Group #20260928 — subgroups

Subgroups — #20260928

Order 6208.

140 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 43 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	C₂×#13095496	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690806	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690739	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690732	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690731	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₉₇	—	2	1	no	—
10	C₂×C₂²⋊D₉₇	—	2	1	no	—
11	C₂²×Dic₉₇	—	2	1	no	—
12	#13095496	—	2	2	no	—
13	C₂×#11715895	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
15	C₂²⋊Dic₉₇	—	2	1	no	—
16	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
19	D₈×C₉₇	—	2	2	no	—
20	C₂×C₈×C₉₇	C₂²	1	1	yes	characteristic
21	C₂³×C₉₇	—	2	1	no	—
22	C₂³×C₉₇	—	2	1	no	—
23	C₂²×D₉₇	—	2	1	no	—
24	D₄×C₉₇	—	2	1	no	—
25	D₄×C₉₇	—	2	1	no	—
26	D₄×C₉₇	—	2	1	no	—
27	D₄×C₉₇	D₄	1	2	yes	complemented
28	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₂²⋊D₉₇	—	4	2	no	—
30	C₂²⋊D₉₇	—	4	2	no	—
31	C₂×Dic₉₇	—	2	1	no	—
32	C₂×Dic₉₇	—	4	2	no	—
33	C₂×Dic₉₇	D₄	1	1	yes	characteristic
34	C₂×Dic₉₇	D₄	1	1	yes	characteristic
35	D₃₈₈	—	2	1	no	—
36	D₃₈₈	—	2	1	no	—
37	C₈×C₉₇	—	2	1	no	—
38	#11715895	—	2	1	no	—
39	C₂²×C₉₇	—	2	2	no	—
40	C₂²×C₉₇	—	4	1	no	—
41	C₂²×C₉₇	—	4	1	no	—
42	C₂²×C₉₇	—	4	1	no	—
43	C₂²×C₉₇	—	4	1	no	—
44	C₂²×C₉₇	—	4	1	no	—
45	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×D₉₇	—	4	1	no	—
47	C₂×D₉₇	—	4	1	no	—
48	C₂×D₉₇	—	4	1	no	—
49	C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
50	C₄×C₉₇	#41	1	1	yes	characteristic
51	Dic₉₇	—	2	1	no	—
52	Dic₉₇	—	4	1	no	—
53	Dic₉₇	D₈	1	2	yes	complemented
54	C₂×C₉₇	—	4	2	no	—
55	C₂×C₉₇	—	8	1	no	—
56	C₂×C₉₇	#121	1	1	yes	characteristic
57	C₂×C₉₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	D₉₇	—	8	1	no	—
60	C₉₇	#459	1	1	yes	characteristic, complemented, Sylow
61	#459	—	97	1	no	maximal, Sylow
62	C₂×D₈	—	97	1	no	—
63	C₂×D₈	D₉₇	1	1	yes	characteristic, complemented
64	#127	—	97	1	no	—
65	C₄×D₄	—	97	1	no	—
66	C₄⋊C₈	—	97	1	no	—
67	#102	—	97	1	no	—
68	#102	—	97	1	no	—
69	C₂×D₄	—	97	1	no	—
70	C₂×D₄	—	194	1	no	—
71	C₂×D₄	—	194	1	no	—
72	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
73	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
74	C₂²×C₄	—	194	1	no	—
75	D₈	—	2	2	no	—
76	D₈	—	194	2	no	—
77	C₂×C₈	—	97	1	no	—
78	C₂×C₈	C₂×D₉₇	1	1	yes	characteristic
79	C₄⋊C₄	—	97	1	no	—
80	C₂²⋊C₄	—	194	1	no	—
81	C₄²	—	97	1	no	—
82	C₂³	—	2	1	no	—
83	C₂³	—	2	1	no	—
84	C₂³	—	194	1	no	—
85	D₄	—	2	1	no	—
86	D₄	—	2	1	no	—
87	D₄	—	2	1	no	—
88	D₄	—	388	2	no	—
89	D₄	—	388	2	no	—
90	D₄	—	194	1	no	—
91	D₄	—	194	1	no	—
92	D₄	C₂²⋊D₉₇	1	2	yes	complemented
93	C₂×C₄	—	97	1	no	—
94	C₂×C₄	—	97	1	no	—
95	C₂×C₄	—	388	2	no	—
96	C₂×C₄	—	194	1	no	—
97	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	194	1	no	—
100	C₂²	—	2	2	no	—
101	C₂²	—	4	1	no	—
102	C₂²	—	4	1	no	—
103	C₂²	—	4	1	no	—
104	C₂²	—	4	1	no	—
105	C₂²	—	4	1	no	—
106	C₂²	—	388	1	no	—
107	C₂²	—	388	1	no	—
108	C₂²	—	388	1	no	—
109	C₂²	D₄×D₉₇	1	1	yes	characteristic
110	C₄	—	97	2	no	—
111	C₄	—	388	1	no	—
112	C₄	—	194	1	no	—
113	C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
114	C₄	#13095510	1	1	yes	characteristic
115	C₂	—	4	2	no	—
116	C₂	—	8	1	no	—
117	C₂	—	776	1	no	—
118	C₂	#15690782	1	1	yes	characteristic
119	C₂	D₈×D₉₇	1	1	yes	characteristic
120	C₂	#15690834	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13095496

