Subgroups — #27140038

Order 12416.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095494	C₂	1	3	yes	maximal
2	#20260456	C₂	1	3	yes	maximal, complemented
3	C₈²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13095494	C₄	1	6	yes	—
5	C₄²×D₉₇	C₂²	1	1	yes	characteristic
6	#15690720	C₄	1	6	yes	—
7	C₄×#11715895	C₂²	1	3	yes	—
8	C₄×C₈×C₉₇	C₂²	1	3	yes	—
9	C₂×C₄×D₉₇	C₂×C₄	1	3	yes	—
10	#13095494	—	2	12	no	—
11	C₂×#11715895	C₂×C₄	1	6	yes	—
12	C₄×Dic₉₇	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₉₇	C₂×C₄	1	6	yes	—
14	C₄²×C₉₇	C₂³	1	1	yes	characteristic
15	C₂²×D₉₇	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
17	C₂×Dic₉₇	C₄²	1	3	yes	—
18	C₄×D₉₇	—	2	6	no	—
19	C₈×C₉₇	M₁₆	1	12	yes	complemented
20	#11715895	—	2	6	no	—
21	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₉₇	—	2	3	no	—
23	C₄×C₉₇	C₂×M₁₆	1	6	yes	—
24	Dic₉₇	—	2	3	no	—
25	C₂×C₉₇	C₄×M₁₆	1	3	yes	—
26	D₉₇	—	4	1	no	—
27	#1455	—	97	1	no	maximal, Sylow
28	C₉₇	#1455	1	1	yes	characteristic, complemented, Sylow
29	C₄×M₁₆	—	97	3	no	—
30	#322	—	97	3	no	—
31	C₈²	D₉₇	1	1	yes	characteristic, complemented
32	C₂×M₁₆	—	97	6	no	—
33	C₂×C₄²	—	97	1	no	—
34	#97	—	97	6	no	—
35	C₄×C₈	—	97	3	no	—
36	C₄×C₈	C₂×D₉₇	1	3	yes	—
37	C₂²×C₄	—	97	3	no	—
38	M₁₆	—	194	12	no	—
39	C₂×C₈	—	97	6	no	—
40	C₂×C₈	C₄×D₉₇	1	6	yes	—
41	C₄²	—	97	3	no	—
42	C₄²	C₂²×D₉₇	1	1	yes	characteristic
43	C₂³	—	97	1	no	—
44	C₂×C₄	—	97	3	no	—
45	C₂×C₄	—	194	6	no	—
46	C₂×C₄	C₂×C₄×D₉₇	1	3	yes	—
47	C₈	—	194	6	no	—
48	C₈	#13095494	1	12	yes	complemented
49	C₂²	—	194	3	no	—
50	C₂²	C₄²×D₉₇	1	1	yes	characteristic
51	C₄	—	194	3	no	—
52	C₄	C₂×#13095494	1	6	yes	—
53	C₂	—	388	1	no	—
54	C₂	C₄×#13095494	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	C₈²×C₉₇	C₂
prime radical	C₂²×D₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₈²	D₉₇
hypocenter	C₉₇	#1455
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #20260456	C₂	semidirect
3. C₈²×C₉₇	C₂	semidirect
19. C₈×C₉₇	M₁₆	semidirect
28. C₉₇	#1455	semidirect
31. C₈²	D₉₇	semidirect
48. C₈	#13095494	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095494	2
#20260456	2
C₈²×C₉₇	2
#1455	97