Subgroups — C₂₃×#60716 #19208342

Order 5888.

1942 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3511	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3185	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3184	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3173	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3167	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3141	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3060	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2952	C₂	1	4	yes	maximal, complemented
11	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	4	yes	—
13	C₂×C₂₃×#131	C₂²	1	2	yes	—
14	C₂₃×#556	—	2	2	no	—
15	C₂₃×#555	—	2	2	no	—
16	C₂₃×#554	—	2	2	no	—
17	C₂₃×#553	—	2	4	no	—
18	C₂₃×#548	—	2	4	no	—
19	C₂₃×#547	—	2	2	no	—
20	C₂₃×#542	C₂²	1	8	yes	—
21	C₂₃×#537	C₂²	1	4	yes	—
22	C₂₃×#533	C₂²	1	4	yes	—
23	C₂₃×#532	C₂²	1	2	yes	—
24	C₂₃×#532	C₂²	1	4	yes	—
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₂₃×#520	C₂²	1	4	yes	—
27	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#498	C₂²	1	2	yes	—
29	C₂₃×#498	C₂²	1	4	yes	—
30	C₂₃×#498	C₂²	1	2	yes	—
31	C₂₃×#495	C₂²	1	4	yes	—
32	C₂₃×#494	C₂²	1	1	yes	characteristic
33	C₂₃×#493	C₂²	1	1	yes	characteristic
34	C₂₃×#492	C₂²	1	2	yes	—
35	C₂₃×#489	C₂²	1	8	yes	—
36	C₂₃×#487	C₂²	1	4	yes	—
37	C₂₃×#486	C₂²	1	4	yes	—
38	C₂₃×#470	C₂²	1	8	yes	—
39	C₂₃×#469	C₂²	1	8	yes	—
40	C₂₃×#465	C₂²	1	16	yes	—
41	C₂₃×#442	C₂²	1	8	yes	—
42	C₂₃×#441	C₂²	1	8	yes	—
43	C₄×SD₁₆×C₂₃	C₂²	1	16	yes	—
44	C₂×C₂₃×#106	C₂²	1	4	yes	—
45	C₂₃×#417	C₂²	1	8	yes	—
46	C₂₃×#405	C₂²	1	4	yes	—
47	C₂₃×#405	C₂²	1	2	yes	—
48	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
49	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
51	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
52	C₂×#41×C₂₃	—	2	4	no	—
53	C₂×#41×C₂₃	C₂³	1	4	yes	—
54	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	2	no	—
56	C₂₃×#135	—	2	16	no	—
57	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
58	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
59	C₂×D₈×C₂₃	C₂³	1	4	yes	—
60	C₂₃×#131	—	2	8	no	—
61	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
63	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
65	C₂₃×C₄⋊Q₈	—	2	2	no	—
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
69	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
70	C₂₃×#127	C₂³	1	2	yes	—
71	C₂₃×#126	—	2	8	no	—
72	C₂₃×#125	—	2	4	no	—
73	C₂₃×#124	—	2	8	no	—
74	C₂₃×#124	—	2	4	no	—
75	C₂₃×#124	C₂³	1	4	yes	—
76	C₂₃×#123	—	2	8	no	—
77	C₂₃×C₂²⋊Q₈	—	2	4	no	—
78	C₂₃×C₂²⋊Q₈	—	2	4	no	—
79	C₂₃×C₂²⋊Q₈	—	2	8	no	—
80	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
81	C₂₃×#121	—	2	4	no	—
82	C₂₃×#121	—	2	8	no	—
83	C₂₃×#121	—	2	2	no	—
84	C₂₃×#121	—	2	2	no	—
85	C₂₃×#121	—	2	4	no	—
86	C₂₃×#121	C₂³	1	8	yes	—
87	C₄×Q₈×C₂₃	—	2	4	no	—
88	C₄×Q₈×C₂₃	D₄	1	2	yes	—
89	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
90	C₄×D₄×C₂₃	—	2	8	no	—
91	C₄×D₄×C₂₃	—	2	2	no	—
92	C₄×D₄×C₂₃	—	2	4	no	—
93	C₄×D₄×C₂₃	—	2	2	no	—
94	C₄×D₄×C₂₃	D₄	1	2	yes	—
95	C₄×D₄×C₂₃	D₄	1	4	yes	—
96	C₄×D₄×C₂₃	C₂³	1	8	yes	—
97	C₂₃×#117	—	2	4	no	—
98	C₂₃×#117	—	2	4	no	—
99	C₂₃×#117	C₂³	1	1	yes	characteristic
100	C₂₃×#117	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
106	C₂₃×#106	C₂³	1	8	yes	—
107	C₂₃×#106	C₂³	1	8	yes	—
108	C₂₃×#103	C₂³	1	16	yes	—
109	C₂₃×#102	C₂³	1	16	yes	—
110	C₂₃×#97	C₂³	1	4	yes	—
111	C₂₃×#97	C₂³	1	2	yes	—
112	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₂₃	C₂³	1	2	yes	—
114	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
