Subgroups — D₂₃×#533 #15686682

Order 2944.

1168 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092973	C₂	1	2	yes	maximal, complemented
3	#13092972	C₂	1	2	yes	maximal, complemented
4	#13092971	C₂	1	4	yes	maximal, complemented
5	#13092970	C₂	1	2	yes	maximal, complemented
6	#13092969	C₂	1	4	yes	maximal, complemented
7	C₄⋊Q₈×D₂₃	C₂	1	2	yes	maximal, complemented
8	#125×D₂₃	C₂	1	2	yes	maximal, complemented
9	C₂²⋊Q₈×D₂₃	C₂	1	4	yes	maximal
10	C₄×Q₈×D₂₃	C₂	1	4	yes	maximal, complemented
11	#117×D₂₃	C₂	1	2	yes	maximal
12	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₂₃	C₂²	1	4	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₂₃×#125	C₂²	1	2	yes	complemented
19	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
21	C₂₃×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	#10625117	C₂²	1	4	yes	—
24	Q₈×Dic₂₃	C₂²	1	4	yes	complemented
25	#10625115	C₂²	1	4	yes	complemented
26	#10625097	C₂²	1	2	yes	—
27	#10625095	C₂²	1	4	yes	—
28	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	#10625068	C₂²	1	4	yes	—
31	#10625067	C₂²	1	8	yes	—
32	#10625063	C₂²	1	4	yes	—
33	C₄⋊C₄×D₂₃	C₂²	1	4	yes	—
34	C₄⋊C₄×D₂₃	C₂²	1	8	yes	—
35	C₄⋊C₄×D₂₃	C₂²	1	4	yes	—
36	#10625061	C₂²	1	4	yes	complemented
37	#10625060	C₂²	1	8	yes	complemented
38	#10625059	C₂²	1	4	yes	complemented
39	#10625059	C₂²	1	4	yes	complemented
40	#10625058	C₂²	1	4	yes	complemented
41	#10625058	C₂²	1	8	yes	complemented
42	C₂²⋊C₄×D₂₃	C₂²	1	4	yes	—
43	#10625049	C₂²	1	8	yes	—
44	#10625048	C₂²	1	4	yes	—
45	#10625043	C₂²	1	4	yes	—
46	C₄²×D₂₃	C₂²	1	4	yes	—
47	C₄²×D₂₃	C₂²	1	2	yes	—
48	C₄²×D₂₃	C₂²	1	1	yes	characteristic
49	C₄²×D₂₃	C₂²	1	1	yes	characteristic
50	C₄²×D₂₃	C₂²	1	2	yes	—
51	C₄²×D₂₃	C₂²	1	1	yes	characteristic
52	C₄²×D₂₃	C₂²	1	1	yes	characteristic
53	#10625041	C₂²	1	2	yes	complemented
54	#10625040	C₂²	1	2	yes	complemented
55	C₄×Dic₄₆	C₂²	1	8	yes	complemented
56	C₄×Dic₄₆	C₂²	1	4	yes	complemented
57	C₂³×D₂₃	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₃	C₂³	1	2	yes	—
62	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
63	Q₈×D₂₃	—	2	16	no	—
64	C₂×C₄×D₂₃	—	2	8	no	—
65	C₂×C₄×D₂₃	Q₈	1	8	yes	complemented
66	C₂×C₄×D₂₃	C₂³	1	2	yes	—
67	C₂×C₄×D₂₃	C₂³	1	2	yes	—
68	C₂×C₄×D₂₃	C₂³	1	2	yes	—
69	C₂×C₄×D₂₃	C₂³	1	8	yes	—
70	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₂₃	C₂³	1	2	yes	—
75	C₂×Dic₄₆	C₂³	1	4	yes	—
76	C₂×Dic₄₆	C₂³	1	8	yes	—
77	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
78	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
79	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
81	C₄²×C₂₃	C₂³	1	2	yes	—
82	C₄²×C₂₃	C₂³	1	4	yes	—
83	C₄²×C₂₃	C₂³	1	1	yes	characteristic
