Subgroups — C₂₃×#62542 #19210168

Order 5888.

1754 conjugacy classes of subgroups.

The subgroup lattice has 690 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	1	yes	maximal, characteristic
2	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3369	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3331	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3316	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3229	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3210	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3185	C₂	1	2	yes	maximal, complemented
11	C₂₃×#3167	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#3141	C₂	1	2	yes	maximal, complemented
13	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
15	Q₈²×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#557	—	2	1	no	—
17	C₂₃×#556	C₂²	1	2	yes	—
18	C₂₃×#555	—	2	1	no	—
19	C₂₃×#554	—	2	1	no	—
20	C₂₃×#554	C₂²	1	4	yes	—
21	C₂₃×#554	C₂²	1	2	yes	—
22	C₂₃×#553	—	2	2	no	—
23	D₄×Q₈×C₂₃	—	2	2	no	—
24	D₄×Q₈×C₂₃	—	2	1	no	—
25	C₂₃×#548	C₂²	1	4	yes	—
26	C₂₃×#547	C₂²	1	2	yes	—
27	C₂₃×#544	—	2	1	no	—
28	C₂₃×#544	—	2	2	no	—
29	C₂₃×#542	—	2	2	no	—
30	C₂₃×#537	—	2	1	no	—
31	C₂₃×#533	C₂²	1	1	yes	characteristic
32	C₂₃×#533	C₂²	1	2	yes	—
33	C₂₃×#533	C₂²	1	2	yes	—
34	C₂₃×#532	C₂²	1	2	yes	—
35	C₂₃×#532	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
39	C₂₃×#520	—	2	1	no	—
40	C₂₃×#520	—	2	1	no	—
41	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#498	C₂²	1	1	yes	characteristic
43	C₂₃×#498	C₂²	1	2	yes	—
44	C₂₃×#494	C₂²	1	2	yes	—
45	C₂₃×#492	C₂²	1	1	yes	characteristic
46	C₂₃×#489	C₂²	1	8	yes	—
47	C₂₃×#487	C₂²	1	1	yes	characteristic
48	C₂₃×#487	C₂²	1	2	yes	—
49	C₂₃×#487	C₂²	1	2	yes	—
50	C₂₃×#487	C₂²	1	2	yes	—
51	C₂₃×#487	C₂²	1	4	yes	—
52	C₂₃×#486	C₂²	1	1	yes	characteristic
53	C₂₃×#486	C₂²	1	2	yes	—
54	C₂₃×#483	C₂²	1	8	yes	—
55	C₂₃×#481	C₂²	1	4	yes	—
56	C₂₃×#476	C₂²	1	4	yes	—
57	C₂₃×#475	C₂²	1	8	yes	—
58	C₂₃×#470	C₂²	1	8	yes	—
59	C₂₃×#465	C₂²	1	4	yes	—
60	C₂₃×#464	C₂²	1	8	yes	—
61	C₂₃×#462	C₂²	1	4	yes	—
62	C₂₃×#452	C₂²	1	8	yes	—
63	C₂₃×#451	C₂²	1	4	yes	—
64	C₂₃×#446	C₂²	1	2	yes	—
65	C₂₃×#441	C₂²	1	8	yes	—
66	C₄×SD₁₆×C₂₃	C₂²	1	4	yes	—
67	C₂₃×#436	C₂²	1	2	yes	—
68	C₂₃×#435	C₂²	1	4	yes	—
69	C₂₃×#431	C₂²	1	1	yes	characteristic
70	C₂₃×#431	C₂²	1	2	yes	—
71	C₂₃×#417	C₂²	1	8	yes	—
72	C₂×C₂₃×#103	C₂²	1	4	yes	—
73	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
74	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
75	C₂×#41×C₂₃	—	2	1	no	—
76	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
77	C₂×#41×C₂₃	C₂³	1	4	yes	—
78	C₂²×Q₈×C₂₃	—	2	1	no	—
79	C₂²×Q₈×C₂₃	—	2	2	no	—
80	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
81	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
82	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
83	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
84	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
85	C₂₃×C₄⋊Q₈	—	2	2	no	—
86	C₂₃×C₄⋊Q₈	—	2	4	no	—
87	C₂₃×C₄⋊Q₈	—	2	2	no	—
88	C₂₃×C₄⋊Q₈	—	2	1	no	—
89	C₂₃×C₄⋊Q₈	—	2	1	no	—
90	C₂₃×C₄⋊Q₈	—	2	1	no	—
91	C₂₃×C₄⋊Q₈	—	2	2	no	—
92	C₂₃×C₄⋊Q₈	—	2	1	no	—
93	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
