Subgroups — #19152348

Order 5696.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15668010	C₂	1	2	yes	maximal, complemented
2	#15668005	C₂	1	4	yes	maximal, complemented
3	#15668004	C₂	1	2	yes	maximal, complemented
4	#15667993	C₂	1	2	yes	maximal
5	#15667989	C₂	1	4	yes	maximal, complemented
6	#124×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₁₇₈	C₂²	1	2	yes	—
8	C₂²×Dic₈₉	C₂²	1	2	yes	—
9	C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	#13083553	C₂²	1	4	yes	—
12	#13083553	C₂²	1	4	yes	—
13	#13083553	C₂²	1	4	yes	—
14	C₂²⋊Dic₈₉	C₂²	1	4	yes	—
15	C₂²⋊Dic₈₉	C₂²	1	2	yes	—
16	C₄×Dic₈₉	C₂²	1	2	yes	—
17	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
20	C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂³×C₈₉	C₂³	1	2	yes	—
22	Q₈×C₈₉	—	4	1	no	—
23	D₄×C₈₉	—	4	1	no	—
24	C₂×C₄×C₈₉	C₂³	1	2	yes	—
25	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₈₉	C₂³	1	2	yes	—
27	C₂×Dic₈₉	—	4	2	no	—
28	C₂×Dic₈₉	C₂³	1	4	yes	—
29	C₂×Dic₈₉	C₂³	1	4	yes	—
30	Dic₁₇₈	—	4	2	no	—
31	C₂²×C₈₉	—	2	2	no	—
32	C₂²×C₈₉	—	2	4	no	—
33	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
34	C₄×C₈₉	—	2	2	no	—
35	C₄×C₈₉	—	2	1	no	—
36	C₄×C₈₉	—	2	2	no	—
37	Dic₈₉	—	2	4	no	—
38	Dic₈₉	—	2	4	no	—
39	C₂×C₈₉	—	4	2	no	—
40	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂×C₈₉	2₋¹⁺⁴	1	2	yes	—
42	C₈₉	#563	1	1	yes	characteristic, complemented, Sylow
43	#563	—	89	1	no	maximal, Sylow
44	C₄⋊Q₈	—	89	2	no	—
45	#126	—	89	4	no	—
46	#125	—	89	2	no	—
47	#124	D₈₉	1	1	yes	characteristic
48	#123	—	89	2	no	—
49	C₂²⋊Q₈	—	89	4	no	—
50	C₂×Q₈	—	89	2	no	—
51	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
52	C₂×D₄	C₂×D₈₉	1	1	yes	characteristic
53	C₂²×C₄	—	89	2	no	—
54	C₄⋊C₄	—	89	2	no	—
55	C₄⋊C₄	—	89	4	no	—
56	C₄⋊C₄	—	89	4	no	—
57	C₄⋊C₄	—	89	4	no	—
58	C₄⋊C₄	—	89	2	no	—
59	C₂²⋊C₄	—	89	4	no	—
60	C₂²⋊C₄	—	89	2	no	—
61	C₂²⋊C₄	C₂×D₈₉	1	4	yes	—
62	C₄²	—	89	2	no	—
63	C₄²	C₂×D₈₉	1	1	yes	characteristic
64	C₂³	C₂²×D₈₉	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	356	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	89	4	no	—
69	C₂×C₄	—	89	4	no	—
70	C₂×C₄	—	356	2	no	—
71	C₂×C₄	C₂²×D₈₉	1	2	yes	—
72	C₂×C₄	C₂²×D₈₉	1	2	yes	—
73	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₈₉	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	178	4	no	—
81	C₄	—	178	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#15668057	1	1	yes	characteristic
84	C₂	#15668059	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	#124×C₈₉	C₂
hypercenter	#124	D₈₉
hypocenter	C₈₉	#563
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
1. #15668010	C₂	semidirect
2. #15668005	C₂	semidirect
3. #15668004	C₂	semidirect
5. #15667989	C₂	semidirect
42. C₈₉	#563	semidirect