Subgroups — C₂₃×#60870 #19208496

Order 5888.

1690 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#542	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3244	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3243	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3165	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3140	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3095	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3094	C₂	1	2	yes	maximal
9	C₂₃×#3078	C₂	1	2	yes	maximal, complemented
10	C₂₃×#3077	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#544	—	2	1	no	—
13	C₂₃×#544	—	2	1	no	—
14	C₂₃×#544	C₂²	1	4	yes	complemented
15	C₂₃×#544	C₂²	1	4	yes	complemented
16	C₂₃×#543	—	2	1	no	—
17	C₂₃×#543	—	2	1	no	—
18	C₂₃×#542	—	2	4	no	—
19	C₂₃×#542	C₂²	1	8	yes	—
20	C₂₃×#533	—	2	4	no	—
21	C₂₃×#533	C₂²	1	2	yes	—
22	C₂₃×#533	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#126	—	2	1	no	—
25	C₂×C₂₃×#124	—	2	1	no	—
26	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#123	—	2	1	no	—
28	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
29	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#117	C₂²	1	2	yes	—
34	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
35	C₂₃×#501	C₂²	1	8	yes	complemented
36	C₂₃×#500	C₂²	1	4	yes	complemented
37	C₂₃×#498	C₂²	1	4	yes	complemented
38	C₂₃×#489	C₂²	1	4	yes	complemented
39	C₂₃×#489	C₂²	1	4	yes	complemented
40	C₂₃×#488	C₂²	1	8	yes	complemented
41	C₂₃×#485	C₂²	1	8	yes	complemented
42	C₂₃×#484	C₂²	1	4	yes	complemented
43	C₂₃×#483	C₂²	1	4	yes	complemented
44	C₂₃×#482	C₂²	1	8	yes	—
45	C₂₃×#481	C₂²	1	4	yes	—
46	C₂₃×#480	C₂²	1	4	yes	—
47	C₂₃×#479	C₂²	1	8	yes	complemented
48	C₂₃×#478	C₂²	1	8	yes	complemented
49	C₂₃×#477	C₂²	1	4	yes	complemented
50	C₂₃×#476	C₂²	1	4	yes	complemented
51	C₂₃×#475	C₂²	1	4	yes	complemented
52	C₂₃×#474	C₂²	1	4	yes	complemented
53	C₂₃×#433	C₂²	1	8	yes	complemented
54	C₂₃×#428	C₂²	1	4	yes	—
55	C₂₃×#428	C₂²	1	4	yes	—
56	C₂₃×#423	C₂²	1	2	yes	—
57	C₂₃×#419	C₂²	1	4	yes	—
58	C₂₃×#418	C₂²	1	4	yes	—
59	C₂₃×#407	C₂²	1	2	yes	—
60	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	—	2	1	no	—
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
68	C₂₃×C₄⋊Q₈	—	2	4	no	—
69	C₂₃×C₄⋊Q₈	—	2	4	no	—
70	C₂₃×C₄⋊Q₈	—	2	2	no	—
71	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
72	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
76	C₂₃×#126	—	2	8	no	—
77	C₂₃×#125	—	2	4	no	—
78	C₂₃×#125	—	2	4	no	—
79	C₂₃×#125	C₂³	1	4	yes	—
80	C₂₃×#125	C₂³	1	4	yes	—
81	C₂₃×#124	—	2	8	no	—
82	C₂₃×#124	—	2	2	no	—
83	C₂₃×#124	C₂³	1	4	yes	—
84	C₂₃×#123	—	2	8	no	—
85	C₂₃×C₂²⋊Q₈	—	2	2	no	—
86	C₂₃×C₂²⋊Q₈	—	2	2	no	—
87	C₂₃×C₂²⋊Q₈	—	2	2	no	—
88	C₂₃×C₂²⋊Q₈	—	2	2	no	—
89	C₂₃×C₂²⋊Q₈	—	2	2	no	—
90	C₂₃×C₂²⋊Q₈	—	2	8	no	—
91	C₂₃×C₂²⋊Q₈	—	2	4	no	—
92	C₂₃×C₂²⋊Q₈	—	2	4	no	—
93	C₂₃×C₂²⋊Q₈	—	2	4	no	—
94	C₂₃×C₂²⋊Q₈	—	2	4	no	—
95	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
96	C₂₃×#121	—	2	2	no	—
97	C₂₃×#121	C₂³	1	4	yes	—
98	C₄×Q₈×C₂₃	—	2	2	no	—
99	C₄×Q₈×C₂₃	—	2	2	no	—
100	C₄×Q₈×C₂₃	—	2	2	no	—
101	C₄×Q₈×C₂₃	—	2	2	no	—
102	C₄×Q₈×C₂₃	—	2	2	no	—
