Subgroups — C₂₃×#62589 #19210215

Order 5888.

1702 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3518	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3363	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3359	C₂	1	2	yes	maximal, complemented
5	C₂₃×#3329	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3328	C₂	1	6	yes	maximal, complemented
7	C₂₃×#3237	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3218	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3172	C₂	1	3	yes	maximal, complemented
10	C₂₃×#3139	C₂	1	3	yes	maximal, complemented
11	C₂₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₂₃	—	2	1	no	—
13	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#557	C₂²	1	2	yes	—
15	C₂₃×#556	—	2	3	no	—
16	C₂₃×#556	C₂²	1	3	yes	—
17	C₂₃×#555	C₂²	1	3	yes	—
18	C₂₃×#554	—	2	3	no	—
19	D₄×Q₈×C₂₃	C₂²	1	6	yes	—
20	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#548	—	2	1	no	—
22	C₂₃×#544	—	2	3	no	—
23	C₂₃×#542	—	2	3	no	—
24	C₂₃×#533	C₂²	1	3	yes	—
25	C₂₃×#533	C₂²	1	3	yes	—
26	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
27	C₂₃×#529	C₂²	1	3	yes	—
28	C₂₃×#520	—	2	1	no	—
29	C₂₃×#519	—	2	1	no	—
30	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#501	C₂²	1	3	yes	—
32	C₂₃×#497	C₂²	1	3	yes	—
33	C₂₃×#491	C₂²	1	3	yes	—
34	C₂₃×#491	C₂²	1	6	yes	—
35	C₂₃×#489	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	6	yes	—
37	C₂₃×#487	C₂²	1	6	yes	—
38	C₂₃×#485	C₂²	1	12	yes	—
39	C₂₃×#477	C₂²	1	6	yes	—
40	C₂₃×#475	C₂²	1	6	yes	—
41	C₂₃×#470	C₂²	1	6	yes	—
42	C₂₃×#468	C₂²	1	6	yes	—
43	C₂₃×#462	C₂²	1	6	yes	—
44	C₂₃×#461	C₂²	1	6	yes	—
45	C₂₃×#451	C₂²	1	6	yes	—
46	C₂₃×#448	C₂²	1	6	yes	—
47	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
48	C₄×Q₁₆×C₂₃	C₂²	1	6	yes	—
49	C₄×SD₁₆×C₂₃	C₂²	1	6	yes	—
50	C₂₃×#436	C₂²	1	6	yes	—
51	C₂₃×#432	C₂²	1	3	yes	—
52	C₂₃×#419	C₂²	1	12	yes	—
53	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
54	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	1	no	—
56	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
57	C₂×Q₁₆×C₂₃	C₂³	1	6	yes	—
58	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
59	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
60	C₂₃×C₄⋊Q₈	—	2	6	no	—
61	C₂₃×C₄⋊Q₈	—	2	3	no	—
62	C₂₃×C₄⋊Q₈	—	2	3	no	—
63	C₂₃×C₄⋊Q₈	—	2	3	no	—
64	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
66	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
67	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₂₃×#126	—	2	6	no	—
70	C₂₃×#126	—	2	3	no	—
71	C₂₃×#125	—	2	6	no	—
72	C₂₃×#125	—	2	3	no	—
73	C₂₃×#125	C₂³	1	3	yes	—
74	C₂₃×#124	—	2	3	no	—
75	C₂₃×#124	—	2	6	no	—
76	C₂₃×#124	C₂³	1	1	yes	characteristic
77	C₂₃×#123	—	2	3	no	—
78	C₂₃×C₂²⋊Q₈	—	2	3	no	—
79	C₂₃×C₂²⋊Q₈	—	2	3	no	—
80	C₂₃×C₂²⋊Q₈	—	2	3	no	—
81	C₂₃×C₂²⋊Q₈	—	2	3	no	—
82	C₂₃×C₂²⋊Q₈	—	2	12	no	—
83	C₂₃×C₂²⋊Q₈	C₂³	1	6	yes	—
84	C₂₃×C₂²⋊Q₈	C₂³	1	3	yes	—
85	C₂₃×#121	—	2	3	no	—
86	C₂₃×#121	C₂³	1	3	yes	—
87	C₄×Q₈×C₂₃	—	2	1	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	3	no	—
90	C₄×Q₈×C₂₃	—	2	3	no	—
91	C₄×Q₈×C₂₃	—	2	3	no	—
92	C₄×Q₈×C₂₃	—	2	3	no	—
93	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₂₃	C₂³	1	6	yes	—
