Group #20261322 — subgroups

Subgroups — #20261322

Order 6208.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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	#15690805	C₂	1	2	yes	maximal, complemented
2	#15690796	C₂	1	2	yes	maximal, complemented
3	#15690794	C₂	1	2	yes	maximal, complemented
4	#15690789	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690782	C₂	1	2	yes	maximal, complemented
6	#15690778	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₉₇	C₂	1	2	yes	maximal, complemented
8	C₂×#13095488	C₂	1	1	yes	maximal, characteristic, complemented
9	#121×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
14	C₂²×Dic₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	C₂²	1	2	yes	complemented
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	C₂²⋊Dic₉₇	C₂²	1	2	yes	—
20	C₂²⋊Dic₉₇	C₂²	1	2	yes	complemented
21	#13095488	C₂²	1	2	yes	—
22	#13095488	C₂²	1	2	yes	—
23	#13095488	C₂²	1	2	yes	—
24	#13095488	C₂²	1	2	yes	—
25	C₄×Dic₉₇	C₂²	1	2	yes	complemented
26	C₂×D₄×C₉₇	C₂²	1	2	yes	—
27	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	complemented
31	C₂³×C₉₇	C₂³	1	2	yes	—
32	C₂³×C₉₇	C₂³	1	1	yes	characteristic
33	C₂²×D₉₇	—	2	2	no	—
34	C₂²×D₉₇	—	4	1	no	—
35	C₂²×D₉₇	—	4	1	no	—
36	C₂²×D₉₇	C₂³	1	2	yes	—
37	D₄×C₉₇	—	2	4	no	—
38	D₄×C₉₇	—	4	1	no	—
39	C₂×C₄×C₉₇	—	4	1	no	—
40	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂³	1	2	yes	—
42	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
43	C₂²⋊D₉₇	—	4	2	no	—
44	C₂²⋊D₉₇	—	4	2	no	—
45	C₂×Dic₉₇	—	4	1	no	—
46	C₂×Dic₉₇	—	4	2	no	—
47	C₂×Dic₉₇	C₂³	1	2	yes	—
48	C₂×Dic₉₇	C₂³	1	2	yes	—
49	C₂×Dic₉₇	C₂³	1	2	yes	—
50	Dic₁₉₄	—	4	1	no	—
51	C₂²×C₉₇	—	2	1	no	—
52	C₂²×C₉₇	—	2	1	no	—
53	C₂²×C₉₇	—	2	2	no	—
54	C₂²×C₉₇	—	2	2	no	—
55	C₂²×C₉₇	—	2	2	no	—
56	C₂²×C₉₇	#41	1	2	yes	—
57	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
58	C₂×D₉₇	—	2	2	no	—
59	C₂×D₉₇	—	2	2	no	—
60	C₂×D₉₇	—	2	2	no	—
61	C₂×D₉₇	—	4	4	no	—
62	C₄×C₉₇	—	2	2	no	—
63	C₄×C₉₇	—	2	1	no	—
64	C₄×C₉₇	—	2	1	no	—
65	Dic₉₇	—	2	2	no	—
66	Dic₉₇	—	2	2	no	—
