Subgroups — C₂₃×#61701 #19209327

Order 5888.

1778 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3506	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3342	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3340	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3304	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3302	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3190	C₂	1	4	yes	maximal, complemented
10	C₂₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#3103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#3091	C₂	1	4	yes	maximal, complemented
13	C₂₃×#2972	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#559	—	2	2	no	—
15	C₂₃×#556	—	2	2	no	—
16	C₂₃×#556	—	2	2	no	—
17	C₂₃×#556	C₂²	1	4	yes	—
18	C₂₃×#556	C₂²	1	4	yes	—
19	C₂₃×#556	C₂²	1	2	yes	—
20	C₂₃×#555	C₂²	1	2	yes	—
21	C₂₃×#553	—	2	2	no	—
22	C₂₃×#550	C₂²	1	2	yes	—
23	C₂₃×#547	C₂²	1	2	yes	—
24	C₂₃×#538	—	2	2	no	—
25	C₂₃×#535	—	2	1	no	—
26	C₂₃×#533	C₂²	1	1	yes	characteristic
27	C₂₃×#532	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#126	—	2	2	no	—
30	C₂×C₂₃×#124	—	2	1	no	—
31	C₂₃×#525	C₂²	1	4	yes	—
32	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
33	C₂₃×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
35	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#117	C₂²	1	2	yes	—
38	C₂₃×#500	C₂²	1	2	yes	complemented
39	C₂₃×#498	C₂²	1	2	yes	complemented
40	C₂₃×#487	C₂²	1	2	yes	complemented
41	C₂₃×#486	C₂²	1	2	yes	complemented
42	C₂₃×#485	C₂²	1	8	yes	complemented
43	C₂₃×#485	C₂²	1	8	yes	complemented
44	C₂₃×#484	C₂²	1	2	yes	—
45	C₂₃×#483	C₂²	1	4	yes	—
46	C₂₃×#482	C₂²	1	8	yes	complemented
47	C₂₃×#481	C₂²	1	2	yes	—
48	C₂₃×#480	C₂²	1	4	yes	—
49	C₂₃×#477	C₂²	1	2	yes	complemented
50	C₂₃×#475	C₂²	1	4	yes	complemented
51	C₂₃×#475	C₂²	1	2	yes	complemented
52	C₂₃×#474	C₂²	1	4	yes	complemented
53	C₂₃×#467	C₂²	1	8	yes	complemented
54	C₂₃×#465	C₂²	1	8	yes	complemented
55	C₂₃×#462	C₂²	1	2	yes	complemented
56	C₂₃×#460	C₂²	1	2	yes	complemented
57	C₂₃×#452	C₂²	1	8	yes	complemented
58	C₂₃×#451	C₂²	1	2	yes	—
59	C₂₃×#448	C₂²	1	2	yes	—
60	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
61	C₄×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
62	C₈×D₄×C₂₃	C₂²	1	4	yes	—
63	C₈×D₄×C₂₃	C₂²	1	4	yes	—
64	C₂₃×#431	C₂²	1	1	yes	characteristic
65	C₂₃×#431	C₂²	1	1	yes	characteristic
66	C₂₃×#427	C₂²	1	8	yes	complemented
67	C₂₃×#416	C₂²	1	8	yes	complemented
68	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
69	C₂×#41×C₂₃	—	2	1	no	—
70	C₂×#41×C₂₃	C₂³	1	4	yes	—
71	C₂²×Q₈×C₂₃	—	2	1	no	—
72	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
74	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
77	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×#127	—	2	2	no	—
86	C₂₃×#127	C₂³	1	1	yes	characteristic
87	C₂₃×#126	—	2	4	no	—
88	C₂₃×#126	—	2	8	no	—
89	C₂₃×#126	—	2	8	no	—
90	C₂₃×#125	—	2	1	no	—
91	C₂₃×#124	—	2	4	no	—
92	C₂₃×#124	—	2	4	no	—
93	C₂₃×#124	—	2	2	no	—
94	C₂₃×#124	—	2	4	no	—
95	C₂₃×#124	—	2	1	no	—
