Group #20260880 — subgroups

Subgroups — #20260880

Order 6208.

162 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 47 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×#13095498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13095496	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690784	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690734	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690731	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#11715895	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₉₇	—	2	1	no	—
10	#13095498	—	2	2	no	—
11	#13095496	—	2	2	no	—
12	C₂×#11715895	C₂²	1	1	yes	characteristic
13	C₂×#11715895	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
15	#13095488	—	2	1	no	—
16	#41×C₉₇	—	2	2	no	—
17	#41×C₉₇	—	2	2	no	—
18	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₉₇	—	2	1	no	—
20	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₉₇	—	2	1	no	—
22	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
23	C₂³×C₉₇	—	2	1	no	—
24	C₂³×C₉₇	D₄	1	1	yes	characteristic
25	C₂²×D₉₇	—	2	1	no	—
26	Q₈×C₉₇	—	2	1	no	—
27	Q₈×C₉₇	D₄	1	2	yes	complemented
28	D₄×C₉₇	—	2	4	no	—
29	D₄×C₉₇	—	2	1	no	—
30	D₄×C₉₇	D₄	1	2	yes	complemented
31	C₂×C₄×C₉₇	—	2	4	no	—
32	C₂×C₄×C₉₇	—	2	1	no	—
33	C₂×C₄×C₉₇	—	2	1	no	—
34	C₂×C₄×C₉₇	—	2	1	no	—
35	C₂×C₄×C₉₇	D₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
37	C₂²⋊D₉₇	—	4	2	no	—
38	C₂×Dic₉₇	—	2	1	no	—
39	D₃₈₈	—	2	1	no	—
40	D₃₈₈	—	2	1	no	—
41	#11715895	—	2	1	no	—
42	#11715895	—	2	1	no	—
43	C₂²×C₉₇	—	2	1	no	—
44	C₂²×C₉₇	—	2	2	no	—
45	C₂²×C₉₇	—	2	1	no	—
46	C₂²×C₉₇	—	2	1	no	—
47	C₂²×C₉₇	—	4	1	no	—
48	C₂²×C₉₇	—	4	1	no	—
49	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×D₉₇	—	4	1	no	—
51	C₂×D₉₇	—	4	1	no	—
52	C₂×D₉₇	—	4	1	no	—
53	C₄×C₉₇	—	2	1	no	—
54	C₄×C₉₇	—	2	2	no	—
55	C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
56	C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	Dic₉₇	—	4	1	no	—
58	C₂×C₉₇	—	4	1	no	—
59	C₂×C₉₇	—	4	2	no	—
60	C₂×C₉₇	#120	1	1	yes	characteristic
61	C₂×C₉₇	#135	1	1	yes	characteristic
62	C₂×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	D₉₇	—	8	1	no	—
64	C₉₇	#449	1	1	yes	characteristic, complemented, Sylow
65	#449	—	97	1	no	maximal, Sylow
66	C₂×#41	D₉₇	1	1	yes	characteristic, complemented
67	C₂×SD₁₆	—	97	1	no	—
68	C₂×D₈	—	97	1	no	—
69	#121	—	97	1	no	—
70	#103	—	97	1	no	—
71	#102	—	97	1	no	—
72	C₂²⋊C₈	—	97	1	no	—
73	#41	—	2	2	no	—
74	#41	—	2	2	no	—
75	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
76	C₂×D₄	—	2	1	no	—
77	C₂×D₄	—	97	1	no	—
78	C₂×D₄	—	194	1	no	—
79	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
82	SD₁₆	—	194	2	no	—
83	D₈	—	194	2	no	—
84	C₂×C₈	—	97	1	no	—
85	C₂×C₈	—	97	1	no	—
86	C₄⋊C₄	—	97	1	no	—
87	C₂²⋊C₄	—	194	1	no	—
88	C₂³	—	2	1	no	—
89	C₂³	—	194	1	no	—
90	C₂³	C₂²⋊D₉₇	1	1	yes	characteristic
91	Q₈	—	2	1	no	—
92	Q₈	C₂²⋊D₉₇	1	2	yes	complemented
93	D₄	—	2	4	no	—
94	D₄	—	2	1	no	—
95	D₄	—	388	2	no	—
96	D₄	—	194	1	no	—
97	D₄	—	194	1	no	—
98	D₄	C₂²⋊D₉₇	1	2	yes	complemented
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	194	1	no	—
104	C₂×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
105	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
106	C₈	—	194	1	no	—
107	C₈	—	194	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	388	1	no	—
115	C₂²	—	388	1	no	—
116	C₂²	—	388	1	no	—
117	C₂²	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	2	2	no	—
120	C₄	—	388	1	no	—
121	C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
122	C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	—	4	2	no	—
125	C₂	—	776	1	no	—
126	C₂	#15690779	1	1	yes	characteristic
127	C₂	#15690830	1	1	yes	characteristic
128	C₂	#15690835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13095498

