Subgroups — #27158880

Order 12416.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11715906	C₂	1	6	yes	maximal
2	C₄²×#87604	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715906	C₂²	1	4	yes	—
4	C₄⋊#11715906	—	2	12	no	—
5	C₂×C₄×#87604	C₄	1	3	yes	—
6	C₂×C₄×#87604	C₂²	1	3	yes	—
7	C₄²×D₉₇	C₄	1	1	yes	characteristic
8	C₂×#11715906	—	2	12	no	—
9	C₂²×#87604	C₂×C₄	1	3	yes	—
10	C₂²×#87604	C₂³	1	1	yes	characteristic
11	C₄×#87604	—	2	6	no	—
12	C₄×#87604	D₄	1	6	yes	—
13	C₄×#87604	Q₈	1	6	yes	—
14	C₂×C₄×D₉₇	C₂×C₄	1	3	yes	—
15	C₄×Dic₉₇	C₈	1	3	yes	complemented
16	C₄²×C₉₇	C₈	1	1	yes	characteristic, complemented
17	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×#87604	—	2	6	no	—
19	C₂×#87604	C₄⋊C₄	1	6	yes	—
20	C₂×#87604	C₂×D₄	1	3	yes	—
21	C₂×#87604	C₂×Q₈	1	3	yes	—
22	#11715906	—	4	4	no	—
23	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
24	C₂×Dic₉₇	C₂×C₈	1	3	yes	—
25	C₄×D₉₇	C₄⋊C₄	1	6	yes	—
26	C₄×D₉₇	M₁₆	1	6	yes	—
27	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
28	C₂×D₉₇	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×D₉₇	C₂×M₁₆	1	3	yes	—
30	#87604	—	2	6	no	—
31	#87604	#127	1	1	yes	characteristic
32	#87604	C₄⋊Q₈	1	3	yes	—
33	C₄×C₉₇	C₄⋊C₈	1	6	yes	complemented
34	Dic₉₇	C₄⋊C₈	1	6	yes	complemented
35	C₂×C₉₇	C₂×C₄⋊C₈	1	3	yes	—
36	D₉₇	#384	1	1	yes	characteristic
37	D₉₇	#423	1	3	yes	—
38	#1842	—	97	1	no	maximal, Sylow
39	C₉₇	#1842	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄⋊C₈	—	97	6	no	—
41	C₄³	—	97	1	no	—
42	C₂²×C₈	—	97	4	no	—
43	C₂×C₄²	—	97	3	no	—
44	C₂×C₄²	—	97	1	no	—
45	C₂×C₄²	—	97	3	no	—
46	C₄⋊C₈	—	194	12	no	—
47	C₂²×C₄	—	97	1	no	—
48	C₂²×C₄	—	97	3	no	—
49	C₂²×C₄	—	97	3	no	—
50	C₂×C₈	—	194	12	no	—
51	C₄²	—	97	6	no	—
52	C₄²	—	97	6	no	—
53	C₄²	—	97	3	no	—
54	C₄²	—	194	6	no	—
55	C₄²	#11715906	1	1	yes	characteristic, complemented
56	C₂³	—	97	1	no	—
57	C₂×C₄	—	97	3	no	—
58	C₂×C₄	—	97	3	no	—
59	C₂×C₄	—	97	3	no	—
60	C₂×C₄	—	97	6	no	—
61	C₂×C₄	—	97	6	no	—
62	C₂×C₄	—	97	6	no	—
63	C₂×C₄	—	194	6	no	—
64	C₂×C₄	C₂×#11715906	1	3	yes	—
65	C₈	—	388	4	no	—
66	C₂²	—	97	3	no	—
67	C₂²	—	97	3	no	—
68	C₂²	C₂²×#11715906	1	1	yes	characteristic
69	C₄	—	97	1	no	—
70	C₄	—	97	3	no	—
71	C₄	—	97	6	no	—
72	C₄	—	194	6	no	—
73	C₄	C₄⋊#11715906	1	6	yes	complemented
74	C₂	—	97	1	no	—
75	C₂	—	97	3	no	—
76	C₂	C₂×C₄⋊#11715906	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₄²×C₉₇	C₈
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂²×#87604	C₂³
hypercenter	C₄²	#11715906
hypocenter	C₉₇	#1842
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
15. C₄×Dic₉₇	C₈	semidirect
16. C₄²×C₉₇	C₈	semidirect
33. C₄×C₉₇	C₄⋊C₈	semidirect
34. Dic₉₇	C₄⋊C₈	semidirect
39. C₉₇	#1842	semidirect
55. C₄²	#11715906	semidirect
73. C₄	C₄⋊#11715906	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715906	2
C₄²×#87604	2
#1842	97