Subgroups — #26018223

Order 11392.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#19151349	C₂	1	1	yes	maximal, characteristic
3	#346×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₅₆	C₄	1	1	yes	characteristic
5	C₄×Dic₁₇₈	—	2	1	no	—
6	#15667946	C₄	1	1	yes	characteristic
7	C₄⋊#10623494	—	2	1	no	—
8	#15667937	—	2	1	no	—
9	C₂×C₁₆×C₈₉	—	2	1	no	—
10	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₇₈	—	2	1	no	—
12	Dic₃₅₆	—	2	1	no	—
13	Dic₃₅₆	C₈	1	2	yes	complemented
14	C₂×#10623494	—	4	1	no	—
15	C₄⋊Dic₈₉	—	2	1	no	—
16	#13083553	—	4	1	no	—
17	C₄×Dic₈₉	—	4	1	no	—
18	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
20	C₄²×C₈₉	D₄	1	1	yes	characteristic
21	C₁₆×C₈₉	—	2	2	no	—
22	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₈₉	D₈	1	1	yes	characteristic
24	C₂×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
25	C₂×Dic₈₉	—	4	1	no	—
26	C₂×Dic₈₉	—	4	1	no	—
27	Dic₁₇₈	—	2	2	no	—
28	Dic₁₇₈	—	4	1	no	—
29	C₈×C₈₉	—	2	1	no	—
30	C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
31	C₈×C₈₉	M₁₆	1	1	yes	characteristic
32	#10623494	—	8	1	no	—
33	C₂²×C₈₉	#102	1	1	yes	characteristic
34	C₄×C₈₉	—	2	1	no	—
35	C₄×C₈₉	—	2	1	no	—
36	C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₈₉	#104	1	1	yes	characteristic
38	Dic₈₉	—	4	2	no	—
39	Dic₈₉	—	8	1	no	—
40	C₂×C₈₉	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₈₉	#360	1	1	yes	characteristic
42	C₂×C₈₉	#362	1	1	yes	characteristic
43	#1334	—	89	1	no	maximal, Sylow
44	C₈₉	#1334	1	1	yes	characteristic, complemented, Sylow
45	C₄×Q₁₆	—	89	1	no	—
46	#346	D₈₉	1	1	yes	characteristic
47	#334	—	89	1	no	—
48	C₂×Q₁₆	—	89	1	no	—
49	C₄×Q₈	—	178	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	89	1	no	—
52	C₄⋊C₈	—	178	1	no	—
53	#103	—	178	1	no	—
54	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
55	C₂×Q₈	—	178	1	no	—
56	Q₁₆	—	89	2	no	—
57	Q₁₆	—	178	1	no	—
58	C₂×C₈	—	356	1	no	—
59	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
61	C₄⋊C₄	—	356	1	no	—
62	C₄⋊C₄	—	178	1	no	—
63	C₄²	—	356	1	no	—
64	C₄²	D₃₅₆	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	356	1	no	—
67	Q₈	—	178	2	no	—
68	C₂×C₄	—	356	1	no	—
69	C₂×C₄	—	356	1	no	—
70	C₂×C₄	#13083551	1	1	yes	characteristic
71	C₂×C₄	D₇₁₂	1	1	yes	characteristic
72	C₂×C₄	#13083563	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	712	1	no	—
75	C₈	C₈×D₈₉	1	1	yes	characteristic
76	C₈	#13083557	1	1	yes	characteristic
77	C₂²	#15667934	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	356	2	no	—
81	C₄	—	712	1	no	—
82	C₄	#15667924	1	1	yes	characteristic
83	C₄	#15667940	1	1	yes	characteristic
84	C₂	#19151327	1	1	yes	characteristic
85	C₂	#19151409	1	1	yes	characteristic
86	C₂	#19151415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15667934
commutator	C₈×C₈₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₈₉
Fitting	#346×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#102
cosocle	C₄×C₈×C₈₉	C₂²
hypercenter	#346	D₈₉
hypocenter	C₈₉	#1334
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₃₅₆	C₈	semidirect
44. C₈₉	#1334	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₅₆	2
#19151349	2
#346×C₈₉	2
#1334	89