84	C₄²×C₂₃	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
86	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
87	C₄⋊Dic₂₃	C₂³	1	4	yes	—
88	C₄⋊Dic₂₃	C₂³	1	8	yes	—
89	C₄⋊Dic₂₃	C₂³	1	4	yes	—
90	Dic₂₃⋊C₄	C₂³	1	8	yes	—
91	Dic₂₃⋊C₄	C₂³	1	8	yes	—
92	Dic₂₃⋊C₄	C₂³	1	8	yes	—
93	Dic₂₃⋊C₄	C₂³	1	8	yes	—
94	C₄×Dic₂₃	C₂³	1	2	yes	—
95	C₄×Dic₂₃	C₂³	1	4	yes	—
96	C₄×Dic₂₃	C₂³	1	2	yes	—
97	C₄×Dic₂₃	C₂³	1	2	yes	—
98	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₂₃	C₂³	1	2	yes	—
100	C₄×Dic₂₃	C₂³	1	2	yes	—
101	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
102	C₄×Dic₂₃	C₂³	1	8	yes	—
103	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂²×D₂₃	—	2	4	no	—
105	C₂²×D₂₃	C₂×Q₈	1	4	yes	—
106	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
107	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
108	Q₈×C₂₃	—	2	8	no	—
109	C₂×C₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	complemented
111	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
113	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
115	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₂₃	—	2	4	no	—
117	C₂×Dic₂₃	C₂×Q₈	1	4	yes	complemented
118	C₂×Dic₂₃	C₂⁴	1	2	yes	—
119	C₂×Dic₂₃	C₂⁴	1	8	yes	—
120	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₂₃	C₂⁴	1	2	yes	—
123	C₄×D₂₃	—	2	4	no	—
124	C₄×D₂₃	—	2	16	no	—
125	C₄×D₂₃	—	2	4	no	—
126	C₄×D₂₃	—	2	4	no	—
127	C₄×D₂₃	C₂×Q₈	1	4	yes	—
128	C₄×D₂₃	C₂×Q₈	1	4	yes	—
129	C₄×D₂₃	C₂×Q₈	1	4	yes	complemented
130	C₄×D₂₃	C₂×Q₈	1	4	yes	complemented
131	C₄×D₂₃	#41	1	8	yes	—
132	Dic₄₆	—	2	8	no	—
133	Dic₄₆	—	2	16	no	—
134	C₂×#533	—	23	1	no	maximal, Sylow
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
137	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
138	C₂×D₂₃	—	2	2	no	—
139	C₂×D₂₃	—	2	8	no	—
140	C₂×D₂₃	C₂²×Q₈	1	2	yes	—
141	C₂×D₂₃	C₂²×Q₈	1	2	yes	—
142	C₂×D₂₃	C₂×#41	1	4	yes	—
143	C₄×C₂₃	—	2	8	no	—
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
146	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
147	C₄×C₂₃	C₂×#41	1	4	yes	—
148	Dic₂₃	—	2	8	no	—
149	Dic₂₃	—	2	2	no	—
150	Dic₂₃	C₂²×Q₈	1	2	yes	—
151	Dic₂₃	C₂²×Q₈	1	2	yes	—
152	Dic₂₃	C₂×#41	1	4	yes	—
153	#533	—	23	2	no	—
154	#533	—	23	2	no	—
155	#533	—	23	4	no	—
156	#533	—	23	4	no	—
157	#533	—	23	2	no	—
158	#533	—	23	1	no	—
159	#533	D₂₃	1	1	yes	characteristic, direct product factor
160	C₂×C₄⋊Q₈	—	23	2	no	—
161	C₂×#125	—	23	2	no	—
162	C₂×C₂²⋊Q₈	—	23	4	no	—
163	C₂×C₄×Q₈	—	23	4	no	—
164	C₂×#117	—	23	2	no	—
165	C₂²×C₄²	—	23	1	no	—
166	C₂×C₂₃	—	2	2	no	—
167	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
168	C₂×C₂₃	C₂²×#41	1	2	yes	—
169	D₂₃	—	2	2	no	—
170	D₂₃	#533	1	4	yes	direct product factor
171	C₂²×Q₈	—	23	4	no	—