96	C₂₃×#123	—	2	4	no	—
97	C₂₃×#123	—	2	4	no	—
98	C₂₃×C₂²⋊Q₈	—	2	4	no	—
99	C₂₃×C₂²⋊Q₈	—	2	1	no	—
100	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
101	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
102	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
103	C₂₃×#121	—	2	2	no	—
104	C₂₃×#121	—	2	2	no	—
105	C₂₃×#121	—	2	1	no	—
106	C₂₃×#121	—	2	4	no	—
107	C₂₃×#121	C₂³	1	4	yes	—
108	C₂₃×#121	C₂³	1	2	yes	—
109	C₂₃×#120	—	2	4	no	—
110	C₄×Q₈×C₂₃	—	2	4	no	—
111	C₄×Q₈×C₂₃	—	2	4	no	—
112	C₄×Q₈×C₂₃	—	2	2	no	—
113	C₄×Q₈×C₂₃	—	2	1	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
116	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
117	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
118	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
119	C₄×D₄×C₂₃	—	2	4	no	—
120	C₄×D₄×C₂₃	—	2	4	no	—
121	C₄×D₄×C₂₃	—	2	2	no	—
122	C₄×D₄×C₂₃	—	2	1	no	—
123	C₄×D₄×C₂₃	D₄	1	2	yes	—
124	C₄×D₄×C₂₃	D₄	1	2	yes	—
125	C₄×D₄×C₂₃	C₂³	1	2	yes	—
126	C₄×D₄×C₂₃	C₂³	1	2	yes	—
127	C₄×D₄×C₂₃	C₂³	1	2	yes	—
128	C₂₃×#117	—	2	4	no	—
129	C₂₃×#117	—	2	1	no	—
130	C₂₃×#117	—	2	2	no	—
131	C₂₃×#117	C₂³	1	4	yes	—
132	C₂₃×#117	C₂³	1	8	yes	—
133	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
135	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
136	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
137	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
139	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
140	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
141	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
143	C₂×C₄²×C₂₃	—	2	2	no	—
144	C₂×C₄²×C₂₃	—	2	1	no	—
145	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
146	C₂₃×#107	C₂³	1	4	yes	—
147	C₂₃×#107	C₂³	1	8	yes	—
148	C₂₃×#106	C₂³	1	4	yes	—
149	C₂₃×#106	C₂³	1	8	yes	—
150	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
151	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
152	C₂₃×#103	C₂³	1	4	yes	—
153	C₂₃×#103	C₂³	1	8	yes	—
154	C₂₃×#103	C₂³	1	4	yes	—
155	C₂₃×#103	C₂³	1	4	yes	—
156	C₂₃×#102	C₂³	1	4	yes	—
157	C₂₃×#102	C₂³	1	8	yes	—
158	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
159	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
160	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
161	C₄×C₈×C₂₃	C₂³	1	2	yes	—
162	C₄×C₈×C₂₃	C₂³	1	2	yes	—
163	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
164	#41×C₂₃	—	4	8	no	—
165	#41×C₂₃	—	4	4	no	—
166	C₂×Q₈×C₂₃	—	2	1	no	—
167	C₂×Q₈×C₂₃	—	2	1	no	—
168	C₂×Q₈×C₂₃	—	2	2	no	—
169	C₂×Q₈×C₂₃	—	2	2	no	—
170	C₂×Q₈×C₂₃	—	2	1	no	—
171	C₂×Q₈×C₂₃	—	4	2	no	—
172	C₂×Q₈×C₂₃	—	4	4	no	—
173	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
174	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
175	C₂×D₄×C₂₃	—	2	2	no	—
176	C₂×D₄×C₂₃	—	2	4	no	—
177	C₂×D₄×C₂₃	—	2	4	no	—
178	C₂×D₄×C₂₃	—	2	4	no	—
179	C₂×D₄×C₂₃	—	2	1	no	—
180	C₂×D₄×C₂₃	—	2	1	no	—
181	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
182	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
