Subgroups — #26018224

Order 11392.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151716	C₂	1	1	yes	maximal, characteristic
2	C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#15668034	C₄	1	1	yes	characteristic
5	#15668018	—	2	1	no	—
6	#15667948	C₄	1	1	yes	characteristic
7	#15667940	—	2	1	no	—
8	C₄×#10623494	—	2	1	no	—
9	M₃₂×C₈₉	—	2	1	no	—
10	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	#13083574	—	2	1	no	—
12	Dic₃₅₆	C₈	1	1	yes	characteristic, complemented
13	#13083563	—	2	1	no	—
14	D₇₁₂	C₈	1	1	yes	characteristic, complemented
15	#13083558	—	2	1	no	—
16	#13083557	—	4	1	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₈×D₈₉	—	4	1	no	—
19	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
21	C₄²×C₈₉	D₄	1	1	yes	characteristic
22	C₁₆×C₈₉	—	4	1	no	—
23	C₂×C₄×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₈₉	—	4	1	no	—
26	D₃₅₆	—	2	1	no	—
27	C₄×D₈₉	—	4	1	no	—
28	Dic₁₇₈	—	2	1	no	—
29	C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
30	C₈×C₈₉	M₁₆	1	1	yes	characteristic
31	C₈×C₈₉	D₈	1	1	yes	characteristic
32	C₈×C₈₉	SD₁₆	1	1	yes	characteristic
33	#10623494	—	2	4	no	—
34	#10623494	—	4	1	no	—
35	C₂²×C₈₉	#104	1	1	yes	characteristic
36	C₂×D₈₉	—	4	1	no	—
37	C₄×C₈₉	—	2	2	no	—
38	C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₈₉	#102	1	1	yes	characteristic
40	Dic₈₉	—	4	1	no	—
41	C₂×C₈₉	—	2	1	no	—
42	C₂×C₈₉	D₄⋊C₈	1	1	yes	characteristic
43	D₈₉	—	8	1	no	—
44	#1335	—	89	1	no	maximal, Sylow
45	C₈₉	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	89	1	no	—
47	#364	D₈₉	1	1	yes	characteristic, complemented
48	C₈²	—	89	1	no	—
49	#135	—	89	1	no	—
50	#131	—	178	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	89	1	no	—
53	#104	—	178	1	no	—
54	C₄×C₈	—	178	1	no	—
55	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
56	#41	—	178	1	no	—
57	Q₁₆	—	89	1	no	—
58	SD₁₆	—	178	1	no	—
59	D₈	—	89	1	no	—
60	M₁₆	—	356	1	no	—
61	M₁₆	—	178	1	no	—
62	C₂×C₈	—	356	1	no	—
63	C₂×C₈	—	178	2	no	—
64	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
66	C₄²	D₃₅₆	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	178	1	no	—
69	D₄	—	356	1	no	—
70	D₄	—	178	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	356	1	no	—
73	C₂×C₄	#13083551	1	1	yes	characteristic
74	C₈	—	356	1	no	—
75	C₈	—	178	4	no	—
76	C₈	C₈×D₈₉	1	1	yes	characteristic
77	C₈	#13083557	1	1	yes	characteristic
78	C₈	#13083559	1	1	yes	characteristic
79	C₈	#13083561	1	1	yes	characteristic
80	C₂²	—	356	1	no	—
81	C₂²	#15667938	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	356	1	no	—
84	C₄	#15667924	1	1	yes	characteristic
85	C₄	#15667932	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	712	1	no	—
88	C₂	#19151325	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13083559
commutator	C₈×C₈₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₈₉
Fitting	#364×C₈₉	C₂
socle	C₂×C₈₉	D₄⋊C₈
prime radical	#15668034	C₄
cosocle	C₄×C₈×C₈₉	C₂²
hypercenter	#364	D₈₉
hypocenter	C₈₉	#1335
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×#10623494	C₂	semidirect
3. #364×C₈₉	C₂	semidirect
12. Dic₃₅₆	C₈	semidirect
14. D₇₁₂	C₈	semidirect
45. C₈₉	#1335	semidirect
47. #364	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151716	2
C₈×#10623494	2
#364×C₈₉	2
#1335	89