172	C₂³×C₄	—	23	1	no	—
173	C₂³×C₄	—	23	2	no	—
174	C₄⋊Q₈	—	23	4	no	—
175	C₄⋊Q₈	—	23	4	no	—
176	C₄⋊Q₈	—	23	4	no	—
177	C₄⋊Q₈	—	23	2	no	—
178	C₄⋊Q₈	C₂×D₂₃	1	2	yes	complemented
179	#125	—	23	8	no	—
180	#125	—	23	4	no	—
181	#125	—	23	2	no	—
182	#125	C₂×D₂₃	1	2	yes	complemented
183	C₂²⋊Q₈	—	23	4	no	—
184	C₂²⋊Q₈	—	23	4	no	—
185	C₂²⋊Q₈	—	23	8	no	—
186	C₂²⋊Q₈	—	23	4	no	—
187	C₂²⋊Q₈	—	23	8	no	—
188	C₂²⋊Q₈	C₂×D₂₃	1	4	yes	—
189	C₄×Q₈	—	23	4	no	—
190	C₄×Q₈	—	23	4	no	—
191	C₄×Q₈	—	23	8	no	—
192	C₄×Q₈	—	23	8	no	—
193	C₄×Q₈	—	23	4	no	—
194	C₄×Q₈	C₂×D₂₃	1	4	yes	complemented
195	#117	—	23	2	no	—
196	#117	—	23	4	no	—
197	#117	—	23	4	no	—
198	#117	—	23	4	no	—
199	#117	C₂×D₂₃	1	2	yes	—
200	C₂×C₄⋊C₄	—	23	4	no	—
201	C₂×C₄⋊C₄	—	23	8	no	—
202	C₂×C₄⋊C₄	—	23	4	no	—
203	C₂×C₂²⋊C₄	—	23	4	no	—
204	C₂×C₄²	—	23	2	no	—
205	C₂×C₄²	—	23	1	no	—
206	C₂×C₄²	—	23	4	no	—
207	C₂×C₄²	—	23	2	no	—
208	C₂×C₄²	—	23	1	no	—
209	C₂×C₄²	—	23	1	no	—
210	C₂×C₄²	—	23	1	no	—
211	C₂×C₄²	—	23	1	no	—
212	C₂×C₄²	—	23	2	no	—
213	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
214	C₂₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	—	23	1	no	—
216	C₂×Q₈	—	23	8	no	—
217	C₂×Q₈	—	23	4	no	—
218	C₂×Q₈	—	46	16	no	—
219	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
220	C₂²×C₄	—	23	1	no	—
221	C₂²×C₄	—	23	8	no	—
222	C₂²×C₄	—	23	1	no	—
223	C₂²×C₄	—	23	2	no	—
224	C₂²×C₄	—	23	1	no	—
225	C₂²×C₄	—	23	1	no	—
226	C₂²×C₄	—	23	1	no	—
227	C₂²×C₄	—	23	2	no	—
228	C₂²×C₄	—	23	2	no	—
229	C₂²×C₄	—	23	2	no	—
230	C₂²×C₄	—	23	2	no	—
231	C₂²×C₄	—	23	8	no	—
232	C₂²×C₄	—	46	8	no	—
233	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
234	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
235	C₄⋊C₄	—	23	8	no	—
236	C₄⋊C₄	—	23	4	no	—
237	C₄⋊C₄	—	23	8	no	—
238	C₄⋊C₄	—	23	8	no	—
239	C₄⋊C₄	—	23	8	no	—
240	C₄⋊C₄	—	23	8	no	—
241	C₄⋊C₄	—	23	4	no	—
242	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
243	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
244	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
245	C₂²⋊C₄	—	23	4	no	—
246	C₂²⋊C₄	—	23	8	no	—
247	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
248	C₄²	—	23	2	no	—
249	C₄²	—	23	2	no	—
250	C₄²	—	23	2	no	—
251	C₄²	—	23	1	no	—
252	C₄²	—	23	2	no	—
253	C₄²	—	23	1	no	—
254	C₄²	—	23	8	no	—
255	C₄²	—	23	2	no	—
256	C₄²	—	23	4	no	—
257	C₄²	C₂²×D₂₃	1	1	yes	characteristic
258	C₄²	C₂²×D₂₃	1	1	yes	characteristic
259	C₄²	C₂²×D₂₃	1	2	yes	—
260	C₄²	C₂²×D₂₃	1	4	yes	—
261	C₂³	—	23	1	no	—
262	C₂³	—	23	4	no	—
263	C₂³	—	23	1	no	—
264	C₂³	—	46	4	no	—
265	C₂³	C₂³×D₂₃	1	1	yes	characteristic
266	Q₈	—	2	8	no	—
267	Q₈	—	46	16	no	—
268	Q₈	—	46	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	23	4	no	—
