Subgroups — C₉₇×#2273 #27138231

Order 12416.

912 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊C₄×C₉₇	C₂	1	3	yes	maximal
2	C₄×C₂²⋊C₄×C₉₇	C₂	1	12	yes	maximal, complemented
3	C₂⁵×C₉₇	C₂²	1	1	yes	characteristic
4	C₂³×C₄×C₉₇	C₂²	1	6	yes	—
5	C₂×C₂²⋊C₄×C₉₇	C₄	1	24	yes	complemented
6	C₂×C₂²⋊C₄×C₉₇	C₂²	1	12	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	8	yes	complemented
8	#95×C₉₇	C₂²	1	8	yes	complemented
9	C₂⁴×C₉₇	—	4	4	no	—
10	C₂⁴×C₉₇	C₂×C₄	1	12	yes	—
11	C₂⁴×C₉₇	C₂³	1	3	yes	—
12	C₂²×C₄×C₉₇	—	4	12	no	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	24	yes	complemented
14	C₂²×C₄×C₉₇	C₂³	1	12	yes	—
15	C₂²⋊C₄×C₉₇	C₂×C₄	1	48	yes	complemented
16	C₄²×C₉₇	—	4	8	no	—
17	C₂³×C₉₇	—	2	12	no	—
18	C₂³×C₉₇	—	4	24	no	—
19	C₂³×C₉₇	C₄²	1	16	yes	complemented
20	C₂³×C₉₇	C₂²×C₄	1	12	yes	—
21	C₂³×C₉₇	C₂²×C₄	1	6	yes	—
22	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₉₇	—	2	24	no	—
24	C₂×C₄×C₉₇	—	4	24	no	—
25	C₂×C₄×C₉₇	C₂²×C₄	1	24	yes	—
26	C₂²×C₉₇	—	2	12	no	—
27	C₂²×C₉₇	—	2	24	no	—
28	C₂²×C₉₇	—	4	16	no	—
29	C₂²×C₉₇	C₂×C₄²	1	12	yes	—
30	C₂²×C₉₇	C₂³×C₄	1	3	yes	—
31	C₂²×C₉₇	H(5,𝔽₂)	1	4	yes	—
32	C₄×C₉₇	—	2	24	no	—
33	C₂×C₉₇	—	2	12	no	—
34	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
35	C₂×C₉₇	#518	1	6	yes	—
36	#2273	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₉₇	#2273	1	1	yes	characteristic, direct product factor, Sylow
38	C₂²×C₂²⋊C₄	C₂×C₉₇	1	3	yes	—
39	C₄×C₂²⋊C₄	C₂×C₉₇	1	12	yes	complemented
40	C₂⁵	C₂²×C₉₇	1	1	yes	characteristic
41	C₂³×C₄	C₂²×C₉₇	1	6	yes	—
42	C₂×C₂²⋊C₄	C₄×C₉₇	1	24	yes	complemented
43	C₂×C₂²⋊C₄	C₂²×C₉₇	1	12	yes	—
44	C₂×C₄²	C₂²×C₉₇	1	8	yes	complemented
45	#95	C₂²×C₉₇	1	8	yes	complemented
46	C₂⁴	—	4	4	no	—
47	C₂⁴	C₂×C₄×C₉₇	1	12	yes	—
48	C₂⁴	C₂³×C₉₇	1	3	yes	—
49	C₂²×C₄	—	4	12	no	—
50	C₂²×C₄	C₂×C₄×C₉₇	1	24	yes	complemented
51	C₂²×C₄	C₂³×C₉₇	1	12	yes	—
52	C₂²⋊C₄	C₂×C₄×C₉₇	1	48	yes	complemented
53	C₄²	—	4	8	no	—
54	C₂³	—	2	12	no	—
55	C₂³	—	4	24	no	—
56	C₂³	C₄²×C₉₇	1	16	yes	complemented
57	C₂³	C₂²×C₄×C₉₇	1	12	yes	—
58	C₂³	C₂²×C₄×C₉₇	1	6	yes	—
59	C₂³	C₂⁴×C₉₇	1	1	yes	characteristic
60	C₂×C₄	—	2	24	no	—
61	C₂×C₄	—	4	24	no	—
62	C₂×C₄	C₂²×C₄×C₉₇	1	24	yes	—
63	C₂²	—	2	12	no	—
64	C₂²	—	2	24	no	—
65	C₂²	—	4	16	no	—
66	C₂²	C₂×C₄²×C₉₇	1	12	yes	—
67	C₂²	C₂³×C₄×C₉₇	1	3	yes	—
68	C₂²	H(5,𝔽₂)×C₉₇	1	4	yes	—
69	C₄	—	2	24	no	—
70	C₂	—	2	12	no	—
71	C₂	C₂²×C₄²×C₉₇	1	1	yes	characteristic
72	C₂	#518×C₉₇	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₉₇	C₂⁴
commutator	C₂	C₂²×C₄²×C₉₇
Frattini, cosocle	C₂³	C₂⁴×C₉₇
prime radical	C₂⁵×C₉₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₂²⋊C₄×C₉₇	C₂	semidirect
5. C₂×C₂²⋊C₄×C₉₇	C₄	semidirect
7. C₂×C₄²×C₉₇	C₂²	semidirect
8. #95×C₉₇	C₂²	semidirect
13. C₂²×C₄×C₉₇	C₂×C₄	semidirect
15. C₂²⋊C₄×C₉₇	C₂×C₄	semidirect
19. C₂³×C₉₇	C₄²	semidirect
36. #2273	C₉₇	direct (N × H)
37. C₉₇	#2273	direct (N × H)
39. C₄×C₂²⋊C₄	C₂×C₉₇	semidirect
42. C₂×C₂²⋊C₄	C₄×C₉₇	semidirect
44. C₂×C₄²	C₂²×C₉₇	semidirect
45. #95	C₂²×C₉₇	semidirect
50. C₂²×C₄	C₂×C₄×C₉₇	semidirect
52. C₂²⋊C₄	C₂×C₄×C₉₇	semidirect
56. C₂³	C₄²×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊C₄×C₉₇	2
C₄×C₂²⋊C₄×C₉₇	2
#2273	97

Subgroups — C97×#2273 #27138231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2273 #27138231