—
178	C₈×C₂₃	—	2	1	no	—
179	C₈×C₂₃	—	4	2	no	—
180	C₈×C₂₃	—	4	2	no	—
181	C₈×C₂₃	—	4	2	no	—
182	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
183	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
184	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
185	#3208	C₂×C₂₃	1	1	yes	characteristic
186	C₂×#492	C₂×C₂₃	1	1	yes	characteristic
187	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
188	#1686	C₂×C₂₃	1	4	yes	complemented
189	#1666	C₂×C₂₃	1	4	yes	complemented
190	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
191	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
192	#1545	C₂×C₂₃	1	1	yes	characteristic
193	C₂²×C₂₃	—	2	1	no	—
194	C₂²×C₂₃	—	2	1	no	—
195	C₂²×C₂₃	—	8	2	no	—
196	C₂²×C₂₃	—	8	2	no	—
197	C₂²×C₂₃	—	8	2	no	—
198	C₂²×C₂₃	—	16	4	no	—
199	C₂²×C₂₃	#460	1	2	yes	—
200	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
201	C₄×C₂₃	—	2	1	no	—
202	C₄×C₂₃	—	2	1	no	—
203	C₄×C₂₃	—	4	2	no	—
204	C₄×C₂₃	—	4	2	no	—
205	C₄×C₂₃	—	8	2	no	—
206	C₄×C₂₃	#460	1	2	yes	—
207	C₄×C₂₃	#471	1	2	yes	—
208	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
209	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₄×C₂₃	#577	1	2	yes	—
211	C₂²×SD₁₆	—	2	1	no	—
212	C₂²×SD₁₆	—	2	1	no	—
213	#533	C₂²×C₂₃	1	1	yes	characteristic
214	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
215	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
216	#498	—	2	1	no	—
217	#492	—	2	1	no	—
218	#492	—	2	1	no	—
219	#492	—	2	1	no	—
220	#492	C₂²×C₂₃	1	2	yes	—
221	#492	C₂²×C₂₃	1	2	yes	—
222	#487	—	2	1	no	—
223	#486	—	2	1	no	—
224	#483	—	2	2	no	—
225	#475	C₂²×C₂₃	1	4	yes	complemented
226	#460	C₂²×C₂₃	1	4	yes	—
227	#450	—	2	2	no	—
228	#446	—	2	1	no	—
229	#446	C₂²×C₂₃	1	2	yes	complemented
230	C₈×Q₈	—	2	1	no	—
231	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
232	#431	C₂²×C₂₃	1	1	yes	characteristic
233	#431	C₂²×C₂₃	1	1	yes	characteristic
234	#424	—	2	1	no	—
235	#423	C₂²×C₂₃	1	1	yes	characteristic
236	#405	—	2	1	no	—
237	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
238	#335	C₂²×C₂₃	1	2	yes	complemented
239	#334	C₂²×C₂₃	1	2	yes	complemented
240	#332	C₂²×C₂₃	1	4	yes	complemented
241	#331	C₂²×C₂₃	1	4	yes	—
242	C₂×C₂₃	—	2	2	no	—
243	C₂×C₂₃	—	16	2	no	—
244	C₂×C₂₃	C₂×#460	1	1	yes	characteristic
245	C₂×C₂₃	#3057	1	1	yes	characteristic
246	C₂×C₂₃	C₂×#471	1	1	yes	characteristic
247	C₂²×Q₈	—	2	1	no	—
248	C₂²×D₄	—	2	1	no	—
249	C₂²×D₄	—	2	1	no	—
250	C₂²×D₄	—	2	1	no	—
251	C₂×SD₁₆	—	2	4	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×SD₁₆	—	2	2	no	—
256	C₂×SD₁₆	—	4	4	no	—
257	C₂×SD₁₆	—	4	4	no	—
258	C₂×M₁₆	—	2	1	no	—
259	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
260	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
261	C₄⋊Q₈	—	2	1	no	—
262	C₄⋊Q₈	—	2	1	no	—
263	C₄⋊Q₈	—	2	1	no	—
264	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
265	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
266	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
267	#127	—	2	1	no	—
268	#127	—	2	1	no	—
269	#127	—	2	1	no	—
270	#127	C₂³×C₂₃	1	2	yes	—
271	#125	—	2	1	no	—
272	C₂²⋊Q₈	—	2	1	no	—
273	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
274	C₄×Q₈	—	2	1	no	—
275	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
276	#117	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
280	#106	—	2	4	no	—
281	C₄⋊C₈	—	2	2	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
285	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
286	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
287	#103	—	2	4	no	—
288	#102	—	2	4	no	—
289	C₂²⋊C₈	—	2	1	no	—
290	C₂²⋊C₈	—	2	1	no	—
291	#97	—	2	2	no	—
292	C₄×C₈	—	2	2	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