183	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₂₃	—	2	1	no	—
185	C₂²×C₄×C₂₃	—	2	1	no	—
186	C₂²×C₄×C₂₃	—	2	2	no	—
187	C₂²×C₄×C₂₃	—	2	2	no	—
188	C₂²×C₄×C₂₃	—	2	4	no	—
189	C₂²×C₄×C₂₃	—	4	4	no	—
190	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
193	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
196	Q₁₆×C₂₃	—	4	4	no	—
197	SD₁₆×C₂₃	—	4	4	no	—
198	C₂×C₈×C₂₃	—	4	8	no	—
199	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
200	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	4	no	—
203	C₄⋊C₄×C₂₃	—	2	8	no	—
204	C₄⋊C₄×C₂₃	—	2	2	no	—
205	C₄⋊C₄×C₂₃	—	2	2	no	—
206	C₄⋊C₄×C₂₃	—	2	2	no	—
207	C₄⋊C₄×C₂₃	—	2	2	no	—
208	C₄⋊C₄×C₂₃	—	2	1	no	—
209	C₄⋊C₄×C₂₃	—	2	2	no	—
210	C₄⋊C₄×C₂₃	—	2	1	no	—
211	C₄⋊C₄×C₂₃	—	2	2	no	—
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	2	no	—
215	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
217	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
218	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
219	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
220	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
221	C₂²⋊C₄×C₂₃	—	2	8	no	—
222	C₂²⋊C₄×C₂₃	—	2	8	no	—
223	C₂²⋊C₄×C₂₃	—	2	2	no	—
224	C₂²⋊C₄×C₂₃	—	2	4	no	—
225	C₂²⋊C₄×C₂₃	—	2	4	no	—
226	C₂²⋊C₄×C₂₃	—	2	4	no	—
227	C₂²⋊C₄×C₂₃	—	2	2	no	—
228	C₂²⋊C₄×C₂₃	—	2	2	no	—
229	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
230	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
231	C₄²×C₂₃	—	2	8	no	—
232	C₄²×C₂₃	—	2	2	no	—
233	C₄²×C₂₃	—	2	2	no	—
234	C₄²×C₂₃	—	2	4	no	—
235	C₄²×C₂₃	—	2	2	no	—
236	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
239	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
240	#61701	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂³×C₂₃	—	2	4	no	—
242	C₂³×C₂₃	—	2	2	no	—
243	C₂³×C₂₃	—	4	2	no	—
244	C₂³×C₂₃	—	4	2	no	—
245	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
246	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₂₃	—	2	2	no	—
248	Q₈×C₂₃	—	2	2	no	—
249	Q₈×C₂₃	—	2	4	no	—
250	Q₈×C₂₃	—	4	2	no	—
251	Q₈×C₂₃	—	4	4	no	—
252	Q₈×C₂₃	—	4	4	no	—
253	Q₈×C₂₃	—	4	2	no	—
254	Q₈×C₂₃	—	8	1	no	—
255	D₄×C₂₃	—	2	4	no	—
256	D₄×C₂₃	—	2	4	no	—
257	D₄×C₂₃	—	2	4	no	—
258	D₄×C₂₃	—	4	4	no	—
259	D₄×C₂₃	—	4	2	no	—
260	D₄×C₂₃	—	8	1	no	—
261	D₄×C₂₃	—	8	4	no	—
262	D₄×C₂₃	—	8	4	no	—
263	C₂×C₄×C₂₃	—	2	8	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	2	no	—
267	C₂×C₄×C₂₃	—	2	2	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	4	no	—
270	C₂×C₄×C₂₃	—	2	2	no	—
271	C₂×C₄×C₂₃	—	4	4	no	—
272	C₂×C₄×C₂₃	—	4	2	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	2	no	—
275	C₂×C₄×C₂₃	—	4	2	no	—
276	C₂×C₄×C₂₃	—	4	4	no	—
277	C₂×C₄×C₂₃	—	4	4	no	—
278	C₂×C₄×C₂₃	—	8	4	no	—
279	C₂×C₄×C₂₃	#135	1	4	yes	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
281	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
285	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
287	C₈×C₂₃	—	2	4	no	—
288	C₈×C₂₃	—	2	4	no	—
289	#3506	C₂×C₂₃	1	1	yes	characteristic
290	C₂×#556	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#3342	C₂×C₂₃	1	4	yes	complemented
