Subgroups — #15686692

Order 2944.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#13092972	C₂	1	2	yes	maximal, complemented
3	#13092965	C₂	1	2	yes	maximal, complemented
4	#13092949	C₂	1	4	yes	maximal, complemented
5	#13092948	C₂	1	4	yes	maximal, complemented
6	#13092946	C₂	1	4	yes	maximal, complemented
7	#13092927	C₂	1	4	yes	maximal, complemented
8	#13092924	C₂	1	2	yes	maximal, complemented
9	#41×Dic₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×#67290	C₂	1	4	yes	maximal, complemented
11	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#41×C₂₃	C₂²	1	2	yes	—
14	C₂×#67290	C₂²	1	4	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#124	C₂²	1	2	yes	complemented
20	C₂₃×#121	C₂²	1	4	yes	—
21	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	#10625117	C₂²	1	4	yes	—
24	Q₈×Dic₂₃	C₂²	1	4	yes	complemented
25	#10625109	C₂²	1	4	yes	—
26	#10625109	C₂²	1	4	yes	—
27	#10625108	C₂²	1	4	yes	—
28	#10625107	C₂²	1	4	yes	complemented
29	#10625106	C₂²	1	4	yes	—
30	D₄×Dic₂₃	C₂²	1	8	yes	—
31	D₄×Dic₂₃	C₂²	1	4	yes	complemented
32	D₄×Dic₂₃	C₂²	1	4	yes	complemented
33	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	—
34	#10625097	C₂²	1	2	yes	—
35	#10625097	C₂²	1	4	yes	complemented
36	#10625095	C₂²	1	4	yes	—
37	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
39	C₂×C₄×Dic₂₃	C₂²	1	4	yes	complemented
40	#10625068	C₂²	1	4	yes	—
41	#10625063	C₂²	1	4	yes	—
42	#10625061	C₂²	1	4	yes	complemented
43	#10625058	C₂²	1	4	yes	complemented
44	#10625054	C₂²	1	8	yes	—
45	(C₂²⋊D₂₃)⋊C₄	C₂²	1	8	yes	—
46	#10625050	C₂²	1	8	yes	complemented
47	#10625048	C₂²	1	8	yes	complemented
48	C₄²×D₂₃	C₂²	1	2	yes	—
49	C₄²×D₂₃	C₂²	1	2	yes	—
50	C₄²×D₂₃	C₂²	1	2	yes	—
51	C₄²×D₂₃	C₂²	1	2	yes	—
52	#10625041	C₂²	1	2	yes	complemented
53	#10625040	C₂²	1	2	yes	complemented
54	C₄×Dic₄₆	C₂²	1	4	yes	complemented
55	C₂³×D₂₃	C₂³	1	1	yes	characteristic
56	#41×C₂₃	—	2	8	no	—
57	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
58	C₂×D₄×C₂₃	C₂³	1	4	yes	—
59	C₂×D₄×C₂₃	C₂³	1	2	yes	—
60	C₂×D₄×C₂₃	C₂³	1	4	yes	—
61	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
65	C₂²×Dic₂₃	C₂³	1	8	yes	—
66	C₂²×Dic₂₃	C₂³	1	2	yes	—
67	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
68	#67290	—	2	16	no	—
69	C₂×C₄×D₂₃	—	2	4	no	—
70	C₂×C₄×D₂₃	—	2	8	no	—
71	C₂×C₄×D₂₃	C₂³	1	2	yes	—
72	C₂×C₄×D₂₃	C₂³	1	2	yes	—
73	C₂×C₄×D₂₃	C₂³	1	2	yes	—
74	C₂×C₄×D₂₃	C₂³	1	2	yes	—
75	C₂×C₄×D₂₃	C₂³	1	2	yes	—
76	C₂×C₄×D₂₃	C₂³	1	2	yes	—
77	C₂×Dic₄₆	C₂³	1	4	yes	—
78	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