115	C₄×C₈×C₂₃	C₂³	1	4	yes	—
116	C₄×C₈×C₂₃	C₂³	1	2	yes	—
117	#41×C₂₃	—	2	4	no	—
118	#41×C₂₃	—	2	16	no	—
119	#41×C₂₃	—	8	8	no	—
120	C₂×Q₈×C₂₃	—	2	4	no	—
121	C₂×Q₈×C₂₃	—	2	4	no	—
122	C₂×Q₈×C₂₃	—	8	4	no	—
123	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	8	no	—
127	C₂×D₄×C₂₃	—	2	2	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
130	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	2	8	no	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₂₃	—	2	8	no	—
140	SD₁₆×C₂₃	—	2	16	no	—
141	D₈×C₂₃	—	2	8	no	—
142	M₁₆×C₂₃	—	2	4	no	—
143	M₁₆×C₂₃	—	2	8	no	—
144	C₂×C₈×C₂₃	—	2	8	no	—
145	C₂×C₈×C₂₃	#41	1	8	yes	complemented
146	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
147	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
148	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	8	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	8	no	—
156	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
158	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
159	C₂²⋊C₄×C₂₃	—	2	8	no	—
160	C₂²⋊C₄×C₂₃	—	2	8	no	—
161	C₂²⋊C₄×C₂₃	—	2	8	no	—
162	C₂²⋊C₄×C₂₃	—	2	4	no	—
163	C₂²⋊C₄×C₂₃	—	2	4	no	—
164	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
166	C₄²×C₂₃	—	2	8	no	—
167	C₄²×C₂₃	C₂×D₄	1	2	yes	—
168	C₄²×C₂₃	C₂×D₄	1	4	yes	—
169	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
170	C₄²×C₂₃	C₂⁴	1	2	yes	—
171	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
172	#60716	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₂₃	—	2	4	no	—
174	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
175	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₂₃	—	2	2	no	—
177	Q₈×C₂₃	—	2	8	no	—
178	Q₈×C₂₃	—	4	8	no	—
179	Q₈×C₂₃	—	8	4	no	—
180	D₄×C₂₃	—	2	4	no	—
181	D₄×C₂₃	—	2	2	no	—
182	D₄×C₂₃	—	2	8	no	—
183	D₄×C₂₃	—	4	8	no	—
184	D₄×C₂₃	—	4	4	no	—
185	D₄×C₂₃	—	4	4	no	—
186	D₄×C₂₃	—	8	8	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	8	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	8	no	—
192	C₂×C₄×C₂₃	—	4	8	no	—
193	C₂×C₄×C₂₃	—	8	8	no	—
194	C₂×C₄×C₂₃	—	8	4	no	—
195	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
196	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
197	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
201	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
202	C₈×C₂₃	—	2	4	no	—
203	C₈×C₂₃	C₂×#41	1	4	yes	—
204	C₈×C₂₃	C₂×#41	1	4	yes	—
205	#3511	C₂×C₂₃	1	2	yes	complemented
206	#3185	C₂×C₂₃	1	2	yes	complemented
207	#3184	C₂×C₂₃	1	2	yes	complemented
208	#3173	C₂×C₂₃	1	8	yes	complemented
209	#3167	C₂×C₂₃	1	4	yes	complemented
210	C₂×#498	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#3147	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#3141	C₂×C₂₃	1	2	yes	complemented
213	#3060	C₂×C₂₃	1	4	yes	complemented
214	#2952	C₂×C₂₃	1	4	yes	complemented
215	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂²×C₂₃	—	2	2	no	—
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	2	4	no	—
219	C₂²×C₂₃	—	4	4	no	—
220	C₂²×C₂₃	—	4	4	no	—
221	C₂²×C₂₃	—	4	4	no	—
222	C₂²×C₂₃	#532	1	2	yes	—
223	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	2	4	no	—
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	4	4	no	—
228	C₄×C₂₃	—	4	8	no	—
229	C₄×C₂₃	#532	1	2	yes	—
230	C₄×C₂₃	#577	1	4	yes	—
231	C₄×C₂₃	C₂²×#41	1	2	yes	—
232	C₂×#135	C₂²×C₂₃	1	4	yes	—
233	C₂×#131	C₂²×C₂₃	1	2	yes	—
234	#556	—	2	2	no	—
235	#555	—	2	2	no	—
236	#554	—	2	2	no	—
237	#553	—	2	4	no	—
238	#548	—	2	4	no	—
239	#547	—	2	2	no	—
240	#542	C₂²×C₂₃	1	8	yes	—
241	#537	C₂²×C₂₃	1	4	yes	—