103	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
104	C₄×D₄×C₂₃	—	2	2	no	—
105	C₄×D₄×C₂₃	C₂³	1	4	yes	—
106	C₂₃×#117	—	2	2	no	—
107	C₂₃×#117	—	2	2	no	—
108	C₂₃×#117	D₄	1	4	yes	—
109	C₂₃×#117	C₂³	1	4	yes	—
110	C₂₃×#117	C₂³	1	4	yes	—
111	C₂₃×#117	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
127	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
128	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
129	C₂×C₄²×C₂₃	—	2	2	no	—
130	C₂×C₄²×C₂₃	—	2	1	no	—
131	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
133	C₂₃×#107	C₂³	1	8	yes	—
134	C₂₃×#107	C₂³	1	8	yes	—
135	C₂₃×#106	C₂³	1	8	yes	—
136	C₂₃×#106	C₂³	1	8	yes	—
137	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
138	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
139	C₂₃×#103	C₂³	1	8	yes	—
140	C₂₃×#103	C₂³	1	8	yes	—
141	C₂₃×#102	C₂³	1	8	yes	—
142	C₂₃×#102	C₂³	1	8	yes	—
143	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
144	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₂₃×#97	C₂³	1	4	yes	—
146	C₄×C₈×C₂₃	C₂³	1	4	yes	—
147	C₂⁴×C₂₃	—	2	1	no	—
148	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	2	no	—
151	C₂×Q₈×C₂₃	—	2	2	no	—
152	C₂×Q₈×C₂₃	—	2	2	no	—
153	C₂×Q₈×C₂₃	—	2	2	no	—
154	C₂×Q₈×C₂₃	—	2	4	no	—
155	C₂×Q₈×C₂₃	—	2	4	no	—
156	C₂×Q₈×C₂₃	—	8	2	no	—
157	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
158	C₂×D₄×C₂₃	—	2	4	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	4	no	—
161	C₂×D₄×C₂₃	—	8	2	no	—
162	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	2	no	—
168	C₂²×C₄×C₂₃	—	2	2	no	—
169	C₂²×C₄×C₂₃	—	2	4	no	—
170	C₂²×C₄×C₂₃	—	8	2	no	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
179	M₁₆×C₂₃	—	2	8	no	—
180	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
181	C₄⋊C₄×C₂₃	—	2	4	no	—
182	C₄⋊C₄×C₂₃	—	2	4	no	—
183	C₄⋊C₄×C₂₃	—	2	4	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	4	no	—
196	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
198	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
199	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
202	C₂²⋊C₄×C₂₃	—	2	4	no	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
210	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
211	C₄²×C₂₃	—	2	4	no	—
212	C₄²×C₂₃	—	2	4	no	—
213	C₄²×C₂₃	—	2	4	no	—
214	C₄²×C₂₃	C₂×D₄	1	2	yes	—
215	C₄²×C₂₃	C₂×D₄	1	2	yes	—
216	C₄²×C₂₃	C₂⁴	1	2	yes	—
217	C₄²×C₂₃	C₂⁴	1	2	yes	—
218	#60870	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₂₃	—	2	2	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	C₂³×C₂₃	—	4	1	no	—
223	C₂³×C₂₃	—	4	2	no	—
224	C₂³×C₂₃	—	4	2	no	—
225	C₂³×C₂₃	—	4	2	no	—
226	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
227	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₂₃	—	2	4	no	—
229	Q₈×C₂₃	—	4	2	no	—
230	Q₈×C₂₃	—	8	2	no	—
231	Q₈×C₂₃	—	8	2	no	—
232	Q₈×C₂₃	—	8	2	no	—
233	Q₈×C₂₃	—	8	2	no	—
234	Q₈×C₂₃	—	8	4	no	—
235	D₄×C₂₃	—	2	4	no	—
236	D₄×C₂₃	—	4	2	no	—
237	D₄×C₂₃	—	8	4	no	—
238	C₂×C₄×C₂₃	—	2	2	no	—
239	C₂×C₄×C₂₃	—	2	4	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	2	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	—	4	2	no	—
