Subgroups — #27159004

Order 12416.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261864	C₂	1	2	yes	maximal
2	#20261828	C₂	1	1	yes	maximal, characteristic
3	#15690909	C₄	1	4	yes	complemented
4	#15690905	C₄	1	1	yes	characteristic
5	#15690905	C₂²	1	1	yes	characteristic
6	C₈×#87604	C₄	1	1	yes	characteristic
7	#13095523	C₈	1	2	yes	—
8	#13095523	C₂×C₄	1	2	yes	—
9	#13095519	C₂×C₄	1	1	yes	characteristic
10	C₄×#87604	C₈	1	1	yes	characteristic
11	C₈×D₉₇	C₈	1	1	yes	characteristic
12	C₂×#87604	C₁₆	1	2	yes	—
13	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
14	#11715897	C₄²	1	1	yes	characteristic
15	#11715897	C₂×C₈	1	1	yes	characteristic
16	C₈×C₉₇	C₁₆	1	1	yes	characteristic
17	#11715895	C₁₆	1	1	yes	characteristic
18	C₂×D₉₇	C₂×C₁₆	1	1	yes	characteristic
19	#87604	C₃₂	1	4	yes	complemented
20	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
21	Dic₉₇	C₄×C₈	1	1	yes	characteristic
22	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
23	D₉₇	C₂×C₃₂	1	2	yes	—
24	C₄×C₃₂	—	97	1	no	maximal, Sylow
25	C₉₇	C₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃₂	—	97	2	no	—
27	C₄×C₁₆	—	97	1	no	—
28	C₂×C₁₆	—	97	1	no	—
29	C₂×C₁₆	—	97	1	no	—
30	C₄×C₈	—	97	1	no	—
31	C₃₂	—	97	4	no	—
32	C₂×C₈	—	97	1	no	—
33	C₂×C₈	—	97	1	no	—
34	C₄²	—	97	1	no	—
35	C₁₆	—	97	2	no	—
36	C₁₆	—	97	2	no	—
37	C₂×C₄	—	97	2	no	—
38	C₂×C₄	—	97	1	no	—
39	C₈	—	97	1	no	—
40	C₈	—	97	1	no	—
41	C₈	—	97	1	no	—
42	C₈	#13095525	1	1	yes	characteristic
43	C₂²	—	97	1	no	—
44	C₄	—	97	4	no	—
45	C₄	—	97	1	no	—
46	C₄	C₂×#13095525	1	1	yes	characteristic
47	C₂	—	97	2	no	—
48	C₂	C₄×#13095525	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	#13095525
commutator, hypocenter	C₉₇	C₄×C₃₂
Frattini	C₄	C₂×#13095525
Fitting	C₈×C₉₇	C₁₆
socle	C₂×C₉₇	C₄×C₁₆
prime radical	C₂×D₉₇	C₂×C₁₆
cosocle	#15690905	C₂²
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
3. #15690909	C₄	semidirect
19. #87604	C₃₂	semidirect
25. C₉₇	C₄×C₃₂	semidirect

Maximal subgroups

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

subgroup	index
#20261864	2
#20261828	2
C₄×C₃₂	97