Subgroups — #27141352

Order 12416.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261138	C₂	1	4	yes	maximal
2	#20261115	C₂	1	1	yes	maximal, characteristic
3	C₈×#11715895	C₂	1	1	yes	maximal, characteristic
4	#494×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₈	—	2	2	no	—
6	#15690810	C₂²	1	2	yes	—
7	#15690743	—	2	8	no	—
8	C₄×#11715895	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₉₇	—	2	2	no	—
10	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
11	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₉₄	—	2	2	no	—
13	Dic₃₈₈	—	4	4	no	—
14	C₂×#11715895	D₄	1	4	yes	—
15	C₄⋊Dic₉₇	—	2	4	no	—
16	C₂×Q₈×C₉₇	—	2	2	no	—
17	Q₁₆×C₉₇	—	4	4	no	—
18	C₂×C₈×C₉₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₉₇	—	2	4	no	—
20	C₄²×C₉₇	C₂³	1	1	yes	characteristic
21	Q₈×C₉₇	—	4	4	no	—
22	C₂×C₄×C₉₇	—	4	2	no	—
23	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
24	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
25	C₂×Dic₉₇	—	4	2	no	—
26	Dic₁₉₄	—	4	4	no	—
27	C₈×C₉₇	D₈	1	4	yes	—
28	#11715895	Q₁₆	1	8	yes	complemented
29	C₂²×C₉₇	#127	1	1	yes	characteristic
30	C₄×C₉₇	—	8	2	no	—
31	C₄×C₉₇	C₂×D₈	1	2	yes	—
32	C₄×C₉₇	C₂×Q₁₆	1	4	yes	—
33	Dic₉₇	—	8	2	no	—
34	C₂×C₉₇	#493	1	1	yes	characteristic
35	C₂×C₉₇	#494	1	2	yes	—
36	#1713	—	97	1	no	maximal, Sylow
37	C₉₇	#1713	1	1	yes	characteristic, complemented, Sylow
38	#500	—	97	4	no	—
39	#494	—	97	1	no	—
40	#494	D₉₇	1	1	yes	characteristic
41	C₈²	—	97	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	194	2	no	—
44	C₄⋊Q₈	—	97	2	no	—
45	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
46	#107	—	194	8	no	—
47	C₄×C₈	—	97	2	no	—
48	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	194	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	388	4	no	—
53	C₂×C₈	—	97	4	no	—
54	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	194	4	no	—
57	C₄²	C₂²×D₉₇	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	388	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	388	2	no	—
62	C₂×C₄	D₄×D₉₇	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
64	C₈	—	97	8	no	—
65	C₈	#13095496	1	4	yes	—
66	C₂²	#15690806	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	776	2	no	—
69	C₄	C₂×#13095496	1	2	yes	—
70	C₄	Q₁₆×D₉₇	1	4	yes	—
71	C₂	#20261111	1	1	yes	characteristic
72	C₂	#20261113	1	2	yes	—

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#494×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#127
hypercenter	#494	D₉₇
hypocenter	C₉₇	#1713
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
28. #11715895	Q₁₆	semidirect
37. C₉₇	#1713	semidirect

Maximal subgroups

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

subgroup	index
#20261138	2
#20261115	2
C₈×#11715895	2
#494×C₉₇	2
#1713	97