253	C₂×C₄×C₂₃	—	8	4	no	—
254	C₂×C₄×C₂₃	—	8	2	no	—
255	C₂×C₄×C₂₃	—	8	2	no	—
256	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
263	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
264	C₈×C₂₃	—	2	8	no	—
265	#3458	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂×#542	C₂×C₂₃	1	1	yes	characteristic
267	#3244	C₂×C₂₃	1	8	yes	complemented
268	#3243	C₂×C₂₃	1	8	yes	complemented
269	#3165	C₂×C₂₃	1	2	yes	complemented
270	#3140	C₂×C₂₃	1	2	yes	complemented
271	#3095	C₂×C₂₃	1	2	yes	complemented
272	#3094	C₂×C₂₃	1	2	yes	—
273	#3078	C₂×C₂₃	1	2	yes	complemented
274	#3077	C₂×C₂₃	1	2	yes	complemented
275	#2921	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₂²×C₂₃	—	2	1	no	—
277	C₂²×C₂₃	—	2	1	no	—
278	C₂²×C₂₃	—	2	2	no	—
279	C₂²×C₂₃	—	2	2	no	—
280	C₂²×C₂₃	—	2	2	no	—
281	C₂²×C₂₃	—	4	2	no	—
282	C₂²×C₂₃	—	4	2	no	—
283	C₂²×C₂₃	—	4	2	no	—
284	C₂²×C₂₃	—	8	2	no	—
285	C₂²×C₂₃	—	8	4	no	—
286	C₂²×C₂₃	#536	1	2	yes	—
287	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	2	2	no	—
290	C₄×C₂₃	—	2	2	no	—
291	C₄×C₂₃	—	2	2	no	—
292	C₄×C₂₃	—	2	2	no	—
293	C₄×C₂₃	—	4	2	no	—
294	C₄×C₂₃	—	4	2	no	—
295	C₄×C₂₃	—	4	4	no	—
296	C₄×C₂₃	—	4	4	no	—
297	C₄×C₂₃	—	4	2	no	—
298	C₄×C₂₃	#536	1	2	yes	—
299	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
300	#544	—	2	1	no	—
301	#544	—	2	1	no	—
302	#544	C₂²×C₂₃	1	4	yes	complemented
303	#544	C₂²×C₂₃	1	4	yes	complemented
304	#543	—	2	1	no	—
305	#543	—	2	1	no	—
306	#542	—	2	4	no	—
307	#542	C₂²×C₂₃	1	8	yes	—
308	#533	—	2	4	no	—
309	#533	C₂²×C₂₃	1	2	yes	—
310	#533	C₂²×C₂₃	1	2	yes	—
311	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#126	—	2	1	no	—
313	C₂×#124	—	2	1	no	—
314	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×#123	—	2	1	no	—
316	C₂×C₂²⋊Q₈	—	2	1	no	—
317	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#117	C₂²×C₂₃	1	2	yes	—
322	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
323	#501	C₂²×C₂₃	1	8	yes	complemented
324	#500	C₂²×C₂₃	1	4	yes	complemented
325	#498	C₂²×C₂₃	1	4	yes	complemented
326	#489	C₂²×C₂₃	1	4	yes	complemented
327	#489	C₂²×C₂₃	1	4	yes	complemented
328	#488	C₂²×C₂₃	1	8	yes	complemented
329	#485	C₂²×C₂₃	1	8	yes	complemented
330	#484	C₂²×C₂₃	1	4	yes	complemented
331	#483	C₂²×C₂₃	1	4	yes	complemented
332	#482	C₂²×C₂₃	1	8	yes	—
333	#481	C₂²×C₂₃	1	4	yes	—
334	#480	C₂²×C₂₃	1	4	yes	—
335	#479	C₂²×C₂₃	1	8	yes	complemented
336	#478	C₂²×C₂₃	1	8	yes	complemented
337	#477	C₂²×C₂₃	1	4	yes	complemented
338	#476	C₂²×C₂₃	1	4	yes	complemented
339	#475	C₂²×C₂₃	1	4	yes	complemented
340	#474	C₂²×C₂₃	1	4	yes	complemented
341	#433	C₂²×C₂₃	1	8	yes	complemented
342	#428	C₂²×C₂₃	1	4	yes	—
343	#428	C₂²×C₂₃	1	4	yes	—
344	#423	C₂²×C₂₃	1	2	yes	—
345	#419	C₂²×C₂₃	1	4	yes	—
346	#418	C₂²×C₂₃	1	4	yes	—
347	#407	C₂²×C₂₃	1	2	yes	—
348	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
349	C₂×C₂₃	—	2	2	no	—
350	C₂×C₂₃	—	4	2	no	—
351	C₂×C₂₃	—	8	2	no	—
352	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
353	C₂×C₂₃	#3585	1	1	yes	characteristic
354	C₂×C₂₃	#3586	1	1	yes	characteristic
355	C₂²×Q₈	—	2	1	no	—
356	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
