Subgroups — #27159015

Order 12416.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261866	C₂	1	2	yes	maximal
2	#20261828	C₂	1	1	yes	maximal, characteristic
3	#15690909	—	2	2	no	—
4	#15690905	C₄	1	1	yes	characteristic
5	#15690905	C₂²	1	1	yes	characteristic
6	C₈×#87604	C₄	1	1	yes	characteristic
7	#13095523	—	2	1	no	—
8	#13095523	D₄	1	1	yes	characteristic
9	#13095523	Q₈	1	1	yes	characteristic
10	#13095519	C₂×C₄	1	1	yes	characteristic
11	C₄×#87604	C₈	1	1	yes	characteristic
12	C₈×D₉₇	C₈	1	1	yes	characteristic
13	C₂×#87604	—	2	1	no	—
14	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
15	#11715897	C₄⋊C₄	1	1	yes	characteristic
16	#11715897	M₁₆	1	1	yes	characteristic
17	C₈×C₉₇	C₁₆	1	1	yes	characteristic
18	#11715895	C₁₆	1	1	yes	characteristic
19	C₂×D₉₇	M₃₂	1	1	yes	characteristic
20	#87604	—	2	2	no	—
21	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
22	Dic₉₇	C₄⋊C₈	1	1	yes	characteristic
23	C₂×C₉₇	C₄⋊C₁₆	1	1	yes	characteristic
24	D₉₇	—	2	1	no	—
25	#1422	—	97	1	no	maximal, Sylow
26	C₉₇	#1422	1	1	yes	characteristic, complemented, Sylow
27	M₆₄	—	97	2	no	—
28	C₄×C₁₆	—	97	1	no	—
29	C₂×C₁₆	—	97	1	no	—
30	C₂×C₁₆	—	97	1	no	—
31	C₄×C₈	—	97	1	no	—
32	C₃₂	—	194	2	no	—
33	C₂×C₈	—	97	1	no	—
34	C₂×C₈	—	97	1	no	—
35	C₄²	—	97	1	no	—
36	C₁₆	—	97	1	no	—
37	C₁₆	—	97	1	no	—
38	C₁₆	—	194	1	no	—
39	C₂×C₄	—	97	1	no	—
40	C₂×C₄	—	194	1	no	—
41	C₈	—	97	1	no	—
42	C₈	—	97	1	no	—
43	C₈	—	97	1	no	—
44	C₈	#13095525	1	1	yes	characteristic
45	C₂²	—	97	1	no	—
46	C₄	—	97	1	no	—
47	C₄	—	194	2	no	—
48	C₄	C₂×#13095525	1	1	yes	characteristic
49	C₂	—	194	1	no	—
50	C₂	C₄⋊#13095525	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊#13095525
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₄	C₂×#13095525
Fitting	C₈×C₉₇	C₁₆
socle	C₂×C₉₇	C₄⋊C₁₆
prime radical	C₂×D₉₇	M₃₂
cosocle	#15690905	C₂²
hypercenter	C₈	#13095525
hypocenter	C₉₇	#1422
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
26. C₉₇	#1422	semidirect

Maximal subgroups

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

subgroup	index
#20261866	2
#20261828	2
#1422	97