Subgroups — #27141348

Order 12416.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261086	C₂	1	3	yes	maximal, complemented
2	#20261082	C₂	1	3	yes	maximal
3	C₈²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690810	C₂²	1	3	yes	—
5	#15690807	C₂²	1	1	yes	characteristic
6	#15690735	—	2	6	no	—
7	#15690732	—	2	6	no	—
8	C₄×C₈×C₉₇	C₂²	1	3	yes	—
9	C₂×Dic₁₉₄	—	2	3	no	—
10	C₂×D₃₈₈	—	2	3	no	—
11	C₄⋊Dic₉₇	—	2	3	no	—
12	C₄⋊Dic₉₇	—	2	3	no	—
13	C₂×C₈×C₉₇	D₄	1	6	yes	—
14	C₄²×C₉₇	C₂³	1	1	yes	characteristic
15	C₂²×D₉₇	—	4	1	no	—
16	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
17	C₂×Dic₉₇	—	4	3	no	—
18	D₃₈₈	—	4	6	no	—
19	Dic₁₉₄	—	4	6	no	—
20	C₈×C₉₇	—	2	6	no	—
21	C₂²×C₉₇	#127	1	1	yes	characteristic
22	C₂×D₉₇	—	8	3	no	—
23	C₄×C₉₇	#135	1	6	yes	—
24	Dic₉₇	—	8	3	no	—
25	C₂×C₉₇	#495	1	3	yes	—
26	D₉₇	—	16	1	no	—
27	#1711	—	97	1	no	maximal, Sylow
28	C₉₇	#1711	1	1	yes	characteristic, complemented, Sylow
29	#487	—	97	3	no	—
30	#486	—	97	3	no	—
31	C₈²	D₉₇	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	—	97	3	no	—
33	#127	—	97	1	no	—
34	#103	—	194	6	no	—
35	#102	—	194	6	no	—
36	C₄×C₈	C₂×D₉₇	1	3	yes	—
37	C₂×Q₈	—	194	3	no	—
38	C₂×D₄	—	194	3	no	—
39	C₂×C₈	D₃₈₈	1	6	yes	—
40	C₄⋊C₄	—	194	3	no	—
41	C₄⋊C₄	—	194	3	no	—
42	C₄²	C₂²×D₉₇	1	1	yes	characteristic
43	C₂³	—	388	1	no	—
44	Q₈	—	388	6	no	—
45	D₄	—	388	6	no	—
46	C₂×C₄	—	388	3	no	—
47	C₂×C₄	C₂×D₃₈₈	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	776	3	no	—
50	C₂²	#15690807	1	1	yes	characteristic
51	C₄	—	776	3	no	—
52	C₄	#15690832	1	6	yes	—
53	C₂	—	1552	1	no	—
54	C₂	#20261119	1	3	yes	—

	subgroup	G/H
center	C₂²	#15690807
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	C₈²×C₉₇	C₂
socle	C₂²×C₉₇	#127
prime radical	#15690807	C₂²
hypercenter	C₈²	D₉₇
hypocenter	C₉₇	#1711
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #20261086	C₂	semidirect
3. C₈²×C₉₇	C₂	semidirect
28. C₉₇	#1711	semidirect
31. C₈²	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261086	2
#20261082	2
C₈²×C₉₇	2
#1711	97