Subgroups — #27141343

Order 12416.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261112	C₂	1	1	yes	maximal, characteristic
2	#20261083	C₂	1	4	yes	maximal, complemented
3	C₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₇₆	—	2	2	no	—
6	#15690807	C₂²	1	2	yes	—
7	#15690730	—	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×D₃₈₈	—	2	2	no	—
13	C₂×D₃₈₈	—	2	4	no	—
14	D₇₇₆	—	4	4	no	—
15	C₂×#11715895	D₄	1	4	yes	—
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	C₂²×D₉₇	—	4	2	no	—
22	Q₈×C₉₇	—	4	4	no	—
23	C₂×C₄×C₉₇	—	4	2	no	—
24	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
26	D₃₈₈	—	4	4	no	—
27	D₃₈₈	—	4	8	no	—
28	C₈×C₉₇	D₈	1	4	yes	—
29	#11715895	—	2	4	no	—
30	C₂²×C₉₇	#127	1	1	yes	characteristic
31	C₂×D₉₇	—	8	2	no	—
32	C₂×D₉₇	—	8	4	no	—
33	C₄×C₉₇	—	8	2	no	—
34	C₄×C₉₇	C₂×D₈	1	2	yes	—
35	C₄×C₉₇	#135	1	4	yes	—
36	C₂×C₉₇	#493	1	1	yes	characteristic
37	C₂×C₉₇	#495	1	2	yes	—
38	D₉₇	—	16	2	no	—
39	#1709	—	97	1	no	maximal, Sylow
40	C₉₇	#1709	1	1	yes	characteristic, complemented, Sylow
41	#494	D₉₇	1	1	yes	characteristic, complemented
42	#493	—	97	1	no	—
43	#486	—	97	4	no	—
44	C₈²	—	97	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	194	2	no	—
47	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
48	#127	—	97	2	no	—
49	#102	—	194	8	no	—
50	C₄×C₈	—	97	2	no	—
51	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	194	4	no	—
54	C₂×D₄	—	194	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	388	4	no	—
57	C₂×C₈	—	97	4	no	—
58	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₉₇	1	1	yes	characteristic
61	C₂³	—	388	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	388	4	no	—
64	D₄	—	388	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₉₇	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
68	C₈	—	194	4	no	—
69	C₈	#13095496	1	4	yes	—
70	C₂²	—	776	4	no	—
71	C₂²	—	776	2	no	—
72	C₂²	#15690806	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#13095496	1	2	yes	—
75	C₄	#15690833	1	4	yes	—
76	C₂	—	1552	2	no	—
77	C₂	#20261111	1	1	yes	characteristic
78	C₂	#20261120	1	2	yes	—

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#494×C₉₇	C₂
socle	C₂²×C₉₇	#127
prime radical	#20261112	C₂
hypercenter	#494	D₉₇
hypocenter	C₉₇	#1709
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
2. #20261083	C₂	semidirect
3. C₈×#11715895	C₂	semidirect
4. #494×C₉₇	C₂	semidirect
40. C₉₇	#1709	semidirect
41. #494	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261112	2
#20261083	2
C₈×#11715895	2
#494×C₉₇	2
#1709	97