361	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
362	C₄⋊Q₈	—	2	4	no	—
363	C₄⋊Q₈	—	2	4	no	—
364	C₄⋊Q₈	—	2	2	no	—
365	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
366	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
367	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
368	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
370	#126	—	2	8	no	—
371	#125	—	2	4	no	—
372	#125	—	2	4	no	—
373	#125	C₂³×C₂₃	1	4	yes	—
374	#125	C₂³×C₂₃	1	4	yes	—
375	#124	—	2	8	no	—
376	#124	—	2	2	no	—
377	#124	C₂³×C₂₃	1	4	yes	—
378	#123	—	2	8	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	2	8	no	—
385	C₂²⋊Q₈	—	2	4	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	C₂²⋊Q₈	—	2	4	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
390	#121	—	2	2	no	—
391	#121	C₂³×C₂₃	1	4	yes	—
392	C₄×Q₈	—	2	2	no	—
393	C₄×Q₈	—	2	2	no	—
394	C₄×Q₈	—	2	2	no	—
395	C₄×Q₈	—	2	2	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	C₂³×C₂₃	1	4	yes	—
400	#117	—	2	2	no	—
401	#117	—	2	2	no	—
402	#117	D₄×C₂₃	1	4	yes	—
403	#117	C₂³×C₂₃	1	4	yes	—
404	#117	C₂³×C₂₃	1	4	yes	—
405	#117	C₂³×C₂₃	1	4	yes	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
410	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	—	2	2	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	1	no	—
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
422	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
423	C₂×C₄²	—	2	2	no	—
424	C₂×C₄²	—	2	1	no	—
425	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
427	#107	C₂³×C₂₃	1	8	yes	—
428	#107	C₂³×C₂₃	1	8	yes	—
429	#106	C₂³×C₂₃	1	8	yes	—
430	#106	C₂³×C₂₃	1	8	yes	—
431	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
432	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
433	#103	C₂³×C₂₃	1	8	yes	—
434	#103	C₂³×C₂₃	1	8	yes	—
435	#102	C₂³×C₂₃	1	8	yes	—
436	#102	C₂³×C₂₃	1	8	yes	—
437	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
438	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
439	#97	C₂³×C₂₃	1	4	yes	—
440	C₄×C₈	C₂³×C₂₃	1	4	yes	—
441	C₂₃	#60870	1	1	yes	characteristic, direct product factor, Sylow
442	C₂⁴	—	2	1	no	—
443	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	4	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	8	2	no	—
452	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	8	2	no	—
457	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	8	2	no	—
466	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
474	M₁₆	—	2	8	no	—
475	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
492	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
493	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
494	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
495	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
496	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
505	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
506	C₄²	—	2	4	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₄²	C₂×D₄×C₂₃	1	2	yes	—
510	C₄²	C₂×D₄×C₂₃	1	2	yes	—
511	C₄²	C₂⁴×C₂₃	1	2	yes	—
512	C₄²	C₂⁴×C₂₃	1	2	yes	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	4	1	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	2	no	—