80	C₄²×C₂₃	C₂³	1	2	yes	—
81	C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
84	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
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	4	yes	—
89	C₄⋊Dic₂₃	C₂³	1	8	yes	—
90	Dic₂₃⋊C₄	C₂³	1	8	yes	—
91	Dic₂₃⋊C₄	C₂³	1	8	yes	—
92	C₄×Dic₂₃	C₂³	1	4	yes	—
93	C₄×Dic₂₃	C₂³	1	2	yes	—
94	C₄×Dic₂₃	C₂³	1	2	yes	—
95	C₄×Dic₂₃	C₂³	1	2	yes	—
96	C₄×Dic₂₃	C₂³	1	2	yes	—
97	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₂₃	C₂³	1	2	yes	—
100	C₄×Dic₂₃	C₂³	1	8	yes	—
101	C₂³×C₂₃	C₂⁴	1	4	yes	—
102	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
103	C₂²×D₂₃	—	2	2	no	—
104	C₂²×D₂₃	—	2	4	no	—
105	C₂²×D₂₃	C₂⁴	1	2	yes	—
106	Q₈×C₂₃	—	2	4	no	—
107	D₄×C₂₃	—	2	8	no	—
108	D₄×C₂₃	—	2	8	no	—
109	D₄×C₂₃	—	2	4	no	—
110	C₂×C₄×C₂₃	—	2	4	no	—
111	C₂×C₄×C₂₃	—	2	8	no	—
112	C₂×C₄×C₂₃	#41	1	4	yes	complemented
113	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
117	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₂₃	—	2	16	no	—
119	C₂×Dic₂₃	—	2	2	no	—
120	C₂×Dic₂₃	—	2	4	no	—
121	C₂×Dic₂₃	—	2	16	no	—
122	C₂×Dic₂₃	C₂⁴	1	2	yes	—
123	C₂×Dic₂₃	C₂⁴	1	2	yes	—
124	C₂×Dic₂₃	C₂⁴	1	8	yes	—
125	C₂×Dic₂₃	C₂⁴	1	2	yes	—
126	C₄×D₂₃	—	2	4	no	—
127	C₄×D₂₃	—	2	4	no	—
128	C₄×D₂₃	—	2	4	no	—
129	C₄×D₂₃	—	2	4	no	—
130	C₄×D₂₃	—	2	4	no	—
131	C₄×D₂₃	#41	1	8	yes	complemented
132	Dic₄₆	—	2	8	no	—
133	#3444	—	23	1	no	maximal, Sylow
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	2	8	no	—
136	C₂²×C₂₃	—	2	4	no	—
137	C₂²×C₂₃	C₂×#41	1	2	yes	—
138	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
139	C₂×D₂₃	—	2	8	no	—
140	C₂×D₂₃	—	2	2	no	—
141	C₂×D₂₃	—	2	2	no	—
142	C₂×D₂₃	C₂×#41	1	4	yes	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	4	no	—
145	C₄×C₂₃	C₂×#41	1	2	yes	—
146	C₄×C₂₃	C₂×#41	1	2	yes	—
147	C₄×C₂₃	C₂×#41	1	4	yes	—
148	Dic₂₃	—	2	8	no	—
149	Dic₂₃	—	2	2	no	—
150	Dic₂₃	—	2	2	no	—
151	Dic₂₃	C₂×#41	1	4	yes	—
152	#533	—	23	2	no	—
153	#533	—	23	1	no	—
154	#532	—	23	2	no	—
155	#532	D₂₃	1	1	yes	characteristic, complemented
156	#529	—	23	4	no	—
157	#529	—	23	4	no	—
158	#529	—	23	4	no	—
159	#525	—	23	2	no	—
160	#525	—	23	4	no	—
161	C₄×#41	—	23	2	no	—
162	C₄×#41	—	23	4	no	—
163	C₂²×C₄²	—	23	1	no	—
164	C₂×C₂₃	—	2	2	no	—
165	C₂×C₂₃	—	4	4	no	—
166	C₂×C₂₃	C₂²×#41	1	2	yes	—
167	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
168	D₂₃	—	2	4	no	—
169	C₂×#41	—	23	4	no	—
170	C₂×#41	C₂×D₂₃	1	2	yes	—
171	C₂³×C₄	—	23	2	no	—
172	C₂³×C₄	—	23	1	no	—
173	C₄⋊Q₈	—	23	2	no	—
