Subgroups — #15686786

Order 2944.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093036	C₂	1	4	yes	maximal, complemented
2	#13093028	C₂	1	2	yes	maximal, complemented
3	#13093021	C₂	1	2	yes	maximal, complemented
4	#13093012	C₂	1	4	yes	maximal, complemented
5	#13093004	C₂	1	4	yes	maximal, complemented
6	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092947	C₂	1	8	yes	maximal, complemented
8	#13092924	C₂	1	2	yes	maximal, complemented
9	#41×Dic₂₃	C₂	1	2	yes	maximal, complemented
10	C₂×#10625043	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
16	C₂₃×#124	C₂²	1	2	yes	complemented
17	C₂₃×#121	C₂²	1	4	yes	—
18	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	#10625117	C₂²	1	4	yes	—
21	Q₈×Dic₂₃	C₂²	1	4	yes	complemented
22	#10625110	C₂²	1	8	yes	—
23	#10625109	C₂²	1	4	yes	—
24	#10625108	C₂²	1	4	yes	—
25	#10625106	C₂²	1	4	yes	—
26	#10625106	C₂²	1	8	yes	complemented
27	D₄×Dic₂₃	C₂²	1	8	yes	—
28	D₄×Dic₂₃	C₂²	1	4	yes	complemented
29	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	—
30	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
31	#10625097	C₂²	1	2	yes	—
32	#10625097	C₂²	1	4	yes	complemented
33	#10625095	C₂²	1	4	yes	—
34	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
35	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₂₃	C₂²	1	4	yes	complemented
37	#10625069	C₂²	1	8	yes	—
38	#10625060	C₂²	1	8	yes	complemented
39	#10625057	C₂²	1	8	yes	—
40	#10625056	C₂²	1	8	yes	—
41	#10625050	C₂²	1	8	yes	complemented
42	#10625049	C₂²	1	8	yes	complemented
43	#10625043	C₂²	1	4	yes	—
44	#10625043	C₂²	1	2	yes	—
45	#10625043	C₂²	1	2	yes	—
46	#10625041	C₂²	1	2	yes	complemented
47	#10625040	C₂²	1	2	yes	complemented
48	C₄×Dic₄₆	C₂²	1	4	yes	complemented
49	C₂³×D₂₃	C₂³	1	1	yes	characteristic
50	#41×C₂₃	—	2	8	no	—
51	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂×D₄×C₂₃	C₂³	1	4	yes	—
53	C₂×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂×D₄×C₂₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
59	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
60	C₂²×Dic₂₃	C₂³	1	8	yes	—
61	C₂²×Dic₂₃	C₂³	1	2	yes	—
62	C₂×C₄×D₂₃	—	2	4	no	—
63	C₂×C₄×D₂₃	C₂³	1	2	yes	—
64	C₂×C₄×D₂₃	C₂³	1	2	yes	—
65	C₂×Dic₄₆	C₂³	1	4	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
67	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
68	C₄²×C₂₃	C₂³	1	2	yes	—
69	C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
72	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
73	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
74	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
75	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
77	C₄⋊Dic₂₃	C₂³	1	4	yes	—
78	C₄⋊Dic₂₃	C₂³	1	4	yes	—
79	C₄⋊Dic₂₃	C₂³	1	8	yes	—
80	Dic₂₃⋊C₄	C₂³	1	8	yes	—
81	Dic₂₃⋊C₄	C₂³	1	4	yes	—
82	Dic₂₃⋊C₄	C₂³	1	4	yes	—
83	Dic₂₃⋊C₄	C₂³	1	8	yes	—
84	C₄×Dic₂₃	C₂³	1	4	yes	—
85	C₄×Dic₂₃	C₂³	1	2	yes	—
86	C₄×Dic₂₃	C₂³	1	2	yes	—
87	C₄×Dic₂₃	C₂³	1	8	yes	—
88	C₂³×C₂₃	C₂⁴	1	4	yes	—
89	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
90	C₂²×D₂₃	—	2	2	no	—
91	C₂²×D₂₃	—	4	2	no	—
92	C₂²×D₂₃	C₂⁴	1	2	yes	—
93	Q₈×C₂₃	—	2	4	no	—
94	D₄×C₂₃	—	2	8	no	—
95	D₄×C₂₃	—	2	4	no	—
96	D₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	2	8	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	#41	1	4	yes	—
100	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₂₃	—	4	8	no	—
106	C₂×Dic₂₃	—	2	2	no	—
107	C₂×Dic₂₃	—	4	8	no	—
108	C₂×Dic₂₃	—	4	2	no	—
109	C₂×Dic₂₃	C₂⁴	1	4	yes	—
110	C₂×Dic₂₃	C₂⁴	1	8	yes	—
111	C₂×Dic₂₃	C₂⁴	1	2	yes	—
112	C₄×D₂₃	—	2	4	no	—
113	C₄×D₂₃	—	2	4	no	—
114	Dic₄₆	—	4	4	no	—
115	#3455	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	4	no	—
118	C₂²×C₂₃	—	2	8	no	—
119	C₂²×C₂₃	C₂×#41	1	2	yes	—
120	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
121	C₂×D₂₃	—	2	2	no	—
