Group #34053410 — subgroups

Subgroups — #34053410

Order 19072.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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×#18011917	C₂	1	2	yes	maximal, complemented
2	#23750175	C₂	1	1	yes	maximal, characteristic
3	#23749698	C₂	1	8	yes	maximal, complemented
4	#23749649	C₂	1	1	yes	maximal, characteristic, complemented
5	#23749591	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14558736	C₂²	1	2	yes	—
8	#18011917	—	2	8	no	—
9	C₂×#14558720	C₂²	1	2	yes	—
10	C₂×#14558719	C₂²	1	4	yes	—
11	C₂×C₈×D₁₄₉	C₂²	1	4	yes	—
12	#18011911	C₄	1	1	yes	characteristic
13	#18011908	C₄	1	1	yes	characteristic
14	#18011899	C₄	1	2	yes	—
15	#18011885	C₄	1	4	yes	—
16	C₄×D₅₉₆	C₂²	1	4	yes	—
17	C₄⋊#11924746	C₂²	1	4	yes	—
18	#18011823	C₂²	1	8	yes	—
19	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
22	#97×C₁₄₉	C₂²	1	4	yes	—
23	#14558736	—	2	4	no	—
24	#14558736	—	2	4	no	—
25	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
26	C₂×D₅₉₆	C₂×C₄	1	2	yes	—
27	C₂×D₅₉₆	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
30	#14558720	—	2	4	no	—
31	#14558719	—	2	8	no	—
32	C₂×#11924746	C₂³	1	4	yes	—
33	C₈×D₁₄₉	—	2	8	no	—
34	C₄⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
35	#14558713	C₂×C₄	1	8	yes	—
36	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₄₉	—	2	2	no	—
39	C₂×C₈×C₁₄₉	D₄	1	4	yes	complemented
40	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
41	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
42	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₄₉	C₂³	1	2	yes	—
45	C₂²×D₁₄₉	C₂²×C₄	1	4	yes	—
46	Dic₂₉₈	—	2	4	no	—
47	D₅₉₆	—	2	4	no	—
48	D₅₉₆	—	4	4	no	—
49	C₂²⋊D₁₄₉	—	2	8	no	—
50	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
51	C₄×D₁₄₉	—	2	4	no	—
52	C₄×D₁₄₉	—	2	4	no	—
53	#11924746	—	2	4	no	—
54	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₄₉	—	4	2	no	—
56	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₄₉	#41	1	2	yes	—
61	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
62	C₈×C₁₄₉	—	2	2	no	—
63	C₈×C₁₄₉	C₂×D₄	1	4	yes	—
64	C₂²×C₁₄₉	—	2	2	no	—
65	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
66	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₁₄₉	—	2	4	no	—
68	C₂×D₁₄₉	—	2	8	no	—
69	C₄×C₁₄₉	—	2	2	no	—
70	C₄×C₁₄₉	—	2	2	no	—
71	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
72	C₄×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₄₉	C₂×#41	1	1	yes	characteristic
74	Dic₁₄₉	—	2	4	no	—
