Group #34054079 — subgroups

Subgroups — C₂×#23749815 #34054079

Order 19072.

794 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 165 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×#18011892	C₂	1	3	yes	maximal, complemented
2	#23749815	C₂	1	8	yes	maximal, direct product factor
3	C₂×#18011826	C₂	1	3	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₄₉	C₂²	1	1	yes	characteristic, complemented
6	#18011892	—	2	6	no	—
7	#18011892	C₂²	1	12	yes	complemented
8	C₂×#14558715	—	2	3	no	—
9	C₂×C₂²⋊Dic₁₄₉	—	2	3	no	—
10	C₂×C₂²⋊Dic₁₄₉	C₂²	1	3	yes	complemented
11	#18011826	C₂²	1	12	yes	complemented
12	H(5,𝔽₂)×C₁₄₉	—	2	6	no	—
13	H(5,𝔽₂)×C₁₄₉	C₂²	1	4	yes	—
14	C₂×#41×C₁₄₉	—	2	3	no	—
15	C₂²×D₄×C₁₄₉	—	2	3	no	—
16	C₂²×D₄×C₁₄₉	C₂²	1	3	yes	—
17	C₂²×Dic₁₄₉	—	2	3	no	—
18	C₂²×Dic₁₄₉	—	2	3	no	—
19	C₂²×Dic₁₄₉	—	2	3	no	—
20	C₂²×Dic₁₄₉	C₂³	1	2	yes	complemented
21	#14558715	—	2	12	no	—
22	C₂²⋊Dic₁₄₉	—	2	12	no	—
23	C₂²⋊Dic₁₄₉	—	2	3	no	—
24	C₂²⋊Dic₁₄₉	C₂³	1	6	yes	complemented
25	C₂⁴×C₁₄₉	—	2	1	no	—
26	C₂⁴×C₁₄₉	D₄	1	3	yes	—
27	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
28	#41×C₁₄₉	—	2	12	no	—
29	#41×C₁₄₉	—	2	12	no	—
30	C₂×Q₈×C₁₄₉	—	2	1	no	—
31	C₂×D₄×C₁₄₉	—	2	3	no	—
32	C₂×D₄×C₁₄₉	—	2	12	no	—
33	C₂×D₄×C₁₄₉	—	2	6	no	—
34	C₂×D₄×C₁₄₉	—	2	12	no	—
35	C₂×D₄×C₁₄₉	—	2	6	no	—
36	C₂×D₄×C₁₄₉	D₄	1	6	yes	—
37	C₂×D₄×C₁₄₉	C₂³	1	6	yes	—
38	C₂²×C₄×C₁₄₉	—	2	3	no	—
39	C₂²×C₄×C₁₄₉	D₄	1	3	yes	—
40	C₂×Dic₁₄₉	—	2	6	no	—
41	C₂×Dic₁₄₉	—	2	6	no	—
42	C₂×Dic₁₄₉	—	4	6	no	—
43	C₂×Dic₁₄₉	—	4	3	no	—
44	C₂×Dic₁₄₉	—	4	1	no	—
45	C₂³×C₁₄₉	—	2	4	no	—
46	C₂³×C₁₄₉	—	2	3	no	—
47	C₂³×C₁₄₉	—	2	3	no	—
48	C₂³×C₁₄₉	—	4	2	no	—
49	C₂³×C₁₄₉	—	4	6	no	—
50	C₂³×C₁₄₉	—	8	2	no	—
51	C₂³×C₁₄₉	C₂×D₄	1	3	yes	—
52	C₂³×C₁₄₉	C₂×D₄	1	6	yes	—
53	C₂³×C₁₄₉	C₂×D₄	1	3	yes	—
54	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
55	Q₈×C₁₄₉	—	2	4	no	—
56	D₄×C₁₄₉	—	2	12	no	—
57	D₄×C₁₄₉	—	2	12	no	—
58	D₄×C₁₄₉	—	2	12	no	—
59	C₂×C₄×C₁₄₉	—	2	12	no	—
60	C₂×C₄×C₁₄₉	—	2	3	no	—
61	C₂×C₄×C₁₄₉	—	2	3	no	—
62	C₂×C₄×C₁₄₉	C₂×D₄	1	6	yes	—
63	C₂²×C₁₄₉	—	2	6	no	—
64	C₂²×C₁₄₉	—	2	6	no	—
65	C₂²×C₁₄₉	—	4	3	no	—
66	C₂²×C₁₄₉	—	4	1	no	—
67	C₂²×C₁₄₉	—	4	6	no	—
68	C₂²×C₁₄₉	—	4	6	no	—
69	C₂²×C₁₄₉	—	8	3	no	—
70	C₂²×C₁₄₉	—	8	4	no	—
71	C₂²×C₁₄₉	#120	1	3	yes	—
72	C₂²×C₁₄₉	#120	1	1	yes	characteristic
73	C₂²×C₁₄₉	C₂²×D₄	1	3	yes	—
74	C₄×C₁₄₉	—	2	6	no	—
75	Dic₁₄₉	—	4	6	no	—
76	Dic₁₄₉	—	4	2	no	—
77	C₂×C₁₄₉	—	2	3	no	—
78	C₂×C₁₄₉	—	2	3	no	—
79	C₂×C₁₄₉	—	4	6	no	—
80	C₂×C₁₄₉	#458	1	2	yes	complemented
81	C₂×C₁₄₉	C₂×#120	1	1	yes	characteristic
82	C₁₄₉	C₂×#458	1	1	yes	characteristic, complemented, Sylow
83	C₂×#458	—	149	1	no	maximal, Sylow
84	C₂×H(5,𝔽₂)	D₁₄₉	1	1	yes	characteristic
85	C₂×#123	—	149	3	no	—
86	#458	—	149	8	no	—
87	C₂×#99	—	149	3	no	—
88	H(5,𝔽₂)	—	2	6	no	—
89	H(5,𝔽₂)	C₂×D₁₄₉	1	4	yes	—
90	C₂×#41	—	2	3	no	—
91	C₂²×D₄	—	2	3	no	—
92	C₂²×D₄	C₂×D₁₄₉	1	3	yes	—
93	C₂³×C₄	—	149	1	no	—
94	#123	—	298	6	no	—
95	#123	—	149	12	no	—
96	C₂×C₄⋊C₄	—	298	3	no	—
97	C₂×C₂²⋊C₄	—	298	3	no	—
98	C₂×C₂²⋊C₄	—	149	3	no	—
99	#99	—	149	12	no	—
100	C₂⁴	—	2	1	no	—
101	C₂⁴	C₂²⋊D₁₄₉	1	3	yes	—
102	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
103	#41	—	2	12	no	—
104	#41	—	2	12	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	—	2	12	no	—
108	C₂×D₄	—	2	6	no	—
109	C₂×D₄	—	2	12	no	—
110	C₂×D₄	—	2	6	no	—
111	C₂×D₄	C₂²⋊D₁₄₉	1	6	yes	—
112	C₂×D₄	C₂²×D₁₄₉	1	6	yes	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	—	298	3	no	—
115	C₂²×C₄	—	298	3	no	—
116	C₂²×C₄	—	298	3	no	—
117	C₂²×C₄	—	149	2	no	—
118	C₂²×C₄	C₂²⋊D₁₄₉	1	3	yes	—
119	C₄⋊C₄	—	298	12	no	—
120	C₂²⋊C₄	—	298	12	no	—
121	C₂²⋊C₄	—	298	3	no	—
122	C₂²⋊C₄	—	149	6	no	—
123	C₂³	—	2	4	no	—
124	C₂³	—	2	3	no	—
125	C₂³	—	2	3	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	6	no	—
128	C₂³	—	8	2	no	—
129	C₂³	C₂×C₂²⋊D₁₄₉	1	3	yes	—
130	C₂³	C₂×C₂²⋊D₁₄₉	1	6	yes	—
131	C₂³	C₂×C₂²⋊D₁₄₉	1	3	yes	—
132	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	D₄	—	2	12	no	—
135	D₄	—	2	12	no	—
136	D₄	—	2	12	no	—
137	C₂×C₄	—	2	12	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	298	6	no	—
141	C₂×C₄	—	298	6	no	—
142	C₂×C₄	—	596	6	no	—
143	C₂×C₄	—	596	3	no	—
144	C₂×C₄	—	596	1	no	—
145	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	6	yes	—
146	C₂²	—	2	6	no	—
147	C₂²	—	2	6	no	—
148	C₂²	—	4	3	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	6	no	—
151	C₂²	—	4	6	no	—
152	C₂²	—	8	3	no	—
153	C₂²	—	8	4	no	—
154	C₂²	#18011880	1	3	yes	—
155	C₂²	#18011880	1	1	yes	characteristic
156	C₂²	C₂²×C₂²⋊D₁₄₉	1	3	yes	—
157	C₄	—	2	6	no	—
158	C₄	—	596	6	no	—
159	C₄	—	596	2	no	—
160	C₂	—	2	3	no	—
161	C₂	—	2	3	no	—
162	C₂	—	4	6	no	—
163	C₂	#23749815	1	2	yes	direct product factor
164	C₂	C₂×#18011880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011892

