Subgroups — #27141038

Order 12416.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261101	C₂	1	3	yes	maximal
2	#20260477	C₂	1	3	yes	maximal, complemented
3	#564×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690807	C₂²	1	1	yes	characteristic
5	#15690730	—	2	6	no	—
6	#15690721	C₂²	1	3	yes	—
7	#125×C₉₇	—	2	6	no	—
8	#125×C₉₇	C₂²	1	3	yes	—
9	C₂×D₃₈₈	—	2	3	no	—
10	C₂×#11715895	—	2	3	no	—
11	C₄⋊C₄×C₉₇	—	2	6	no	—
12	C₄⋊C₄×C₉₇	—	2	6	no	—
13	C₄⋊C₄×C₉₇	D₄	1	6	yes	—
14	C₄²×C₉₇	—	2	2	no	—
15	C₄²×C₉₇	C₂³	1	1	yes	characteristic
16	C₂²×D₉₇	—	4	1	no	—
17	C₂×C₄×C₉₇	—	2	6	no	—
18	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
19	D₃₈₈	—	4	6	no	—
20	#11715895	—	4	3	no	—
21	C₂²×C₉₇	#120	1	1	yes	characteristic
22	C₂×D₉₇	—	8	3	no	—
23	C₄×C₉₇	—	2	3	no	—
24	C₄×C₉₇	—	4	6	no	—
25	C₂×C₉₇	#454	1	3	yes	—
26	D₉₇	—	16	1	no	—
27	#1658	—	97	1	no	maximal, Sylow
28	C₉₇	#1658	1	1	yes	characteristic, complemented, Sylow
29	#564	D₉₇	1	1	yes	characteristic, complemented
30	#490	—	97	3	no	—
31	#330	—	97	3	no	—
32	#127	—	97	1	no	—
33	#125	—	2	6	no	—
34	#125	C₂×D₉₇	1	3	yes	—
35	#102	—	194	6	no	—
36	#97	—	97	3	no	—
37	C₂×D₄	—	194	3	no	—
38	C₂×C₈	—	194	3	no	—
39	C₄⋊C₄	—	2	6	no	—
40	C₄⋊C₄	—	2	6	no	—
41	C₄⋊C₄	C₂²⋊D₉₇	1	6	yes	—
42	C₄²	—	2	2	no	—
43	C₄²	C₂²×D₉₇	1	1	yes	characteristic
44	C₂³	—	388	1	no	—
45	D₄	—	388	6	no	—
46	C₂×C₄	—	2	6	no	—
47	C₂×C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
48	C₈	—	388	3	no	—
49	C₂²	—	776	3	no	—
50	C₂²	#15690779	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₄	—	4	6	no	—
53	C₂	—	1552	1	no	—
54	C₂	#20260908	1	3	yes	—

	subgroup	G/H
center	C₂²	#15690779
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#564×C₉₇	C₂
socle	C₂²×C₉₇	#120
prime radical	#15690807	C₂²
hypercenter	#564	D₉₇
hypocenter	C₉₇	#1658
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

Complements

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

N	H ≅ G/N	type
2. #20260477	C₂	semidirect
3. #564×C₉₇	C₂	semidirect
28. C₉₇	#1658	semidirect
29. #564	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261101	2
#20260477	2
#564×C₉₇	2
#1658	97