242	#533	C₂²×C₂₃	1	4	yes	—
243	#532	C₂²×C₂₃	1	2	yes	—
244	#532	C₂²×C₂₃	1	4	yes	—
245	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
246	#520	C₂²×C₂₃	1	4	yes	—
247	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
248	#498	C₂²×C₂₃	1	2	yes	—
249	#498	C₂²×C₂₃	1	4	yes	—
250	#498	C₂²×C₂₃	1	2	yes	—
251	#495	C₂²×C₂₃	1	4	yes	—
252	#494	C₂²×C₂₃	1	1	yes	characteristic
253	#493	C₂²×C₂₃	1	1	yes	characteristic
254	#492	C₂²×C₂₃	1	2	yes	—
255	#489	C₂²×C₂₃	1	8	yes	—
256	#487	C₂²×C₂₃	1	4	yes	—
257	#486	C₂²×C₂₃	1	4	yes	—
258	#470	C₂²×C₂₃	1	8	yes	—
259	#469	C₂²×C₂₃	1	8	yes	—
260	#465	C₂²×C₂₃	1	16	yes	—
261	#442	C₂²×C₂₃	1	8	yes	—
262	#441	C₂²×C₂₃	1	8	yes	—
263	C₄×SD₁₆	C₂²×C₂₃	1	16	yes	—
264	C₂×#106	C₂²×C₂₃	1	4	yes	—
265	#417	C₂²×C₂₃	1	8	yes	—
266	#405	C₂²×C₂₃	1	4	yes	—
267	#405	C₂²×C₂₃	1	2	yes	—
268	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
269	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
270	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
272	C₂×C₂₃	—	2	2	no	—
273	C₂×C₂₃	—	4	2	no	—
274	C₂×C₂₃	—	8	4	no	—
275	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
276	C₂×C₂₃	C₂×#577	1	2	yes	—
277	C₂×#41	—	2	4	no	—
278	C₂×#41	C₂³×C₂₃	1	4	yes	—
279	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
280	C₂²×Q₈	—	2	2	no	—
281	#135	—	2	16	no	—
282	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
283	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
284	C₂×D₈	C₂³×C₂₃	1	4	yes	—
285	#131	—	2	8	no	—
286	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
287	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
288	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
289	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
290	C₄⋊Q₈	—	2	2	no	—
291	C₄⋊Q₈	—	2	2	no	—
292	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
293	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
294	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
295	#127	C₂³×C₂₃	1	2	yes	—
296	#126	—	2	8	no	—
297	#125	—	2	4	no	—
298	#124	—	2	8	no	—
299	#124	—	2	4	no	—
300	#124	C₂³×C₂₃	1	4	yes	—
301	#123	—	2	8	no	—
302	C₂²⋊Q₈	—	2	4	no	—
303	C₂²⋊Q₈	—	2	4	no	—
304	C₂²⋊Q₈	—	2	8	no	—
305	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
306	#121	—	2	4	no	—
307	#121	—	2	8	no	—
308	#121	—	2	2	no	—
309	#121	—	2	2	no	—
310	#121	—	2	4	no	—
311	#121	C₂³×C₂₃	1	8	yes	—
312	C₄×Q₈	—	2	4	no	—
313	C₄×Q₈	D₄×C₂₃	1	2	yes	—
314	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
315	C₄×D₄	—	2	8	no	—
316	C₄×D₄	—	2	2	no	—
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	2	2	no	—
319	C₄×D₄	D₄×C₂₃	1	2	yes	—
320	C₄×D₄	D₄×C₂₃	1	4	yes	—
321	C₄×D₄	C₂³×C₂₃	1	8	yes	—
322	#117	—	2	4	no	—
323	#117	—	2	4	no	—
324	#117	C₂³×C₂₃	1	1	yes	characteristic
325	#117	C₂³×C₂₃	1	2	yes	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	4	no	—
328	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
329	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
330	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
331	#106	C₂³×C₂₃	1	8	yes	—
332	#106	C₂³×C₂₃	1	8	yes	—
333	#103	C₂³×C₂₃	1	16	yes	—
334	#102	C₂³×C₂₃	1	16	yes	—
335	#97	C₂³×C₂₃	1	4	yes	—
336	#97	C₂³×C₂₃	1	2	yes	—
337	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
338	C₄×C₈	C₂³×C₂₃	1	2	yes	—
339	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
340	C₄×C₈	C₂³×C₂₃	1	4	yes	—
341	C₄×C₈	C₂³×C₂₃	1	2	yes	—
342	C₂₃	#60716	1	1	yes	characteristic, direct product factor, Sylow
343	#41	—	2	4	no	—
344	#41	—	2	16	no	—
345	#41	—	8	8	no	—
346	C₂×Q₈	—	2	4	no	—
347	C₂×Q₈	—	2	4	no	—