94	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
95	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
96	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
97	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
99	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
100	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
101	C₂₃×#127	C₂³	1	1	yes	characteristic
102	C₂₃×#126	—	2	4	no	—
103	C₂₃×#125	—	2	2	no	—
104	C₂₃×#125	—	2	4	no	—
105	C₂₃×#125	—	2	2	no	—
106	C₂₃×#125	—	2	2	no	—
107	C₂₃×#125	—	2	1	no	—
108	C₂₃×#124	—	2	4	no	—
109	C₂₃×#124	—	2	1	no	—
110	C₂₃×#124	C₂³	1	4	yes	—
111	C₂₃×#123	—	2	4	no	—
112	C₂₃×C₂²⋊Q₈	—	2	2	no	—
113	C₂₃×C₂²⋊Q₈	—	2	1	no	—
114	C₂₃×C₂²⋊Q₈	—	2	2	no	—
115	C₂₃×C₂²⋊Q₈	—	2	2	no	—
116	C₂₃×C₂²⋊Q₈	—	2	4	no	—
117	C₂₃×C₂²⋊Q₈	—	2	4	no	—
118	C₂₃×C₂²⋊Q₈	—	2	4	no	—
119	C₂₃×C₂²⋊Q₈	—	2	4	no	—
120	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
121	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
122	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
123	C₂₃×#121	—	2	2	no	—
124	C₂₃×#121	—	2	4	no	—
125	C₂₃×#121	—	2	1	no	—
126	C₂₃×#121	—	2	2	no	—
127	C₂₃×#121	—	2	2	no	—
128	C₂₃×#121	C₂³	1	2	yes	—
129	C₄×Q₈×C₂₃	—	2	2	no	—
130	C₄×Q₈×C₂₃	—	2	2	no	—
131	C₄×Q₈×C₂₃	—	2	1	no	—
132	C₄×Q₈×C₂₃	—	2	2	no	—
133	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
134	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
135	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
136	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
137	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×D₄×C₂₃	—	2	4	no	—
139	C₄×D₄×C₂₃	—	2	2	no	—
140	C₄×D₄×C₂₃	—	2	4	no	—
141	C₄×D₄×C₂₃	—	2	2	no	—
142	C₄×D₄×C₂₃	—	2	1	no	—
143	C₄×D₄×C₂₃	—	2	2	no	—
144	C₄×D₄×C₂₃	D₄	1	2	yes	—
145	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
146	C₄×D₄×C₂₃	C₂³	1	2	yes	—
147	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
148	C₄×D₄×C₂₃	C₂³	1	2	yes	—
149	C₂₃×#117	—	2	2	no	—
150	C₂₃×#117	—	2	2	no	—
151	C₂₃×#117	—	2	1	no	—
152	C₂₃×#117	—	2	2	no	—
153	C₂₃×#117	D₄	1	1	yes	characteristic
154	C₂₃×#117	D₄	1	2	yes	—
155	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
162	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
164	C₂₃×#106	C₂³	1	4	yes	—
165	C₂₃×#106	C₂³	1	8	yes	—
166	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
167	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
168	C₂₃×#103	C₂³	1	8	yes	—
169	C₂₃×#103	C₂³	1	8	yes	—
170	C₂₃×#103	C₂³	1	8	yes	—
171	C₂₃×#103	C₂³	1	4	yes	—
172	C₂₃×#102	C₂³	1	4	yes	—
173	C₂₃×#102	C₂³	1	8	yes	—
174	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
175	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
176	C₂₃×#97	C₂³	1	4	yes	—
177	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
178	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₂₃	C₂³	1	2	yes	—
180	C₄×C₈×C₂₃	C₂³	1	2	yes	—
181	#41×C₂₃	—	2	4	no	—
182	#41×C₂₃	—	4	8	no	—
183	#41×C₂₃	—	4	4	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	2	no	—
186	C₂×Q₈×C₂₃	—	2	1	no	—
187	C₂×Q₈×C₂₃	—	2	2	no	—
188	C₂×Q₈×C₂₃	—	2	4	no	—
189	C₂×Q₈×C₂₃	—	2	2	no	—
190	C₂×Q₈×C₂₃	—	4	4	no	—
191	C₂×Q₈×C₂₃	—	8	2	no	—
192	C₂×Q₈×C₂₃	—	8	4	no	—
193	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