95	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₂₃	—	2	6	no	—
98	C₄×D₄×C₂₃	—	2	3	no	—
99	C₄×D₄×C₂₃	D₄	1	3	yes	—
100	C₄×D₄×C₂₃	C₂³	1	3	yes	—
101	C₄×D₄×C₂₃	C₂³	1	3	yes	—
102	C₂₃×#117	—	2	3	no	—
103	C₂₃×#117	—	2	3	no	—
104	C₂₃×#117	—	2	3	no	—
105	C₂₃×#117	D₄	1	3	yes	—
106	C₂₃×#117	C₂³	1	6	yes	—
107	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
109	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
110	C₂₃×#107	C₂³	1	6	yes	—
111	C₂₃×#106	C₂³	1	6	yes	—
112	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
113	C₂₃×#103	C₂³	1	6	yes	—
114	C₂₃×#103	C₂³	1	6	yes	—
115	C₂₃×#103	C₂³	1	12	yes	—
116	C₂₃×#103	C₂³	1	12	yes	—
117	C₂₃×#103	C₂³	1	2	yes	—
118	C₂₃×#102	C₂³	1	2	yes	—
119	C₂₃×C₂²⋊C₈	C₂³	1	6	yes	—
120	C₂₃×#97	C₂³	1	1	yes	characteristic
121	C₂₃×#97	C₂³	1	3	yes	—
122	C₄×C₈×C₂₃	C₂³	1	6	yes	—
123	#41×C₂₃	—	2	4	no	—
124	C₂×Q₈×C₂₃	—	2	1	no	—
125	C₂×Q₈×C₂₃	—	2	3	no	—
126	C₂×Q₈×C₂₃	—	2	3	no	—
127	C₂×Q₈×C₂₃	—	2	6	no	—
128	C₂×Q₈×C₂₃	—	4	12	no	—
129	C₂×Q₈×C₂₃	—	4	4	no	—
130	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₂₃	C₂⁴	1	6	yes	—
132	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
133	C₂×D₄×C₂₃	—	2	3	no	—
134	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
135	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	—	2	6	no	—
137	C₂²×C₄×C₂₃	—	2	3	no	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
141	Q₁₆×C₂₃	—	4	6	no	—
142	SD₁₆×C₂₃	—	4	6	no	—
143	M₁₆×C₂₃	—	4	6	no	—
144	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
145	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
146	C₄⋊C₄×C₂₃	—	2	3	no	—
147	C₄⋊C₄×C₂₃	—	2	3	no	—
148	C₄⋊C₄×C₂₃	—	2	6	no	—
149	C₄⋊C₄×C₂₃	—	2	6	no	—
150	C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₄⋊C₄×C₂₃	—	2	3	no	—
152	C₄⋊C₄×C₂₃	—	2	3	no	—
153	C₄⋊C₄×C₂₃	—	2	3	no	—
154	C₄⋊C₄×C₂₃	—	2	3	no	—
155	C₄⋊C₄×C₂₃	—	2	3	no	—
156	C₄⋊C₄×C₂₃	—	2	6	no	—
157	C₄⋊C₄×C₂₃	—	2	6	no	—
158	C₄⋊C₄×C₂₃	—	2	3	no	—
159	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
160	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
161	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
163	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
164	C₂²⋊C₄×C₂₃	—	2	6	no	—
165	C₂²⋊C₄×C₂₃	—	2	6	no	—
166	C₂²⋊C₄×C₂₃	—	2	3	no	—
167	C₂²⋊C₄×C₂₃	—	2	3	no	—
168	C₂²⋊C₄×C₂₃	—	2	1	no	—
169	C₂²⋊C₄×C₂₃	—	2	1	no	—
170	C₂²⋊C₄×C₂₃	—	2	3	no	—
171	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
172	C₄²×C₂₃	—	2	6	no	—
173	C₄²×C₂₃	—	2	3	no	—
174	C₄²×C₂₃	—	2	6	no	—
175	C₄²×C₂₃	C₂×D₄	1	3	yes	—
176	C₄²×C₂₃	C₂×D₄	1	3	yes	—
177	C₄²×C₂₃	C₂⁴	1	3	yes	—
178	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
179	#62589	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₂₃	—	2	1	no	—
181	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
182	Q₈×C₂₃	—	2	6	no	—
183	Q₈×C₂₃	—	2	2	no	—
184	Q₈×C₂₃	—	2	6	no	—
185	Q₈×C₂₃	—	4	2	no	—
186	Q₈×C₂₃	—	4	6	no	—
187	Q₈×C₂₃	—	8	3	no	—
188	Q₈×C₂₃	—	8	6	no	—
189	Q₈×C₂₃	#137	1	4	yes	—
190	D₄×C₂₃	—	2	6	no	—
191	D₄×C₂₃	—	2	2	no	—
192	D₄×C₂₃	—	8	3	no	—
193	C₂×C₄×C₂₃	—	2	6	no	—
194	C₂×C₄×C₂₃	—	2	6	no	—
