Group #20292550 — subgroups

Subgroups — #20292550

Order 6848.

186 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 63 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×C₄⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Dic₁₀₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₁₀₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₁₀₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₇	—	2	2	no	—
13	Dic₁₀₇⋊C₄	—	2	2	no	—
14	Dic₁₀₇⋊C₄	—	2	2	no	—
15	C₂²⋊Dic₁₀₇	—	2	2	no	—
16	C₂²⋊Dic₁₀₇	—	2	2	no	—
17	C₂⁴×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
21	C₂³×C₁₀₇	—	2	1	no	—
22	C₂³×C₁₀₇	—	2	1	no	—
23	C₂³×C₁₀₇	—	2	1	no	—
24	C₂³×C₁₀₇	—	2	1	no	—
25	C₂³×C₁₀₇	—	2	1	no	—
26	C₂³×C₁₀₇	—	2	1	no	—
27	C₂³×C₁₀₇	Q₈	1	2	yes	—
28	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	—	2	1	no	—
30	C₂×C₄×C₁₀₇	—	2	1	no	—
31	C₂×C₄×C₁₀₇	—	2	1	no	—
32	C₂×C₄×C₁₀₇	—	2	1	no	—
33	C₂×C₄×C₁₀₇	—	2	1	no	—
34	C₂×C₄×C₁₀₇	D₄	1	2	yes	—
35	C₂×Dic₁₀₇	—	2	1	no	—
36	C₂×Dic₁₀₇	—	2	1	no	—
37	C₂×Dic₁₀₇	—	2	1	no	—
38	C₂×Dic₁₀₇	—	2	1	no	—
39	C₂×Dic₁₀₇	—	2	1	no	—
40	C₂×Dic₁₀₇	—	2	1	no	—
41	C₂×Dic₁₀₇	—	2	1	no	—
42	C₂×Dic₁₀₇	—	2	1	no	—
43	C₂×Dic₁₀₇	—	2	1	no	—
44	C₂×Dic₁₀₇	—	2	1	no	—
45	C₂×Dic₁₀₇	D₄	1	2	yes	—
46	C₂×Dic₁₀₇	D₄	1	2	yes	—
47	C₂²×C₁₀₇	—	2	2	no	—
48	C₂²×C₁₀₇	—	2	2	no	—
49	C₂²×C₁₀₇	—	2	2	no	—
50	C₂²×C₁₀₇	—	4	1	no	—
51	C₂²×C₁₀₇	—	4	1	no	—
52	C₂²×C₁₀₇	—	4	1	no	—
53	C₂²×C₁₀₇	—	4	1	no	—
54	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₁₀₇	#41	1	1	yes	characteristic
59	C₂²×C₁₀₇	#41	1	1	yes	characteristic
60	C₂²×C₁₀₇	#41	1	1	yes	characteristic
61	C₄×C₁₀₇	—	2	2	no	—
62	C₄×C₁₀₇	—	4	1	no	—
63	Dic₁₀₇	—	2	2	no	—
64	Dic₁₀₇	—	2	2	no	—
65	Dic₁₀₇	—	4	1	no	—
66	Dic₁₀₇	—	4	1	no	—
67	C₂×C₁₀₇	—	4	2	no	—
68	C₂×C₁₀₇	#121	1	1	yes	characteristic
69	C₂×C₁₀₇	#121	1	1	yes	characteristic
70	C₂×C₁₀₇	#121	1	1	yes	characteristic
71	C₂×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₁₀₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₀₇	#126	1	1	yes	characteristic
75	C₁₀₇	#396	1	1	yes	characteristic, complemented, Sylow
76	#396	—	107	1	no	maximal, Sylow
77	C₂×C₄⋊C₄	—	107	1	no	—
78	C₂×C₄⋊C₄	—	107	1	no	—
79	C₂×C₄⋊C₄	—	107	1	no	—
80	C₂×C₂²⋊C₄	—	107	1	no	—
81	C₂×C₂²⋊C₄	—	107	1	no	—
82	C₂×C₂²⋊C₄	D₁₀₇	1	1	yes	characteristic
83	#95	—	107	1	no	—
84	C₂⁴	C₂×D₁₀₇	1	1	yes	characteristic
85	C₂²×C₄	—	107	1	no	—
86	C₂²×C₄	—	107	1	no	—
87	C₂²×C₄	—	107	1	no	—
88	C₂²×C₄	—	107	1	no	—
89	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
91	C₄⋊C₄	—	214	2	no	—
92	C₄⋊C₄	—	214	2	no	—
93	C₄⋊C₄	—	214	2	no	—
94	C₂²⋊C₄	—	2	2	no	—
95	C₂²⋊C₄	—	214	2	no	—
96	C₂²⋊C₄	—	214	2	no	—
97	C₂³	—	2	1	no	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	1	no	—
101	C₂³	—	2	1	no	—
102	C₂³	—	2	1	no	—
103	C₂³	Dic₂₁₄	1	2	yes	—
104	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	107	2	no	—
111	C₂×C₄	—	107	2	no	—
112	C₂×C₄	—	214	1	no	—
113	C₂×C₄	—	214	1	no	—
114	C₂×C₄	—	214	1	no	—
115	C₂×C₄	—	214	1	no	—
116	C₂×C₄	—	214	1	no	—
117	C₂×C₄	—	214	1	no	—
118	C₂×C₄	—	214	1	no	—
119	C₂×C₄	—	214	1	no	—
120	C₂×C₄	—	214	1	no	—
121	C₂×C₄	—	214	1	no	—
122	C₂×C₄	C₂²⋊D₁₀₇	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	D₄×D₁₀₇	1	1	yes	characteristic
131	C₂²	D₄×D₁₀₇	1	1	yes	characteristic
132	C₂²	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
133	C₂²	C₂×Dic₂₁₄	1	1	yes	characteristic
134	C₂²	#13133817	1	1	yes	characteristic
135	C₂²	#13133817	1	1	yes	characteristic
136	C₂²	#13133818	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	4	1	no	—
139	C₄	—	428	1	no	—
140	C₄	—	428	1	no	—
141	C₄	—	214	2	no	—
142	C₄	—	214	2	no	—
143	C₂	—	4	2	no	—
144	C₂	#16812080	1	1	yes	characteristic
145	C₂	#16812081	1	1	yes	characteristic
146	C₂	#16812082	1	1	yes	characteristic
147	C₂	#16812086	1	1	yes	characteristic
148	C₂	#16812086	1	1	yes	characteristic
149	C₂	#16812088	1	1	yes	characteristic
150	C₂	#16812102	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₇⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₇⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₀₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₀₇

