Subgroups — #27159001

Order 12416.

76 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	—	2	2	no	—
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	—	4	2	no	—
8	C₂×#11715906	—	2	1	no	—
9	C₂×#11715906	D₄	1	1	yes	characteristic
10	C₂×#11715906	Q₈	1	1	yes	characteristic
11	#13095519	C₈	1	1	yes	characteristic
12	C₄×#87604	C₂×C₄	1	1	yes	characteristic
13	C₈×D₉₇	C₈	1	1	yes	characteristic
14	C₂×#87604	C₄⋊C₄	1	1	yes	characteristic
15	C₂×#87604	M₁₆	1	1	yes	characteristic
16	#11715906	—	2	2	no	—
17	#11715906	—	2	1	no	—
18	#11715906	D₈	1	1	yes	characteristic
19	#11715906	Q₁₆	1	1	yes	characteristic
20	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
21	#11715897	—	2	1	no	—
22	C₈×C₉₇	C₁₆	1	1	yes	characteristic, complemented
23	#11715895	C₁₆	1	1	yes	characteristic, complemented
24	C₂×D₉₇	C₄⋊C₈	1	1	yes	characteristic
25	#87604	—	2	1	no	—
26	#87604	#107	1	1	yes	characteristic
27	#87604	#108	1	1	yes	characteristic
28	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
29	Dic₉₇	M₃₂	1	1	yes	characteristic
30	C₂×C₉₇	C₄⋊C₁₆	1	1	yes	characteristic
31	D₉₇	#335	1	1	yes	characteristic
32	D₉₇	#364	1	1	yes	characteristic
33	#1370	—	97	1	no	maximal, Sylow
34	C₉₇	#1370	1	1	yes	characteristic, complemented, Sylow
35	C₄⋊C₁₆	—	97	2	no	—
36	C₈²	—	97	1	no	—
37	C₂×C₁₆	—	194	2	no	—
38	C₄×C₈	—	97	1	no	—
39	C₄×C₈	—	97	1	no	—
40	C₄×C₈	—	97	1	no	—
41	C₂×C₈	—	97	1	no	—
42	C₂×C₈	—	97	1	no	—
43	C₂×C₈	—	97	1	no	—
44	C₂×C₈	—	97	1	no	—
45	C₂×C₈	—	194	1	no	—
46	C₄²	—	97	1	no	—
47	C₁₆	—	388	2	no	—
48	C₂×C₄	—	97	1	no	—
49	C₂×C₄	—	97	1	no	—
50	C₂×C₄	—	97	1	no	—
51	C₈	—	97	1	no	—
52	C₈	—	97	1	no	—
53	C₈	—	97	1	no	—
54	C₈	—	194	1	no	—
55	C₈	—	194	1	no	—
56	C₈	—	194	2	no	—
57	C₈	#13095525	1	1	yes	characteristic, complemented
58	C₂²	—	97	1	no	—
59	C₄	—	97	1	no	—
60	C₄	—	97	1	no	—
61	C₄	—	97	1	no	—
62	C₄	—	194	1	no	—
63	C₄	C₂×#13095525	1	1	yes	characteristic
64	C₂	—	97	1	no	—
65	C₂	—	97	1	no	—
66	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	C₄⋊C₈
cosocle	C₄×#11715906	C₂²
hypercenter	C₈	#13095525
hypocenter	C₉₇	#1370
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
22. C₈×C₉₇	C₁₆	semidirect
23. #11715895	C₁₆	semidirect
34. C₉₇	#1370	semidirect
57. 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
#1370	97