195	C₂×C₄×C₂₃	—	2	1	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	3	no	—
198	C₂×C₄×C₂₃	—	2	3	no	—
199	C₂×C₄×C₂₃	—	2	6	no	—
200	C₂×C₄×C₂₃	—	4	3	no	—
201	C₂×C₄×C₂₃	—	4	6	no	—
202	C₂×C₄×C₂₃	—	8	6	no	—
203	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
204	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
208	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
209	C₈×C₂₃	—	2	2	no	—
210	C₈×C₂₃	—	2	6	no	—
211	#3518	C₂×C₂₃	1	1	yes	characteristic
212	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#3363	C₂×C₂₃	1	2	yes	complemented
214	#3359	C₂×C₂₃	1	2	yes	complemented
215	#3329	C₂×C₂₃	1	6	yes	complemented
216	#3328	C₂×C₂₃	1	6	yes	complemented
217	#3237	C₂×C₂₃	1	3	yes	complemented
218	#3218	C₂×C₂₃	1	3	yes	complemented
219	#3172	C₂×C₂₃	1	3	yes	complemented
220	#3139	C₂×C₂₃	1	3	yes	complemented
221	#2971	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×C₂₃	—	2	3	no	—
223	C₂²×C₂₃	—	2	3	no	—
224	C₂²×C₂₃	—	2	3	no	—
225	C₂²×C₂₃	—	4	1	no	—
226	C₂²×C₂₃	—	4	1	no	—
227	C₂²×C₂₃	—	4	1	no	—
228	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
229	C₄×C₂₃	—	2	1	no	—
230	C₄×C₂₃	—	2	3	no	—
231	C₄×C₂₃	—	2	3	no	—
232	C₄×C₂₃	—	2	1	no	—
233	C₄×C₂₃	—	4	3	no	—
234	C₄×C₂₃	—	4	2	no	—
235	C₄×C₂₃	—	4	6	no	—
236	C₄×C₂₃	—	4	1	no	—
237	C₄×C₂₃	—	4	3	no	—
238	C₄×C₂₃	C₂×#137	1	6	yes	—
239	C₄×C₂₃	#585	1	2	yes	—
240	Q₈²	—	2	1	no	—
241	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
242	#557	C₂²×C₂₃	1	2	yes	—
243	#556	—	2	3	no	—
244	#556	C₂²×C₂₃	1	3	yes	—
245	#555	C₂²×C₂₃	1	3	yes	—
246	#554	—	2	3	no	—
247	D₄×Q₈	C₂²×C₂₃	1	6	yes	—
248	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
249	#548	—	2	1	no	—
250	#544	—	2	3	no	—
251	#542	—	2	3	no	—
252	#533	C₂²×C₂₃	1	3	yes	—
253	#533	C₂²×C₂₃	1	3	yes	—
254	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
255	#529	C₂²×C₂₃	1	3	yes	—
256	#520	—	2	1	no	—
257	#519	—	2	1	no	—
258	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
259	#501	C₂²×C₂₃	1	3	yes	—
260	#497	C₂²×C₂₃	1	3	yes	—
261	#491	C₂²×C₂₃	1	3	yes	—
262	#491	C₂²×C₂₃	1	6	yes	—
263	#489	C₂²×C₂₃	1	1	yes	characteristic
264	#487	C₂²×C₂₃	1	6	yes	—
265	#487	C₂²×C₂₃	1	6	yes	—
266	#485	C₂²×C₂₃	1	12	yes	—
267	#477	C₂²×C₂₃	1	6	yes	—
268	#475	C₂²×C₂₃	1	6	yes	—
269	#470	C₂²×C₂₃	1	6	yes	—
270	#468	C₂²×C₂₃	1	6	yes	—
271	#462	C₂²×C₂₃	1	6	yes	—
272	#461	C₂²×C₂₃	1	6	yes	—
273	#451	C₂²×C₂₃	1	6	yes	—
274	#448	C₂²×C₂₃	1	6	yes	—
275	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
276	C₄×Q₁₆	C₂²×C₂₃	1	6	yes	—
277	C₄×SD₁₆	C₂²×C₂₃	1	6	yes	—
278	#436	C₂²×C₂₃	1	6	yes	—
279	#432	C₂²×C₂₃	1	3	yes	—
280	#419	C₂²×C₂₃	1	12	yes	—
281	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
282	C₂×C₂₃	—	4	3	no	—
283	C₂×C₂₃	—	8	1	no	—
284	C₂×C₂₃	#3500	1	1	yes	characteristic
285	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
286	C₂×C₂₃	#3586	1	1	yes	characteristic
287	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
288	C₂²×Q₈	—	2	1	no	—
289	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
290	C₂×Q₁₆	C₂³×C₂₃	1	6	yes	—
291	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
292	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
293	C₄⋊Q₈	—	2	3	no	—
