Group #34046016 — subgroups

Subgroups — #34046016

Order 19072.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#18011844	C₂	1	2	yes	maximal
2	#23749523	C₂	1	1	yes	maximal, characteristic
3	#23749387	C₂	1	2	yes	maximal
4	#23749380	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924746	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
9	C₂×#14558715	—	2	1	no	—
10	#18011844	—	2	4	no	—
11	#18011818	—	2	1	no	—
12	C₂²×Q₈×C₁₄₉	—	2	1	no	—
13	C₄⋊Q₈×C₁₄₉	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₄₉	—	2	1	no	—
15	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₄₉	—	2	1	no	—
18	C₂²×Dic₁₄₉	—	2	1	no	—
19	C₂×#11924746	—	2	2	no	—
20	C₂×#11924746	—	2	2	no	—
21	C₂×#11924746	Q₈	1	4	yes	complemented
22	C₄⋊Dic₁₄₉	—	2	2	no	—
23	C₄⋊Dic₁₄₉	—	2	1	no	—
24	C₄⋊Dic₁₄₉	Q₈	1	2	yes	complemented
25	#14558715	—	4	2	no	—
26	C₂×Q₈×C₁₄₉	—	2	4	no	—
27	C₂×Q₈×C₁₄₉	—	4	2	no	—
28	C₂×Q₈×C₁₄₉	—	4	2	no	—
29	C₂²×C₄×C₁₄₉	—	2	2	no	—
30	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₄₉	—	2	4	no	—
34	C₄⋊C₄×C₁₄₉	—	2	2	no	—
35	C₄⋊C₄×C₁₄₉	D₄	1	4	yes	—
36	C₄²×C₁₄₉	—	2	2	no	—
37	C₂×Dic₁₄₉	—	2	2	no	—
38	C₂×Dic₁₄₉	—	4	1	no	—
39	C₂×Dic₁₄₉	—	4	1	no	—
40	C₂×Dic₁₄₉	—	4	1	no	—
41	C₂×Dic₁₄₉	—	4	1	no	—
42	C₂×Dic₁₄₉	—	4	1	no	—
43	#11924746	—	2	4	no	—
44	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₄₉	—	4	8	no	—
46	C₂×C₄×C₁₄₉	—	2	1	no	—
47	C₂×C₄×C₁₄₉	—	2	1	no	—
48	C₂×C₄×C₁₄₉	—	2	1	no	—
49	C₂×C₄×C₁₄₉	—	2	1	no	—
50	C₂×C₄×C₁₄₉	—	2	4	no	—
51	C₂×C₄×C₁₄₉	—	4	2	no	—
52	C₂×C₄×C₁₄₉	—	4	2	no	—
53	C₂×C₄×C₁₄₉	D₈	1	2	yes	—
54	C₂×C₄×C₁₄₉	Q₁₆	1	2	yes	—
55	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
59	C₂²×C₁₄₉	#120	1	1	yes	characteristic
60	C₂²×C₁₄₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₄₉	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₄₉	#137	1	1	yes	characteristic
66	C₄×C₁₄₉	—	2	2	no	—
67	C₄×C₁₄₉	—	2	2	no	—
68	C₄×C₁₄₉	—	4	4	no	—
69	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₁₄₉	C₄⋊Q₈	1	2	yes	—
71	Dic₁₄₉	—	4	2	no	—
72	Dic₁₄₉	—	8	1	no	—
73	C₂×C₁₄₉	#395	1	1	yes	characteristic
74	C₂×C₁₄₉	#447	1	1	yes	characteristic
75	C₂×C₁₄₉	#451	1	1	yes	characteristic
76	C₂×C₁₄₉	#474	1	1	yes	characteristic
77	C₂×C₁₄₉	#477	1	1	yes	characteristic
78	C₂×C₁₄₉	#500	1	1	yes	characteristic
79	C₂×C₁₄₉	#501	1	1	yes	characteristic
80	C₁₄₉	#2057	1	1	yes	characteristic, complemented, Sylow
81	#2057	—	149	1	no	maximal, Sylow
82	C₂×C₄⋊Q₈	D₁₄₉	1	1	yes	characteristic
83	C₂×#107	—	149	2	no	—
84	#389	—	149	1	no	—
85	#340	—	149	2	no	—
86	#336	—	149	1	no	—
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	149	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	298	1	no	—
92	C₂×C₄⋊C₄	—	149	1	no	—
93	C₂×C₄⋊C₄	—	149	1	no	—
94	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
95	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
96	#107	—	298	4	no	—
97	#95	—	298	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	298	1	no	—
103	C₂²×C₄	—	298	1	no	—
104	C₂²×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
107	C₂×C₈	—	298	2	no	—
108	C₂×C₈	—	298	2	no	—
109	C₂×C₈	—	149	4	no	—
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	298	1	no	—
113	C₄⋊C₄	—	298	2	no	—
114	C₄⋊C₄	—	596	2	no	—
115	C₄⋊C₄	—	149	2	no	—
116	C₄⋊C₄	C₂²⋊D₁₄₉	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	298	2	no	—
128	C₂×C₄	—	596	1	no	—
129	C₂×C₄	—	596	1	no	—
130	C₂×C₄	—	596	1	no	—
131	C₂×C₄	—	596	1	no	—
132	C₂×C₄	—	596	1	no	—
133	C₂×C₄	#14558721	1	2	yes	—
134	C₂×C₄	#14558726	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
136	C₂×C₄	Q₈×D₁₄₉	1	1	yes	characteristic
137	C₂×C₄	Q₈×D₁₄₉	1	2	yes	—
138	C₂×C₄	#14558737	1	1	yes	characteristic
139	C₈	—	298	4	no	—
140	C₂²	#18011880	1	1	yes	characteristic
141	C₂²	#18011887	1	1	yes	characteristic
142	C₂²	#18011909	1	1	yes	characteristic
143	C₂²	C₂×#14558721	1	1	yes	characteristic
144	C₂²	C₂×#14558726	1	1	yes	characteristic
145	C₂²	#18011937	1	1	yes	characteristic
146	C₂²	#18011942	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	596	2	no	—
151	C₄	—	1192	1	no	—
152	C₄	#18011887	1	2	yes	—
153	C₄	#18011909	1	2	yes	—
154	C₂	#23749544	1	1	yes	characteristic
155	C₂	#23749759	1	1	yes	characteristic
156	C₂	#23749781	1	1	yes	characteristic
157	C₂	#23749902	1	1	yes	characteristic
158	C₂	#23749917	1	1	yes	characteristic
159	C₂	#23750028	1	1	yes	characteristic
160	C₂	#23750031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011844