174	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
175	#127	C₂×D₂₃	1	1	yes	characteristic, complemented
176	#126	—	23	8	no	—
177	#125	—	23	4	no	—
178	#125	—	23	2	no	—
179	#124	—	23	4	no	—
180	#124	C₂×D₂₃	1	2	yes	complemented
181	#123	—	23	4	no	—
182	#123	—	23	8	no	—
183	C₂²⋊Q₈	—	23	4	no	—
184	C₂²⋊Q₈	—	23	4	no	—
185	C₂²⋊Q₈	—	23	4	no	—
186	#121	—	23	4	no	—
187	#121	—	23	4	no	—
188	#121	C₂×D₂₃	1	4	yes	—
189	#120	—	23	4	no	—
190	C₄×Q₈	—	23	4	no	—
191	C₄×Q₈	—	23	4	no	—
192	C₄×Q₈	—	23	4	no	—
193	C₄×D₄	—	23	8	no	—
194	C₄×D₄	—	23	4	no	—
195	C₄×D₄	—	23	4	no	—
196	C₄×D₄	—	23	8	no	—
197	C₄×D₄	—	23	8	no	—
198	C₄×D₄	C₂×D₂₃	1	4	yes	complemented
199	#117	—	23	2	no	—
200	#117	—	23	4	no	—
201	#117	—	23	4	no	—
202	#117	—	23	8	no	—
203	C₂×C₄²	—	23	4	no	—
204	C₂×C₄²	—	23	1	no	—
205	C₂×C₄²	—	23	2	no	—
206	C₂×C₄²	—	23	2	no	—
207	C₂×C₄²	—	23	2	no	—
208	C₂×C₄²	—	23	2	no	—
209	C₂×C₄²	—	23	2	no	—
210	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
211	C₂₃	#3444	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	—	23	1	no	—
213	#41	—	2	8	no	—
214	#41	—	46	16	no	—
215	C₂×Q₈	—	23	4	no	—
216	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
217	C₂×D₄	—	23	8	no	—
218	C₂×D₄	C₂²×D₂₃	1	4	yes	—
219	C₂×D₄	C₂²×D₂₃	1	2	yes	—
220	C₂×D₄	C₂²×D₂₃	1	4	yes	—
221	C₂²×C₄	—	23	2	no	—
222	C₂²×C₄	—	23	2	no	—
223	C₂²×C₄	—	23	1	no	—
224	C₂²×C₄	—	23	8	no	—
225	C₂²×C₄	—	23	2	no	—
226	C₂²×C₄	—	23	2	no	—
227	C₂²×C₄	—	23	2	no	—
228	C₂²×C₄	—	23	2	no	—
229	C₂²×C₄	—	23	2	no	—
230	C₂²×C₄	—	46	8	no	—
231	C₂²×C₄	—	46	4	no	—
232	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
233	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
234	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
235	C₄⋊C₄	—	23	4	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	C₂²×D₂₃	1	4	yes	—
241	C₂²⋊C₄	—	23	4	no	—
242	C₂²⋊C₄	—	23	8	no	—
243	C₂²⋊C₄	—	23	8	no	—
244	C₂²⋊C₄	—	23	8	no	—
245	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
246	C₄²	—	23	2	no	—
247	C₄²	—	23	2	no	—
248	C₄²	—	23	2	no	—
249	C₄²	—	23	2	no	—
250	C₄²	—	23	2	no	—
251	C₄²	—	23	8	no	—
252	C₄²	—	23	1	no	—
253	C₄²	—	23	1	no	—
254	C₄²	—	23	4	no	—
255	C₄²	C₂²×D₂₃	1	1	yes	characteristic
256	C₄²	C₂²×D₂₃	1	1	yes	characteristic
257	C₄²	C₂²×D₂₃	1	2	yes	—
258	C₂³	—	23	2	no	—
259	C₂³	—	46	4	no	—
260	C₂³	—	46	2	no	—
261	C₂³	C₂³×D₂₃	1	4	yes	—
262	C₂³	C₂³×D₂₃	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	46	8	no	—
265	D₄	—	2	8	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	8	no	—