294	C₄⋊Q₈	—	2	6	no	—
295	C₄⋊Q₈	—	2	3	no	—
296	C₄⋊Q₈	—	2	3	no	—
297	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
298	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
299	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
300	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
301	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
302	#126	—	2	6	no	—
303	#126	—	2	3	no	—
304	#125	—	2	6	no	—
305	#125	—	2	3	no	—
306	#125	C₂³×C₂₃	1	3	yes	—
307	#124	—	2	3	no	—
308	#124	—	2	6	no	—
309	#124	C₂³×C₂₃	1	1	yes	characteristic
310	#123	—	2	3	no	—
311	C₂²⋊Q₈	—	2	3	no	—
312	C₂²⋊Q₈	—	2	3	no	—
313	C₂²⋊Q₈	—	2	3	no	—
314	C₂²⋊Q₈	—	2	3	no	—
315	C₂²⋊Q₈	—	2	12	no	—
316	C₂²⋊Q₈	C₂³×C₂₃	1	3	yes	—
317	C₂²⋊Q₈	C₂³×C₂₃	1	6	yes	—
318	#121	—	2	3	no	—
319	#121	C₂³×C₂₃	1	3	yes	—
320	C₄×Q₈	—	2	1	no	—
321	C₄×Q₈	—	2	2	no	—
322	C₄×Q₈	—	2	3	no	—
323	C₄×Q₈	—	2	3	no	—
324	C₄×Q₈	—	2	3	no	—
325	C₄×Q₈	—	2	3	no	—
326	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
327	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
328	C₄×Q₈	C₂³×C₂₃	1	6	yes	—
329	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
330	C₄×D₄	—	2	6	no	—
331	C₄×D₄	—	2	3	no	—
332	C₄×D₄	D₄×C₂₃	1	3	yes	—
333	C₄×D₄	C₂³×C₂₃	1	3	yes	—
334	C₄×D₄	C₂³×C₂₃	1	3	yes	—
335	#117	—	2	3	no	—
336	#117	—	2	3	no	—
337	#117	—	2	3	no	—
338	#117	D₄×C₂₃	1	3	yes	—
339	#117	C₂³×C₂₃	1	6	yes	—
340	C₂×C₄⋊C₄	—	2	3	no	—
341	C₂×C₄⋊C₄	—	2	3	no	—
342	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
343	#107	C₂³×C₂₃	1	6	yes	—
344	#106	C₂³×C₂₃	1	6	yes	—
345	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
346	#103	C₂³×C₂₃	1	6	yes	—
347	#103	C₂³×C₂₃	1	6	yes	—
348	#103	C₂³×C₂₃	1	12	yes	—
349	#103	C₂³×C₂₃	1	12	yes	—
350	#103	C₂³×C₂₃	1	2	yes	—
351	#102	C₂³×C₂₃	1	2	yes	—
352	C₂²⋊C₈	C₂³×C₂₃	1	6	yes	—
353	#97	C₂³×C₂₃	1	1	yes	characteristic
354	#97	C₂³×C₂₃	1	3	yes	—
355	C₄×C₈	C₂³×C₂₃	1	6	yes	—
356	C₂₃	#62589	1	1	yes	characteristic, direct product factor, Sylow
357	#41	—	2	4	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	3	no	—
360	C₂×Q₈	—	2	3	no	—
361	C₂×Q₈	—	2	6	no	—
362	C₂×Q₈	—	4	12	no	—
363	C₂×Q₈	—	4	4	no	—
364	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂×Q₈	C₂⁴×C₂₃	1	6	yes	—
366	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
367	C₂×D₄	—	2	3	no	—
368	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
369	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
370	C₂²×C₄	—	2	6	no	—
371	C₂²×C₄	—	2	3	no	—
372	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
373	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
374	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
375	Q₁₆	—	4	6	no	—
376	SD₁₆	—	4	6	no	—
377	M₁₆	—	4	6	no	—
378	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
379	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
380	C₄⋊C₄	—	2	3	no	—
381	C₄⋊C₄	—	2	3	no	—
382	C₄⋊C₄	—	2	6	no	—
383	C₄⋊C₄	—	2	6	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	3	no	—
386	C₄⋊C₄	—	2	3	no	—
387	C₄⋊C₄	—	2	3	no	—
388	C₄⋊C₄	—	2	3	no	—
389	C₄⋊C₄	—	2	3	no	—
390	C₄⋊C₄	—	2	6	no	—
391	C₄⋊C₄	—	2	6	no	—
392	C₄⋊C₄	—	2	3	no	—