C₂

yes

maximal, characteristic, complemented

#15690806

C₂

yes

maximal, characteristic, complemented

D₄×Dic₉₇

C₂

yes

maximal, characteristic, complemented

#15690739

C₂

yes

maximal, characteristic, complemented

#15690732

C₂

yes

maximal, characteristic, complemented

#15690731

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₂²⋊D₉₇

—

C₂×C₂²⋊D₉₇

—

C₂²×Dic₉₇

—

#13095496

—

C₂×#11715895

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

C₂²

yes

characteristic

D₈×C₉₇

—

C₂×C₈×C₉₇

C₂²

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂²×D₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

D₄

yes

complemented

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

D₄

yes

characteristic

C₂×Dic₉₇

D₄

yes

characteristic

D₃₈₈

—

D₃₈₈

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

C₂×D₄

yes

characteristic

C₄×C₉₇

#41

yes

characteristic

Dic₉₇

—

Dic₉₇

—

Dic₉₇

D₈

yes

complemented

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#121

yes

characteristic

C₂×C₉₇

C₂×D₈

yes

characteristic

C₂×C₉₇

AGL(1,ℤ/8)

yes

characteristic

D₉₇

—

C₉₇

#459

yes

characteristic, complemented, Sylow

#459

—

maximal, Sylow

C₂×D₈

—

C₂×D₈

D₉₇

yes

characteristic, complemented

#127

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

#102

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

194

—

D₈

—

D₈

—

194

—

C₂×C₈

—

C₂×C₈

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

194

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

194

—

D₄

—

D₄

—

D₄

—

D₄

—

388

—

D₄

—

388

—

D₄

—

194

—

D₄

—

194

—

D₄

C₂²⋊D₉₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₈

—

C₈

—

194

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

388

—

107

C₂²

—

388

—

108

C₂²

—

388

—

109

C₂²

D₄×D₉₇

yes

characteristic

110

C₄

—

111

C₄

—

388

—

112

C₄

—

194

—

113

C₄

C₂×C₂²⋊D₉₇

yes

characteristic

114

C₄

#13095510

yes

characteristic

115

C₂

—

116

C₂

—

117

C₂

—

776

—

118

C₂

#15690782

yes

characteristic

119

C₂

D₈×D₉₇

yes

characteristic

120

C₂

#15690834

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₉₇
Fitting	C₂×D₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₄
hypercenter	C₂×D₈	D₉₇
hypocenter	C₉₇	#459
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂³

Frattini

C₂×C₄

C₂²×D₉₇

Fitting

C₂×D₈×C₉₇

C₂

socle

C₂²×C₉₇

C₂×D₄

hypercenter

C₂×D₈

D₉₇

hypocenter

C₉₇

#459

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. C₂×#13095496	C₂	semidirect
2. #15690806	C₂	semidirect
3. D₄×Dic₉₇	C₂	semidirect
4. #15690739	C₂	semidirect
5. #15690732	C₂	semidirect
6. #15690731	C₂	semidirect
7. C₂×D₈×C₉₇	C₂	semidirect
27. D₄×C₉₇	D₄	semidirect
53. Dic₉₇	D₈	semidirect
60. C₉₇	#459	semidirect
63. C₂×D₈	D₉₇	semidirect
92. D₄	C₂²⋊D₉₇	semidirect

H ≅ G/N

type

1. C₂×#13095496

C₂

semidirect

2. #15690806

C₂

semidirect

3. D₄×Dic₉₇

C₂

semidirect

4. #15690739

C₂

semidirect

5. #15690732

C₂

semidirect

6. #15690731

C₂

semidirect

7. C₂×D₈×C₉₇

C₂

semidirect

27. D₄×C₉₇

D₄

semidirect

53. Dic₉₇

D₈

semidirect

60. C₉₇

#459

semidirect

63. C₂×D₈

D₉₇

semidirect

92. D₄

C₂²⋊D₉₇

semidirect

Subgroups — #20260928

Conjugacy classes of subgroups

Complements