Group #20261360 — subgroups

Subgroups — #20261360

Order 6208.

242 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 103 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	#15690790	C₂	1	2	yes	maximal, complemented
2	#15690789	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690788	C₂	1	2	yes	maximal, complemented
4	#15690788	C₂	1	2	yes	maximal, complemented
5	#15690787	C₂	1	2	yes	maximal, complemented
6	#15690786	C₂	1	2	yes	maximal, complemented
7	C₂×#13095488	C₂	1	1	yes	maximal, characteristic
8	C₂²⋊C₄×D₉₇	C₂	1	2	yes	maximal
9	C₂²⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	C₂²	1	2	yes	complemented
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic, complemented
13	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₉₇	C₂²	1	2	yes	—
15	C₂×C₄×D₉₇	C₂²	1	2	yes	—
16	C₄⋊Dic₉₇	C₂²	1	2	yes	complemented
17	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic, complemented
18	#13095490	C₂²	1	2	yes	complemented
19	#13095490	C₂²	1	2	yes	complemented
20	#13095490	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
22	#13095488	C₂²	1	2	yes	—
23	#13095488	C₂²	1	2	yes	—
24	#13095488	C₂²	1	2	yes	—
25	#13095488	C₂²	1	2	yes	—
26	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
29	C₄⋊C₄×C₉₇	C₂²	1	2	yes	complemented
30	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
31	C₂³×C₉₇	C₂³	1	1	yes	characteristic
32	C₂²×D₉₇	—	4	1	no	—
33	C₂²×D₉₇	—	4	1	no	—
34	C₂²×D₉₇	Q₈	1	4	yes	complemented
35	C₂²×D₉₇	C₂³	1	2	yes	—
36	Q₈×C₉₇	—	4	1	no	—
37	C₂×C₄×C₉₇	—	4	1	no	—
38	C₂×C₄×C₉₇	C₂³	1	2	yes	—
39	C₂×C₄×C₉₇	C₂³	1	2	yes	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₉₇	—	4	1	no	—
43	C₂×Dic₉₇	C₂³	1	2	yes	—
44	C₂×Dic₉₇	C₂³	1	2	yes	—
45	C₂×Dic₉₇	C₂³	1	2	yes	—
46	C₄×D₉₇	—	4	2	no	—
47	C₄×D₉₇	—	4	2	no	—
48	Dic₁₉₄	—	4	1	no	—
49	Dic₁₉₄	—	4	2	no	—
50	C₂²×C₉₇	—	2	1	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
54	C₂×D₉₇	—	2	2	no	—
55	C₂×D₉₇	—	2	2	no	—
56	C₂×D₉₇	—	4	4	no	—
57	C₂×D₉₇	C₂×Q₈	1	4	yes	—
58	C₄×C₉₇	—	2	2	no	—
59	C₄×C₉₇	—	2	1	no	—
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	—	2	1	no	—
62	Dic₉₇	—	2	2	no	—
63	Dic₉₇	—	2	2	no	—
64	Dic₉₇	—	2	2	no	—
65	C₂×C₉₇	—	2	2	no	—
66	C₂×C₉₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
68	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
69	D₉₇	—	2	4	no	—
70	C₉₇	#543	1	1	yes	characteristic, complemented, Sylow
71	#543	—	97	1	no	maximal, Sylow
72	C₂²⋊Q₈	—	97	2	no	—
73	C₂²⋊Q₈	—	97	1	no	—
74	C₂²⋊Q₈	—	97	2	no	—
75	C₂²⋊Q₈	—	97	2	no	—
76	C₂²⋊Q₈	—	97	2	no	—
77	C₂²⋊Q₈	—	97	2	no	—
78	C₂²⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
79	C₂×C₂²⋊C₄	—	97	1	no	—
80	C₂×C₂²⋊C₄	—	97	2	no	—
81	C₂⁴	—	97	1	no	—
82	C₂×Q₈	—	97	1	no	—
83	C₂×Q₈	—	97	2	no	—
84	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
85	C₂²×C₄	—	97	1	no	—
86	C₂²×C₄	—	97	2	no	—
87	C₂²×C₄	—	97	2	no	—
88	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
89	C₄⋊C₄	—	97	2	no	—
90	C₄⋊C₄	—	97	2	no	—
91	C₄⋊C₄	—	97	2	no	—
92	C₄⋊C₄	—	97	1	no	—
93	C₄⋊C₄	—	97	2	no	—
94	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
95	C₄⋊C₄	C₂×D₉₇	1	2	yes	complemented
96	C₂²⋊C₄	—	97	2	no	—
97	C₂²⋊C₄	—	97	2	no	—
98	C₂²⋊C₄	—	97	2	no	—
99	C₂²⋊C₄	—	97	2	no	—
100	C₂²⋊C₄	—	97	2	no	—
101	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
102	C₂³	—	97	4	no	—
103	C₂³	—	97	2	no	—
104	C₂³	—	388	1	no	—
105	C₂³	—	388	1	no	—
106	C₂³	C₂²×D₉₇	1	1	yes	characteristic
107	Q₈	—	4	1	no	—
108	Q₈	—	388	1	no	—
109	Q₈	—	388	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	97	2	no	—
112	C₂×C₄	—	97	2	no	—
113	C₂×C₄	—	97	2	no	—
114	C₂×C₄	—	388	2	no	—
115	C₂×C₄	—	388	2	no	—
116	C₂×C₄	—	388	1	no	—
117	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₉₇	1	2	yes	—
120	C₂×C₄	C₂²×D₉₇	1	2	yes	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	97	4	no	—
124	C₂²	—	388	4	no	—
125	C₂²	—	194	2	no	—
126	C₂²	—	194	2	no	—
127	C₂²	Q₈×D₉₇	1	2	yes	—
128	C₂²	C₂³×D₉₇	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	194	2	no	—
134	C₄	—	194	2	no	—
135	C₄	—	194	2	no	—
136	C₂	—	2	2	no	—
137	C₂	—	194	4	no	—
138	C₂	C₂×Q₈×D₉₇	1	1	yes	characteristic
139	C₂	#15690855	1	1	yes	characteristic
140	C₂	#15690856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690790