C₂

yes

maximal, characteristic, complemented

C₂×#13095496

C₂

yes

maximal, characteristic, complemented

#15690784

C₂

yes

maximal, characteristic, complemented

#15690734

C₂

yes

maximal, characteristic, complemented

#15690731

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11715895

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₂²⋊D₉₇

—

#13095498

—

#13095496

—

C₂×#11715895

C₂²

yes

characteristic

C₂×#11715895

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

#13095488

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

D₄

yes

characteristic

C₂²×D₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

D₄

yes

complemented

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

D₄

yes

complemented

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

D₄

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

D₃₈₈

—

D₃₈₈

—

#11715895

—

#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

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

characteristic

C₄×C₉₇

C₂×D₄

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

D₉₇

—

C₉₇

#449

yes

characteristic, complemented, Sylow

#449

—

maximal, Sylow

C₂×#41

D₉₇

yes

characteristic, complemented

C₂×SD₁₆

—

C₂×D₈

—

#121

—

#103

—

#102

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₂×D₉₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

SD₁₆

—

194

—

D₈

—

194

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

194

—

C₂³

—

C₂³

—

194

—

C₂³

C₂²⋊D₉₇

yes

characteristic

Q₈

—

Q₈

C₂²⋊D₉₇

yes

complemented

D₄

—

D₄

—

D₄

—

388

—

D₄

—

194

—

D₄

—

194

—

D₄

C₂²⋊D₉₇

yes

complemented

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

194

—

104

C₂×C₄

C₂²⋊D₉₇

yes

characteristic

105

C₂×C₄

C₂²×D₉₇

yes

characteristic

106

C₈

—

194

—

107

C₈

—

194

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

388

—

115

C₂²

—

388

—

116

C₂²

—

388

—

117

C₂²

C₂×C₂²⋊D₉₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

388

—

121

C₄

C₂×C₂²⋊D₉₇

yes

characteristic

122

C₄

C₂×C₂²⋊D₉₇

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

776

—

126

C₂

#15690779

yes

characteristic

127

C₂

#15690830

yes

characteristic

128

C₂

#15690835

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₉₇
Fitting	C₂×#41×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₄
hypercenter	C₂×#41	D₉₇
hypocenter	C₉₇	#449
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂³

Frattini

C₂×C₄

C₂²×D₉₇

Fitting

C₂×#41×C₉₇

C₂

socle

C₂²×C₉₇

C₂×D₄

hypercenter

C₂×#41

D₉₇

hypocenter

C₉₇

#449

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#13095498	C₂	semidirect
2. C₂×#13095496	C₂	semidirect
3. #15690784	C₂	semidirect
4. #15690734	C₂	semidirect
5. #15690731	C₂	semidirect
6. C₂²⋊#11715895	C₂	semidirect
7. C₂×#41×C₉₇	C₂	semidirect
27. Q₈×C₉₇	D₄	semidirect
30. D₄×C₉₇	D₄	semidirect
64. C₉₇	#449	semidirect
66. C₂×#41	D₉₇	semidirect
92. Q₈	C₂²⋊D₉₇	semidirect
98. D₄	C₂²⋊D₉₇	semidirect

H ≅ G/N

type

1. C₂×#13095498

C₂

semidirect

2. C₂×#13095496

C₂

semidirect

3. #15690784

C₂

semidirect

4. #15690734

C₂

semidirect

5. #15690731

C₂

semidirect

6. C₂²⋊#11715895

C₂

semidirect

7. C₂×#41×C₉₇

C₂

semidirect

27. Q₈×C₉₇

D₄

semidirect

30. D₄×C₉₇

D₄

semidirect

64. C₉₇

#449

semidirect

66. C₂×#41

D₉₇

semidirect

92. Q₈

C₂²⋊D₉₇

semidirect

98. D₄

C₂²⋊D₉₇

semidirect

Subgroups — #20260880

Conjugacy classes of subgroups

Complements