Subgroups — #27157668

Order 12416.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11715897	C₂	1	6	yes	maximal
2	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715897	C₂²	1	4	yes	—
4	C₄⋊#11715897	—	2	12	no	—
5	C₂×C₄×Dic₉₇	C₄	1	3	yes	—
6	C₂×C₄×Dic₉₇	C₂²	1	3	yes	—
7	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
8	C₂×#11715897	—	2	12	no	—
9	C₂²×Dic₉₇	C₂×C₄	1	3	yes	—
10	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
11	C₄×Dic₉₇	—	2	6	no	—
12	C₄×Dic₉₇	D₄	1	6	yes	—
13	C₄×Dic₉₇	Q₈	1	6	yes	—
14	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
15	C₄²×C₉₇	C₈	1	4	yes	complemented
16	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₄×C₉₇	C₄⋊C₄	1	6	yes	—
18	C₂×C₄×C₉₇	C₂×C₈	1	6	yes	—
19	C₂×C₄×C₉₇	M₁₆	1	6	yes	—
20	C₂×Dic₉₇	—	2	6	no	—
21	C₂×Dic₉₇	C₄⋊C₄	1	6	yes	—
22	C₂×Dic₉₇	C₂×D₄	1	3	yes	—
23	C₂×Dic₉₇	C₂×Q₈	1	3	yes	—
24	#11715897	—	4	4	no	—
25	C₂²×C₉₇	C₂×C₄⋊C₄	1	3	yes	—
26	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
27	C₂²×C₉₇	C₂×M₁₆	1	3	yes	—
28	C₄×C₉₇	C₄⋊C₈	1	12	yes	complemented
29	Dic₉₇	—	2	6	no	—
30	Dic₉₇	#127	1	1	yes	characteristic
31	Dic₉₇	C₄⋊Q₈	1	3	yes	—
32	C₂×C₉₇	#384	1	1	yes	characteristic
33	C₂×C₉₇	C₂×C₄⋊C₈	1	3	yes	—
34	C₂×C₉₇	#423	1	3	yes	—
35	#1842	—	97	1	no	maximal, Sylow
36	C₉₇	#1842	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₄⋊C₈	—	97	6	no	—
38	C₄³	—	97	1	no	—
39	C₂²×C₈	—	97	4	no	—
40	C₂×C₄²	—	97	3	no	—
41	C₂×C₄²	—	97	3	no	—
42	C₂×C₄²	#87604	1	1	yes	characteristic
43	C₄⋊C₈	—	194	12	no	—
44	C₂²×C₄	—	97	1	no	—
45	C₂²×C₄	—	97	3	no	—
46	C₂²×C₄	C₂×#87604	1	3	yes	—
47	C₂×C₈	—	194	12	no	—
48	C₄²	—	97	6	no	—
49	C₄²	—	97	6	no	—
50	C₄²	—	194	6	no	—
51	C₄²	#11715897	1	4	yes	complemented
52	C₂³	C₂²×#87604	1	1	yes	characteristic
53	C₂×C₄	—	97	3	no	—
54	C₂×C₄	—	97	3	no	—
55	C₂×C₄	—	97	6	no	—
56	C₂×C₄	—	194	6	no	—
57	C₂×C₄	C₄⋊#87604	1	6	yes	—
58	C₂×C₄	C₂×#11715897	1	6	yes	—
59	C₂×C₄	#13095521	1	6	yes	—
60	C₈	—	388	4	no	—
61	C₂²	C₂×C₄⋊#87604	1	3	yes	—
62	C₂²	C₂²×#11715897	1	1	yes	characteristic
63	C₂²	C₂×#13095521	1	3	yes	—
64	C₄	—	97	1	no	—
65	C₄	—	97	3	no	—
66	C₄	—	194	6	no	—
67	C₄	C₄⋊#11715897	1	12	yes	complemented
68	C₂	#20261681	1	1	yes	characteristic
69	C₂	C₂×C₄⋊#11715897	1	3	yes	—
70	C₂	#20261739	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87604
commutator	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×C₄²×C₉₇	C₄
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×Dic₉₇	C₂³
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	#1842
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
15. C₄²×C₉₇	C₈	semidirect
28. C₄×C₉₇	C₄⋊C₈	semidirect
36. C₉₇	#1842	semidirect
51. C₄²	#11715897	semidirect
67. C₄	C₄⋊#11715897	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715897	2
C₄²×Dic₉₇	2
#1842	97