Subgroups — #27158995

Order 12416.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13095525	C₂	1	2	yes	maximal
2	C₈×#11715906	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095525	C₄	1	4	yes	—
4	C₄×#11715906	C₄	1	1	yes	characteristic
5	C₄×#11715906	C₂²	1	1	yes	characteristic
6	C₈×#87604	C₄	1	1	yes	characteristic
7	#13095525	—	2	4	no	—
8	C₂×#11715906	C₈	1	2	yes	—
9	C₂×#11715906	C₂×C₄	1	2	yes	—
10	#13095519	C₈	1	1	yes	characteristic
11	C₄×#87604	C₂×C₄	1	1	yes	characteristic
12	C₈×D₉₇	C₈	1	1	yes	characteristic
13	C₂×#87604	C₄²	1	1	yes	characteristic
14	C₂×#87604	C₂×C₈	1	1	yes	characteristic
15	#11715906	—	2	2	no	—
16	#11715906	M₁₆	1	4	yes	—
17	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
18	#11715897	C₁₆	1	2	yes	complemented
19	C₈×C₉₇	C₁₆	1	1	yes	characteristic, complemented
20	#11715895	C₁₆	1	1	yes	characteristic, complemented
21	C₂×D₉₇	C₄×C₈	1	1	yes	characteristic
22	#87604	#97	1	2	yes	—
23	#87604	M₃₂	1	2	yes	—
24	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
25	Dic₉₇	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
27	D₉₇	#322	1	1	yes	characteristic
28	D₉₇	#346	1	1	yes	characteristic
29	#1312	—	97	1	no	maximal, Sylow
30	C₉₇	#1312	1	1	yes	characteristic, complemented, Sylow
31	C₄×C₁₆	—	97	2	no	—
32	C₈²	—	97	1	no	—
33	C₂×C₁₆	—	97	4	no	—
34	C₄×C₈	—	97	1	no	—
35	C₄×C₈	—	97	1	no	—
36	C₄×C₈	—	97	1	no	—
37	C₂×C₈	—	97	1	no	—
38	C₂×C₈	—	97	2	no	—
39	C₂×C₈	—	97	1	no	—
40	C₂×C₈	—	97	2	no	—
41	C₄²	—	97	1	no	—
42	C₁₆	—	194	4	no	—
43	C₂×C₄	—	97	1	no	—
44	C₂×C₄	—	97	1	no	—
45	C₂×C₄	—	97	1	no	—
46	C₈	—	97	4	no	—
47	C₈	—	97	2	no	—
48	C₈	—	97	1	no	—
49	C₈	—	194	2	no	—
50	C₈	#13095525	1	1	yes	characteristic, complemented
51	C₂²	—	97	1	no	—
52	C₄	—	97	2	no	—
53	C₄	—	97	2	no	—
54	C₄	—	97	1	no	—
55	C₄	C₂×#13095525	1	1	yes	characteristic
56	C₂	—	97	1	no	—
57	C₂	—	97	1	no	—
58	C₂	C₄×#13095525	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×#13095525
commutator, socle	C₂×C₉₇	C₄×C₁₆
Fitting	C₈×C₉₇	C₁₆
prime radical	C₂×D₉₇	C₄×C₈
cosocle	C₄×#11715906	C₂²
hypercenter	C₈	#13095525
hypocenter	C₉₇	#1312
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
18. #11715897	C₁₆	semidirect
19. C₈×C₉₇	C₁₆	semidirect
20. #11715895	C₁₆	semidirect
30. C₉₇	#1312	semidirect
50. C₈	#13095525	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13095525	2
C₈×#11715906	2
#1312	97