Subgroups — #36442101

Order 22784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15667918	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667918	—	2	2	no	—
4	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
6	C₈²×C₈₉	C₄	1	1	yes	characteristic
7	#15667918	—	4	2	no	—
8	C₂×C₁₆×C₈₉	—	2	1	no	—
9	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₈₉	C₈	1	1	yes	characteristic
12	C₄×C₈×C₈₉	C₈	1	1	yes	characteristic
13	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₈₉	—	2	1	no	—
15	C₂×C₈×C₈₉	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₈₉	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
19	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₈₉	—	2	2	no	—
21	C₁₆×C₈₉	—	2	1	no	—
22	C₁₆×C₈₉	D₈	1	1	yes	characteristic
23	C₁₆×C₈₉	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
27	C₈×C₈₉	—	2	2	no	—
28	C₈×C₈₉	—	2	1	no	—
29	C₈×C₈₉	—	2	1	no	—
30	C₈×C₈₉	C₃₂	1	2	yes	complemented
31	C₈×C₈₉	#107	1	1	yes	characteristic
32	C₈×C₈₉	#108	1	1	yes	characteristic
33	C₂²×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₈₉	—	2	1	no	—
35	C₄×C₈₉	#335	1	1	yes	characteristic
36	C₄×C₈₉	#364	1	1	yes	characteristic
37	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₈₉	M₆₄	1	1	yes	characteristic
39	#7458	—	89	1	no	maximal, Sylow
40	C₂×C₈₉	#1370	1	1	yes	characteristic
41	C₂×C₈₉	C₄⋊C₃₂	1	1	yes	characteristic
42	C₂×C₈₉	#1422	1	1	yes	characteristic
43	C₄⋊C₃₂	—	89	2	no	—
44	C₈×C₁₆	D₈₉	1	1	yes	characteristic
45	C₈₉	#7458	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃₂	—	178	2	no	—
47	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
48	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
49	C₈²	Dic₈₉	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
52	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
53	C₄×C₈	#10623494	1	1	yes	characteristic
54	C₄×C₈	#10623494	1	1	yes	characteristic
55	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
56	C₃₂	—	356	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	#13083549	1	1	yes	characteristic
59	C₂×C₈	#13083549	1	1	yes	characteristic
60	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
61	C₂×C₈	#13083558	1	1	yes	characteristic
62	C₄²	C₂×#10623494	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	D₇₁₂	1	1	yes	characteristic
66	C₁₆	Dic₃₅₆	1	1	yes	characteristic
67	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
68	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
69	C₂×C₄	#15667952	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	#15667918	1	2	yes	complemented
74	C₈	#15667946	1	1	yes	characteristic
75	C₈	#15667948	1	1	yes	characteristic
76	C₂²	C₄⋊#13083549	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#19151351	1	1	yes	characteristic
79	C₄	#19151419	1	1	yes	characteristic
80	C₄	C₂×#15667918	1	1	yes	characteristic
81	C₄	#19151432	1	1	yes	characteristic
82	C₂	#26018322	1	1	yes	characteristic
83	C₂	C₄⋊#15667918	1	1	yes	characteristic
84	C₂	#26018450	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₃₅₆
commutator	C₄×C₈₉	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#7458
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
30. C₈×C₈₉	C₃₂	semidirect
45. C₈₉	#7458	semidirect
73. C₈	#15667918	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15667918	2
C₈×C₁₆×C₈₉	2
#7458	89