393	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
394	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
395	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
396	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
397	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
398	C₂²⋊C₄	—	2	6	no	—
399	C₂²⋊C₄	—	2	6	no	—
400	C₂²⋊C₄	—	2	3	no	—
401	C₂²⋊C₄	—	2	3	no	—
402	C₂²⋊C₄	—	2	1	no	—
403	C₂²⋊C₄	—	2	1	no	—
404	C₂²⋊C₄	—	2	3	no	—
405	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
406	C₄²	—	2	6	no	—
407	C₄²	—	2	3	no	—
408	C₄²	—	2	6	no	—
409	C₄²	C₂×D₄×C₂₃	1	3	yes	—
410	C₄²	C₂×D₄×C₂₃	1	3	yes	—
411	C₄²	C₂⁴×C₂₃	1	3	yes	—
412	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
413	C₂³	—	2	1	no	—
414	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
415	Q₈	—	2	6	no	—
416	Q₈	—	2	2	no	—
417	Q₈	—	2	6	no	—
418	Q₈	—	4	2	no	—
419	Q₈	—	4	6	no	—
420	Q₈	—	8	6	no	—
421	Q₈	—	8	3	no	—
422	Q₈	C₂₃×#137	1	4	yes	—
423	D₄	—	2	6	no	—
424	D₄	—	2	2	no	—
425	D₄	—	8	3	no	—
426	C₂×C₄	—	2	6	no	—
427	C₂×C₄	—	2	6	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	3	no	—
431	C₂×C₄	—	2	3	no	—
432	C₂×C₄	—	2	6	no	—
433	C₂×C₄	—	4	3	no	—
434	C₂×C₄	—	4	6	no	—
435	C₂×C₄	—	8	6	no	—
436	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
437	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
438	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
439	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
440	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
441	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	2	6	no	—
444	C₂²	—	2	3	no	—
445	C₂²	—	2	3	no	—
446	C₂²	—	2	3	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	1	no	—
450	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
451	C₄	—	2	3	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	2	3	no	—
455	C₄	—	4	6	no	—
456	C₄	—	4	1	no	—
457	C₄	—	4	3	no	—
458	C₄	—	4	3	no	—
459	C₄	—	4	2	no	—
460	C₄	C₂×C₂₃×#137	1	6	yes	—
461	C₄	C₂₃×#585	1	2	yes	—
462	C₂	—	4	3	no	—
463	C₂	—	8	1	no	—
464	C₂	C₂₃×#3500	1	1	yes	characteristic
465	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
466	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3518	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3503	C₂	semidirect
3. C₂₃×#3363	C₂	semidirect
4. C₂₃×#3359	C₂	semidirect
5. C₂₃×#3329	C₂	semidirect
6. C₂₃×#3328	C₂	semidirect
7. C₂₃×#3237	C₂	semidirect
8. C₂₃×#3218	C₂	semidirect
9. C₂₃×#3172	C₂	semidirect
10. C₂₃×#3139	C₂	semidirect
11. C₂₃×#2971	C₂	semidirect
179. #62589	C₂₃	direct (N × H)
212. #3503	C₂×C₂₃	semidirect
213. #3363	C₂×C₂₃	semidirect
214. #3359	C₂×C₂₃	semidirect
215. #3329	C₂×C₂₃	semidirect
216. #3328	C₂×C₂₃	semidirect
217. #3237	C₂×C₂₃	semidirect
218. #3218	C₂×C₂₃	semidirect
219. #3172	C₂×C₂₃	semidirect
220. #3139	C₂×C₂₃	semidirect
221. #2971	C₂×C₂₃	semidirect
356. C₂₃	#62589	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×#3518	2
C₂₃×#3503	2
C₂₃×#3363	2
C₂₃×#3359	2
C₂₃×#3329	2
C₂₃×#3328	2
C₂₃×#3237	2
C₂₃×#3218	2
C₂₃×#3172	2
C₂₃×#3139	2
C₂₃×#2971	2
#62589	23

Subgroups — C23×#62589 #19210215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#62589 #19210215