Subgroups — #27140035

Order 12416.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260786	C₂	1	3	yes	maximal
2	#20260457	C₂	1	3	yes	maximal, complemented
3	C₈²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₉₇	C₂²	1	1	yes	characteristic
5	#15690739	C₄	1	6	yes	—
6	#15690722	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	C₂×#11715895	C₂×C₄	1	6	yes	—
11	C₄×Dic₉₇	C₂×C₄	1	3	yes	—
12	C₂×C₈×C₉₇	C₂×C₄	1	6	yes	—
13	C₄²×C₉₇	C₂³	1	1	yes	characteristic
14	C₂²×D₉₇	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
16	C₂×Dic₉₇	C₄²	1	3	yes	—
17	C₄×D₉₇	—	2	6	no	—
18	C₈×C₉₇	—	2	6	no	—
19	#11715895	—	2	6	no	—
20	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
21	C₂×D₉₇	—	2	3	no	—
22	C₄×C₉₇	#131	1	6	yes	—
23	Dic₉₇	—	2	3	no	—
24	C₂×C₉₇	#405	1	3	yes	—
25	D₉₇	—	4	1	no	—
26	#1454	—	97	1	no	maximal, Sylow
27	C₉₇	#1454	1	1	yes	characteristic, complemented, Sylow
28	#431	—	97	3	no	—
29	#322	—	97	3	no	—
30	C₈²	D₉₇	1	1	yes	characteristic, complemented
31	C₂×C₄²	—	97	1	no	—
32	C₄⋊C₈	—	97	6	no	—
33	C₂²⋊C₈	—	97	6	no	—
34	C₄×C₈	—	97	3	no	—
35	C₄×C₈	C₂×D₉₇	1	3	yes	—
36	C₂²×C₄	—	97	3	no	—
37	C₂×C₈	—	97	6	no	—
38	C₂×C₈	C₄×D₉₇	1	6	yes	—
39	C₄²	—	97	3	no	—
40	C₄²	C₂²×D₉₇	1	1	yes	characteristic
41	C₂³	—	97	1	no	—
42	C₂×C₄	—	97	3	no	—
43	C₂×C₄	—	194	6	no	—
44	C₂×C₄	C₂×C₄×D₉₇	1	3	yes	—
45	C₈	—	2	6	no	—
46	C₈	—	194	6	no	—
47	C₂²	—	194	3	no	—
48	C₂²	C₄²×D₉₇	1	1	yes	characteristic
49	C₄	—	194	3	no	—
50	C₄	#15690816	1	6	yes	—
51	C₂	—	388	1	no	—
52	C₂	#20260689	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	#1454
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. #20260457	C₂	semidirect
3. C₈²×C₉₇	C₂	semidirect
27. C₉₇	#1454	semidirect
30. C₈²	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260786	2
#20260457	2
C₈²×C₉₇	2
#1454	97