Subgroups — #19152328

Order 5696.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15668011	C₂	1	4	yes	maximal
2	#15668010	C₂	1	4	yes	maximal
3	C₄×Dic₁₇₈	C₂	1	2	yes	maximal
4	#15667977	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₇₈	C₂²	1	4	yes	—
7	C₂×Dic₁₇₈	C₂²	1	2	yes	—
8	C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	#13083553	C₂²	1	8	yes	—
10	#13083553	C₂²	1	4	yes	—
11	C₄×Dic₈₉	C₂²	1	4	yes	—
12	C₄×Dic₈₉	C₂²	1	4	yes	—
13	C₂×Q₈×C₈₉	C₂²	1	2	yes	—
14	C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
15	C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	Q₈×C₈₉	—	4	2	no	—
17	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₈₉	C₂³	1	4	yes	—
19	C₂×C₄×C₈₉	C₂³	1	2	yes	—
20	C₂×Dic₈₉	C₂³	1	4	yes	—
21	C₂×Dic₈₉	C₂³	1	4	yes	—
22	Dic₁₇₈	—	4	4	no	—
23	Dic₁₇₈	Q₈	1	8	yes	complemented
24	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
25	C₄×C₈₉	—	2	4	no	—
26	C₄×C₈₉	—	2	1	no	—
27	C₄×C₈₉	C₂×Q₈	1	4	yes	—
28	Dic₈₉	—	2	4	no	—
29	Dic₈₉	C₂×Q₈	1	8	yes	—
30	C₂×C₈₉	C₂²×Q₈	1	2	yes	—
31	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
32	C₈₉	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	89	1	no	maximal, Sylow
34	C₄⋊Q₈	—	89	4	no	—
35	C₄⋊Q₈	—	89	4	no	—
36	C₄⋊Q₈	D₈₉	1	1	yes	characteristic
37	C₄×Q₈	—	89	4	no	—
38	C₄×Q₈	—	89	2	no	—
39	C₂×Q₈	—	89	2	no	—
40	C₂×Q₈	—	89	4	no	—
41	C₂×Q₈	C₂×D₈₉	1	2	yes	—
42	C₄⋊C₄	—	89	8	no	—
43	C₄⋊C₄	—	89	4	no	—
44	C₄⋊C₄	—	89	2	no	—
45	C₄⋊C₄	C₂×D₈₉	1	4	yes	—
46	C₄²	—	89	4	no	—
47	C₄²	—	89	4	no	—
48	C₄²	C₂×D₈₉	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	89	8	no	—
51	Q₈	—	356	4	no	—
52	C₂×C₄	—	89	4	no	—
53	C₂×C₄	—	89	4	no	—
54	C₂×C₄	C₂²×D₈₉	1	4	yes	—
55	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₈₉	1	2	yes	—
57	C₂²	C₂³×D₈₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	89	8	no	—
61	C₄	—	178	4	no	—
62	C₄	Q₈×D₈₉	1	4	yes	—
63	C₂	C₂×Q₈×D₈₉	1	2	yes	—
64	C₂	#15668057	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₄⋊Q₈×C₈₉	C₂
hypercenter	C₄⋊Q₈	D₈₉
hypocenter	C₈₉	Q₈²
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. Dic₁₇₈	Q₈	semidirect
32. C₈₉	Q₈²	semidirect