195	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
196	C₂×D₄×C₂₃	—	2	2	no	—
197	C₂×D₄×C₂₃	—	2	4	no	—
198	C₂×D₄×C₂₃	—	2	1	no	—
199	C₂×D₄×C₂₃	—	2	1	no	—
200	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
201	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₂₃	—	2	2	no	—
204	C₂²×C₄×C₂₃	—	2	4	no	—
205	C₂²×C₄×C₂₃	—	2	2	no	—
206	C₂²×C₄×C₂₃	—	2	2	no	—
207	C₂²×C₄×C₂₃	—	2	4	no	—
208	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
214	Q₁₆×C₂₃	—	4	8	no	—
215	SD₁₆×C₂₃	—	4	4	no	—
216	M₁₆×C₂₃	—	4	4	no	—
217	C₂×C₈×C₂₃	—	4	2	no	—
218	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
219	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
220	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	1	no	—
223	C₄⋊C₄×C₂₃	—	2	2	no	—
224	C₄⋊C₄×C₂₃	—	2	2	no	—
225	C₄⋊C₄×C₂₃	—	2	2	no	—
226	C₄⋊C₄×C₂₃	—	2	4	no	—
227	C₄⋊C₄×C₂₃	—	2	4	no	—
228	C₄⋊C₄×C₂₃	—	2	2	no	—
229	C₄⋊C₄×C₂₃	—	2	2	no	—
230	C₄⋊C₄×C₂₃	—	2	2	no	—
231	C₄⋊C₄×C₂₃	—	2	4	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	1	no	—
235	C₄⋊C₄×C₂₃	—	2	2	no	—
236	C₄⋊C₄×C₂₃	—	2	4	no	—
237	C₄⋊C₄×C₂₃	—	2	4	no	—
238	C₄⋊C₄×C₂₃	—	2	4	no	—
239	C₄⋊C₄×C₂₃	—	2	2	no	—
240	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
241	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
243	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
245	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
246	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
247	C₂²⋊C₄×C₂₃	—	2	4	no	—
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	4	no	—
250	C₂²⋊C₄×C₂₃	—	2	4	no	—
251	C₂²⋊C₄×C₂₃	—	2	4	no	—
252	C₂²⋊C₄×C₂₃	—	2	2	no	—
253	C₂²⋊C₄×C₂₃	—	2	2	no	—
254	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
255	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
256	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
257	C₄²×C₂₃	—	2	4	no	—
258	C₄²×C₂₃	—	2	4	no	—
259	C₄²×C₂₃	—	2	2	no	—
260	C₄²×C₂₃	C₂×D₄	1	2	yes	—
261	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
262	C₄²×C₂₃	C₂×D₄	1	2	yes	—
263	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
264	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
265	C₄²×C₂₃	C₂⁴	1	2	yes	—
266	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
267	#62542	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂³×C₂₃	—	2	2	no	—
269	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
270	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
271	Q₈×C₂₃	—	2	2	no	—
272	Q₈×C₂₃	—	2	4	no	—
273	Q₈×C₂₃	—	2	8	no	—
274	Q₈×C₂₃	—	4	2	no	—
275	Q₈×C₂₃	—	4	4	no	—
276	Q₈×C₂₃	—	4	2	no	—
277	Q₈×C₂₃	—	8	2	no	—
278	Q₈×C₂₃	—	8	4	no	—
279	Q₈×C₂₃	—	8	2	no	—
280	D₄×C₂₃	—	2	4	no	—
281	D₄×C₂₃	—	2	2	no	—
282	D₄×C₂₃	—	2	4	no	—
283	D₄×C₂₃	—	4	4	no	—
284	D₄×C₂₃	—	4	2	no	—
285	D₄×C₂₃	—	4	2	no	—
286	D₄×C₂₃	—	8	4	no	—
287	C₂×C₄×C₂₃	—	2	4	no	—
288	C₂×C₄×C₂₃	—	2	2	no	—
289	C₂×C₄×C₂₃	—	2	4	no	—
290	C₂×C₄×C₂₃	—	2	2	no	—
291	C₂×C₄×C₂₃	—	2	4	no	—
292	C₂×C₄×C₂₃	—	2	2	no	—
293	C₂×C₄×C₂₃	—	2	2	no	—
294	C₂×C₄×C₂₃	—	2	4	no	—
295	C₂×C₄×C₂₃	—	2	4	no	—
296	C₂×C₄×C₂₃	—	2	2	no	—
297	C₂×C₄×C₂₃	—	4	4	no	—
298	C₂×C₄×C₂₃	—	8	4	no	—
299	C₂×C₄×C₂₃	—	8	2	no	—
300	C₂×C₄×C₂₃	—	8	2	no	—
301	C₂×C₄×C₂₃	—	8	4	no	—