75	C₂×C₁₄₉	—	2	2	no	—
76	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₁₄₉	#568	1	2	yes	—
78	D₁₄₉	—	4	4	no	—
79	C₁₄₉	#2932	1	1	yes	characteristic, complemented, Sylow
80	#2932	—	149	1	no	maximal, Sylow
81	C₂×#131	—	149	2	no	—
82	#532	—	149	1	no	—
83	#435	—	149	8	no	—
84	#423	—	149	1	no	—
85	#408	—	149	2	no	—
86	C₂×#97	D₁₄₉	1	1	yes	characteristic, complemented
87	C₂×#41	—	149	2	no	—
88	#131	—	298	8	no	—
89	C₂×M₁₆	—	149	2	no	—
90	C₂×M₁₆	—	149	4	no	—
91	C₂²×C₈	—	149	4	no	—
92	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
94	C₄⋊Q₈	—	149	1	no	—
95	#127	—	149	1	no	—
96	#124	—	149	2	no	—
97	#121	—	149	4	no	—
98	C₄×D₄	—	149	4	no	—
99	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
100	C₄⋊C₈	—	149	4	no	—
101	C₂²⋊C₈	—	149	8	no	—
102	#97	C₂×D₁₄₉	1	4	yes	—
103	#41	—	298	4	no	—
104	#41	—	298	4	no	—
105	C₂×Q₈	—	149	2	no	—
106	C₂×D₄	—	149	4	no	—
107	C₂×D₄	—	149	2	no	—
108	C₂×D₄	—	149	4	no	—
109	C₂²×C₄	—	149	4	no	—
110	C₂²×C₄	C₄×D₁₄₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
112	M₁₆	—	298	4	no	—
113	M₁₆	—	298	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	298	8	no	—
116	C₂×C₈	—	149	4	no	—
117	C₂×C₈	D₅₉₆	1	4	yes	complemented
118	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
119	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
120	C₄⋊C₄	—	149	4	no	—
121	C₂²⋊C₄	—	149	8	no	—
122	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
123	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
124	C₄²	C₂²×D₁₄₉	1	2	yes	—
125	C₂³	—	149	4	no	—
126	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
127	Q₈	—	298	4	no	—
128	D₄	—	298	8	no	—
129	D₄	—	298	4	no	—
130	D₄	—	596	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	298	4	no	—
133	C₂×C₄	—	298	4	no	—
134	C₂×C₄	—	149	4	no	—
135	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
138	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
139	C₂×C₄	#14558736	1	2	yes	—
140	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	298	4	no	—
143	C₈	C₂×D₅₉₆	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	298	4	no	—
146	C₂²	—	298	8	no	—
147	C₂²	C₄×D₅₉₆	1	2	yes	—
148	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	298	4	no	—
152	C₄	C₄×D₅₉₆	1	2	yes	—
153	C₄	C₂²×D₅₉₆	1	1	yes	characteristic
154	C₄	C₂×#14558736	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	596	4	no	—
157	C₂	C₂×C₄×D₅₉₆	1	1	yes	characteristic
158	C₂	#23750391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011917

