Subgroups — #27156869

Order 12416.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261640	C₂	1	2	yes	maximal
2	#20261109	C₂	1	1	yes	maximal, characteristic
3	#15690869	—	2	2	no	—
4	C₂×#13095498	—	2	2	no	—
5	#15690807	C₂²	1	1	yes	characteristic
6	C₄×#11715895	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈×C₉₇	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#87604	—	4	2	no	—
9	C₂×D₃₈₈	—	4	1	no	—
10	C₂×D₃₈₈	D₄	1	2	yes	—
11	#13095498	—	4	4	no	—
12	C₂×#11715895	—	2	1	no	—
13	C₂×Q₈×C₉₇	—	2	1	no	—
14	C₄⋊C₄×C₉₇	—	4	1	no	—
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×D₉₇	—	2	2	no	—
17	C₂×#87604	—	8	2	no	—
18	Q₈×C₉₇	—	4	2	no	—
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	4	1	no	—
21	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	D₃₈₈	—	4	2	no	—
23	D₃₈₈	—	8	2	no	—
24	#11715895	—	2	2	no	—
25	C₂²×C₉₇	#99	1	1	yes	characteristic
26	C₂×D₉₇	—	4	2	no	—
27	C₂×D₉₇	—	8	2	no	—
28	#87604	—	16	2	no	—
29	C₄×C₉₇	—	2	2	no	—
30	C₄×C₉₇	—	2	1	no	—
31	C₄×C₉₇	—	8	1	no	—
32	C₂×C₉₇	—	2	1	no	—
33	C₂×C₉₇	#353	1	1	yes	characteristic
34	D₉₇	—	8	2	no	—
35	#1409	—	97	1	no	maximal, Sylow
36	C₉₇	#1409	1	1	yes	characteristic, complemented, Sylow
37	#492	—	97	1	no	—
38	#353	—	97	2	no	—
39	C₂×SD₁₆	—	194	2	no	—
40	C₄⋊Q₈	#87604	1	1	yes	characteristic, complemented
41	#127	—	97	1	no	—
42	#99	—	194	2	no	—
43	C₄×C₈	—	97	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	97	2	no	—
46	C₂×D₄	—	388	1	no	—
47	SD₁₆	—	388	4	no	—
48	C₂×C₈	—	194	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	388	2	no	—
51	C₄²	C₂×#87604	1	1	yes	characteristic
52	C₂³	—	194	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	776	2	no	—
55	D₄	—	388	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	776	2	no	—
59	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
60	C₈	—	194	2	no	—
61	C₂²	—	776	2	no	—
62	C₂²	—	388	2	no	—
63	C₂²	#15690868	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	1552	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	776	2	no	—
70	C₂	#20261639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20261639
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄²	C₂×#87604
Fitting	C₄⋊Q₈×C₉₇	C₄
socle	C₂×C₉₇	#353
prime radical, cosocle	#15690807	C₂²
hypercenter	C₄⋊Q₈	#87604
hypocenter	C₉₇	#1409
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×#11715895	C₄	semidirect
7. C₄⋊Q₈×C₉₇	C₄	semidirect
36. C₉₇	#1409	semidirect
40. C₄⋊Q₈	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261640	2
#20261109	2
#1409	97