302	C₂×C₄×C₂₃	#137	1	4	yes	—
303	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
304	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
307	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
312	C₈×C₂₃	—	2	2	no	—
313	C₈×C₂₃	—	2	2	no	—
314	C₈×C₂₃	—	2	4	no	—
315	#3511	C₂×C₂₃	1	1	yes	characteristic
316	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#3369	C₂×C₂₃	1	4	yes	complemented
318	#3331	C₂×C₂₃	1	4	yes	complemented
319	#3316	C₂×C₂₃	1	8	yes	complemented
320	#3229	C₂×C₂₃	1	2	yes	complemented
321	#3227	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#3210	C₂×C₂₃	1	2	yes	complemented
323	#3209	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#3185	C₂×C₂₃	1	2	yes	complemented
325	#3167	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#3141	C₂×C₂₃	1	2	yes	complemented
327	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
328	#2965	C₂×C₂₃	1	1	yes	characteristic, complemented
329	C₂²×C₂₃	—	2	1	no	—
330	C₂²×C₂₃	—	2	1	no	—
331	C₂²×C₂₃	—	2	2	no	—
332	C₂²×C₂₃	—	2	2	no	—
333	C₂²×C₂₃	—	2	2	no	—
334	C₂²×C₂₃	—	4	2	no	—
335	C₂²×C₂₃	—	4	2	no	—
336	C₂²×C₂₃	—	4	2	no	—
337	C₂²×C₂₃	C₂×#137	1	2	yes	—
338	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
339	C₄×C₂₃	—	2	2	no	—
340	C₄×C₂₃	—	2	1	no	—
341	C₄×C₂₃	—	2	2	no	—
342	C₄×C₂₃	—	2	2	no	—
343	C₄×C₂₃	—	2	1	no	—
344	C₄×C₂₃	—	4	4	no	—
345	C₄×C₂₃	—	4	2	no	—
346	C₄×C₂₃	—	4	2	no	—
347	C₄×C₂₃	—	4	4	no	—
348	C₄×C₂₃	—	4	2	no	—
349	C₄×C₂₃	C₂×#137	1	2	yes	—
350	C₄×C₂₃	C₂×#137	1	2	yes	—
351	C₄×C₂₃	#577	1	2	yes	—
352	C₄×C₂₃	#585	1	2	yes	—
353	Q₈²	C₂²×C₂₃	1	2	yes	—
354	#557	—	2	1	no	—
355	#556	C₂²×C₂₃	1	2	yes	—
356	#555	—	2	1	no	—
357	#554	—	2	1	no	—
358	#554	C₂²×C₂₃	1	2	yes	—
359	#554	C₂²×C₂₃	1	4	yes	—
360	#553	—	2	2	no	—
361	D₄×Q₈	—	2	2	no	—
362	D₄×Q₈	—	2	1	no	—
363	#548	C₂²×C₂₃	1	4	yes	—
364	#547	C₂²×C₂₃	1	2	yes	—
365	#544	—	2	1	no	—
366	#544	—	2	2	no	—
367	#542	—	2	2	no	—
368	#537	—	2	1	no	—
369	#533	C₂²×C₂₃	1	1	yes	characteristic
370	#533	C₂²×C₂₃	1	2	yes	—
371	#533	C₂²×C₂₃	1	2	yes	—
372	#532	C₂²×C₂₃	1	2	yes	—
373	#532	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
377	#520	—	2	1	no	—
378	#520	—	2	1	no	—
379	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
380	#498	C₂²×C₂₃	1	1	yes	characteristic
381	#498	C₂²×C₂₃	1	2	yes	—
382	#494	C₂²×C₂₃	1	2	yes	—
383	#492	C₂²×C₂₃	1	1	yes	characteristic
384	#489	C₂²×C₂₃	1	8	yes	—
385	#487	C₂²×C₂₃	1	4	yes	—
386	#487	C₂²×C₂₃	1	2	yes	—
387	#487	C₂²×C₂₃	1	2	yes	—
388	#487	C₂²×C₂₃	1	2	yes	—
389	#487	C₂²×C₂₃	1	1	yes	characteristic
390	#486	C₂²×C₂₃	1	1	yes	characteristic
391	#486	C₂²×C₂₃	1	2	yes	—
392	#483	C₂²×C₂₃	1	8	yes	—
393	#481	C₂²×C₂₃	1	4	yes	—
394	#476	C₂²×C₂₃	1	4	yes	—
395	#475	C₂²×C₂₃	1	8	yes	—
396	#470	C₂²×C₂₃	1	8	yes	—
397	#465	C₂²×C₂₃	1	4	yes	—
398	#464	C₂²×C₂₃	1	8	yes	—
399	#462	C₂²×C₂₃	1	4	yes	—
400	#452	C₂²×C₂₃	1	8	yes	—
401	#451	C₂²×C₂₃	1	4	yes	—
402	#446	C₂²×C₂₃	1	2	yes	—
403	#441	C₂²×C₂₃	1	8	yes	—
404	C₄×SD₁₆	C₂²×C₂₃	1	4	yes	—
405	#436	C₂²×C₂₃	1	2	yes	—
406	#435	C₂²×C₂₃	1	4	yes	—
407	#431	C₂²×C₂₃	1	1	yes	characteristic
408	#431	C₂²×C₂₃	1	2	yes	—
409	#417	C₂²×C₂₃	1	8	yes	—
410	C₂×#103	C₂²×C₂₃	1	4	yes	—
411	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
