Subgroups — #26020372

Order 11392.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#10623494	C₂	1	2	yes	maximal
2	#19151353	C₂	1	4	yes	maximal
3	#376×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10623494	C₂²	1	4	yes	—
5	C₄⋊#10623494	—	2	4	no	—
6	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
8	#95×C₈₉	C₄	1	4	yes	—
9	C₂×#10623494	—	2	4	no	—
10	C₂×#10623494	—	2	8	no	—
11	C₂²×C₄×C₈₉	C₂×C₄	1	4	yes	—
12	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
14	C₄²×C₈₉	—	2	4	no	—
15	C₄²×C₈₉	D₄	1	2	yes	—
16	C₄²×C₈₉	Q₈	1	2	yes	—
17	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈₉	—	2	4	no	—
19	C₂×C₄×C₈₉	—	2	4	no	—
20	C₂×C₄×C₈₉	—	2	8	no	—
21	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₈₉	#41	1	4	yes	—
25	#10623494	—	4	4	no	—
26	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₈₉	#117	1	2	yes	—
28	C₂²×C₈₉	#131	1	4	yes	—
29	C₄×C₈₉	—	2	4	no	—
30	C₄×C₈₉	—	4	4	no	—
31	C₄×C₈₉	#124	1	2	yes	—
32	C₄×C₈₉	#125	1	2	yes	—
33	C₂×C₈₉	#382	1	1	yes	characteristic
34	C₂×C₈₉	#424	1	2	yes	—
35	C₂×C₈₉	#433	1	4	yes	—
36	#1832	—	89	1	no	maximal, Sylow
37	C₈₉	#1832	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄⋊C₈	—	89	2	no	—
39	#376	D₈₉	1	1	yes	characteristic
40	#336	—	89	4	no	—
41	C₂²×C₈	—	89	4	no	—
42	C₂×C₄²	C₂×D₈₉	1	2	yes	—
43	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
44	C₄⋊C₈	—	178	4	no	—
45	#95	Dic₈₉	1	4	yes	—
46	C₂²×C₄	C₂×Dic₈₉	1	4	yes	—
47	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
48	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
49	C₂×C₈	—	178	4	no	—
50	C₂×C₈	—	178	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	Dic₁₇₈	1	2	yes	—
53	C₄²	D₃₅₆	1	2	yes	—
54	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	8	no	—
58	C₂×C₄	C₄⋊Dic₈₉	1	4	yes	—
59	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
61	C₂×C₄	#13083574	1	4	yes	—
62	C₈	—	356	4	no	—
63	C₂²	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
64	C₂²	#15667971	1	2	yes	—
65	C₂²	#15668019	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	4	4	no	—
68	C₄	#15668000	1	2	yes	—
69	C₄	#15668004	1	2	yes	—
70	C₂	#19151464	1	1	yes	characteristic
71	C₂	#19151625	1	2	yes	—
72	C₂	#19151659	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂²×C₄
Fitting	#376×C₈₉	C₂
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#376	D₈₉
hypocenter	C₈₉	#1832
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
37. C₈₉	#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#10623494	2
#19151353	2
#376×C₈₉	2
#1832	89