292	#3340	C₂×C₂₃	1	4	yes	complemented
293	#3304	C₂×C₂₃	1	4	yes	complemented
294	#3302	C₂×C₂₃	1	4	yes	complemented
295	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#3226	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#3190	C₂×C₂₃	1	4	yes	complemented
298	#3104	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#3103	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#3091	C₂×C₂₃	1	4	yes	complemented
301	#2972	C₂×C₂₃	1	1	yes	characteristic, complemented
302	C₂²×C₂₃	—	2	1	no	—
303	C₂²×C₂₃	—	2	1	no	—
304	C₂²×C₂₃	—	2	4	no	—
305	C₂²×C₂₃	—	2	4	no	—
306	C₂²×C₂₃	—	2	4	no	—
307	C₂²×C₂₃	—	4	2	no	—
308	C₂²×C₂₃	—	4	2	no	—
309	C₂²×C₂₃	—	4	2	no	—
310	C₂²×C₂₃	—	8	4	no	—
311	C₂²×C₂₃	C₂×#135	1	2	yes	—
312	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₂₃	—	2	1	no	—
314	C₄×C₂₃	—	2	1	no	—
315	C₄×C₂₃	—	2	1	no	—
316	C₄×C₂₃	—	2	4	no	—
317	C₄×C₂₃	—	4	2	no	—
318	C₄×C₂₃	—	4	2	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	—	4	4	no	—
323	C₄×C₂₃	C₂×#135	1	2	yes	—
324	C₄×C₂₃	C₂×#135	1	2	yes	—
325	C₄×C₂₃	#585	1	2	yes	—
326	#559	—	2	2	no	—
327	#556	—	2	2	no	—
328	#556	—	2	2	no	—
329	#556	C₂²×C₂₃	1	4	yes	—
330	#556	C₂²×C₂₃	1	2	yes	—
331	#556	C₂²×C₂₃	1	4	yes	—
332	#555	C₂²×C₂₃	1	2	yes	—
333	#553	—	2	2	no	—
334	#550	C₂²×C₂₃	1	2	yes	—
335	#547	C₂²×C₂₃	1	2	yes	—
336	#538	—	2	2	no	—
337	#535	—	2	1	no	—
338	#533	C₂²×C₂₃	1	1	yes	characteristic
339	#532	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#126	—	2	2	no	—
342	C₂×#124	—	2	1	no	—
343	#525	C₂²×C₂₃	1	4	yes	—
344	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
345	#520	—	2	1	no	—
346	C₂×C₄×Q₈	—	2	1	no	—
347	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#117	C₂²×C₂₃	1	2	yes	—
350	#500	C₂²×C₂₃	1	2	yes	complemented
351	#498	C₂²×C₂₃	1	2	yes	complemented
352	#487	C₂²×C₂₃	1	2	yes	complemented
353	#486	C₂²×C₂₃	1	2	yes	complemented
354	#485	C₂²×C₂₃	1	8	yes	complemented
355	#485	C₂²×C₂₃	1	8	yes	complemented
356	#484	C₂²×C₂₃	1	2	yes	—
357	#483	C₂²×C₂₃	1	4	yes	—
358	#482	C₂²×C₂₃	1	8	yes	complemented
359	#481	C₂²×C₂₃	1	2	yes	—
360	#480	C₂²×C₂₃	1	4	yes	—
361	#477	C₂²×C₂₃	1	2	yes	complemented
362	#475	C₂²×C₂₃	1	2	yes	complemented
363	#475	C₂²×C₂₃	1	4	yes	complemented
364	#474	C₂²×C₂₃	1	4	yes	complemented
365	#467	C₂²×C₂₃	1	8	yes	complemented
366	#465	C₂²×C₂₃	1	8	yes	complemented
367	#462	C₂²×C₂₃	1	2	yes	complemented
368	#460	C₂²×C₂₃	1	2	yes	complemented
369	#452	C₂²×C₂₃	1	8	yes	complemented
370	#451	C₂²×C₂₃	1	2	yes	—
371	#448	C₂²×C₂₃	1	2	yes	—
372	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
373	C₄×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
374	C₈×D₄	C₂²×C₂₃	1	4	yes	—
375	C₈×D₄	C₂²×C₂₃	1	4	yes	—
376	#431	C₂²×C₂₃	1	1	yes	characteristic
377	#431	C₂²×C₂₃	1	1	yes	characteristic
378	#427	C₂²×C₂₃	1	8	yes	complemented
379	#416	C₂²×C₂₃	1	8	yes	complemented
380	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
381	C₂×C₂₃	—	2	2	no	—
382	C₂×C₂₃	—	4	4	no	—
383	C₂×C₂₃	—	8	2	no	—
384	C₂×C₂₃	#3485	1	1	yes	characteristic