271	C₂×C₄	—	23	1	no	—
272	C₂×C₄	—	23	1	no	—
273	C₂×C₄	—	23	2	no	—
274	C₂×C₄	—	23	4	no	—
275	C₂×C₄	—	23	4	no	—
276	C₂×C₄	—	23	4	no	—
277	C₂×C₄	—	23	4	no	—
278	C₂×C₄	—	23	8	no	—
279	C₂×C₄	—	23	2	no	—
280	C₂×C₄	—	23	8	no	—
281	C₂×C₄	—	46	4	no	—
282	C₂×C₄	—	46	4	no	—
283	C₂×C₄	—	46	16	no	—
284	C₂×C₄	—	46	4	no	—
285	C₂×C₄	—	46	4	no	—
286	C₂×C₄	Q₈×D₂₃	1	4	yes	complemented
287	C₂×C₄	C₂³×D₂₃	1	2	yes	—
288	C₂×C₄	C₂³×D₂₃	1	8	yes	—
289	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₂₃	1	2	yes	—
291	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	23	2	no	—
294	C₂²	—	23	4	no	—
295	C₂²	—	23	2	no	—
296	C₂²	—	46	8	no	—
297	C₂²	—	46	2	no	—
298	C₂²	C₂×Q₈×D₂₃	1	2	yes	—
299	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	8	no	—
302	C₄	—	23	2	no	—
303	C₄	—	23	2	no	—
304	C₄	—	23	4	no	—
305	C₄	—	46	2	no	—
306	C₄	—	46	8	no	—
307	C₄	C₂×Q₈×D₂₃	1	2	yes	—
308	C₄	C₂×Q₈×D₂₃	1	2	yes	—
309	C₄	#41×D₂₃	1	4	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	23	4	no	—
312	C₂	—	46	2	no	—
313	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
314	C₂	C₂×#41×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂₃×#533	C₂
prime radical	C₂³×D₂₃	C₂³
hypercenter	#533	D₂₃
hypocenter	C₂₃	C₂×#533
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #13092974	C₂	semidirect
2. #13092973	C₂	semidirect
3. #13092972	C₂	semidirect
4. #13092971	C₂	semidirect
5. #13092970	C₂	semidirect
6. #13092969	C₂	semidirect
7. C₄⋊Q₈×D₂₃	C₂	semidirect
8. #125×D₂₃	C₂	semidirect
10. C₄×Q₈×D₂₃	C₂	semidirect
13. C₂₃×#533	C₂	semidirect
17. C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂₃×#125	C₂²	semidirect
20. C₄×Q₈×C₂₃	C₂²	semidirect
24. Q₈×Dic₂₃	C₂²	semidirect
25. #10625115	C₂²	semidirect
36. #10625061	C₂²	semidirect
37. #10625060	C₂²	semidirect
38. #10625059	C₂²	semidirect
39. #10625059	C₂²	semidirect
40. #10625058	C₂²	semidirect
41. #10625058	C₂²	semidirect
53. #10625041	C₂²	semidirect
54. #10625040	C₂²	semidirect
55. C₄×Dic₄₆	C₂²	semidirect
56. C₄×Dic₄₆	C₂²	semidirect
65. C₂×C₄×D₂₃	Q₈	semidirect
110. C₂×C₄×C₂₃	C₂×Q₈	semidirect
117. C₂×Dic₂₃	C₂×Q₈	semidirect
129. C₄×D₂₃	C₂×Q₈	semidirect
130. C₄×D₂₃	C₂×Q₈	semidirect
159. #533	D₂₃	direct (N × H)
170. D₂₃	#533	direct (N × H)
178. C₄⋊Q₈	C₂×D₂₃	semidirect
182. #125	C₂×D₂₃	semidirect
194. C₄×Q₈	C₂×D₂₃	semidirect
214. C₂₃	C₂×#533	semidirect
286. C₂×C₄	Q₈×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#13092974	2
#13092973	2
#13092972	2
#13092971	2
#13092970	2
#13092969	2
C₄⋊Q₈×D₂₃	2
#125×D₂₃	2
C₂²⋊Q₈×D₂₃	2
C₄×Q₈×D₂₃	2
#117×D₂₃	2
C₂×C₄²×D₂₃	2
C₂₃×#533	2
C₂×#533	23

Subgroups — D23×#533 #15686682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#533 #15686682