Group #20294954 — subgroups

Subgroups — #20294954

Order 6976.

170 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 57 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×#13182008	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812606	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812606	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812606	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	#13182008	—	2	2	no	—
13	C₂²⋊Dic₁₀₉	—	2	2	no	—
14	C₂²⋊Dic₁₀₉	—	2	2	no	—
15	C₂⁴×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
19	C₂³×C₁₀₉	—	2	1	no	—
20	C₂³×C₁₀₉	—	2	1	no	—
21	C₂³×C₁₀₉	—	2	1	no	—
22	C₂³×C₁₀₉	—	2	1	no	—
23	C₂³×C₁₀₉	—	2	1	no	—
24	C₂³×C₁₀₉	—	2	1	no	—
25	C₂³×C₁₀₉	D₄	1	2	yes	—
26	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×Dic₁₀₉	—	2	1	no	—
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	—	2	1	no	—
36	C₂×Dic₁₀₉	—	2	1	no	—
37	C₂×Dic₁₀₉	—	2	1	no	—
38	C₂×Dic₁₀₉	—	2	1	no	—
39	C₂×Dic₁₀₉	—	2	1	no	—
40	C₂×Dic₁₀₉	—	2	1	no	—
41	C₂×Dic₁₀₉	—	2	1	no	—
42	C₂×Dic₁₀₉	—	2	1	no	—
43	C₂×Dic₁₀₉	—	2	1	no	—
44	C₂×Dic₁₀₉	D₄	1	2	yes	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	2	2	no	—
47	C₂²×C₁₀₉	—	2	2	no	—
48	C₂²×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	—	4	1	no	—
50	C₂²×C₁₀₉	—	4	1	no	—
51	C₂²×C₁₀₉	—	4	1	no	—
52	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₁₀₉	#41	1	1	yes	characteristic
55	C₂²×C₁₀₉	#41	1	1	yes	characteristic
56	C₂²×C₁₀₉	#41	1	1	yes	characteristic
57	C₂²×C₁₀₉	#41	1	1	yes	characteristic
58	C₂²×C₁₀₉	#41	1	1	yes	characteristic
59	C₄×C₁₀₉	—	4	1	no	—
60	C₄×C₁₀₉	—	4	1	no	—
61	Dic₁₀₉	—	2	2	no	—
62	Dic₁₀₉	—	4	1	no	—
63	Dic₁₀₉	—	4	1	no	—
64	Dic₁₀₉	—	4	1	no	—
65	C₂×C₁₀₉	—	4	2	no	—
66	C₂×C₁₀₉	#121	1	1	yes	characteristic
67	C₂×C₁₀₉	#123	1	1	yes	characteristic
68	C₂×C₁₀₉	#123	1	1	yes	characteristic
69	C₂×C₁₀₉	#123	1	1	yes	characteristic
70	C₂×C₁₀₉	#124	1	1	yes	characteristic
71	C₂×C₁₀₉	#126	1	1	yes	characteristic
72	C₂×C₁₀₉	#126	1	1	yes	characteristic
73	C₁₀₉	#397	1	1	yes	characteristic, complemented, Sylow
74	#397	—	109	1	no	maximal, Sylow
75	C₂×C₄⋊C₄	—	109	1	no	—
76	C₂×C₂²⋊C₄	—	109	1	no	—
77	C₂×C₂²⋊C₄	—	109	1	no	—
78	C₂×C₂²⋊C₄	D₁₀₉	1	1	yes	characteristic
79	#95	—	109	1	no	—
80	#95	—	109	1	no	—
81	#95	—	109	1	no	—
82	C₂⁴	C₂×D₁₀₉	1	1	yes	characteristic
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
89	C₄⋊C₄	—	218	2	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₂²⋊C₄	—	218	2	no	—
92	C₂²⋊C₄	—	218	2	no	—
93	C₂³	—	2	1	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	2	1	no	—
97	C₂³	—	2	1	no	—
98	C₂³	—	2	1	no	—
99	C₂³	C₂²⋊D₁₀₉	1	2	yes	—
100	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	109	2	no	—
108	C₂×C₄	—	218	1	no	—
109	C₂×C₄	—	218	1	no	—
110	C₂×C₄	—	218	1	no	—
111	C₂×C₄	—	218	1	no	—
112	C₂×C₄	—	218	1	no	—
113	C₂×C₄	—	218	1	no	—
114	C₂×C₄	—	218	1	no	—
115	C₂×C₄	—	218	1	no	—
116	C₂×C₄	—	218	1	no	—
117	C₂×C₄	—	218	1	no	—
118	C₂×C₄	—	218	1	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
127	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
128	C₂²	#13182028	1	1	yes	characteristic
129	C₂²	#13182028	1	1	yes	characteristic
130	C₂²	#13182028	1	1	yes	characteristic
131	C₂²	#13182029	1	1	yes	characteristic
132	C₂²	#13182029	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	436	1	no	—
136	C₄	—	436	1	no	—
137	C₄	—	436	1	no	—
138	C₄	—	218	2	no	—
139	C₂	—	4	2	no	—
140	C₂	#16812671	1	1	yes	characteristic
141	C₂	#16812680	1	1	yes	characteristic
142	C₂	#16812681	1	1	yes	characteristic
143	C₂	#16812682	1	1	yes	characteristic
144	C₂	#16812685	1	1	yes	characteristic
145	C₂	#16812692	1	1	yes	characteristic
146	C₂	#16812692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182008