67	Dic₉₇	—	2	2	no	—
68	C₂×C₉₇	—	2	2	no	—
69	C₂×C₉₇	—	4	2	no	—
70	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
73	D₉₇	—	4	2	no	—
74	C₉₇	#538	1	1	yes	characteristic, complemented, Sylow
75	#538	—	97	1	no	maximal, Sylow
76	#126	—	97	2	no	—
77	#124	—	97	2	no	—
78	#123	—	97	2	no	—
79	C₂²⋊Q₈	—	97	1	no	—
80	#121	—	97	2	no	—
81	#121	D₉₇	1	1	yes	characteristic, complemented
82	#120	—	97	2	no	—
83	C₄×D₄	—	97	2	no	—
84	C₂×C₂²⋊C₄	—	97	1	no	—
85	C₂⁴	—	97	1	no	—
86	C₂×Q₈	—	97	1	no	—
87	C₂×D₄	—	97	2	no	—
88	C₂×D₄	—	97	2	no	—
89	C₂×D₄	C₂×D₉₇	1	2	yes	—
90	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic, complemented
91	C₂²×C₄	—	97	1	no	—
92	C₂²×C₄	—	97	2	no	—
93	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
94	C₄⋊C₄	—	97	2	no	—
95	C₄⋊C₄	—	97	2	no	—
96	C₄⋊C₄	—	97	1	no	—
97	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic, complemented
98	C₂²⋊C₄	—	97	2	no	—
99	C₂²⋊C₄	—	97	2	no	—
100	C₂²⋊C₄	—	97	2	no	—
101	C₂²⋊C₄	—	97	2	no	—
102	C₂²⋊C₄	—	97	2	no	—
103	C₂²⋊C₄	—	97	2	no	—
104	C₂²⋊C₄	C₂×D₉₇	1	2	yes	complemented
105	C₄²	—	97	2	no	—
106	C₂³	—	97	2	no	—
107	C₂³	—	388	1	no	—
108	C₂³	—	388	1	no	—
109	C₂³	—	194	2	no	—
110	C₂³	C₂²×D₉₇	1	2	yes	—
111	C₂³	C₂²×D₉₇	1	1	yes	characteristic
112	Q₈	—	388	1	no	—
113	D₄	—	2	4	no	—
114	D₄	—	4	1	no	—
115	D₄	—	388	2	no	—
116	D₄	—	388	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	97	2	no	—
119	C₂×C₄	—	97	2	no	—
120	C₂×C₄	—	97	2	no	—
121	C₂×C₄	—	388	1	no	—
122	C₂×C₄	—	388	2	no	—
123	C₂×C₄	C₂²×D₉₇	1	2	yes	—
124	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	388	4	no	—
132	C₂²	—	194	2	no	—
133	C₂²	—	194	2	no	—
134	C₂²	—	194	2	no	—
135	C₂²	#13095510	1	2	yes	—
136	C₂²	C₂³×D₉₇	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	194	2	no	—
141	C₄	—	194	2	no	—
142	C₄	—	194	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	4	2	no	—
145	C₂	—	388	2	no	—
146	C₂	C₂×#13095510	1	1	yes	characteristic
147	C₂	#15690855	1	1	yes	characteristic
148	C₂	#15690856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690805