412	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
413	C₂×C₂₃	—	2	2	no	—
414	C₂×C₂₃	—	4	2	no	—
415	C₂×C₂₃	—	8	2	no	—
416	C₂×C₂₃	#3500	1	1	yes	characteristic
417	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
418	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
419	C₂×#41	—	2	1	no	—
420	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
421	C₂×#41	C₂³×C₂₃	1	4	yes	—
422	C₂²×Q₈	—	2	1	no	—
423	C₂²×Q₈	—	2	2	no	—
424	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
425	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
426	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
427	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
428	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	—	2	4	no	—
431	C₄⋊Q₈	—	2	1	no	—
432	C₄⋊Q₈	—	2	2	no	—
433	C₄⋊Q₈	—	2	1	no	—
434	C₄⋊Q₈	—	2	2	no	—
435	C₄⋊Q₈	—	2	1	no	—
436	C₄⋊Q₈	—	2	1	no	—
437	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
438	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
439	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
440	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
441	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
442	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
443	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
444	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
445	#127	C₂³×C₂₃	1	1	yes	characteristic
446	#126	—	2	4	no	—
447	#125	—	2	2	no	—
448	#125	—	2	4	no	—
449	#125	—	2	2	no	—
450	#125	—	2	1	no	—
451	#125	—	2	2	no	—
452	#124	—	2	4	no	—
453	#124	—	2	1	no	—
454	#124	C₂³×C₂₃	1	4	yes	—
455	#123	—	2	4	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	2	no	—
458	C₂²⋊Q₈	—	2	1	no	—
459	C₂²⋊Q₈	—	2	2	no	—
460	C₂²⋊Q₈	—	2	4	no	—
461	C₂²⋊Q₈	—	2	4	no	—
462	C₂²⋊Q₈	—	2	4	no	—
463	C₂²⋊Q₈	—	2	4	no	—
464	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
465	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
466	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
467	#121	—	2	2	no	—
468	#121	—	2	4	no	—
469	#121	—	2	2	no	—
470	#121	—	2	1	no	—
471	#121	—	2	2	no	—
472	#121	C₂³×C₂₃	1	2	yes	—
473	C₄×Q₈	—	2	2	no	—
474	C₄×Q₈	—	2	2	no	—
475	C₄×Q₈	—	2	2	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
478	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
479	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
480	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
481	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
482	C₄×D₄	—	2	2	no	—
483	C₄×D₄	—	2	4	no	—
484	C₄×D₄	—	2	4	no	—
485	C₄×D₄	—	2	2	no	—
486	C₄×D₄	—	2	2	no	—
487	C₄×D₄	—	2	1	no	—
488	C₄×D₄	D₄×C₂₃	1	2	yes	—
489	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
490	C₄×D₄	C₂³×C₂₃	1	2	yes	—
491	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
492	C₄×D₄	C₂³×C₂₃	1	2	yes	—
493	#117	—	2	2	no	—
494	#117	—	2	1	no	—
495	#117	—	2	2	no	—
496	#117	—	2	2	no	—
497	#117	D₄×C₂₃	1	1	yes	characteristic
498	#117	D₄×C₂₃	1	2	yes	—
499	C₂×C₄⋊C₄	—	2	1	no	—
500	C₂×C₄⋊C₄	—	2	2	no	—
501	C₂×C₄⋊C₄	—	2	1	no	—
502	C₂×C₄⋊C₄	—	2	1	no	—
503	C₂×C₄⋊C₄	—	2	2	no	—
504	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
505	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
