Subgroups — C₂₃×#19244 #19166870

Order 5888.

594 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3246	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3141	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1704	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1698	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1684	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1678	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1540	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#533	C₂²	1	1	yes	characteristic
13	C₂₃×#532	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₃×#500	—	2	1	no	—
16	C₂₃×#489	—	2	2	no	—
17	C₂₃×#487	—	2	1	no	—
18	C₂₃×#487	C₂²	1	2	yes	—
19	C₂₃×#486	C₂²	1	2	yes	—
20	C₂₃×#484	—	2	2	no	—
21	C₂₃×#477	C₂²	1	2	yes	—
22	C₂₃×#477	C₂²	1	2	yes	—
23	C₂₃×#476	—	2	1	no	—
24	C₂₃×#475	C₂²	1	2	yes	—
25	C₂₃×#465	—	2	2	no	—
26	C₂₃×#461	—	2	1	no	—
27	C₂₃×#460	C₂²	1	2	yes	—
28	C₂₃×#446	—	2	2	no	—
29	C₂₃×#446	C₂²	1	2	yes	—
30	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
31	C₂₃×#431	C₂²	1	1	yes	characteristic
32	C₂₃×#424	—	2	1	no	—
33	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂₃×#417	—	2	1	no	—
36	C₂₃×#417	—	2	1	no	—
37	C₂₃×#405	—	2	1	no	—
38	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#335	C₂²	1	2	yes	—
40	C₂₃×#334	C₂²	1	2	yes	—
41	C₂₃×#333	C₂²	1	2	yes	—
42	C₂₃×#333	C₂²	1	2	yes	—
43	C₂₃×#332	C₂²	1	1	yes	characteristic
44	C₂₃×#332	C₂²	1	1	yes	characteristic
45	C₂₃×#331	C₂²	1	1	yes	characteristic
46	C₂₃×#331	C₂²	1	1	yes	characteristic
47	C₂×#41×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	—	2	1	no	—
49	C₂×SD₁₆×C₂₃	—	2	2	no	—
50	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
51	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₂₃	—	2	1	no	—
53	C₂²×C₈×C₂₃	—	2	1	no	—
54	C₂₃×C₄⋊Q₈	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
59	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
61	C₂₃×#127	C₂³	1	1	yes	characteristic
62	C₂₃×#125	—	2	1	no	—
63	C₂₃×#124	—	2	1	no	—
64	C₂₃×C₂²⋊Q₈	—	2	1	no	—
65	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
66	C₂₃×#121	—	2	1	no	—
67	C₂₃×#121	—	2	1	no	—
68	C₄×Q₈×C₂₃	—	2	1	no	—
69	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
70	C₄×D₄×C₂₃	—	2	1	no	—
71	C₄×D₄×C₂₃	—	2	1	no	—
72	C₂₃×#117	—	2	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×#107	—	2	2	no	—
77	C₂₃×#107	—	2	2	no	—
78	C₂₃×#106	—	2	2	no	—
79	C₂₃×C₄⋊C₈	—	2	2	no	—
80	C₂₃×C₄⋊C₈	—	2	2	no	—
81	C₂₃×C₄⋊C₈	—	2	1	no	—
82	C₂₃×C₄⋊C₈	—	2	1	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	1	yes	characteristic
86	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
88	C₂₃×#103	—	2	2	no	—
89	C₂₃×#103	—	2	2	no	—
90	C₂₃×#103	—	2	2	no	—
91	C₂₃×#103	—	2	2	no	—
92	C₂₃×#103	—	2	2	no	—
93	C₂₃×#102	—	2	2	no	—
94	C₂₃×#102	—	2	2	no	—
95	C₂₃×#102	—	2	2	no	—
96	C₂₃×C₂²⋊C₈	—	2	1	no	—
97	C₂₃×#97	—	2	2	no	—
98	C₂₃×#97	D₄	1	2	yes	—
99	C₄×C₈×C₂₃	—	2	2	no	—
