Subgroups — #36441164

Order 22784.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018322	C₂	1	2	yes	maximal
2	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13083549	—	2	2	no	—
4	#500×C₈₉	—	2	2	no	—
5	#494×C₈₉	C₄	1	2	yes	—
6	C₈²×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×#13083549	—	4	2	no	—
8	C₂×Q₁₆×C₈₉	—	4	2	no	—
9	C₄⋊Q₈×C₈₉	—	2	2	no	—
10	#107×C₈₉	—	4	2	no	—
11	#107×C₈₉	—	4	2	no	—
12	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₈₉	D₄	1	2	yes	—
14	#13083549	—	8	2	no	—
15	C₂×Q₈×C₈₉	—	4	2	no	—
16	Q₁₆×C₈₉	—	8	4	no	—
17	C₂×C₈×C₈₉	—	2	1	no	—
18	C₂×C₈×C₈₉	D₈	1	2	yes	—
19	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
20	C₄⋊C₄×C₈₉	—	4	4	no	—
21	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
22	Q₈×C₈₉	—	8	4	no	—
23	C₂×C₄×C₈₉	—	8	2	no	—
24	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
25	C₂×C₄×C₈₉	#102	1	2	yes	—
26	C₈×C₈₉	—	2	2	no	—
27	C₈×C₈₉	—	2	2	no	—
28	C₈×C₈₉	SD₃₂	1	2	yes	—
29	C₈×C₈₉	Q₃₂	1	2	yes	—
30	C₂²×C₈₉	#331	1	1	yes	characteristic
31	C₄×C₈₉	—	2	1	no	—
32	C₄×C₈₉	—	16	2	no	—
33	C₄×C₈₉	#358	1	2	yes	—
34	C₄×C₈₉	#362	1	2	yes	—
35	#7098	—	89	1	no	maximal, Sylow
36	C₂×C₈₉	#1357	1	1	yes	characteristic
37	C₂×C₈₉	#1362	1	2	yes	—
38	#1713	D₈₉	1	1	yes	characteristic
39	#1370	—	89	2	no	—
40	C₈₉	#7098	1	1	yes	characteristic, complemented, Sylow
41	#500	—	2	2	no	—
42	#494	Dic₈₉	1	2	yes	—
43	C₄⋊C₁₆	—	178	2	no	—
44	C₈²	C₂×D₈₉	1	1	yes	characteristic
45	C₂×Q₁₆	—	4	2	no	—
46	C₄⋊Q₈	—	2	2	no	—
47	C₂×C₁₆	—	356	2	no	—
48	#107	—	4	2	no	—
49	#107	—	4	2	no	—
50	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
51	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
52	C₂×Q₈	—	4	2	no	—
53	Q₁₆	—	8	4	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	#13083559	1	2	yes	—
56	C₂×C₈	#13083562	1	2	yes	—
57	C₄⋊C₄	—	4	4	no	—
58	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
59	C₁₆	—	712	2	no	—
60	Q₈	—	8	4	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	#15667929	1	1	yes	characteristic
63	C₂×C₄	#15667933	1	2	yes	—
64	C₈	—	2	2	no	—
65	C₈	—	2	2	no	—
66	C₈	#15667955	1	2	yes	—
67	C₈	#15667958	1	2	yes	—
68	C₂²	#19151342	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	16	2	no	—
71	C₄	#19151402	1	2	yes	—
72	C₄	#19151414	1	2	yes	—
73	C₂	#26018289	1	1	yes	characteristic
74	C₂	#26018301	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151342
commutator	C₄×C₈×C₈₉	C₂×C₄
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1713	C₂
socle, prime radical	C₂²×C₈₉	#331
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#7098
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
40. C₈₉	#7098	semidirect

Maximal subgroups

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

subgroup	index
#26018322	2
C₈₉×#1713	2
#7098	89