Subgroups — #36441166

Order 22784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018316	C₂	1	2	yes	maximal
2	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13083549	—	2	2	no	—
4	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
6	C₈²×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×#13083549	—	4	2	no	—
8	C₂×C₁₆×C₈₉	—	2	1	no	—
9	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
11	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
13	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
14	#13083549	—	8	2	no	—
15	C₂×C₈×C₈₉	—	2	1	no	—
16	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
18	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
19	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
20	C₁₆×C₈₉	—	2	2	no	—
21	C₁₆×C₈₉	—	2	1	no	—
22	C₁₆×C₈₉	C₁₆	1	2	yes	complemented
23	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
25	C₂×C₄×C₈₉	#108	1	1	yes	characteristic
26	C₈×C₈₉	—	2	1	no	—
27	C₈×C₈₉	—	2	2	no	—
28	C₈×C₈₉	—	4	1	no	—
29	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
30	C₈×C₈₉	M₃₂	1	1	yes	characteristic
31	C₂²×C₈₉	#334	1	1	yes	characteristic
32	C₄×C₈₉	—	2	1	no	—
33	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₈₉	#364	1	1	yes	characteristic
35	C₄×C₈₉	#365	1	2	yes	—
36	#7099	—	89	1	no	maximal, Sylow
37	C₂×C₈₉	#1367	1	1	yes	characteristic
38	C₂×C₈₉	#1368	1	1	yes	characteristic
39	C₂×C₈₉	#1369	1	1	yes	characteristic
40	#1367	—	89	2	no	—
41	#1311	D₈₉	1	1	yes	characteristic
42	C₈₉	#7099	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊C₁₆	—	178	2	no	—
44	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
45	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
46	C₈²	C₂×D₈₉	1	1	yes	characteristic
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	356	2	no	—
49	C₂×C₁₆	#10623494	1	1	yes	characteristic
50	C₂×C₁₆	#10623494	1	1	yes	characteristic
51	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
52	C₄×C₈	D₃₅₆	1	1	yes	characteristic
53	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
56	C₂×C₈	#13083558	1	1	yes	characteristic
57	C₂×C₈	#13083563	1	2	yes	—
58	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	1	no	—
61	C₁₆	—	712	2	no	—
62	C₁₆	#13083549	1	2	yes	complemented
63	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
64	C₂×C₄	#15667944	1	1	yes	characteristic
65	C₂×C₄	#15667948	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	2	no	—
68	C₈	—	4	1	no	—
69	C₈	C₂×#13083549	1	1	yes	characteristic
70	C₈	#15667952	1	1	yes	characteristic
71	C₂²	#19151349	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₄⋊#13083549	1	1	yes	characteristic
74	C₄	#19151419	1	1	yes	characteristic
75	C₄	#19151421	1	2	yes	—
76	C₂	#26018316	1	1	yes	characteristic
77	C₂	#26018318	1	1	yes	characteristic
78	C₂	#26018320	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667944
commutator	C₈×C₈₉	C₂×C₁₆
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1311	C₂
socle, prime radical	C₂²×C₈₉	#334
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#7099
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
22. C₁₆×C₈₉	C₁₆	semidirect
42. C₈₉	#7099	semidirect
62. C₁₆	#13083549	semidirect

Maximal subgroups

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

subgroup	index
#26018316	2
C₈₉×#1311	2
#7099	89