C₂

yes

maximal, characteristic

C₁₀₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

C₂²

yes

characteristic

C₂²×Dic₁₀₇

C₂²

yes

characteristic

C₄⋊Dic₁₀₇

—

Dic₁₀₇⋊C₄

—

Dic₁₀₇⋊C₄

—

C₂²⋊Dic₁₀₇

—

C₂²⋊Dic₁₀₇

—

C₂⁴×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

Q₈

yes

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

D₄

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

D₄

yes

—

C₂×Dic₁₀₇

D₄

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×C₁₀₇

C₂×D₄

yes

characteristic

C₂²×C₁₀₇

C₂×Q₈

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₄×C₁₀₇

—

C₄×C₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₇

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₇

C₂²⋊Q₈

yes

C₂×C₁₀₇

yes

C₁₀₇

yes

characteristic, complemented, Sylow

#396

—

107

maximal, Sylow

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₂²⋊C₄

—

107

—

C₂×C₂²⋊C₄

—

107

—

C₂×C₂²⋊C₄

D₁₀₇

yes

characteristic

#95

—

107

—

C₂⁴

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₂²⋊C₄

—

C₂²⋊C₄

—

214

—

C₂²⋊C₄

—

214

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₂₁₄

yes

—

104

C₂³

C₂²×D₁₀₇

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

107

—

111

C₂×C₄

—

107

—

112

C₂×C₄

—

214

—

113

C₂×C₄

—

214

—

114

C₂×C₄

—

214

—

115

C₂×C₄

—

214

—

116

C₂×C₄

—

214

—

117

C₂×C₄

—

214

—

118

C₂×C₄

—

214

—

119

C₂×C₄

—

214

—

120

C₂×C₄

—

214

—

121

C₂×C₄

—

214

—

122

C₂×C₄

C₂²⋊D₁₀₇

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

D₄×D₁₀₇

yes

characteristic

131

C₂²

D₄×D₁₀₇

yes

characteristic

132

C₂²

C₂×C₂²⋊D₁₀₇

yes

characteristic

133

C₂²

C₂×Dic₂₁₄

yes

characteristic

134

C₂²

#13133817

yes

characteristic

135

C₂²

#13133817

yes

characteristic

136

C₂²

#13133818

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

428

—

140

C₄

—

428

—

141

C₄

—

214

—

142

C₄

—

214

—

143

C₂

—

144

C₂

#16812080

yes

characteristic

145

C₂

#16812081

yes

characteristic

146

C₂

#16812082

yes

characteristic

147

C₂

#16812086

yes

characteristic

148

C₂

#16812086

yes

characteristic

149

C₂

#16812088

yes

characteristic

150

C₂

#16812102

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₇
commutator, socle	C₂³×C₁₀₇	C₂³
Fitting	C₂×C₂²⋊C₄×C₁₀₇	C₂
hypercenter	C₂×C₂²⋊C₄	D₁₀₇
hypocenter	C₁₀₇	#396
LCS	G ▷ C₂³×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₇

commutator, socle

C₂³×C₁₀₇

C₂³

Fitting

C₂×C₂²⋊C₄×C₁₀₇

C₂

hypercenter

C₂×C₂²⋊C₄

D₁₀₇

C₁₀₇

G ▷ C₂³×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₀₇	C₂	semidirect
2. C₂×Dic₁₀₇⋊C₄	C₂	semidirect
3. C₂×Dic₁₀₇⋊C₄	C₂	semidirect
6. C₁₀₇⋊#95	C₂	semidirect
75. C₁₀₇	#396	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₁₀₇

C₂

semidirect

2. C₂×Dic₁₀₇⋊C₄

C₂

semidirect

3. C₂×Dic₁₀₇⋊C₄

C₂

semidirect

6. C₁₀₇⋊#95

C₂

semidirect

75. C₁₀₇

#396

semidirect

Subgroups — #20292550

Conjugacy classes of subgroups

Complements