Subgroups — C₉₇×#1409 #27137367

Order 12416.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#492×C₉₇	C₂	1	1	yes	maximal, characteristic
2	#353×C₉₇	C₂	1	2	yes	maximal
3	C₂×SD₁₆×C₉₇	—	2	2	no	—
4	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic, complemented
5	#127×C₉₇	C₂²	1	1	yes	characteristic
6	#99×C₉₇	—	2	2	no	—
7	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₈×C₉₇	—	2	1	no	—
9	C₂×D₄×C₉₇	—	4	1	no	—
10	C₂×D₄×C₉₇	D₄	1	2	yes	—
11	SD₁₆×C₉₇	—	4	4	no	—
12	C₂×C₈×C₉₇	—	2	1	no	—
13	C₄⋊C₄×C₉₇	—	4	1	no	—
14	C₂²⋊C₄×C₉₇	—	4	2	no	—
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₉₇	—	2	2	no	—
17	Q₈×C₉₇	—	4	2	no	—
18	D₄×C₉₇	—	4	2	no	—
19	D₄×C₉₇	—	8	2	no	—
20	C₂×C₄×C₉₇	—	2	1	no	—
21	C₂×C₄×C₉₇	—	4	1	no	—
22	C₂×C₄×C₉₇	—	8	2	no	—
23	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₈×C₉₇	—	2	2	no	—
25	C₂²×C₉₇	—	4	2	no	—
26	C₂²×C₉₇	—	8	2	no	—
27	C₂²×C₉₇	#99	1	1	yes	characteristic
28	C₄×C₉₇	—	2	2	no	—
29	C₄×C₉₇	—	2	1	no	—
30	C₄×C₉₇	—	8	1	no	—
31	C₄×C₉₇	—	16	2	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	—	8	2	no	—
34	C₂×C₉₇	#353	1	1	yes	characteristic
35	#1409	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₉₇	#1409	1	1	yes	characteristic, direct product factor, Sylow
37	#492	C₂×C₉₇	1	1	yes	characteristic
38	#353	C₂×C₉₇	1	2	yes	—
39	C₂×SD₁₆	—	2	2	no	—
40	C₄⋊Q₈	C₄×C₉₇	1	1	yes	characteristic, complemented
41	#127	C₂²×C₉₇	1	1	yes	characteristic
42	#99	—	2	2	no	—
43	C₄×C₈	C₄×C₉₇	1	1	yes	characteristic, complemented
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	C₂×D₄	D₄×C₉₇	1	2	yes	—
47	SD₁₆	—	4	4	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	4	2	no	—
51	C₄²	C₂×C₄×C₉₇	1	1	yes	characteristic
52	C₂³	—	2	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	C₂²⋊C₄×C₉₇	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	8	2	no	—
63	C₂²	#99×C₉₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	16	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	8	2	no	—
70	C₂	#353×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₉₇	#353
commutator	C₄²	C₂×C₄×C₉₇
Frattini, cosocle	#127	C₂²×C₉₇
prime radical	#127×C₉₇	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉₇ ◁ C₂²×C₉₇ ◁ C₂×C₄×C₉₇ ◁ #127×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄⋊Q₈×C₉₇	C₄	semidirect
7. C₄×C₈×C₉₇	C₄	semidirect
35. #1409	C₉₇	direct (N × H)
36. C₉₇	#1409	direct (N × H)
40. C₄⋊Q₈	C₄×C₉₇	semidirect
43. C₄×C₈	C₄×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
#492×C₉₇	2
#353×C₉₇	2
#1409	97

Subgroups — C97×#1409 #27137367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1409 #27137367