268	D₄	—	46	16	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	23	2	no	—
272	C₂×C₄	—	23	8	no	—
273	C₂×C₄	—	23	2	no	—
274	C₂×C₄	—	23	2	no	—
275	C₂×C₄	—	23	8	no	—
276	C₂×C₄	—	46	4	no	—
277	C₂×C₄	—	46	4	no	—
278	C₂×C₄	—	46	4	no	—
279	C₂×C₄	—	46	16	no	—
280	C₂×C₄	—	46	2	no	—
281	C₂×C₄	—	46	4	no	—
282	C₂×C₄	—	46	4	no	—
283	C₂×C₄	—	46	4	no	—
284	C₂×C₄	#67290	1	4	yes	complemented
285	C₂×C₄	C₂³×D₂₃	1	2	yes	—
286	C₂×C₄	C₂³×D₂₃	1	4	yes	—
287	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₂₃	1	2	yes	—
289	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	23	4	no	—
294	C₂²	—	46	8	no	—
295	C₂²	—	46	2	no	—
296	C₂²	—	46	2	no	—
297	C₂²	C₂×#67290	1	2	yes	—
298	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	4	no	—
301	C₄	—	23	4	no	—
302	C₄	—	46	2	no	—
303	C₄	—	46	8	no	—
304	C₄	—	46	2	no	—
305	C₄	C₂×#67290	1	2	yes	—
306	C₄	C₂×#67290	1	2	yes	—
307	C₄	#41×D₂₃	1	4	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	4	4	no	—
310	C₂	—	46	4	no	—
311	C₂	C₂×#41×D₂₃	1	2	yes	—
312	C₂	C₂²×#67290	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂₃×#532	C₂
hypercenter	#532	D₂₃
hypocenter	C₂₃	#3444
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #13092974	C₂	semidirect
2. #13092972	C₂	semidirect
3. #13092965	C₂	semidirect
4. #13092949	C₂	semidirect
5. #13092948	C₂	semidirect
6. #13092946	C₂	semidirect
7. #13092927	C₂	semidirect
8. #13092924	C₂	semidirect
9. #41×Dic₂₃	C₂	semidirect
10. C₄×#67290	C₂	semidirect
11. C₂×C₄²×D₂₃	C₂	semidirect
12. C₂₃×#532	C₂	semidirect
17. C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂₃×#127	C₂²	semidirect
19. C₂₃×#124	C₂²	semidirect
21. C₄×D₄×C₂₃	C₂²	semidirect
24. Q₈×Dic₂₃	C₂²	semidirect
28. #10625107	C₂²	semidirect
31. D₄×Dic₂₃	C₂²	semidirect
32. D₄×Dic₂₃	C₂²	semidirect
35. #10625097	C₂²	semidirect
39. C₂×C₄×Dic₂₃	C₂²	semidirect
42. #10625061	C₂²	semidirect
43. #10625058	C₂²	semidirect
46. #10625050	C₂²	semidirect
47. #10625048	C₂²	semidirect
52. #10625041	C₂²	semidirect
53. #10625040	C₂²	semidirect
54. C₄×Dic₄₆	C₂²	semidirect
112. C₂×C₄×C₂₃	#41	semidirect
131. C₄×D₂₃	#41	semidirect
155. #532	D₂₃	semidirect
174. C₄⋊Q₈	C₂×D₂₃	semidirect
175. #127	C₂×D₂₃	semidirect
180. #124	C₂×D₂₃	semidirect
198. C₄×D₄	C₂×D₂₃	semidirect
211. C₂₃	#3444	semidirect
284. C₂×C₄	#67290	semidirect

Maximal subgroups

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

subgroup	index
#13092974	2
#13092972	2
#13092965	2
#13092949	2
#13092948	2
#13092946	2
#13092927	2
#13092924	2
#41×Dic₂₃	2
C₄×#67290	2
C₂×C₄²×D₂₃	2
C₂₃×#532	2
#3444	23