Subgroups — #26026362

Order 11392.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668012	C₂	1	3	yes	maximal, complemented
2	C₂×#15668000	C₂	1	3	yes	maximal
3	#19151535	C₂	1	8	yes	maximal, complemented
4	#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Dic₁₇₈	C₂²	1	3	yes	—
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	#15668012	—	2	12	no	—
8	#15668000	—	2	12	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	12	yes	—
10	C₂×#13083551	C₂²	1	12	yes	—
11	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
12	#95×C₈₉	C₂²	1	4	yes	—
13	C₂³×D₈₉	C₂³	1	2	yes	—
14	C₂×Dic₁₇₈	—	2	6	no	—
15	C₂×Dic₁₇₈	—	2	12	no	—
16	C₂×D₃₅₆	—	2	6	no	—
17	C₂²×Dic₈₉	C₂³	1	6	yes	—
18	C₄⋊Dic₈₉	—	2	24	no	—
19	#13083551	—	2	24	no	—
20	C₂²×C₄×C₈₉	C₂³	1	6	yes	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₄²×C₈₉	D₄	1	12	yes	—
23	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
24	C₂²×D₈₉	—	2	6	no	—
25	C₂²×D₈₉	—	2	8	no	—
26	C₂×C₄×C₈₉	—	2	12	no	—
27	C₂×C₄×C₈₉	C₂×D₄	1	12	yes	—
28	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
29	C₂×Dic₈₉	—	2	6	no	—
30	C₂×Dic₈₉	—	2	12	no	—
31	D₃₅₆	—	4	4	no	—
32	Dic₁₇₈	—	4	12	no	—
33	C₂²×C₈₉	C₂²×D₄	1	3	yes	—
34	C₂²×C₈₉	2₋¹⁺⁴	1	4	yes	—
35	C₂×D₈₉	—	4	2	no	—
36	C₂×D₈₉	—	4	12	no	—
37	C₄×C₈₉	—	2	12	no	—
38	C₄×C₈₉	#127	1	4	yes	—
39	Dic₈₉	—	4	6	no	—
40	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
41	C₂×C₈₉	#536	1	6	yes	—
42	D₈₉	—	8	2	no	—
43	#2658	—	89	1	no	maximal, Sylow
44	C₈₉	#2658	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄⋊Q₈	—	89	3	no	—
46	C₂×#124	—	89	3	no	—
47	#399	—	89	8	no	—
48	#376	D₈₉	1	1	yes	characteristic, complemented
49	C₂²×Q₈	—	89	3	no	—
50	C₂²×D₄	—	89	1	no	—
51	C₄⋊Q₈	—	178	12	no	—
52	#124	—	178	12	no	—
53	C₂×C₄⋊C₄	—	89	12	no	—
54	C₂×C₂²⋊C₄	—	89	12	no	—
55	C₂×C₄²	C₂×D₈₉	1	3	yes	—
56	#95	C₂×D₈₉	1	4	yes	—
57	C₂⁴	—	89	2	no	—
58	C₂×Q₈	—	178	6	no	—
59	C₂×Q₈	—	178	12	no	—
60	C₂×D₄	—	178	6	no	—
61	C₂²×C₄	—	89	6	no	—
62	C₂²×C₄	C₂²×D₈₉	1	6	yes	—
63	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
64	C₄⋊C₄	—	178	24	no	—
65	C₂²⋊C₄	—	178	24	no	—
66	C₄²	D₃₅₆	1	12	yes	—
67	C₂³	—	178	6	no	—
68	C₂³	—	178	8	no	—
69	C₂³	C₂³×D₈₉	1	1	yes	characteristic
70	Q₈	—	356	12	no	—
71	D₄	—	356	4	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	178	6	no	—
74	C₂×C₄	—	178	12	no	—
75	C₂×C₄	C₂×D₃₅₆	1	12	yes	—
76	C₂×C₄	C₂×D₃₅₆	1	6	yes	—
77	C₂²	—	356	2	no	—
78	C₂²	—	356	12	no	—
79	C₂²	C₂²×D₃₅₆	1	3	yes	—
80	C₂²	#15668059	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	356	6	no	—
83	C₄	#15668009	1	4	yes	—
84	C₂	—	712	2	no	—
85	C₂	C₂×#15668009	1	1	yes	characteristic
86	C₂	#19152174	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#376×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#376	D₈₉
hypocenter	C₈₉	#2658
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #376

Complements

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

N	H ≅ G/N	type
1. C₂×#15668012	C₂	semidirect
3. #19151535	C₂	semidirect
4. #376×C₈₉	C₂	semidirect
44. C₈₉	#2658	semidirect
48. #376	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668012	2
C₂×#15668000	2
#19151535	2
#376×C₈₉	2
#2658	89