296	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
298	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
299	C₂₃	#19202	1	1	yes	characteristic, direct product factor, Sylow
300	C₂⁴	—	4	1	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	4	1	no	—
304	C₂×Q₈	—	4	2	no	—
305	C₂×Q₈	—	4	2	no	—
306	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	1	no	—
312	C₂×D₄	—	4	1	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	8	2	no	—
316	C₂×D₄	—	8	2	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
320	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
321	SD₁₆	—	4	4	no	—
322	SD₁₆	—	4	4	no	—
323	SD₁₆	—	4	4	no	—
324	SD₁₆	—	4	4	no	—
325	SD₁₆	—	8	4	no	—
326	M₁₆	—	4	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	SD₁₆×C₂₃	1	4	yes	complemented
332	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₄⋊C₄	—	4	1	no	—
345	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
346	C₂²⋊C₄	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
349	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
350	C₄²	#41×C₂₃	1	1	yes	characteristic
351	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
352	C₂³	—	4	2	no	—
353	C₂³	—	8	2	no	—
354	C₂³	—	8	2	no	—
355	C₂³	—	8	2	no	—
356	C₂³	C₂₃×#121	1	1	yes	characteristic
357	Q₈	—	2	4	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	4	2	no	—
361	Q₈	—	8	2	no	—
362	D₄	—	4	2	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	4	no	—
365	D₄	—	8	2	no	—
366	D₄	—	8	2	no	—
367	D₄	—	8	2	no	—
368	D₄	—	8	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	2	no	—
375	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
376	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
377	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
378	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
379	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
380	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
383	C₈	—	2	1	no	—
384	C₈	—	2	1	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
389	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	8	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	16	4	no	—
396	C₂²	C₂₃×#460	1	2	yes	—
397	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₂₃×#460	1	2	yes	—
404	C₄	C₂₃×#471	1	2	yes	—
405	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
406	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
407	C₄	C₂₃×#577	1	2	yes	—
408	C₂	—	2	2	no	—
409	C₂	—	16	2	no	—
410	C₂	C₂×C₂₃×#460	1	1	yes	characteristic
411	C₂	C₂₃×#3057	1	1	yes	characteristic
412	C₂	C₂×C₂₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3229	C₂	semidirect
4. C₂₃×#2960	C₂	semidirect
5. C₂₃×#1686	C₂	semidirect
6. C₂₃×#1666	C₂	semidirect
7. C₂₃×#1564	C₂	semidirect
8. C₂₃×#1546	C₂	semidirect
24. C₂₃×#475	C₂²	semidirect
28. C₂₃×#446	C₂²	semidirect
30. C₈×Q₈×C₂₃	C₂²	semidirect
37. C₂₃×#335	C₂²	semidirect
38. C₂₃×#334	C₂²	semidirect
39. C₂₃×#332	C₂²	semidirect
124. C₂×C₈×C₂₃	SD₁₆	semidirect
145. #19202	C₂₃	direct (N × H)
184. #3229	C₂×C₂₃	semidirect
187. #2960	C₂×C₂₃	semidirect
188. #1686	C₂×C₂₃	semidirect
189. #1666	C₂×C₂₃	semidirect
190. #1564	C₂×C₂₃	semidirect
191. #1546	C₂×C₂₃	semidirect
225. #475	C₂²×C₂₃	semidirect
229. #446	C₂²×C₂₃	semidirect
231. C₈×Q₈	C₂²×C₂₃	semidirect
238. #335	C₂²×C₂₃	semidirect
239. #334	C₂²×C₂₃	semidirect
240. #332	C₂²×C₂₃	semidirect
299. C₂₃	#19202	direct (N × H)
331. C₂×C₈	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3229	2
C₂₃×#3208	2
C₂×C₂₃×#492	2
C₂₃×#2960	2
C₂₃×#1686	2
C₂₃×#1666	2
C₂₃×#1564	2
C₂₃×#1546	2
C₂₃×#1545	2
#19202	23

Subgroups — C23×#19202 #19166828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19202 #19166828