506	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
507	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
508	#106	C₂³×C₂₃	1	4	yes	—
509	#106	C₂³×C₂₃	1	8	yes	—
510	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
511	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
512	#103	C₂³×C₂₃	1	8	yes	—
513	#103	C₂³×C₂₃	1	8	yes	—
514	#103	C₂³×C₂₃	1	8	yes	—
515	#103	C₂³×C₂₃	1	4	yes	—
516	#102	C₂³×C₂₃	1	4	yes	—
517	#102	C₂³×C₂₃	1	8	yes	—
518	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
519	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
520	#97	C₂³×C₂₃	1	4	yes	—
521	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
522	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
523	C₄×C₈	C₂³×C₂₃	1	2	yes	—
524	C₄×C₈	C₂³×C₂₃	1	2	yes	—
525	C₂₃	#62542	1	1	yes	characteristic, direct product factor, Sylow
526	#41	—	2	4	no	—
527	#41	—	4	8	no	—
528	#41	—	4	4	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	2	no	—
533	C₂×Q₈	—	2	4	no	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	4	4	no	—
536	C₂×Q₈	—	8	2	no	—
537	C₂×Q₈	—	8	4	no	—
538	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
539	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
540	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
541	C₂×D₄	—	2	2	no	—
542	C₂×D₄	—	2	4	no	—
543	C₂×D₄	—	2	1	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
546	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	4	no	—
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
554	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
556	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
558	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
559	Q₁₆	—	4	8	no	—
560	SD₁₆	—	4	4	no	—
561	M₁₆	—	4	4	no	—
562	C₂×C₈	—	4	2	no	—
563	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
564	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
565	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	4	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	4	no	—
582	C₄⋊C₄	—	2	4	no	—
583	C₄⋊C₄	—	2	4	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
586	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
587	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
588	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
589	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
590	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
591	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
592	C₂²⋊C₄	—	2	4	no	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	—	2	4	no	—
595	C₂²⋊C₄	—	2	4	no	—
596	C₂²⋊C₄	—	2	4	no	—
597	C₂²⋊C₄	—	2	2	no	—
598	C₂²⋊C₄	—	2	2	no	—
599	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
600	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
601	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
602	C₄²	—	2	4	no	—
603	C₄²	—	2	4	no	—
604	C₄²	—	2	2	no	—
605	C₄²	C₂×D₄×C₂₃	1	2	yes	—
606	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
607	C₄²	C₂×D₄×C₂₃	1	2	yes	—
608	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
609	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
610	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
611	C₄²	C₂⁴×C₂₃	1	2	yes	—
612	C₂³	—	2	2	no	—