C₂

yes

maximal

#23749523

C₂

yes

maximal, characteristic

#23749387

C₂

yes

maximal

#23749380

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

—

—

C₂²×Q₈×C₁₄₉

—

C₄⋊Q₈×C₁₄₉

—

C₂×C₄⋊C₄×C₁₄₉

—

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₂×#11924746

—

C₂×#11924746

—

C₂×#11924746

Q₈

yes

complemented

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

Q₈

yes

complemented

#14558715

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

D₈

yes

—

C₂×C₄×C₁₄₉

Q₁₆

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

C₂×Q₈

yes

C₂×C₄×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₄₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₄₉

C₂×D₈

yes

characteristic

C₂²×C₁₄₉

C₂×Q₁₆

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

C₄⋊Q₈

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2057

—

149

maximal, Sylow

C₂×C₄⋊Q₈

D₁₄₉

yes

characteristic

C₂×#107

—

149

—

#389

—

149

—

#340

—

149

—

#336

—

149

—

C₂²×Q₈

—

C₂²×C₈

—

149

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#107

—

298

—

#95

—

298

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

298

—

103

C₂²×C₄

—

298

—

104

C₂²×C₄

C₂²⋊D₁₄₉

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₄₉

yes

characteristic

106

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

107

C₂×C₈

—

298

—

108

C₂×C₈

—

298

—

109

C₂×C₈

—

149

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

298

—

113

C₄⋊C₄

—

298

—

114

C₄⋊C₄

—

596

—

115

C₄⋊C₄

—

149

—

116

C₄⋊C₄

C₂²⋊D₁₄₉

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

298

—

128

C₂×C₄

—

596

—

129

C₂×C₄

—

596

—

130

C₂×C₄

—

596

—

131

C₂×C₄

—

596

—

132

C₂×C₄

—

596

—

133

C₂×C₄

#14558721

yes

—

134

C₂×C₄

#14558726

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

136

C₂×C₄

Q₈×D₁₄₉

yes

characteristic

137

C₂×C₄

Q₈×D₁₄₉

yes

—

138

C₂×C₄

#14558737

yes

characteristic

139

C₈

—

298

—

140

C₂²

#18011880

yes

characteristic

141

C₂²

#18011887

yes

characteristic

142

C₂²

#18011909

yes

characteristic

143

C₂²

C₂×#14558721

yes

characteristic

144

C₂²

C₂×#14558726

yes

characteristic

145

C₂²

#18011937

yes

characteristic

146

C₂²

#18011942

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

596

—

151

C₄

—

1192

—

152

C₄

#18011887

yes

—

153

C₄

#18011909

yes

—

154

C₂

#23749544

yes

characteristic

155

C₂

#23749759

yes

characteristic

156

C₂

#23749781

yes

characteristic

157

C₂

#23749902

yes

characteristic

158

C₂

#23749917

yes

characteristic

159

C₂

#23750028

yes

characteristic

160

C₂

#23750031

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₂×C₄⋊Q₈×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₄₉
hypocenter	C₁₄₉	#2057
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₂×C₄⋊Q₈×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₄₉

hypocenter

C₁₄₉

#2057

LCS

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. C₂×#11924746	Q₈	semidirect
24. C₄⋊Dic₁₄₉	Q₈	semidirect
80. C₁₄₉	#2057	semidirect

H ≅ G/N

type

21. C₂×#11924746

Q₈

semidirect

24. C₄⋊Dic₁₄₉

Q₈

semidirect

80. C₁₄₉

#2057

semidirect

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

subgroup	index
C₂×#18011844	2
#23749523	2
#23749387	2
#23749380	2
C₂×C₄⋊Q₈×C₁₄₉	2
#2057	149

subgroup

index

C₂×C₄⋊Q₈×C₁₄₉

149

Subgroups — #34046016

Conjugacy classes of subgroups

Complements

Maximal subgroups