Subgroups — #459×C₉₇ #20260326

Order 6208.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×C₉₇	C₂	1	2	yes	maximal, complemented
2	#127×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#102×C₉₇	C₂	1	2	yes	maximal, complemented
6	C₂×D₄×C₉₇	—	2	1	no	—
7	C₂×D₄×C₉₇	—	2	1	no	—
8	C₂×D₄×C₉₇	C₂²	1	2	yes	—
9	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₉₇	—	2	1	no	—
11	D₈×C₉₇	—	2	4	no	—
12	C₂×C₈×C₉₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₉₇	—	2	1	no	—
15	C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂³×C₉₇	—	2	1	no	—
17	C₂³×C₉₇	—	2	2	no	—
18	D₄×C₉₇	—	2	2	no	—
19	D₄×C₉₇	—	2	1	no	—
20	D₄×C₉₇	—	2	2	no	—
21	D₄×C₉₇	—	4	2	no	—
22	D₄×C₉₇	—	4	2	no	—
23	D₄×C₉₇	D₄	1	2	yes	complemented
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	—	4	2	no	—
26	C₂×C₄×C₉₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₈×C₉₇	—	2	2	no	—
30	C₂²×C₉₇	—	2	2	no	—
31	C₂²×C₉₇	—	4	1	no	—
32	C₂²×C₉₇	—	4	1	no	—
33	C₂²×C₉₇	—	4	2	no	—
34	C₂²×C₉₇	—	4	2	no	—
35	C₂²×C₉₇	—	4	2	no	—
36	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₉₇	—	2	1	no	—
38	C₄×C₉₇	—	4	1	no	—
39	C₄×C₉₇	D₈	1	2	yes	complemented
40	C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₉₇	#41	1	1	yes	characteristic
42	C₂×C₉₇	—	4	2	no	—
43	C₂×C₉₇	—	8	2	no	—
44	C₂×C₉₇	#121	1	1	yes	characteristic
45	C₂×C₉₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₉₇	#459	1	1	yes	characteristic, direct product factor, Sylow
48	#459	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×D₈	C₂×C₉₇	1	2	yes	complemented
50	#127	C₂×C₉₇	1	1	yes	characteristic, complemented
51	C₄×D₄	C₂×C₉₇	1	1	yes	characteristic, complemented
52	C₄⋊C₈	C₂×C₉₇	1	1	yes	characteristic, complemented
53	#102	C₂×C₉₇	1	2	yes	complemented
54	C₂×D₄	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	C₂²×C₉₇	1	2	yes	—
57	C₂×D₄	C₂²×C₉₇	1	1	yes	characteristic
58	C₂²×C₄	—	2	1	no	—
59	D₈	—	2	4	no	—
60	C₂×C₈	C₂²×C₉₇	1	2	yes	—
61	C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
62	C₂²⋊C₄	—	2	1	no	—
63	C₄²	C₂²×C₉₇	1	1	yes	characteristic
64	C₂³	—	2	1	no	—
65	C₂³	—	2	2	no	—
66	D₄	—	2	2	no	—
67	D₄	—	2	1	no	—
68	D₄	—	2	2	no	—
69	D₄	—	4	2	no	—
70	D₄	—	4	2	no	—
71	D₄	D₄×C₉₇	1	2	yes	complemented
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	D₄×C₉₇	1	1	yes	characteristic
75	C₂×C₄	D₄×C₉₇	1	1	yes	characteristic
76	C₂×C₄	C₂³×C₉₇	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂×D₄×C₉₇	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	D₈×C₉₇	1	2	yes	complemented
88	C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
89	C₄	#41×C₉₇	1	1	yes	characteristic
90	C₂	—	4	2	no	—
91	C₂	—	8	2	no	—
92	C₂	#121×C₉₇	1	1	yes	characteristic
93	C₂	C₂×D₈×C₉₇	1	1	yes	characteristic
94	C₂	AGL(1,ℤ/8)×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×C₉₇	C₂	semidirect
2. #127×C₉₇	C₂	semidirect
3. C₄×D₄×C₉₇	C₂	semidirect
4. C₄⋊C₈×C₉₇	C₂	semidirect
5. #102×C₉₇	C₂	semidirect
23. D₄×C₉₇	D₄	semidirect
39. C₄×C₉₇	D₈	semidirect
47. C₉₇	#459	direct (N × H)
48. #459	C₉₇	direct (N × H)
49. C₂×D₈	C₂×C₉₇	semidirect
50. #127	C₂×C₉₇	semidirect
51. C₄×D₄	C₂×C₉₇	semidirect
52. C₄⋊C₈	C₂×C₉₇	semidirect
53. #102	C₂×C₉₇	semidirect
71. D₄	D₄×C₉₇	semidirect
87. C₄	D₈×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×C₉₇	2
#127×C₉₇	2
C₄×D₄×C₉₇	2
C₄⋊C₈×C₉₇	2
#102×C₉₇	2
#459	97

Subgroups — #459×C97 #20260326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #459×C₉₇ #20260326