100	C₄×C₈×C₂₃	—	2	1	no	—
101	C₄×C₈×C₂₃	C₂³	1	2	yes	—
102	#41×C₂₃	—	4	2	no	—
103	#41×C₂₃	—	4	2	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	4	1	no	—
108	C₂×Q₈×C₂₃	—	8	2	no	—
109	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×D₄×C₂₃	—	2	1	no	—
111	C₂×D₄×C₂₃	—	2	1	no	—
112	C₂×D₄×C₂₃	—	2	1	no	—
113	C₂×D₄×C₂₃	—	4	1	no	—
114	C₂²×C₄×C₂₃	—	4	1	no	—
115	C₂²×C₄×C₂₃	—	4	1	no	—
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
119	SD₁₆×C₂₃	—	8	2	no	—
120	M₁₆×C₂₃	—	2	2	no	—
121	M₁₆×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	2	1	no	—
125	C₂×C₈×C₂₃	—	2	1	no	—
126	C₂×C₈×C₂₃	—	2	1	no	—
127	C₂×C₈×C₂₃	—	4	2	no	—
128	C₂×C₈×C₂₃	—	8	1	no	—
129	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	1	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	—	2	2	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	4	1	no	—
141	C₄⋊C₄×C₂₃	—	4	1	no	—
142	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
143	C₂²⋊C₄×C₂₃	—	2	2	no	—
144	C₂²⋊C₄×C₂₃	—	4	1	no	—
145	C₂²⋊C₄×C₂₃	—	4	1	no	—
146	C₄²×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₂₃	#41	1	1	yes	characteristic
150	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
151	#19244	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₂₃	—	4	1	no	—
153	C₂³×C₂₃	#121	1	1	yes	characteristic
154	Q₈×C₂₃	—	2	2	no	—
155	Q₈×C₂₃	—	2	2	no	—
156	Q₈×C₂₃	—	4	1	no	—
157	Q₈×C₂₃	—	4	1	no	—
158	Q₈×C₂₃	—	8	1	no	—
159	Q₈×C₂₃	—	8	2	no	—
160	Q₈×C₂₃	—	8	2	no	—
161	D₄×C₂₃	—	4	1	no	—
162	D₄×C₂₃	—	4	1	no	—
163	D₄×C₂₃	—	4	1	no	—
164	D₄×C₂₃	—	4	1	no	—
165	D₄×C₂₃	—	8	1	no	—
166	D₄×C₂₃	—	8	2	no	—
167	C₂×C₄×C₂₃	—	2	1	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	4	1	no	—
172	C₂×C₄×C₂₃	—	4	1	no	—
173	C₂×C₄×C₂₃	—	4	2	no	—
174	C₂×C₄×C₂₃	—	8	2	no	—
175	C₂×C₄×C₂₃	—	8	2	no	—
176	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
177	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
178	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₂₃	#135	1	2	yes	—
180	C₂×C₄×C₂₃	#137	1	2	yes	—
181	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
184	C₈×C₂₃	—	2	2	no	—
185	C₈×C₂₃	—	2	2	no	—
186	C₈×C₂₃	—	4	1	no	—
187	C₈×C₂₃	—	4	1	no	—
188	C₈×C₂₃	—	4	1	no	—
189	C₈×C₂₃	—	4	2	no	—
190	C₈×C₂₃	—	4	1	no	—
191	#3246	C₂×C₂₃	1	1	yes	characteristic
192	#3230	C₂×C₂₃	1	1	yes	characteristic, complemented
193	#3141	C₂×C₂₃	1	1	yes	characteristic
194	#2962	C₂×C₂₃	1	1	yes	characteristic, complemented
195	#1704	C₂×C₂₃	1	2	yes	complemented
196	#1698	C₂×C₂₃	1	2	yes	complemented
197	#1684	C₂×C₂₃	1	2	yes	complemented
198	#1678	C₂×C₂₃	1	2	yes	complemented
199	#1565	C₂×C₂₃	1	1	yes	characteristic, complemented
200	C₂×#333	C₂×C₂₃	1	1	yes	characteristic, complemented
201	#1540	C₂×C₂₃	1	1	yes	characteristic
202	C₂²×C₂₃	—	2	1	no	—
203	C₂²×C₂₃	—	2	1	no	—
204	C₂²×C₂₃	—	8	1	no	—
205	C₂²×C₂₃	—	8	1	no	—
206	C₂²×C₂₃	—	8	1	no	—
207	C₂²×C₂₃	#464	1	2	yes	—
208	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
209	C₄×C₂₃	—	2	1	no	—