122	C₂×D₂₃	—	2	4	no	—
123	C₂×D₂₃	—	4	4	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	C₂×#41	1	2	yes	—
128	C₄×C₂₃	C₂×#41	1	2	yes	—
129	Dic₂₃	—	2	2	no	—
130	Dic₂₃	—	2	8	no	—
131	Dic₂₃	—	2	4	no	—
132	#542	—	23	4	no	—
133	#541	—	23	2	no	—
134	#540	—	23	2	no	—
135	#539	—	23	4	no	—
136	#538	—	23	4	no	—
137	#533	—	23	1	no	—
138	#532	D₂₃	1	1	yes	characteristic, complemented
139	#529	—	23	8	no	—
140	#525	—	23	2	no	—
141	C₄×#41	—	23	2	no	—
142	C₂×#117	—	23	1	no	—
143	C₂×C₂₃	—	2	2	no	—
144	C₂×C₂₃	—	4	4	no	—
145	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
146	C₂×C₂₃	#585	1	2	yes	—
147	D₂₃	—	4	2	no	—
148	C₂×#41	C₂×D₂₃	1	2	yes	—
149	C₂³×C₄	—	23	1	no	—
150	C₄⋊Q₈	—	23	2	no	—
151	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
152	#127	C₂×D₂₃	1	1	yes	characteristic, complemented
153	#126	—	23	8	no	—
154	#126	—	23	8	no	—
155	#125	—	23	8	no	—
156	#125	—	23	2	no	—
157	#124	—	23	8	no	—
158	#124	C₂×D₂₃	1	2	yes	complemented
159	#123	—	23	4	no	—
160	#123	—	23	8	no	—
161	#123	—	23	8	no	—
162	C₂²⋊Q₈	—	23	4	no	—
163	C₂²⋊Q₈	—	23	4	no	—
164	C₂²⋊Q₈	—	23	8	no	—
165	#121	—	23	4	no	—
166	#121	—	23	8	no	—
167	#121	C₂×D₂₃	1	4	yes	—
168	#120	—	23	4	no	—
169	C₄×Q₈	—	23	4	no	—
170	C₄×Q₈	—	23	4	no	—
171	C₄×D₄	—	23	8	no	—
172	C₄×D₄	—	23	4	no	—
173	C₄×D₄	—	23	8	no	—
174	C₄×D₄	C₂×D₂₃	1	4	yes	complemented
175	#117	—	23	2	no	—
176	#117	—	23	4	no	—
177	#117	—	23	4	no	—
178	#117	—	23	2	no	—
179	#117	—	23	2	no	—
180	C₂×C₄⋊C₄	—	23	2	no	—
181	C₂×C₂²⋊C₄	—	23	2	no	—
182	C₂×C₄²	—	23	1	no	—
183	C₂×C₄²	—	23	4	no	—
184	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
185	C₂₃	#3455	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	—	23	1	no	—
187	#41	—	2	8	no	—
188	C₂×Q₈	—	23	4	no	—
189	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
190	C₂×D₄	—	23	8	no	—
191	C₂×D₄	C₂²×D₂₃	1	4	yes	—
192	C₂×D₄	C₂²×D₂₃	1	2	yes	—
193	C₂×D₄	C₂²×D₂₃	1	4	yes	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	2	no	—
196	C₂²×C₄	—	23	8	no	—
197	C₂²×C₄	—	23	2	no	—
198	C₂²×C₄	—	23	2	no	—
199	C₂²×C₄	—	46	4	no	—
200	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
201	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
203	C₄⋊C₄	—	23	4	no	—
204	C₄⋊C₄	—	23	4	no	—
205	C₄⋊C₄	—	23	4	no	—
206	C₄⋊C₄	—	23	8	no	—
207	C₄⋊C₄	—	23	8	no	—
208	C₄⋊C₄	—	23	8	no	—
209	C₄⋊C₄	—	23	4	no	—
210	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
211	C₂²⋊C₄	—	23	4	no	—
212	C₂²⋊C₄	—	23	8	no	—
213	C₂²⋊C₄	—	23	8	no	—
214	C₂²⋊C₄	—	23	4	no	—
215	C₂²⋊C₄	—	23	4	no	—
216	C₂²⋊C₄	—	23	8	no	—
217	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
218	C₄²	—	23	2	no	—
219	C₄²	—	23	8	no	—
220	C₄²	—	23	2	no	—
221	C₄²	—	23	4	no	—
222	C₄²	C₂²×D₂₃	1	1	yes	characteristic
223	C₄²	C₂²×D₂₃	1	1	yes	characteristic
224	C₄²	C₂²×D₂₃	1	2	yes	—
225	C₂³	—	23	2	no	—
226	C₂³	—	46	2	no	—
227	C₂³	—	92	2	no	—
228	C₂³	C₂³×D₂₃	1	4	yes	—
229	C₂³	C₂³×D₂₃	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	92	4	no	—
232	D₄	—	2	8	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	92	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	23	4	no	—
239	C₂×C₄	—	23	8	no	—
240	C₂×C₄	—	23	2	no	—
241	C₂×C₄	—	46	2	no	—
242	C₂×C₄	—	46	4	no	—
243	C₂×C₄	—	46	4	no	—
244	C₂×C₄	—	92	2	no	—
245	C₂×C₄	—	92	8	no	—
246	C₂×C₄	#67290	1	4	yes	—
247	C₂×C₄	C₂³×D₂₃	1	4	yes	—
248	C₂×C₄	C₂³×D₂₃	1	2	yes	—
249	C₂×C₄	C₂³×D₂₃	1	2	yes	—
250	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	46	2	no	—
256	C₂²	—	46	4	no	—
257	C₂²	—	92	4	no	—
258	C₂²	C₂×#67290	1	2	yes	—
259	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	46	2	no	—
264	C₄	—	46	4	no	—
265	C₄	—	46	8	no	—
266	C₄	C₂×#67290	1	2	yes	—
267	C₄	C₂×#67290	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	—	92	2	no	—
271	C₂	C₂²×#67290	1	1	yes	characteristic
272	C₂	#13093253	1	2	yes	—