C₂

yes

maximal, complemented

#15690796

C₂

yes

maximal, complemented

#15690794

C₂

yes

maximal, complemented

#15690789

C₂

yes

maximal, characteristic, complemented

#15690782

C₂

yes

maximal, complemented

#15690778

C₂

yes

maximal, complemented

D₄×Dic₉₇

C₂

yes

maximal, complemented

C₂×#13095488

C₂

yes

maximal, characteristic, complemented

#121×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂²×Dic₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

C₂²

yes

complemented

C₄⋊Dic₉₇

C₂²

yes

complemented

C₄⋊Dic₉₇

C₂²

yes

characteristic, complemented

#13095490

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

complemented

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

complemented

C₂×D₄×C₉₇

C₂²

yes

—

C₂×D₄×C₉₇

C₂²

yes

characteristic, complemented

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₉₇

C₂²

yes

complemented

C₂³×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂³

yes

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#41

yes

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#41

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

D₉₇

—

C₉₇

#538

yes

characteristic, complemented, Sylow

—

—

—

—

C₂²⋊Q₈

—

#121

—

#121

D₉₇

yes

characteristic, complemented

#120

—

C₄×D₄

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₉₇

yes

—

C₂×D₄

C₂×D₉₇

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₉₇

yes

complemented

105

C₄²

—

106

C₂³

—

107

C₂³

—

388

—

108

C₂³

—

388

—

109

C₂³

—

194

—

110

C₂³

C₂²×D₉₇

yes

—

111

C₂³

C₂²×D₉₇

yes

characteristic

112

Q₈

—

388

—

113

D₄

—

114

D₄

—

115

D₄

—

388

—

116

D₄

—

388

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

388

—

122

C₂×C₄

—

388

—

123

C₂×C₄

C₂²×D₉₇

yes

—

124

C₂×C₄

C₂²×D₉₇

yes

characteristic

125

C₂×C₄

C₂²×D₉₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

388

—

132

C₂²

—

194

—

133

C₂²

—

194

—

134

C₂²

—

194

—

135

C₂²

#13095510

yes

—

136

C₂²

C₂³×D₉₇

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

194

—

141

C₄

—

194

—

142

C₄

—

194

—

143

C₂

—

144

C₂

—

145

C₂

—

388

—

146

C₂

C₂×#13095510

yes

characteristic

147

C₂

#15690855

yes

characteristic

148

C₂

#15690856

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#121×C₉₇	C₂
hypercenter	#121	D₉₇
hypocenter	C₉₇	#538
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

#121×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #15690805	C₂	semidirect
2. #15690796	C₂	semidirect
3. #15690794	C₂	semidirect
4. #15690789	C₂	semidirect
5. #15690782	C₂	semidirect
6. #15690778	C₂	semidirect
7. D₄×Dic₉₇	C₂	semidirect
8. C₂×#13095488	C₂	semidirect
9. #121×C₉₇	C₂	semidirect
11. C₂×Dic₁₉₄	C₂²	semidirect
15. C₂²×Dic₉₇	C₂²	semidirect
16. C₄⋊Dic₉₇	C₂²	semidirect
17. C₄⋊Dic₉₇	C₂²	semidirect
18. #13095490	C₂²	semidirect
20. C₂²⋊Dic₉₇	C₂²	semidirect
25. C₄×Dic₉₇	C₂²	semidirect
27. C₂×D₄×C₉₇	C₂²	semidirect
29. C₄⋊C₄×C₉₇	C₂²	semidirect
30. C₂²⋊C₄×C₉₇	C₂²	semidirect
74. C₉₇	#538	semidirect
81. #121	D₉₇	semidirect
90. C₂×D₄	C₂×D₉₇	semidirect
97. C₄⋊C₄	C₂×D₉₇	semidirect
104. C₂²⋊C₄	C₂×D₉₇	semidirect

H ≅ G/N

type

1. #15690805

C₂

semidirect

2. #15690796

C₂

semidirect

3. #15690794

C₂

semidirect

4. #15690789

C₂

semidirect

5. #15690782

C₂

semidirect

6. #15690778

C₂

semidirect

7. D₄×Dic₉₇

C₂

semidirect

8. C₂×#13095488

C₂

semidirect

9. #121×C₉₇

C₂

semidirect

11. C₂×Dic₁₉₄

C₂²

semidirect

15. C₂²×Dic₉₇

C₂²

semidirect

16. C₄⋊Dic₉₇

C₂²

semidirect

17. C₄⋊Dic₉₇

C₂²

semidirect

18. #13095490

C₂²

semidirect

20. C₂²⋊Dic₉₇

C₂²

semidirect

25. C₄×Dic₉₇

C₂²

semidirect

27. C₂×D₄×C₉₇

C₂²

semidirect

29. C₄⋊C₄×C₉₇

C₂²

semidirect

30. C₂²⋊C₄×C₉₇

C₂²

semidirect

74. C₉₇

#538

semidirect

81. #121

D₉₇

semidirect

90. C₂×D₄

C₂×D₉₇

semidirect

97. C₄⋊C₄

C₂×D₉₇

semidirect

104. C₂²⋊C₄

C₂×D₉₇

semidirect

Subgroups — #20261322

Conjugacy classes of subgroups

Complements