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₀₉

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, characteristic, complemented

#16812606

C₂

yes

maximal, characteristic, complemented

#16812606

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

#13182008

—

C₂²⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₂⁴×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

characteristic, complemented, Sylow

#397

—

109

maximal, Sylow

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

D₁₀₉

yes

characteristic

#95

—

109

—

#95

—

109

—

#95

—

109

—

C₂⁴

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₀₉

yes

—

100

C₂³

C₂²×D₁₀₉

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

218

—

109

C₂×C₄

—

218

—

110

C₂×C₄

—

218

—

111

C₂×C₄

—

218

—

112

C₂×C₄

—

218

—

113

C₂×C₄

—

218

—

114

C₂×C₄

—

218

—

115

C₂×C₄

—

218

—

116

C₂×C₄

—

218

—

117

C₂×C₄

—

218

—

118

C₂×C₄

—

218

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

D₄×D₁₀₉

yes

characteristic

127

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

128

C₂²

#13182028

yes

characteristic

129

C₂²

#13182028

yes

characteristic

130

C₂²

#13182028

yes

characteristic

131

C₂²

#13182029

yes

characteristic

132

C₂²

#13182029

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

436

—

136

C₄

—

436

—

137

C₄

—

436

—

138

C₄

—

218

—

139

C₂

—

140

C₂

#16812671

yes

characteristic

141

C₂

#16812680

yes

characteristic

142

C₂

#16812681

yes

characteristic

143

C₂

#16812682

yes

characteristic

144

C₂

#16812685

yes

characteristic

145

C₂

#16812692

yes

characteristic

146

C₂

#16812692

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator, socle	C₂³×C₁₀₉	C₂³
Fitting	C₂×C₂²⋊C₄×C₁₀₉	C₂
hypercenter	C₂×C₂²⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#397
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator, socle

C₂³×C₁₀₉

C₂³

Fitting

C₂×C₂²⋊C₄×C₁₀₉

C₂

hypercenter

C₂×C₂²⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×#13182008	C₂	semidirect
4. #16812606	C₂	semidirect
5. #16812606	C₂	semidirect
6. #16812606	C₂	semidirect
73. C₁₀₉	#397	semidirect

H ≅ G/N

type

1. C₂×#13182008

C₂

semidirect

4. #16812606

C₂

semidirect

5. #16812606

C₂

semidirect

6. #16812606

C₂

semidirect

73. C₁₀₉

#397

semidirect

Subgroups — #20294954

Conjugacy classes of subgroups

Complements