385	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
386	C₂×C₂₃	#3586	1	1	yes	characteristic
387	C₂×#41	—	2	1	no	—
388	C₂×#41	C₂³×C₂₃	1	4	yes	—
389	C₂²×Q₈	—	2	1	no	—
390	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
391	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
393	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
394	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
395	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
396	C₄⋊Q₈	—	2	1	no	—
397	C₄⋊Q₈	—	2	1	no	—
398	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
400	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
401	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
403	#127	—	2	2	no	—
404	#127	C₂³×C₂₃	1	1	yes	characteristic
405	#126	—	2	8	no	—
406	#126	—	2	8	no	—
407	#126	—	2	4	no	—
408	#125	—	2	1	no	—
409	#124	—	2	4	no	—
410	#124	—	2	4	no	—
411	#124	—	2	2	no	—
412	#124	—	2	4	no	—
413	#124	—	2	1	no	—
414	#123	—	2	4	no	—
415	#123	—	2	4	no	—
416	C₂²⋊Q₈	—	2	4	no	—
417	C₂²⋊Q₈	—	2	1	no	—
418	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
419	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
420	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
421	#121	—	2	2	no	—
422	#121	—	2	2	no	—
423	#121	—	2	1	no	—
424	#121	—	2	4	no	—
425	#121	C₂³×C₂₃	1	2	yes	—
426	#121	C₂³×C₂₃	1	4	yes	—
427	#120	—	2	4	no	—
428	C₄×Q₈	—	2	4	no	—
429	C₄×Q₈	—	2	4	no	—
430	C₄×Q₈	—	2	2	no	—
431	C₄×Q₈	—	2	1	no	—
432	C₄×Q₈	—	2	1	no	—
433	C₄×Q₈	—	2	1	no	—
434	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
435	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
436	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
437	C₄×D₄	—	2	4	no	—
438	C₄×D₄	—	2	4	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	2	1	no	—
441	C₄×D₄	D₄×C₂₃	1	2	yes	—
442	C₄×D₄	D₄×C₂₃	1	2	yes	—
443	C₄×D₄	C₂³×C₂₃	1	2	yes	—
444	C₄×D₄	C₂³×C₂₃	1	2	yes	—
445	C₄×D₄	C₂³×C₂₃	1	2	yes	—
446	#117	—	2	4	no	—
447	#117	—	2	1	no	—
448	#117	—	2	2	no	—
449	#117	C₂³×C₂₃	1	8	yes	—
450	#117	C₂³×C₂₃	1	4	yes	—
451	C₂×C₄⋊C₄	—	2	2	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	2	no	—
455	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
458	C₂×C₂²⋊C₄	—	2	2	no	—
459	C₂×C₂²⋊C₄	—	2	2	no	—
460	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
461	C₂×C₄²	—	2	2	no	—
462	C₂×C₄²	—	2	1	no	—
463	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
464	#107	C₂³×C₂₃	1	4	yes	—
465	#107	C₂³×C₂₃	1	8	yes	—
466	#106	C₂³×C₂₃	1	4	yes	—
467	#106	C₂³×C₂₃	1	8	yes	—
468	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
469	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
470	#103	C₂³×C₂₃	1	4	yes	—
471	#103	C₂³×C₂₃	1	8	yes	—
472	#103	C₂³×C₂₃	1	4	yes	—
473	#103	C₂³×C₂₃	1	4	yes	—
474	#102	C₂³×C₂₃	1	4	yes	—
475	#102	C₂³×C₂₃	1	8	yes	—
476	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
477	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
478	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
479	C₄×C₈	C₂³×C₂₃	1	2	yes	—
480	C₄×C₈	C₂³×C₂₃	1	2	yes	—