348	C₂×Q₈	—	8	4	no	—
349	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
350	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
356	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
357	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	8	no	—
361	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
363	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
365	Q₁₆	—	2	8	no	—
366	SD₁₆	—	2	16	no	—
367	D₈	—	2	8	no	—
368	M₁₆	—	2	4	no	—
369	M₁₆	—	2	8	no	—
370	C₂×C₈	—	2	8	no	—
371	C₂×C₈	#41×C₂₃	1	8	yes	complemented
372	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
373	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
374	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	4	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
383	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
384	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₂²⋊C₄	—	2	8	no	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
391	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
392	C₄²	—	2	8	no	—
393	C₄²	C₂×D₄×C₂₃	1	2	yes	—
394	C₄²	C₂×D₄×C₂₃	1	4	yes	—
395	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
396	C₄²	C₂⁴×C₂₃	1	2	yes	—
397	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
398	C₂³	—	2	4	no	—
399	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
400	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
401	Q₈	—	2	2	no	—
402	Q₈	—	2	8	no	—
403	Q₈	—	4	8	no	—
404	Q₈	—	8	4	no	—
405	D₄	—	2	4	no	—
406	D₄	—	2	8	no	—
407	D₄	—	2	2	no	—
408	D₄	—	4	8	no	—
409	D₄	—	4	4	no	—
410	D₄	—	4	4	no	—
411	D₄	—	8	8	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	8	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	—	8	8	no	—
419	C₂×C₄	—	8	4	no	—
420	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
421	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
426	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	C₂×#41×C₂₃	1	4	yes	—
429	C₈	C₂×#41×C₂₃	1	4	yes	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	4	no	—
435	C₂²	—	4	4	no	—
436	C₂²	C₂₃×#532	1	2	yes	—
437	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	2	no	—
441	C₄	—	4	4	no	—
442	C₄	—	4	8	no	—
443	C₄	C₂₃×#532	1	2	yes	—
444	C₄	C₂₃×#577	1	4	yes	—
445	C₄	C₂²×#41×C₂₃	1	2	yes	—
446	C₂	—	2	2	no	—
447	C₂	—	4	2	no	—
448	C₂	—	8	4	no	—
449	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
450	C₂	C₂×C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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
1. C₂₃×#3511	C₂	semidirect
2. C₂₃×#3185	C₂	semidirect
3. C₂₃×#3184	C₂	semidirect
4. C₂₃×#3173	C₂	semidirect
5. C₂₃×#3167	C₂	semidirect
6. C₂×C₂₃×#498	C₂	semidirect
7. C₂₃×#3147	C₂	semidirect
8. C₂₃×#3141	C₂	semidirect
9. C₂₃×#3060	C₂	semidirect
10. C₂₃×#2952	C₂	semidirect
11. C₄×C₂₃×#131	C₂	semidirect
145. C₂×C₈×C₂₃	#41	semidirect
172. #60716	C₂₃	direct (N × H)
205. #3511	C₂×C₂₃	semidirect
206. #3185	C₂×C₂₃	semidirect
207. #3184	C₂×C₂₃	semidirect
208. #3173	C₂×C₂₃	semidirect
209. #3167	C₂×C₂₃	semidirect
210. C₂×#498	C₂×C₂₃	semidirect
211. #3147	C₂×C₂₃	semidirect
212. #3141	C₂×C₂₃	semidirect
213. #3060	C₂×C₂₃	semidirect
214. #2952	C₂×C₂₃	semidirect
215. C₄×#131	C₂×C₂₃	semidirect
342. C₂₃	#60716	direct (N × H)
371. C₂×C₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3511	2
C₂₃×#3185	2
C₂₃×#3184	2
C₂₃×#3173	2
C₂₃×#3167	2
C₂×C₂₃×#498	2
C₂₃×#3147	2
C₂₃×#3141	2
C₂₃×#3060	2
C₂₃×#2952	2
C₄×C₂₃×#131	2
#60716	23

Subgroups — C23×#60716 #19208342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60716 #19208342