Subgroups — #533×C₉₇ #20260400

Order 6208.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₉₇	C₂	1	2	yes	maximal, complemented
2	#125×C₉₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Q₈×C₉₇	C₂	1	4	yes	maximal
4	C₄×Q₈×C₉₇	C₂	1	4	yes	maximal, complemented
5	#117×C₉₇	C₂	1	2	yes	maximal
6	C₂×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×C₉₇	C₂²	1	4	yes	—
8	C₂²×C₄×C₉₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
11	C₄⋊C₄×C₉₇	C₂²	1	8	yes	—
12	C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
13	C₂²⋊C₄×C₉₇	C₂²	1	4	yes	—
14	C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₄²×C₉₇	C₂²	1	2	yes	—
17	C₄²×C₉₇	C₂²	1	4	yes	—
18	C₂³×C₉₇	C₂³	1	1	yes	characteristic
19	Q₈×C₉₇	—	2	8	no	—
20	C₂×C₄×C₉₇	—	2	4	no	—
21	C₂×C₄×C₉₇	Q₈	1	4	yes	complemented
22	C₂×C₄×C₉₇	C₂³	1	2	yes	—
23	C₂×C₄×C₉₇	C₂³	1	8	yes	—
24	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₂³	1	2	yes	—
26	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₂²×C₉₇	—	2	2	no	—
28	C₂²×C₉₇	C₂×Q₈	1	2	yes	—
29	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
30	C₄×C₉₇	—	2	2	no	—
31	C₄×C₉₇	—	2	8	no	—
32	C₄×C₉₇	C₂×Q₈	1	2	yes	—
33	C₄×C₉₇	C₂×Q₈	1	2	yes	—
34	C₄×C₉₇	#41	1	4	yes	—
35	C₂×C₉₇	—	2	2	no	—
36	C₂×C₉₇	C₂²×Q₈	1	1	yes	characteristic
37	C₂×C₉₇	C₂×#41	1	2	yes	—
38	C₉₇	#533	1	1	yes	characteristic, direct product factor, Sylow
39	#533	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₄⋊Q₈	C₂×C₉₇	1	2	yes	complemented
41	#125	C₂×C₉₇	1	2	yes	complemented
42	C₂²⋊Q₈	C₂×C₉₇	1	4	yes	—
43	C₄×Q₈	C₂×C₉₇	1	4	yes	complemented
44	#117	C₂×C₉₇	1	2	yes	—
45	C₂×C₄²	C₂×C₉₇	1	1	yes	characteristic
46	C₂×Q₈	C₂²×C₉₇	1	4	yes	—
47	C₂²×C₄	C₂²×C₉₇	1	2	yes	—
48	C₂²×C₄	C₂²×C₉₇	1	1	yes	characteristic
49	C₄⋊C₄	C₂²×C₉₇	1	4	yes	—
50	C₄⋊C₄	C₂²×C₉₇	1	8	yes	—
51	C₄⋊C₄	C₂²×C₉₇	1	4	yes	—
52	C₂²⋊C₄	C₂²×C₉₇	1	4	yes	—
53	C₄²	C₂²×C₉₇	1	2	yes	—
54	C₄²	C₂²×C₉₇	1	1	yes	characteristic
55	C₄²	C₂²×C₉₇	1	1	yes	characteristic
56	C₄²	C₂²×C₉₇	1	4	yes	—
57	C₂³	C₂³×C₉₇	1	1	yes	characteristic
58	Q₈	—	2	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	Q₈×C₉₇	1	4	yes	complemented
61	C₂×C₄	C₂³×C₉₇	1	2	yes	—
62	C₂×C₄	C₂³×C₉₇	1	8	yes	—
63	C₂×C₄	C₂³×C₉₇	1	1	yes	characteristic
64	C₂×C₄	C₂³×C₉₇	1	1	yes	characteristic
65	C₂×C₄	C₂³×C₉₇	1	2	yes	—
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Q₈×C₉₇	1	2	yes	—
68	C₂²	C₂⁴×C₉₇	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	8	no	—
71	C₄	C₂×Q₈×C₉₇	1	2	yes	—
72	C₄	C₂×Q₈×C₉₇	1	2	yes	—
73	C₄	#41×C₉₇	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Q₈×C₉₇	1	1	yes	characteristic
76	C₂	C₂×#41×C₉₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₉₇
socle	C₂²×C₉₇	C₂⁴
prime radical	C₂³×C₉₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₉₇	C₂	semidirect
2. #125×C₉₇	C₂	semidirect
4. C₄×Q₈×C₉₇	C₂	semidirect
21. C₂×C₄×C₉₇	Q₈	semidirect
38. C₉₇	#533	direct (N × H)
39. #533	C₉₇	direct (N × H)
40. C₄⋊Q₈	C₂×C₉₇	semidirect
41. #125	C₂×C₉₇	semidirect
43. C₄×Q₈	C₂×C₉₇	semidirect
60. C₂×C₄	Q₈×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₉₇	2
#125×C₉₇	2
C₂²⋊Q₈×C₉₇	2
C₄×Q₈×C₉₇	2
#117×C₉₇	2
C₂×C₄²×C₉₇	2
#533	97

Subgroups — #533×C97 #20260400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #533×C₉₇ #20260400