Subgroups — #27158751

Order 12416.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261828	C₂	1	1	yes	maximal, characteristic
2	C₈×#11715906	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#87604	C₂	1	1	yes	maximal, characteristic
4	#15690905	C₄	1	2	yes	—
5	C₄×#11715906	C₄	1	2	yes	—
6	#15690883	C₄	1	1	yes	characteristic
7	C₈×#87604	C₂²	1	1	yes	characteristic
8	C₁₆×D₉₇	C₄	1	1	yes	characteristic
9	C₂×#11715906	C₈	1	4	yes	—
10	#13095523	C₈	1	4	yes	complemented
11	#13095519	C₂×C₄	1	1	yes	characteristic
12	C₄×#87604	C₂×C₄	1	1	yes	characteristic
13	#13095514	C₈	1	2	yes	complemented
14	C₈×D₉₇	C₂×C₄	1	1	yes	characteristic
15	#13095486	C₈	1	1	yes	characteristic, complemented
16	C₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
17	C₂×#87604	C₂×C₈	1	2	yes	—
18	#11715906	—	2	4	no	—
19	C₄×D₉₇	C₄²	1	1	yes	characteristic
20	#11715897	C₂×C₈	1	2	yes	—
21	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
22	#11715895	C₂×C₈	1	1	yes	characteristic
23	C₂×D₉₇	C₄×C₈	1	1	yes	characteristic
24	#87604	M₃₂	1	4	yes	—
25	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
26	Dic₉₇	C₄×C₈	1	1	yes	characteristic
27	C₂×C₉₇	C₈²	1	1	yes	characteristic
28	D₉₇	#346	1	2	yes	—
29	#1311	—	97	1	no	maximal, Sylow
30	C₉₇	#1311	1	1	yes	characteristic, complemented, Sylow
31	C₄×C₁₆	—	97	1	no	—
32	C₄×C₁₆	—	97	1	no	—
33	C₈²	—	97	1	no	—
34	C₂×C₁₆	—	97	1	no	—
35	C₂×C₁₆	—	97	2	no	—
36	C₂×C₁₆	—	97	1	no	—
37	C₄×C₈	—	97	1	no	—
38	C₄×C₈	—	97	2	no	—
39	C₂×C₈	—	97	1	no	—
40	C₂×C₈	—	97	1	no	—
41	C₂×C₈	—	97	4	no	—
42	C₄²	—	97	1	no	—
43	C₁₆	—	97	2	no	—
44	C₁₆	—	97	4	no	—
45	C₁₆	—	97	1	no	—
46	C₁₆	#11715906	1	1	yes	characteristic, complemented
47	C₂×C₄	—	97	2	no	—
48	C₂×C₄	—	97	1	no	—
49	C₈	—	97	2	no	—
50	C₈	—	97	1	no	—
51	C₈	—	194	4	no	—
52	C₈	C₂×#11715906	1	1	yes	characteristic
53	C₂²	—	97	1	no	—
54	C₄	—	97	4	no	—
55	C₄	—	97	1	no	—
56	C₄	C₄×#11715906	1	1	yes	characteristic
57	C₂	—	97	2	no	—
58	C₂	C₈×#11715906	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×#11715906
commutator, socle	C₂×C₉₇	C₈²
Fitting	C₁₆×C₉₇	C₈
prime radical	C₂×D₉₇	C₄×C₈
cosocle	C₈×#87604	C₂²
hypercenter	C₁₆	#11715906
hypocenter	C₉₇	#1311
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
10. #13095523	C₈	semidirect
13. #13095514	C₈	semidirect
15. #13095486	C₈	semidirect
16. C₁₆×C₉₇	C₈	semidirect
30. C₉₇	#1311	semidirect
46. C₁₆	#11715906	semidirect

Maximal subgroups

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

subgroup	index
#20261828	2
C₈×#11715906	2
C₁₆×#87604	2
#1311	97