Group #27141403 — subgroups

Subgroups — #27141403

Order 12416.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261111	C₂	1	2	yes	maximal, complemented
2	#20261108	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261081	C₂	1	2	yes	maximal, complemented
4	#20260456	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095500	—	2	2	no	—
7	C₂×#13095496	—	2	4	no	—
8	#15690810	C₂²	1	1	yes	characteristic
9	#15690807	C₂²	1	1	yes	characteristic
10	#15690731	—	2	4	no	—
11	C₄×#11715895	C₂²	1	2	yes	—
12	C₂×D₈×C₉₇	—	2	2	no	—
13	#127×C₉₇	C₂²	1	2	yes	—
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₄	—	2	1	no	—
16	C₂×D₃₈₈	—	2	2	no	—
17	C₂×D₃₈₈	—	2	1	no	—
18	#13095500	—	4	4	no	—
19	#13095496	—	4	8	no	—
20	C₂×#11715895	D₄	1	4	yes	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	C₂×D₄×C₉₇	—	2	2	no	—
23	C₂×D₄×C₉₇	—	2	2	no	—
24	C₂×D₄×C₉₇	—	2	2	no	—
25	D₈×C₉₇	—	4	4	no	—
26	C₂×C₈×C₉₇	D₄	1	2	yes	—
27	C₄²×C₉₇	C₂³	1	1	yes	characteristic
28	C₂³×C₉₇	—	4	2	no	—
29	C₂²×D₉₇	—	4	1	no	—
30	D₄×C₉₇	—	4	4	no	—
31	D₄×C₉₇	—	4	4	no	—
32	D₄×C₉₇	—	4	4	no	—
33	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₉₇	—	4	1	no	—
36	D₃₈₈	—	4	2	no	—
37	D₃₈₈	—	4	4	no	—
38	Dic₁₉₄	—	4	2	no	—
39	C₈×C₉₇	D₈	1	4	yes	complemented
40	#11715895	—	2	4	no	—
41	C₂²×C₉₇	—	8	2	no	—
42	C₂²×C₉₇	—	8	2	no	—
43	C₂²×C₉₇	—	8	2	no	—
44	C₂²×C₉₇	#127	1	1	yes	characteristic
45	C₂×D₉₇	—	8	1	no	—
46	C₂×D₉₇	—	8	2	no	—
47	C₄×C₉₇	C₂×D₈	1	2	yes	—
48	C₄×C₉₇	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₉₇	—	8	1	no	—
50	C₂×C₉₇	—	16	2	no	—
51	C₂×C₉₇	#493	1	1	yes	characteristic
52	C₂×C₉₇	#496	1	2	yes	—
53	D₉₇	—	16	1	no	—
54	#1721	—	97	1	no	maximal, Sylow
55	C₉₇	#1721	1	1	yes	characteristic, complemented, Sylow
56	#493	—	97	2	no	—
57	#493	D₉₇	1	1	yes	characteristic, complemented
58	#492	—	97	1	no	—
59	#486	—	97	2	no	—
60	#322	—	97	1	no	—
61	C₂×SD₁₆	—	194	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	194	4	no	—
64	C₄⋊Q₈	—	97	1	no	—
65	#127	—	97	1	no	—
66	#127	C₂×D₉₇	1	2	yes	—
67	#102	—	194	4	no	—
68	C₄×C₈	—	97	2	no	—
69	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
70	C₂×Q₈	—	194	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	194	2	no	—
75	C₂×D₄	—	194	1	no	—
76	SD₁₆	—	388	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	388	8	no	—
79	C₂×C₈	—	97	4	no	—
80	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
81	C₄⋊C₄	—	194	2	no	—
82	C₄²	C₂²×D₉₇	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	388	1	no	—
85	Q₈	—	388	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	388	2	no	—
90	D₄	—	388	4	no	—
91	C₂×C₄	—	388	1	no	—
92	C₂×C₄	D₄×D₉₇	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
94	C₈	—	194	4	no	—
95	C₈	#13095496	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	776	2	no	—
100	C₂²	—	776	1	no	—
101	C₂²	#15690806	1	1	yes	characteristic
102	C₄	—	776	1	no	—
103	C₄	C₂×#13095496	1	2	yes	—
104	C₄	#15690834	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	1552	1	no	—
107	C₂	#20261111	1	1	yes	characteristic
108	C₂	#20261123	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261111

C₂

yes

maximal, complemented

#20261108

C₂

yes

maximal, characteristic, complemented

#20261081

C₂

yes

maximal, complemented

#20260456

C₂

yes

maximal, characteristic, complemented

#493×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095500

—

C₂×#13095496

—

#15690810

C₂²

yes

characteristic

#15690807

C₂²

yes

characteristic

#15690731

—

C₄×#11715895

C₂²

yes

—

C₂×D₈×C₉₇

—

#127×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

#13095500

—

#13095496

—

C₂×#11715895

D₄

yes

—

C₄⋊Dic₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

D₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

C₂²×D₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

Dic₁₉₄

—

C₈×C₉₇

D₈

yes

complemented

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#127

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

C₂×D₈

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#493

yes

characteristic

C₂×C₉₇

#496

yes

—

D₉₇

—

#1721

—

maximal, Sylow

C₉₇

#1721

yes

characteristic, complemented, Sylow

#493

—

#493

D₉₇

yes

characteristic, complemented

#492

—

#486

—

#322

—

C₂×SD₁₆

—

194

—

C₂×D₈

—

C₂×D₈

—

194

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₉₇

yes

—

#102

—

194

—

C₄×C₈

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

194

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

SD₁₆

—

388

—

D₈

—

D₈

—

388

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₄⋊C₄

—

194

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

C₂³

—

388

—

Q₈

—

388

—

D₄

—

D₄

—

D₄

—

D₄

—

388

—

D₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

D₄×D₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₈

—

194

—

C₈

#13095496

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

776

—

100

C₂²

—

776

—

101

C₂²

#15690806

yes

characteristic

102

C₄

—

776

—

103

C₄

C₂×#13095496

yes

—

104

C₄

#15690834

yes

—

105

C₂

—

106

C₂

—

1552

—

107

C₂

#20261111

yes

characteristic

108

C₂

#20261123

yes

—

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#493×C₉₇	C₂
socle	C₂²×C₉₇	#127
hypercenter	#493	D₉₇
hypocenter	C₉₇	#1721
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#15690806

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#493×C₉₇

C₂

C₂²×C₉₇

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20261111	C₂	semidirect
2. #20261108	C₂	semidirect
3. #20261081	C₂	semidirect
4. #20260456	C₂	semidirect
5. #493×C₉₇	C₂	semidirect
39. C₈×C₉₇	D₈	semidirect
55. C₉₇	#1721	semidirect
57. #493	D₉₇	semidirect
95. C₈	#13095496	semidirect

H ≅ G/N

type

1. #20261111

C₂

semidirect

2. #20261108

C₂

semidirect

3. #20261081

C₂

semidirect

4. #20260456

C₂

semidirect

5. #493×C₉₇

C₂

semidirect

39. C₈×C₉₇

D₈

semidirect

55. C₉₇

#1721

semidirect

57. #493

D₉₇

semidirect

95. C₈

#13095496

semidirect

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

subgroup	index
#20261111	2
#20261108	2
#20261081	2
#20260456	2
#493×C₉₇	2
#1721	97

subgroup

index

#493×C₉₇

Subgroups — #27141403

Conjugacy classes of subgroups

Complements

Maximal subgroups