Subgroups — #27158831

Order 12416.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261843	C₂	1	2	yes	maximal, complemented
2	C₂×#15690885	C₂	1	1	yes	maximal, characteristic
3	#15690900	—	2	2	no	—
4	#15690885	—	2	2	no	—
5	C₂×#13095514	C₄	1	1	yes	characteristic
6	C₂×#13095514	C₂²	1	1	yes	characteristic
7	C₂×C₈×D₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	2	1	no	—
9	#13095514	D₄	1	2	yes	—
10	C₂×C₄×D₉₇	C₈	1	1	yes	characteristic
11	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
12	C₈×D₉₇	—	2	1	no	—
13	C₈×D₉₇	—	2	1	no	—
14	C₂×C₈×C₉₇	C₈	1	1	yes	characteristic
15	C₂²×D₉₇	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
17	C₂×Dic₉₇	C₁₆	1	1	yes	characteristic
18	C₄×D₉₇	—	2	1	no	—
19	C₄×D₉₇	—	2	1	no	—
20	C₈×C₉₇	—	2	1	no	—
21	#11715895	C₂²⋊C₄	1	1	yes	characteristic
22	#11715895	M₁₆	1	1	yes	characteristic
23	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
24	C₂×D₉₇	—	2	1	no	—
25	C₂×D₉₇	—	4	1	no	—
26	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
27	C₄×C₉₇	M₃₂	1	1	yes	characteristic
28	Dic₉₇	—	2	1	no	—
29	C₂×C₉₇	—	2	1	no	—
30	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
31	D₉₇	—	4	1	no	—
32	#1400	—	97	1	no	maximal, Sylow
33	C₉₇	#1400	1	1	yes	characteristic, complemented, Sylow
34	C₂×M₃₂	—	97	1	no	—
35	M₆₄	—	97	2	no	—
36	C₂²×C₈	—	97	1	no	—
37	M₃₂	—	194	2	no	—
38	C₂×C₁₆	—	97	1	no	—
39	C₂×C₁₆	—	97	1	no	—
40	C₃₂	—	194	2	no	—
41	C₂²×C₄	—	97	1	no	—
42	C₂×C₈	—	97	1	no	—
43	C₂×C₈	—	194	1	no	—
44	C₂×C₈	—	194	1	no	—
45	C₂×C₈	#11715906	1	1	yes	characteristic
46	C₁₆	—	97	2	no	—
47	C₁₆	—	194	1	no	—
48	C₂³	—	97	1	no	—
49	C₂×C₄	—	97	1	no	—
50	C₂×C₄	—	194	1	no	—
51	C₂×C₄	—	194	1	no	—
52	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	97	1	no	—
55	C₈	—	97	1	no	—
56	C₂²	—	388	1	no	—
57	C₂²	—	194	1	no	—
58	C₂²	#15690905	1	1	yes	characteristic
59	C₄	—	194	1	no	—
60	C₄	C₂²⋊#11715906	1	1	yes	characteristic
61	C₄	#15690907	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	388	1	no	—
64	C₂	#20261832	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11715906
commutator	C₂²×C₉₇	C₂×C₁₆
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₂×C₈×C₉₇	C₈
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	C₂²×D₉₇	C₁₆
cosocle	C₂×#13095514	C₂²
hypercenter	C₂×C₈	#11715906
hypocenter	C₉₇	#1400
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #20261843	C₂	semidirect
33. C₉₇	#1400	semidirect

Maximal subgroups

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

subgroup	index
#20261843	2
C₂×#15690885	2
#1400	97