	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	#3455
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. #13093036	C₂	semidirect
2. #13093028	C₂	semidirect
3. #13093021	C₂	semidirect
4. #13093012	C₂	semidirect
5. #13093004	C₂	semidirect
6. #13092974	C₂	semidirect
7. #13092947	C₂	semidirect
8. #13092924	C₂	semidirect
9. #41×Dic₂₃	C₂	semidirect
10. C₂×#10625043	C₂	semidirect
11. C₂₃×#532	C₂	semidirect
14. C₂₃×C₄⋊Q₈	C₂²	semidirect
15. C₂₃×#127	C₂²	semidirect
16. C₂₃×#124	C₂²	semidirect
18. C₄×D₄×C₂₃	C₂²	semidirect
21. Q₈×Dic₂₃	C₂²	semidirect
26. #10625106	C₂²	semidirect
28. D₄×Dic₂₃	C₂²	semidirect
32. #10625097	C₂²	semidirect
36. C₂×C₄×Dic₂₃	C₂²	semidirect
38. #10625060	C₂²	semidirect
41. #10625050	C₂²	semidirect
42. #10625049	C₂²	semidirect
46. #10625041	C₂²	semidirect
47. #10625040	C₂²	semidirect
48. C₄×Dic₄₆	C₂²	semidirect
138. #532	D₂₃	semidirect
151. C₄⋊Q₈	C₂×D₂₃	semidirect
152. #127	C₂×D₂₃	semidirect
158. #124	C₂×D₂₃	semidirect
174. C₄×D₄	C₂×D₂₃	semidirect
185. C₂₃	#3455	semidirect

Maximal subgroups

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

subgroup	index
#13093036	2
#13093028	2
#13093021	2
#13093012	2
#13093004	2
#13092974	2
#13092947	2
#13092924	2
#41×Dic₂₃	2
C₂×#10625043	2
C₂₃×#532	2
#3455	23