Subgroups — #19152334

Order 5696.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15667999	C₂	1	4	yes	maximal, complemented
2	#15667993	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₁₇₈	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₈₉	C₂	1	4	yes	maximal, complemented
5	#127×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂²×Dic₈₉	C₂²	1	4	yes	—
8	C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	#13083553	C₂²	1	4	yes	—
10	C₂²⋊Dic₈₉	C₂²	1	8	yes	—
11	C₂²⋊Dic₈₉	C₂²	1	4	yes	—
12	C₄×Dic₈₉	C₂²	1	4	yes	—
13	C₂×D₄×C₈₉	C₂²	1	4	yes	—
14	C₂×D₄×C₈₉	C₂²	1	2	yes	—
15	C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈₉	C₂³	1	4	yes	—
17	D₄×C₈₉	—	2	8	no	—
18	D₄×C₈₉	—	4	2	no	—
19	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₈₉	C₂³	1	2	yes	—
21	C₂×Dic₈₉	—	4	4	no	—
22	C₂×Dic₈₉	C₂³	1	4	yes	—
23	C₂×Dic₈₉	C₂³	1	4	yes	—
24	Dic₁₇₈	—	2	4	no	—
25	C₂²×C₈₉	—	2	8	no	—
26	C₂²×C₈₉	—	2	4	no	—
27	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
28	C₄×C₈₉	—	2	1	no	—
29	C₄×C₈₉	#41	1	4	yes	—
30	Dic₈₉	—	2	4	no	—
31	Dic₈₉	—	2	4	no	—
32	C₂×C₈₉	—	4	4	no	—
33	C₂×C₈₉	C₂×#41	1	2	yes	—
34	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
35	C₈₉	#559	1	1	yes	characteristic, complemented, Sylow
36	#559	—	89	1	no	maximal, Sylow
37	#127	D₈₉	1	1	yes	characteristic
38	#124	—	89	4	no	—
39	#123	—	89	4	no	—
40	C₄×Q₈	—	89	2	no	—
41	C₄×D₄	—	89	4	no	—
42	C₂×Q₈	—	89	2	no	—
43	C₂×D₄	C₂×D₈₉	1	4	yes	—
44	C₂×D₄	C₂×D₈₉	1	2	yes	—
45	C₂²×C₄	—	89	4	no	—
46	C₄⋊C₄	—	89	2	no	—
47	C₄⋊C₄	—	89	4	no	—
48	C₂²⋊C₄	—	89	8	no	—
49	C₂²⋊C₄	—	89	4	no	—
50	C₄²	—	89	4	no	—
51	C₄²	C₂×D₈₉	1	1	yes	characteristic
52	C₂³	C₂²×D₈₉	1	4	yes	—
53	Q₈	—	178	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	89	4	no	—
57	C₂×C₄	—	89	4	no	—
58	C₂×C₄	—	356	4	no	—
59	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₈₉	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₈₉	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	178	4	no	—
66	C₄	—	178	4	no	—
67	C₄	#13083573	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#13083573	1	2	yes	—
70	C₂	#15668057	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#127×C₈₉	C₂
hypercenter	#127	D₈₉
hypocenter	C₈₉	#559
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #15667999	C₂	semidirect
2. #15667993	C₂	semidirect
3. C₄×Dic₁₇₈	C₂	semidirect
4. D₄×Dic₈₉	C₂	semidirect
35. C₈₉	#559	semidirect