210	C₄×C₂₃	—	2	1	no	—
211	C₄×C₂₃	—	2	1	no	—
212	C₄×C₂₃	—	4	2	no	—
213	C₄×C₂₃	—	4	2	no	—
214	C₄×C₂₃	—	8	1	no	—
215	C₄×C₂₃	—	8	2	no	—
216	C₄×C₂₃	#464	1	2	yes	—
217	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
218	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
219	C₄×C₂₃	#577	1	1	yes	characteristic
220	C₄×C₂₃	#578	1	1	yes	characteristic
221	#533	C₂²×C₂₃	1	1	yes	characteristic
222	#532	C₂²×C₂₃	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
224	#500	—	2	1	no	—
225	#489	—	2	2	no	—
226	#487	—	2	1	no	—
227	#487	C₂²×C₂₃	1	2	yes	—
228	#486	C₂²×C₂₃	1	2	yes	—
229	#484	—	2	2	no	—
230	#477	C₂²×C₂₃	1	2	yes	—
231	#477	C₂²×C₂₃	1	2	yes	—
232	#476	—	2	1	no	—
233	#475	C₂²×C₂₃	1	2	yes	—
234	#465	—	2	2	no	—
235	#461	—	2	1	no	—
236	#460	C₂²×C₂₃	1	2	yes	—
237	#446	—	2	2	no	—
238	#446	C₂²×C₂₃	1	2	yes	—
239	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
240	#431	C₂²×C₂₃	1	1	yes	characteristic
241	#424	—	2	1	no	—
242	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
244	#417	—	2	1	no	—
245	#417	—	2	1	no	—
246	#405	—	2	1	no	—
247	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
248	#335	C₂²×C₂₃	1	2	yes	—
249	#334	C₂²×C₂₃	1	2	yes	—
250	#333	C₂²×C₂₃	1	2	yes	—
251	#333	C₂²×C₂₃	1	2	yes	—
252	#332	C₂²×C₂₃	1	1	yes	characteristic
253	#332	C₂²×C₂₃	1	1	yes	characteristic
254	#331	C₂²×C₂₃	1	1	yes	characteristic
255	#331	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×C₂₃	—	2	2	no	—
257	C₂×C₂₃	—	16	1	no	—
258	C₂×C₂₃	C₂×#464	1	1	yes	characteristic
259	C₂×C₂₃	#3063	1	1	yes	characteristic
260	C₂×C₂₃	#3068	1	1	yes	characteristic
261	C₂×#41	—	2	1	no	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
265	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
272	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
273	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
274	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
275	#127	C₂³×C₂₃	1	1	yes	characteristic
276	#125	—	2	1	no	—
277	#124	—	2	1	no	—
278	C₂²⋊Q₈	—	2	1	no	—
279	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
280	#121	—	2	1	no	—
281	#121	—	2	1	no	—
282	C₄×Q₈	—	2	1	no	—
283	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	—	2	1	no	—
286	#117	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
290	#107	—	2	2	no	—
291	#107	—	2	2	no	—
292	#106	—	2	2	no	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	2	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
299	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
300	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
301	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
302	#103	—	2	2	no	—
303	#103	—	2	2	no	—
304	#103	—	2	2	no	—
305	#103	—	2	2	no	—
306	#103	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	#97	—	2	2	no	—
312	#97	D₄×C₂₃	1	2	yes	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	2	1	no	—
315	C₄×C₈	C₂³×C₂₃	1	2	yes	—
316	C₂₃	#19244	1	1	yes	characteristic, direct product factor, Sylow
