Subgroups — #36585417

Order 22784.

604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉⋊#2104	C₂	1	7	yes	maximal
2	#26019983	C₂	1	7	yes	maximal
3	C₈₉×#2104	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667968	C₂²	1	7	yes	—
5	#19151365	C₂²	1	7	yes	—
6	#19151364	C₂²	1	14	yes	—
7	#342×C₈₉	C₂²	1	7	yes	—
8	C₂³×Dic₈₉	C₂³	1	1	yes	characteristic
9	#15667968	—	2	14	no	—
10	#15667968	—	2	14	no	—
11	C₂×#13083553	—	2	7	no	—
12	C₂×C₂²⋊Dic₈₉	C₂³	1	7	yes	—
13	C₂×C₄×Dic₈₉	—	2	7	no	—
14	C₂×C₂²⋊C₄×C₈₉	C₂³	1	7	yes	—
15	C₂²×Dic₈₉	—	2	7	no	—
16	C₂²×Dic₈₉	#41	1	7	yes	—
17	C₂²×Dic₈₉	#41	1	7	yes	—
18	#13083553	—	4	14	no	—
19	C₂²⋊Dic₈₉	—	2	14	no	—
20	C₄×Dic₈₉	—	4	7	no	—
21	C₄×Dic₈₉	—	4	7	no	—
22	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	—	2	7	no	—
24	C₂²⋊C₄×C₈₉	—	2	14	no	—
25	C₂³×C₈₉	—	8	1	no	—
26	C₂³×C₈₉	C₂×#41	1	7	yes	—
27	C₂×C₄×C₈₉	—	4	7	no	—
28	C₂×C₄×C₈₉	—	8	7	no	—
29	C₂×Dic₈₉	—	4	7	no	—
30	C₂×Dic₈₉	—	8	7	no	—
31	C₂×Dic₈₉	#126	1	7	yes	—
32	C₂×Dic₈₉	#126	1	7	yes	—
33	C₂×Dic₈₉	#126	1	14	yes	—
34	C₂²×C₈₉	—	4	7	no	—
35	C₂²×C₈₉	C₂×#126	1	7	yes	—
36	C₄×C₈₉	—	8	7	no	—
37	Dic₈₉	—	8	7	no	—
38	Dic₈₉	#401	1	1	yes	characteristic
39	Dic₈₉	#401	1	7	yes	—
40	#23847	—	89	1	no	maximal, Sylow
41	C₂×C₈₉	—	2	7	no	—
42	C₂×C₈₉	C₂×#401	1	1	yes	characteristic
43	#2104	—	89	7	no	—
44	#2104	D₈₉	1	1	yes	characteristic
45	#1740	—	89	7	no	—
46	C₈₉	#23847	1	1	yes	characteristic, complemented, Sylow
47	C₂×#117	—	89	7	no	—
48	#342	—	89	14	no	—
49	#342	—	89	7	no	—
50	#342	C₂×D₈₉	1	7	yes	—
51	C₂³×C₄	—	89	1	no	—
52	#117	—	178	14	no	—
53	#117	—	178	14	no	—
54	C₂×C₄⋊C₄	—	178	7	no	—
55	C₂×C₂²⋊C₄	—	89	7	no	—
56	C₂×C₂²⋊C₄	C₂²×D₈₉	1	7	yes	—
57	C₂×C₄²	—	178	7	no	—
58	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
59	C₂²×C₄	—	2	7	no	—
60	C₂²×C₄	—	89	7	no	—
61	C₂²×C₄	—	89	7	no	—
62	C₂²×C₄	—	178	7	no	—
63	C₄⋊C₄	—	356	14	no	—
64	C₂²⋊C₄	—	2	14	no	—
65	C₂²⋊C₄	—	178	14	no	—
66	C₄²	—	356	7	no	—
67	C₄²	—	356	7	no	—
68	C₂³	—	8	1	no	—
69	C₂³	#41×D₈₉	1	7	yes	—
70	C₂×C₄	—	4	7	no	—
71	C₂×C₄	—	8	7	no	—
72	C₂×C₄	—	89	7	no	—
73	C₂×C₄	—	89	7	no	—
74	C₂×C₄	—	89	14	no	—
75	C₂×C₄	—	356	7	no	—
76	C₂×C₄	—	712	7	no	—
77	C₂²	—	4	7	no	—
78	C₂²	#126×D₈₉	1	7	yes	—
79	C₄	—	8	7	no	—
80	C₄	—	89	1	no	—
81	C₄	—	89	7	no	—
82	C₄	—	712	7	no	—
83	C₂	—	2	7	no	—
84	C₂	#401×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#401×D₈₉
commutator, prime radical, cosocle	C₂⁴×C₈₉	C₂⁴
Frattini	C₂⁴	C₂³×D₈₉
Fitting	C₈₉×#2104	C₂
socle	C₂×C₈₉	C₂×#401
hypercenter	#2104	D₈₉
hypocenter	C₈₉	#23847
LCS	G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2104

Complements

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

N	H ≅ G/N	type
46. C₈₉	#23847	semidirect

Maximal subgroups

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

subgroup	index
C₈₉⋊#2104	2
#26019983	2
C₈₉×#2104	2
#23847	89