Group #34055515 — subgroups

Subgroups — #34055515

Order 19072.

410 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 125 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×#18011890	C₂	1	2	yes	maximal
2	#23749956	C₂	1	4	yes	maximal, complemented
3	#23749883	C₂	1	4	yes	maximal, complemented
4	#23749783	C₂	1	4	yes	maximal, complemented
5	#407×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
7	C₂×Dic₅₉₆	C₂²	1	4	yes	complemented
8	#18011890	—	2	2	no	—
9	#18011890	—	2	2	no	—
10	#18011890	C₂²	1	4	yes	—
11	#18011890	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
13	C₂×#14558715	—	2	2	no	—
14	C₂×C₂²⋊Dic₁₄₉	—	2	2	no	—
15	#18011843	C₂²	1	4	yes	complemented
16	#18011836	C₂²	1	8	yes	complemented
17	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
19	C₂²⋊C₈×C₁₄₉	C₂²	1	4	yes	—
20	C₂×Dic₂₉₈	—	2	2	no	—
21	C₂×Dic₂₉₈	—	4	4	no	—
22	C₂×Dic₂₉₈	C₂³	1	4	yes	—
23	C₂²×Dic₁₄₉	—	2	2	no	—
24	C₂²×Dic₁₄₉	—	2	2	no	—
25	Dic₅₉₆	—	4	4	no	—
26	C₄⋊Dic₁₄₉	—	2	2	no	—
27	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
28	#14558715	—	2	4	no	—
29	#14558715	—	2	4	no	—
30	C₂²⋊Dic₁₄₉	—	2	4	no	—
31	C₂²⋊Dic₁₄₉	—	2	4	no	—
32	C₂⁴×C₁₄₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₄₉	—	4	2	no	—
34	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
35	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₄₉	—	4	2	no	—
39	C₂×C₈×C₁₄₉	C₂³	1	4	yes	—
40	Dic₂₉₈	—	2	8	no	—
41	Dic₂₉₈	—	4	4	no	—
42	C₂×Dic₁₄₉	—	2	4	no	—
43	C₂×Dic₁₄₉	—	2	4	no	—
44	C₂×Dic₁₄₉	—	4	4	no	—
45	C₂×Dic₁₄₉	—	4	4	no	—
46	C₂³×C₁₄₉	—	2	4	no	—
47	C₂³×C₁₄₉	—	4	1	no	—
48	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	C₂³×C₁₄₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₄₉	—	2	4	no	—
51	C₂×C₄×C₁₄₉	—	2	2	no	—
52	C₂×C₄×C₁₄₉	—	4	2	no	—
53	C₂×C₄×C₁₄₉	—	4	1	no	—
54	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
57	C₈×C₁₄₉	—	2	4	no	—
58	C₂²×C₁₄₉	—	2	2	no	—
59	C₂²×C₁₄₉	—	2	2	no	—
60	C₂²×C₁₄₉	—	2	1	no	—
61	C₂²×C₁₄₉	—	2	2	no	—
62	C₂²×C₁₄₉	—	4	4	no	—
63	C₂²×C₁₄₉	#137	1	4	yes	—
64	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₄₉	—	2	1	no	—
66	C₄×C₁₄₉	—	2	2	no	—
67	C₄×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
68	Dic₁₄₉	—	4	4	no	—
69	Dic₁₄₉	—	4	4	no	—
70	C₂×C₁₄₉	—	2	4	no	—
71	C₂×C₁₄₉	—	4	1	no	—
72	C₂×C₁₄₉	#534	1	1	yes	characteristic
73	C₂×C₁₄₉	C₂×#137	1	2	yes	—
74	C₁₄₉	#3195	1	1	yes	characteristic, complemented, Sylow
75	#3195	—	149	1	no	maximal, Sylow
76	C₂×C₂²⋊Q₈	—	149	2	no	—
77	#483	—	149	4	no	—
78	#470	—	149	4	no	—
79	#451	—	149	4	no	—
80	#407	D₁₄₉	1	1	yes	characteristic
81	C₂²×Q₈	—	149	2	no	—
82	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
83	C₂×Q₁₆	—	149	4	no	—
84	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
85	C₂²⋊Q₈	—	298	2	no	—
86	C₂²⋊Q₈	—	298	2	no	—
87	C₂²⋊Q₈	—	149	4	no	—
88	C₂²⋊Q₈	—	149	4	no	—
89	C₂×C₄⋊C₄	—	298	2	no	—
90	C₂×C₄⋊C₄	—	149	2	no	—
91	C₂×C₂²⋊C₄	—	298	2	no	—
92	#106	—	149	4	no	—
93	#103	—	149	8	no	—
94	C₂²⋊C₈	C₂×D₁₄₉	1	4	yes	—
95	C₂⁴	D₅₉₆	1	1	yes	characteristic
96	C₂×Q₈	—	298	2	no	—
97	C₂×Q₈	—	596	4	no	—
98	C₂×Q₈	—	149	4	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	—	298	2	no	—
101	C₂²×C₄	—	298	2	no	—
102	C₂²×C₄	D₅₉₆	1	2	yes	—
103	C₂²×C₄	D₅₉₆	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
106	Q₁₆	—	596	4	no	—
107	M₁₆	—	4	2	no	—
108	C₂×C₈	C₂²×D₁₄₉	1	4	yes	—
109	C₄⋊C₄	—	298	4	no	—
110	C₄⋊C₄	—	298	4	no	—
111	C₄⋊C₄	—	298	2	no	—
112	C₄⋊C₄	—	149	4	no	—
113	C₂²⋊C₄	—	298	4	no	—
114	C₂²⋊C₄	—	298	4	no	—
115	C₂³	—	2	4	no	—
116	C₂³	—	4	1	no	—
117	C₂³	C₂×D₅₉₆	1	1	yes	characteristic
118	C₂³	C₂×D₅₉₆	1	2	yes	—
119	Q₈	—	298	8	no	—
120	Q₈	—	596	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	298	4	no	—
126	C₂×C₄	—	298	4	no	—
127	C₂×C₄	—	596	4	no	—
128	C₂×C₄	—	596	4	no	—
129	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
130	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
131	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#18011944	1	4	yes	—
139	C₂²	C₂²×D₅₉₆	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	596	4	no	—
143	C₄	—	596	4	no	—
144	C₄	#18011956	1	2	yes	—
145	C₂	—	2	4	no	—
146	C₂	—	4	1	no	—
147	C₂	#23750187	1	1	yes	characteristic
148	C₂	C₂×#18011944	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011890