520	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
521	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	4	2	no	—
524	Q₈	—	8	4	no	—
525	Q₈	—	8	2	no	—
526	Q₈	—	8	2	no	—
527	Q₈	—	8	2	no	—
528	Q₈	—	8	2	no	—
529	D₄	—	2	4	no	—
530	D₄	—	4	2	no	—
531	D₄	—	8	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	8	4	no	—
548	C₂×C₄	—	8	2	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
555	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
556	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
557	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
558	C₈	—	2	8	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	8	2	no	—
568	C₂²	—	8	4	no	—
569	C₂²	C₂₃×#536	1	2	yes	—
570	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	4	no	—
580	C₄	—	4	4	no	—
581	C₄	C₂₃×#536	1	2	yes	—
582	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
583	C₂	—	2	2	no	—
584	C₂	—	4	2	no	—
585	C₂	—	8	2	no	—
586	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
587	C₂	C₂₃×#3585	1	1	yes	characteristic
588	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	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3458	C₂	semidirect
3. C₂₃×#3244	C₂	semidirect
4. C₂₃×#3243	C₂	semidirect
5. C₂₃×#3165	C₂	semidirect
6. C₂₃×#3140	C₂	semidirect
7. C₂₃×#3095	C₂	semidirect
9. C₂₃×#3078	C₂	semidirect
10. C₂₃×#3077	C₂	semidirect
11. C₂₃×#2921	C₂	semidirect
14. C₂₃×#544	C₂²	semidirect
15. C₂₃×#544	C₂²	semidirect
35. C₂₃×#501	C₂²	semidirect
36. C₂₃×#500	C₂²	semidirect
37. C₂₃×#498	C₂²	semidirect
38. C₂₃×#489	C₂²	semidirect
39. C₂₃×#489	C₂²	semidirect
40. C₂₃×#488	C₂²	semidirect
41. C₂₃×#485	C₂²	semidirect
42. C₂₃×#484	C₂²	semidirect
43. C₂₃×#483	C₂²	semidirect
47. C₂₃×#479	C₂²	semidirect
48. C₂₃×#478	C₂²	semidirect
49. C₂₃×#477	C₂²	semidirect
50. C₂₃×#476	C₂²	semidirect
51. C₂₃×#475	C₂²	semidirect
52. C₂₃×#474	C₂²	semidirect
53. C₂₃×#433	C₂²	semidirect
218. #60870	C₂₃	direct (N × H)
265. #3458	C₂×C₂₃	semidirect
267. #3244	C₂×C₂₃	semidirect
268. #3243	C₂×C₂₃	semidirect
269. #3165	C₂×C₂₃	semidirect
270. #3140	C₂×C₂₃	semidirect
271. #3095	C₂×C₂₃	semidirect
273. #3078	C₂×C₂₃	semidirect
274. #3077	C₂×C₂₃	semidirect
275. #2921	C₂×C₂₃	semidirect
302. #544	C₂²×C₂₃	semidirect
303. #544	C₂²×C₂₃	semidirect
323. #501	C₂²×C₂₃	semidirect
324. #500	C₂²×C₂₃	semidirect
325. #498	C₂²×C₂₃	semidirect
326. #489	C₂²×C₂₃	semidirect
327. #489	C₂²×C₂₃	semidirect
328. #488	C₂²×C₂₃	semidirect
329. #485	C₂²×C₂₃	semidirect
330. #484	C₂²×C₂₃	semidirect
331. #483	C₂²×C₂₃	semidirect
335. #479	C₂²×C₂₃	semidirect
336. #478	C₂²×C₂₃	semidirect
337. #477	C₂²×C₂₃	semidirect
338. #476	C₂²×C₂₃	semidirect
339. #475	C₂²×C₂₃	semidirect
340. #474	C₂²×C₂₃	semidirect
341. #433	C₂²×C₂₃	semidirect
441. C₂₃	#60870	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×#3458	2
C₂×C₂₃×#542	2
C₂₃×#3244	2
C₂₃×#3243	2
C₂₃×#3165	2
C₂₃×#3140	2
C₂₃×#3095	2
C₂₃×#3094	2
C₂₃×#3078	2
C₂₃×#3077	2
C₂₃×#2921	2
#60870	23

Subgroups — C23×#60870 #19208496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60870 #19208496