481	C₂₃	#61701	1	1	yes	characteristic, direct product factor, Sylow
482	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
483	#41	—	4	8	no	—
484	#41	—	4	4	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	2	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	4	2	no	—
491	C₂×Q₈	—	4	4	no	—
492	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
493	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
501	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
502	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	4	4	no	—
509	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
510	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
511	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
512	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
514	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
515	Q₁₆	—	4	4	no	—
516	SD₁₆	—	4	4	no	—
517	C₂×C₈	—	4	8	no	—
518	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
519	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	8	no	—
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	2	no	—
534	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
535	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
536	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
537	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
538	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
539	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
540	C₂²⋊C₄	—	2	8	no	—
541	C₂²⋊C₄	—	2	8	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	4	no	—
544	C₂²⋊C₄	—	2	4	no	—
545	C₂²⋊C₄	—	2	4	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
549	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
550	C₄²	—	2	8	no	—
551	C₄²	—	2	2	no	—
552	C₄²	—	2	2	no	—
553	C₄²	—	2	4	no	—
554	C₄²	—	2	2	no	—
555	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
558	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
559	C₂³	—	2	4	no	—
560	C₂³	—	2	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	—	4	2	no	—
563	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
564	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	4	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	4	4	no	—
570	Q₈	—	4	4	no	—
571	Q₈	—	4	2	no	—
572	Q₈	—	8	1	no	—
573	D₄	—	2	4	no	—
574	D₄	—	2	4	no	—
575	D₄	—	2	4	no	—
576	D₄	—	4	4	no	—
577	D₄	—	4	2	no	—
578	D₄	—	8	1	no	—
579	D₄	—	8	4	no	—
580	D₄	—	8	4	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	4	no	—
596	C₂×C₄	—	8	4	no	—
597	C₂×C₄	C₂₃×#135	1	4	yes	—
598	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
599	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
600	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
601	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
602	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
603	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
604	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
605	C₈	—	2	4	no	—
606	C₈	—	2	4	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	2	4	no	—
