Subgroups — #27156759

Order 12416.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095514	C₂	1	2	yes	maximal
2	C₈×#11715895	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095514	—	2	2	no	—
4	C₄×#11715895	C₄	1	1	yes	characteristic
5	C₄×#11715895	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₉₇	C₄	1	1	yes	characteristic
7	#13095514	—	4	2	no	—
8	C₂×#11715895	C₈	1	1	yes	characteristic
9	C₂×#11715895	C₈	1	1	yes	characteristic
10	C₂×#11715895	D₄	1	1	yes	characteristic
11	C₂×#11715895	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₉₇	—	2	1	no	—
13	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
17	C₈×C₉₇	—	2	2	no	—
18	#11715895	—	2	1	no	—
19	#11715895	—	2	1	no	—
20	#11715895	C₁₆	1	2	yes	complemented
21	#11715895	SD₁₆	1	2	yes	—
22	C₂²×C₉₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₉₇	—	2	1	no	—
24	C₄×C₉₇	#106	1	1	yes	characteristic
25	C₄×C₉₇	#108	1	1	yes	characteristic
26	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₉₇	M₃₂	1	1	yes	characteristic
28	C₂×C₉₇	#334	1	1	yes	characteristic
29	C₂×C₉₇	C₄⋊C₁₆	1	1	yes	characteristic
30	C₂×C₉₇	#364	1	1	yes	characteristic
31	#1367	—	97	1	no	maximal, Sylow
32	C₉₇	#1367	1	1	yes	characteristic, complemented, Sylow
33	C₄⋊C₁₆	—	97	2	no	—
34	C₈²	—	97	1	no	—
35	C₂×C₁₆	—	194	2	no	—
36	C₄×C₈	—	97	1	no	—
37	C₄×C₈	—	97	1	no	—
38	C₄×C₈	#87604	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	97	1	no	—
41	C₂×C₈	—	97	1	no	—
42	C₂×C₈	—	97	1	no	—
43	C₂×C₈	—	97	1	no	—
44	C₄²	C₂×#87604	1	1	yes	characteristic
45	C₁₆	—	388	2	no	—
46	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
47	C₂×C₄	#13095519	1	1	yes	characteristic
48	C₂×C₄	#13095520	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	97	2	no	—
51	C₈	—	97	2	no	—
52	C₈	—	194	1	no	—
53	C₈	—	194	1	no	—
54	C₂²	#15690877	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#15690878	1	1	yes	characteristic
57	C₄	#15690881	1	1	yes	characteristic
58	C₄	#15690883	1	1	yes	characteristic
59	C₄	#15690885	1	1	yes	characteristic
60	C₂	#20261597	1	1	yes	characteristic
61	C₂	#20261656	1	1	yes	characteristic
62	C₂	#20261658	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#87604
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₄²	C₂×#87604
Fitting	C₄×C₈×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₄⋊C₈
cosocle	C₄×#11715895	C₂²
hypercenter	C₄×C₈	#87604
hypocenter	C₉₇	#1367
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
20. #11715895	C₁₆	semidirect
32. C₉₇	#1367	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095514	2
C₈×#11715895	2
#1367	97