C₂

yes

maximal, complemented

#23749815

C₂

yes

maximal, direct product factor

C₂×#18011826

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₁₄₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₄₉

C₂²

yes

characteristic, complemented

#18011892

—

#18011892

C₂²

yes

complemented

C₂×#14558715

—

C₂×C₂²⋊Dic₁₄₉

—

C₂×C₂²⋊Dic₁₄₉

C₂²

yes

complemented

#18011826

C₂²

yes

complemented

H(5,𝔽₂)×C₁₄₉

—

H(5,𝔽₂)×C₁₄₉

C₂²

yes

—

C₂×#41×C₁₄₉

—

C₂²×D₄×C₁₄₉

—

C₂²×D₄×C₁₄₉

C₂²

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

C₂³

yes

complemented

#14558715

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

C₂³

yes

complemented

C₂⁴×C₁₄₉

—

C₂⁴×C₁₄₉

D₄

yes

—

C₂⁴×C₁₄₉

C₂³

yes

characteristic

#41×C₁₄₉

—

#41×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

D₄

yes

—

C₂×D₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

—

C₂³×C₁₄₉

C₂×D₄

yes

—

C₂³×C₁₄₉

C₂×D₄

yes

—

C₂³×C₁₄₉

C₂⁴

yes

characteristic

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#120

yes

—

C₂²×C₁₄₉

#120

yes

characteristic

C₂²×C₁₄₉

C₂²×D₄

yes

—

C₄×C₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

C₂×#458

—

149

maximal, Sylow

C₂×H(5,𝔽₂)