610	C₂²	—	2	4	no	—
611	C₂²	—	2	4	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	8	4	no	—
616	C₂²	C₂×C₂₃×#135	1	2	yes	—
617	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
618	C₄	—	2	1	no	—
619	C₄	—	2	4	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	1	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	4	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	2	no	—
628	C₄	C₂×C₂₃×#135	1	2	yes	—
629	C₄	C₂×C₂₃×#135	1	2	yes	—
630	C₄	C₂₃×#585	1	2	yes	—
631	C₂	—	2	2	no	—
632	C₂	—	4	4	no	—
633	C₂	—	8	2	no	—
634	C₂	C₂₃×#3485	1	1	yes	characteristic
635	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
636	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×#3506	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#556	C₂	semidirect
3. C₂₃×#3342	C₂	semidirect
4. C₂₃×#3340	C₂	semidirect
5. C₂₃×#3304	C₂	semidirect
6. C₂₃×#3302	C₂	semidirect
7. C₂₃×#3229	C₂	semidirect
8. C₂₃×#3226	C₂	semidirect
9. C₂₃×#3190	C₂	semidirect
10. C₂₃×#3104	C₂	semidirect
11. C₂₃×#3103	C₂	semidirect
12. C₂₃×#3091	C₂	semidirect
13. C₂₃×#2972	C₂	semidirect
38. C₂₃×#500	C₂²	semidirect
39. C₂₃×#498	C₂²	semidirect
40. C₂₃×#487	C₂²	semidirect
41. C₂₃×#486	C₂²	semidirect
42. C₂₃×#485	C₂²	semidirect
43. C₂₃×#485	C₂²	semidirect
46. C₂₃×#482	C₂²	semidirect
49. C₂₃×#477	C₂²	semidirect
50. C₂₃×#475	C₂²	semidirect
51. C₂₃×#475	C₂²	semidirect
52. C₂₃×#474	C₂²	semidirect
53. C₂₃×#467	C₂²	semidirect
54. C₂₃×#465	C₂²	semidirect
55. C₂₃×#462	C₂²	semidirect
56. C₂₃×#460	C₂²	semidirect
57. C₂₃×#452	C₂²	semidirect
60. C₄×Q₁₆×C₂₃	C₂²	semidirect
61. C₄×SD₁₆×C₂₃	C₂²	semidirect
66. C₂₃×#427	C₂²	semidirect
67. C₂₃×#416	C₂²	semidirect
240. #61701	C₂₃	direct (N × H)
290. C₂×#556	C₂×C₂₃	semidirect
291. #3342	C₂×C₂₃	semidirect
292. #3340	C₂×C₂₃	semidirect
293. #3304	C₂×C₂₃	semidirect
294. #3302	C₂×C₂₃	semidirect
295. #3229	C₂×C₂₃	semidirect
296. #3226	C₂×C₂₃	semidirect
297. #3190	C₂×C₂₃	semidirect
298. #3104	C₂×C₂₃	semidirect
299. #3103	C₂×C₂₃	semidirect
300. #3091	C₂×C₂₃	semidirect
301. #2972	C₂×C₂₃	semidirect
350. #500	C₂²×C₂₃	semidirect
351. #498	C₂²×C₂₃	semidirect
352. #487	C₂²×C₂₃	semidirect
353. #486	C₂²×C₂₃	semidirect
354. #485	C₂²×C₂₃	semidirect
355. #485	C₂²×C₂₃	semidirect
358. #482	C₂²×C₂₃	semidirect
361. #477	C₂²×C₂₃	semidirect
362. #475	C₂²×C₂₃	semidirect
363. #475	C₂²×C₂₃	semidirect
364. #474	C₂²×C₂₃	semidirect
365. #467	C₂²×C₂₃	semidirect
366. #465	C₂²×C₂₃	semidirect
367. #462	C₂²×C₂₃	semidirect
368. #460	C₂²×C₂₃	semidirect
369. #452	C₂²×C₂₃	semidirect
372. C₄×Q₁₆	C₂²×C₂₃	semidirect
373. C₄×SD₁₆	C₂²×C₂₃	semidirect
378. #427	C₂²×C₂₃	semidirect
379. #416	C₂²×C₂₃	semidirect
481. C₂₃	#61701	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×#3506	2
C₂×C₂₃×#556	2
C₂₃×#3342	2
C₂₃×#3340	2
C₂₃×#3304	2
C₂₃×#3302	2
C₂₃×#3229	2
C₂₃×#3226	2
C₂₃×#3190	2
C₂₃×#3104	2
C₂₃×#3103	2
C₂₃×#3091	2
C₂₃×#2972	2
#61701	23

Subgroups — C23×#61701 #19209327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61701 #19209327