Subgroups — #26018446

Order 11392.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#19151428	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₅₆	—	2	1	no	—
5	Dic₇₁₂	—	2	1	no	—
6	Dic₇₁₂	C₄	1	2	yes	—
7	#15667948	—	2	1	no	—
8	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	C₃₂×C₈₉	—	2	1	no	—
10	C₂×Dic₁₇₈	—	4	1	no	—
11	Dic₃₅₆	—	2	2	no	—
12	Dic₃₅₆	—	4	1	no	—
13	#13083558	—	4	1	no	—
14	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
15	C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₈₉	D₈	1	1	yes	characteristic
18	C₂×Dic₈₉	—	8	1	no	—
19	Dic₁₇₈	—	4	2	no	—
20	Dic₁₇₈	—	8	1	no	—
21	C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₈₉	SD₁₆	1	1	yes	characteristic
23	#10623494	—	8	1	no	—
24	C₂²×C₈₉	SD₃₂	1	1	yes	characteristic
25	C₄×C₈₉	#102	1	1	yes	characteristic
26	C₄×C₈₉	D₁₆	1	1	yes	characteristic
27	Dic₈₉	—	8	2	no	—
28	C₂×C₈₉	—	2	1	no	—
29	C₂×C₈₉	#357	1	1	yes	characteristic
30	#1420	—	89	1	no	maximal, Sylow
31	C₈₉	#1420	1	1	yes	characteristic, complemented, Sylow
32	C₂×Q₃₂	—	89	1	no	—
33	M₆₄	D₈₉	1	1	yes	characteristic
34	#368	—	89	1	no	—
35	C₂×Q₁₆	—	178	1	no	—
36	Q₃₂	—	89	2	no	—
37	Q₃₂	—	178	1	no	—
38	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
39	#108	—	178	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	356	1	no	—
42	Q₁₆	—	356	1	no	—
43	Q₁₆	—	178	2	no	—
44	M₁₆	—	356	1	no	—
45	C₂×C₈	D₃₅₆	1	1	yes	characteristic
46	C₁₆	C₄×D₈₉	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
48	Q₈	—	356	2	no	—
49	Q₈	—	712	1	no	—
50	C₂×C₄	—	712	1	no	—
51	C₂×C₄	D₇₁₂	1	1	yes	characteristic
52	C₈	—	712	1	no	—
53	C₈	#13083551	1	1	yes	characteristic
54	C₈	#13083563	1	1	yes	characteristic
55	C₂²	#15667957	1	1	yes	characteristic
56	C₄	—	712	2	no	—
57	C₄	#15667934	1	1	yes	characteristic
58	C₄	D₁₄₂₄	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#19151400	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151400
commutator	C₁₆×C₈₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₈₉
Fitting	M₆₄×C₈₉	C₂
socle	C₂×C₈₉	#357
prime radical	C₂²×C₈₉	SD₃₂
cosocle	C₂×C₁₆×C₈₉	C₂²
hypercenter	M₆₄	D₈₉
hypocenter	C₈₉	#1420
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
31. C₈₉	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₁₂	2
#19151428	2
M₆₄×C₈₉	2
#1420	89