C₂

yes

maximal, complemented

#15690789

C₂

yes

maximal, characteristic, complemented

#15690788

C₂

yes

maximal, complemented

#15690788

C₂

yes

maximal, complemented

#15690787

C₂

yes

maximal, complemented

#15690786

C₂

yes

maximal, complemented

C₂×#13095488

C₂

yes

maximal, characteristic

C₂²⋊C₄×D₉₇

C₂

yes

maximal

C₂²⋊Q₈×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂²

yes

complemented

C₂×Dic₁₉₄

C₂²

yes

characteristic, complemented

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

complemented

C₄⋊Dic₉₇

C₂²

yes

characteristic, complemented

#13095490

C₂²

yes

complemented

#13095490

C₂²

yes

complemented

#13095490

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

C₂×Q₈×C₉₇

C₂²

yes

characteristic, complemented

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₉₇

C₂²

yes

complemented

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

Q₈

yes

complemented

C₂²×D₉₇

C₂³

yes

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×Q₈

yes

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₂×Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×Q₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

D₉₇

—

C₉₇

#543

yes

characteristic, complemented, Sylow

#543

—

maximal, Sylow

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₉₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₉₇

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic, complemented

C₄⋊C₄

C₂×D₉₇

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

C₂×D₉₇

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

388

—

105

C₂³

—

388

—

106

C₂³

C₂²×D₉₇

yes

characteristic

107

Q₈

—

108

Q₈

—

388

—

109

Q₈

—

388

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

388

—

115

C₂×C₄

—

388

—

116

C₂×C₄

—

388

—

117

C₂×C₄

C₂²×D₉₇

yes

characteristic

118

C₂×C₄

C₂²×D₉₇

yes

characteristic

119

C₂×C₄

C₂²×D₉₇

yes

—

120

C₂×C₄

C₂²×D₉₇

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

388

—

125

C₂²

—

194

—

126

C₂²

—

194

—

127

C₂²

Q₈×D₉₇

yes

—

128

C₂²

C₂³×D₉₇

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

194

—

134

C₄

—

194

—

135

C₄

—

194

—

136

C₂

—

137

C₂

—

194

—

138

C₂

C₂×Q₈×D₉₇

yes

characteristic

139

C₂

#15690855

yes

characteristic

140

C₂

#15690856

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₂²⋊Q₈×C₉₇	C₂
hypercenter	C₂²⋊Q₈	D₉₇
hypocenter	C₉₇	#543
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

C₂²⋊Q₈×C₉₇

C₂

hypercenter

C₂²⋊Q₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #15690790	C₂	semidirect
2. #15690789	C₂	semidirect
3. #15690788	C₂	semidirect
4. #15690788	C₂	semidirect
5. #15690787	C₂	semidirect
6. #15690786	C₂	semidirect
9. C₂²⋊Q₈×C₉₇	C₂	semidirect
11. C₂×Dic₁₉₄	C₂²	semidirect
12. C₂×Dic₁₉₄	C₂²	semidirect
16. C₄⋊Dic₉₇	C₂²	semidirect
17. C₄⋊Dic₉₇	C₂²	semidirect
18. #13095490	C₂²	semidirect
19. #13095490	C₂²	semidirect
20. #13095490	C₂²	semidirect
26. C₂×Q₈×C₉₇	C₂²	semidirect
28. C₄⋊C₄×C₉₇	C₂²	semidirect
29. C₄⋊C₄×C₉₇	C₂²	semidirect
34. C₂²×D₉₇	Q₈	semidirect
70. C₉₇	#543	semidirect
78. C₂²⋊Q₈	D₉₇	semidirect
84. C₂×Q₈	C₂×D₉₇	semidirect
94. C₄⋊C₄	C₂×D₉₇	semidirect
95. C₄⋊C₄	C₂×D₉₇	semidirect

H ≅ G/N

type

1. #15690790

C₂

semidirect

2. #15690789

C₂

semidirect

3. #15690788

C₂

semidirect

4. #15690788

C₂

semidirect

5. #15690787

C₂

semidirect

6. #15690786

C₂

semidirect

9. C₂²⋊Q₈×C₉₇

C₂

semidirect

11. C₂×Dic₁₉₄

C₂²

semidirect

12. C₂×Dic₁₉₄

C₂²

semidirect

16. C₄⋊Dic₉₇

C₂²

semidirect

17. C₄⋊Dic₉₇

C₂²

semidirect

18. #13095490

C₂²

semidirect

19. #13095490

C₂²

semidirect

20. #13095490

C₂²

semidirect

26. C₂×Q₈×C₉₇

C₂²

semidirect

28. C₄⋊C₄×C₉₇

C₂²

semidirect

29. C₄⋊C₄×C₉₇

C₂²

semidirect

34. C₂²×D₉₇

Q₈

semidirect

70. C₉₇

#543

semidirect

78. C₂²⋊Q₈

D₉₇

semidirect

84. C₂×Q₈

C₂×D₉₇

semidirect

94. C₄⋊C₄

C₂×D₉₇

semidirect

95. C₄⋊C₄

C₂×D₉₇

semidirect

Subgroups — #20261360

Conjugacy classes of subgroups

Complements