D₁₄₉

yes

characteristic

C₂×#123

—

149

—

#458

—

149

—

C₂×#99

—

149

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₁₄₉

yes

—

C₂×#41

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₄₉

yes

—

C₂³×C₄

—

149

—

#123

—

298

—

#123

—

149

—

C₂×C₄⋊C₄

—

298

—

C₂×C₂²⋊C₄

—

298

—

C₂×C₂²⋊C₄

—

149

—

#99

—

149

—

100

C₂⁴

—

101

C₂⁴

C₂²⋊D₁₄₉

yes

—

102

C₂⁴

C₂²×D₁₄₉

yes

characteristic

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂²⋊D₁₄₉

yes

—

112

C₂×D₄

C₂²×D₁₄₉

yes

—

113

C₂²×C₄

—

114

C₂²×C₄

—

298

—

115

C₂²×C₄

—

298

—

116

C₂²×C₄

—

298

—

117

C₂²×C₄

—

149

—

118

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

119

C₄⋊C₄

—

298

—

120

C₂²⋊C₄

—

298

—

121

C₂²⋊C₄

—

298

—

122

C₂²⋊C₄

—

149

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂×C₂²⋊D₁₄₉

yes

—

130

C₂³

C₂×C₂²⋊D₁₄₉

yes

—

131

C₂³

C₂×C₂²⋊D₁₄₉

yes

—

132

C₂³

C₂³×D₁₄₉

yes

characteristic

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

298

—

141

C₂×C₄

—

298

—

142

C₂×C₄

—

596

—

143

C₂×C₄

—

596

—

144

C₂×C₄

—

596

—

145

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#18011880

yes

—

155

C₂²

#18011880

yes

characteristic

156

C₂²

C₂²×C₂²⋊D₁₄₉

yes

—

157

C₄

—

158

C₄

—

596

—

159

C₄

—

596

—

160

C₂

—

161

C₂

—

162

C₂

—

163

C₂

#23749815

yes

direct product factor

164

C₂

C₂×#18011880

yes

characteristic

	subgroup	G/H
center	C₂²	#18011880
commutator, cosocle	C₂³×C₁₄₉	C₂⁴
Frattini	C₂³	C₂³×D₁₄₉
Fitting, prime radical	C₂×H(5,𝔽₂)×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#120
hypercenter	C₂×H(5,𝔽₂)	D₁₄₉
hypocenter	C₁₄₉	C₂×#458
LCS	G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂²

#18011880

commutator, cosocle

C₂³×C₁₄₉

C₂⁴

Frattini

C₂³

C₂³×D₁₄₉

Fitting, prime radical

C₂×H(5,𝔽₂)×C₁₄₉

C₂

socle

C₂²×C₁₄₉

#120

hypercenter

C₂×H(5,𝔽₂)

D₁₄₉

hypocenter

C₁₄₉

C₂×#458

LCS

G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. C₂×#18011892	C₂	semidirect
2. #23749815	C₂	direct (N × H)
3. C₂×#18011826	C₂	semidirect
5. C₂³×Dic₁₄₉	C₂²	semidirect
7. #18011892	C₂²	semidirect
10. C₂×C₂²⋊Dic₁₄₉	C₂²	semidirect
11. #18011826	C₂²	semidirect
20. C₂²×Dic₁₄₉	C₂³	semidirect
24. C₂²⋊Dic₁₄₉	C₂³	semidirect
80. C₂×C₁₄₉	#458	semidirect
82. C₁₄₉	C₂×#458	semidirect
163. C₂	#23749815	direct (N × H)

H ≅ G/N

type

1. C₂×#18011892

C₂

semidirect

2. #23749815

C₂

direct (N × H)

3. C₂×#18011826

C₂

semidirect

5. C₂³×Dic₁₄₉

C₂²

semidirect

7. #18011892

C₂²

semidirect

10. C₂×C₂²⋊Dic₁₄₉

C₂²

semidirect

11. #18011826

C₂²

semidirect

20. C₂²×Dic₁₄₉

C₂³

semidirect

24. C₂²⋊Dic₁₄₉

C₂³

semidirect

80. C₂×C₁₄₉

#458

semidirect

82. C₁₄₉

C₂×#458

semidirect

163. C₂

#23749815

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#18011892	2
#23749815	2
C₂×#18011826	2
C₂×H(5,𝔽₂)×C₁₄₉	2
C₂×#458	149

subgroup

index

C₂×H(5,𝔽₂)×C₁₄₉

149

Subgroups — C₂×#23749815 #34054079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23749815 #34054079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23749815 #34054079