613	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
614	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
615	Q₈	—	2	2	no	—
616	Q₈	—	2	4	no	—
617	Q₈	—	2	8	no	—
618	Q₈	—	4	2	no	—
619	Q₈	—	4	4	no	—
620	Q₈	—	4	2	no	—
621	Q₈	—	8	2	no	—
622	Q₈	—	8	4	no	—
623	Q₈	—	8	2	no	—
624	D₄	—	2	4	no	—
625	D₄	—	2	2	no	—
626	D₄	—	2	4	no	—
627	D₄	—	4	4	no	—
628	D₄	—	4	2	no	—
629	D₄	—	4	2	no	—
630	D₄	—	8	4	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	4	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	4	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	4	no	—
639	C₂×C₄	—	2	4	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	4	4	no	—
642	C₂×C₄	—	8	4	no	—
643	C₂×C₄	—	8	2	no	—
644	C₂×C₄	—	8	2	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	C₂₃×#137	1	4	yes	—
647	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
648	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
649	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
650	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
651	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
652	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
654	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
655	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
656	C₈	—	2	2	no	—
657	C₈	—	2	2	no	—
658	C₈	—	2	4	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	2	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	C₂×C₂₃×#137	1	2	yes	—
668	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
669	C₄	—	2	2	no	—
670	C₄	—	2	1	no	—
671	C₄	—	2	2	no	—
672	C₄	—	2	2	no	—
673	C₄	—	2	1	no	—
674	C₄	—	4	4	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	4	no	—
678	C₄	—	4	2	no	—
679	C₄	C₂×C₂₃×#137	1	2	yes	—
680	C₄	C₂×C₂₃×#137	1	2	yes	—
681	C₄	C₂₃×#577	1	2	yes	—
682	C₄	C₂₃×#585	1	2	yes	—
683	C₂	—	2	2	no	—
684	C₂	—	4	2	no	—
685	C₂	—	8	2	no	—
686	C₂	C₂₃×#3500	1	1	yes	characteristic
687	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
688	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3511	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×#3369	C₂	semidirect
4. C₂₃×#3331	C₂	semidirect
5. C₂₃×#3316	C₂	semidirect
6. C₂₃×#3229	C₂	semidirect
7. C₂₃×#3227	C₂	semidirect
8. C₂₃×#3210	C₂	semidirect
9. C₂₃×#3209	C₂	semidirect
10. C₂₃×#3185	C₂	semidirect
11. C₂₃×#3167	C₂	semidirect
12. C₂₃×#3141	C₂	semidirect
13. C₂×C₂₃×#487	C₂	semidirect
14. C₂₃×#2965	C₂	semidirect
267. #62542	C₂₃	direct (N × H)
316. #3503	C₂×C₂₃	semidirect
317. #3369	C₂×C₂₃	semidirect
318. #3331	C₂×C₂₃	semidirect
319. #3316	C₂×C₂₃	semidirect
320. #3229	C₂×C₂₃	semidirect
321. #3227	C₂×C₂₃	semidirect
322. #3210	C₂×C₂₃	semidirect
323. #3209	C₂×C₂₃	semidirect
324. #3185	C₂×C₂₃	semidirect
325. #3167	C₂×C₂₃	semidirect
326. #3141	C₂×C₂₃	semidirect
327. C₂×#487	C₂×C₂₃	semidirect
328. #2965	C₂×C₂₃	semidirect
525. C₂₃	#62542	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×#3511	2
C₂₃×#3503	2
C₂₃×#3369	2
C₂₃×#3331	2
C₂₃×#3316	2
C₂₃×#3229	2
C₂₃×#3227	2
C₂₃×#3210	2
C₂₃×#3209	2
C₂₃×#3185	2
C₂₃×#3167	2
C₂₃×#3141	2
C₂×C₂₃×#487	2
C₂₃×#2965	2
#62542	23

Subgroups — C23×#62542 #19210168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#62542 #19210168