C₂

yes

maximal

#23749956

C₂

yes

maximal, complemented

#23749883

C₂

yes

maximal, complemented

#23749783

C₂

yes

maximal, complemented

#407×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

—

C₂×Dic₅₉₆

C₂²

yes

—

—

C₂²

yes

—

#18011890

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558715

—

C₂×C₂²⋊Dic₁₄₉

—

#18011843

C₂²

yes

complemented

#18011836

C₂²

yes

complemented

C₂³×C₄×C₁₄₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂²⋊C₈×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₂³

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

Dic₅₉₆

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂³

yes

—

#14558715

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₂⁴×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

C₂³

yes

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂³×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#137

yes

—

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

H(5,𝔽₂)

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#534

yes

characteristic

C₂×C₁₄₉

C₂×#137

yes

—

C₁₄₉

#3195

yes

characteristic, complemented, Sylow

#3195

—

149

maximal, Sylow

C₂×C₂²⋊Q₈

—

149

—

#483

—

149

—

#470

—

149

—

#451

—

149

—

#407

D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

C₂³×C₄

C₂×D₁₄₉

yes

characteristic

C₂×Q₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₂²⋊Q₈

—

298

—

C₂²⋊Q₈

—

298

—

C₂²⋊Q₈

—

149

—

C₂²⋊Q₈

—

149

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₂²⋊C₄

—

298

—

#106

—

149

—

#103

—

149

—

C₂²⋊C₈

C₂×D₁₄₉

yes

—

C₂⁴

D₅₉₆

yes

characteristic

C₂×Q₈

—

298

—

C₂×Q₈

—

596

—

C₂×Q₈

—

149

—

C₂²×C₄

—

100

C₂²×C₄

—

298

—

101

C₂²×C₄

—

298

—

102

C₂²×C₄

D₅₉₆

yes

—

103

C₂²×C₄

D₅₉₆

yes

characteristic

104

C₂²×C₄

C₂²×D₁₄₉

yes

—

105

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

106

Q₁₆

—

596

—

107

M₁₆

—

108

C₂×C₈

C₂²×D₁₄₉

yes

—

109

C₄⋊C₄

—

298

—

110

C₄⋊C₄

—

298

—

111

C₄⋊C₄

—

298

—

112

C₄⋊C₄

—

149

—

113

C₂²⋊C₄

—

298

—

114

C₂²⋊C₄

—

298

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×D₅₉₆

yes

characteristic

118

C₂³

C₂×D₅₉₆

yes

—

119

Q₈

—

298

—

120

Q₈

—

596

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

298

—

126

C₂×C₄

—

298

—

127

C₂×C₄

—

596

—

128

C₂×C₄

—

596

—

129

C₂×C₄

C₂×D₅₉₆

yes

characteristic

130

C₂×C₄

C₂×D₅₉₆

yes

—

131

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#18011944

yes

—

139

C₂²

C₂²×D₅₉₆

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

596

—

143

C₄

—

596

—

144

C₄

#18011956

yes

—

145

C₂

—

146

C₂

—

147

C₂

#23750187

yes

characteristic

148

C₂

C₂×#18011944

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₉₆
commutator, cosocle	C₂×C₄×C₁₄₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	#407×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂²×D₄
prime radical	C₂⁴×C₁₄₉	D₄
hypercenter	#407	D₁₄₉
hypocenter	C₁₄₉	#3195
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂²×D₅₉₆

commutator, cosocle

C₂×C₄×C₁₄₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting

#407×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂²×D₄

prime radical

C₂⁴×C₁₄₉

D₄

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. #23749956	C₂	semidirect
3. #23749883	C₂	semidirect
4. #23749783	C₂	semidirect
7. C₂×Dic₅₉₆	C₂²	semidirect
15. #18011843	C₂²	semidirect
16. #18011836	C₂²	semidirect
74. C₁₄₉	#3195	semidirect

H ≅ G/N

type

2. #23749956

C₂

semidirect

3. #23749883

C₂

semidirect

4. #23749783

C₂

semidirect

7. C₂×Dic₅₉₆

C₂²

semidirect

15. #18011843

C₂²

semidirect

16. #18011836

C₂²

semidirect

74. C₁₄₉

#3195

semidirect

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

subgroup	index
C₂×#18011890	2
#23749956	2
#23749883	2
#23749783	2
#407×C₁₄₉	2
#3195	149

subgroup

index

#407×C₁₄₉

149

Subgroups — #34055515

Conjugacy classes of subgroups

Complements

Maximal subgroups