317	#41	—	4	2	no	—
318	#41	—	4	2	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	4	1	no	—
323	C₂×Q₈	—	8	2	no	—
324	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
333	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
334	SD₁₆	—	8	2	no	—
335	M₁₆	—	2	2	no	—
336	M₁₆	—	2	2	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	4	2	no	—
343	C₂×C₈	—	8	1	no	—
344	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
345	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	4	1	no	—
360	C₂²⋊C₄	—	4	1	no	—
361	C₄²	—	2	2	no	—
362	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₄²	#41×C₂₃	1	1	yes	characteristic
365	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
366	C₂³	—	4	1	no	—
367	C₂³	C₂₃×#121	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	2	2	no	—
370	Q₈	—	4	1	no	—
371	Q₈	—	4	1	no	—
372	Q₈	—	8	1	no	—
373	Q₈	—	8	2	no	—
374	Q₈	—	8	2	no	—
375	D₄	—	4	1	no	—
376	D₄	—	4	1	no	—
377	D₄	—	4	1	no	—
378	D₄	—	4	1	no	—
379	D₄	—	8	1	no	—
380	D₄	—	8	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
391	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
392	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
393	C₂×C₄	C₂₃×#135	1	2	yes	—
394	C₂×C₄	C₂₃×#137	1	2	yes	—
395	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	4	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	1	no	—
404	C₈	—	4	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	8	1	no	—
408	C₂²	—	8	1	no	—
409	C₂²	—	8	1	no	—
410	C₂²	C₂₃×#464	1	2	yes	—
411	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	2	1	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	2	no	—
417	C₄	—	8	1	no	—
418	C₄	—	8	2	no	—
419	C₄	C₂₃×#464	1	2	yes	—
420	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
421	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
422	C₄	C₂₃×#577	1	1	yes	characteristic
423	C₄	C₂₃×#578	1	1	yes	characteristic
424	C₂	—	2	2	no	—
425	C₂	—	16	1	no	—
426	C₂	C₂×C₂₃×#464	1	1	yes	characteristic
427	C₂	C₂₃×#3063	1	1	yes	characteristic
428	C₂	C₂₃×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	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
2. C₂₃×#3230	C₂	semidirect
4. C₂₃×#2962	C₂	semidirect
5. C₂₃×#1704	C₂	semidirect
6. C₂₃×#1698	C₂	semidirect
7. C₂₃×#1684	C₂	semidirect
8. C₂₃×#1678	C₂	semidirect
9. C₂₃×#1565	C₂	semidirect
10. C₂×C₂₃×#333	C₂	semidirect
151. #19244	C₂₃	direct (N × H)
192. #3230	C₂×C₂₃	semidirect
194. #2962	C₂×C₂₃	semidirect
195. #1704	C₂×C₂₃	semidirect
196. #1698	C₂×C₂₃	semidirect
197. #1684	C₂×C₂₃	semidirect
198. #1678	C₂×C₂₃	semidirect
199. #1565	C₂×C₂₃	semidirect
200. C₂×#333	C₂×C₂₃	semidirect
316. C₂₃	#19244	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×#3246	2
C₂₃×#3230	2
C₂₃×#3141	2
C₂₃×#2962	2
C₂₃×#1704	2
C₂₃×#1698	2
C₂₃×#1684	2
C₂₃×#1678	2
C₂₃×#1565	2
C₂×C₂₃×#333	2
C₂₃×#1540	2
#19244	23

Subgroups — C23×#19244 #19166870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19244 #19166870