C₂

yes

maximal, complemented

#23750175

C₂

yes

maximal, characteristic

#23749698

C₂

yes

maximal, complemented

#23749649

C₂

yes

maximal, characteristic, complemented

#23749591

C₂

yes

maximal, complemented

C₂×#97×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×#14558736

C₂²

yes

—

#18011917

—

C₂×#14558720

C₂²

yes

—

C₂×#14558719

C₂²

yes

—

C₂×C₈×D₁₄₉

C₂²

yes

—

#18011911

C₄

yes

characteristic

#18011908

C₄

yes

characteristic

#18011899

C₄

yes

—

#18011885

C₄

yes

—

C₄×D₅₉₆

C₂²

yes

—

C₄⋊#11924746

C₂²

yes

—

#18011823

C₂²

yes

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#97×C₁₄₉

C₂²

yes

—

#14558736

—

#14558736

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

C₂×D₅₉₆

C₂×C₄

yes

—

C₂×D₅₉₆

C₂×C₄

yes

—

C₂×C₂²⋊D₁₄₉

C₂×C₄

yes

—

C₂×C₄×D₁₄₉

C₂³

yes

—

#14558720

—

#14558719

—

C₂×#11924746

C₂³

yes

—

C₈×D₁₄₉

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558713

C₂×C₄

yes

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

complemented

C₂×C₈×C₁₄₉

C₂³

yes

—

C₂×C₈×C₁₄₉

C₂³

yes

—

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂³

yes

—

C₂²×D₁₄₉

C₂²×C₄

yes

—

Dic₂₉₈

—

D₅₉₆

—

D₅₉₆

—

C₂²⋊D₁₄₉

—

C₂×Dic₁₄₉

C₂²×C₄

yes

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

—

C₈×C₁₄₉

C₂×D₄

yes

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₄×D₄

yes

—

C₂²×C₁₄₉

C₂³×C₄

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×D₄

yes

—

C₄×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

C₂×#41

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₄₉

#568

yes

—

D₁₄₉

—

C₁₄₉

#2932

yes

characteristic, complemented, Sylow

#2932

—

149

maximal, Sylow

C₂×#131

—

149

—

#532

—

149

—

#435

—

149

—

#423

—

149

—

#408

—

149

—

C₂×#97

D₁₄₉

yes

characteristic, complemented

C₂×#41

—

149

—

#131

—

298

—

C₂×M₁₆

—

149

—

C₂×M₁₆

—

149

—

C₂²×C₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₄⋊Q₈

—

149

—

#127

—

149

—

#124

—

149

—

#121

—

149

—

C₄×D₄

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

100

C₄⋊C₈

—

149

—

101

C₂²⋊C₈

—

149

—

102

#97

C₂×D₁₄₉

yes

—

103

#41

—

298

—

104

#41

—

298

—

105

C₂×Q₈

—

149

—

106

C₂×D₄

—

149

—

107

C₂×D₄

—

149

—

108

C₂×D₄

—

149

—

109

C₂²×C₄

—

149

—

110

C₂²×C₄

C₄×D₁₄₉

yes

—

111

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

112

M₁₆

—

298

—

113

M₁₆

—

298

—

114

C₂×C₈

—

115

C₂×C₈

—

298

—

116

C₂×C₈

—

149

—

117

C₂×C₈

D₅₉₆

yes

complemented

118

C₂×C₈

C₂²×D₁₄₉

yes

—

119

C₂×C₈

C₂²×D₁₄₉

yes

—

120

C₄⋊C₄

—

149

—

121

C₂²⋊C₄

—

149

—

122

C₄²

C₄×D₁₄₉

yes

characteristic

123

C₄²

C₄×D₁₄₉

yes

characteristic

124

C₄²

C₂²×D₁₄₉

yes

—

125

C₂³

—

149

—

126

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

127

Q₈

—

298

—

128

D₄

—

298

—

129

D₄

—

298

—

130

D₄

—

596

—

131

C₂×C₄

—

132

C₂×C₄

—

298

—

133

C₂×C₄

—

298

—

134

C₂×C₄

—

149

—

135

C₂×C₄

C₂×C₄×D₁₄₉

yes

—

136

C₂×C₄

C₂×C₄×D₁₄₉

yes

—

137

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

138

C₂×C₄

C₂×D₅₉₆

yes

—

139

C₂×C₄

#14558736

yes

—

140

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

141

C₈

—

142

C₈

—

298

—

143

C₈

C₂×D₅₉₆

yes

—

144

C₂²

—

145

C₂²

—

298

—

146

C₂²

—

298

—

147

C₂²

C₄×D₅₉₆

yes

—

148

C₂²

C₂²×C₄×D₁₄₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

298

—

152

C₄

C₄×D₅₉₆

yes

—

153

C₄

C₂²×D₅₉₆

yes

characteristic

154

C₄

C₂×#14558736

yes

characteristic

155

C₂

—

156

C₂

—

596

—

157

C₂

C₂×C₄×D₅₉₆

yes

characteristic

158

C₂

#23750391

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₂×#97×C₁₄₉	C₂
prime radical	#23750175	C₂
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	C₂×#97	D₁₄₉
hypocenter	C₁₄₉	#2932
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₄₉

commutator, socle

C₂²×C₁₄₉

C₂³×C₄

Fitting

C₂×#97×C₁₄₉

C₂

prime radical

#23750175

C₂

cosocle

C₂×C₄×C₁₄₉

C₂⁴

hypercenter

C₂×#97

D₁₄₉

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#18011917	C₂	semidirect
3. #23749698	C₂	semidirect
4. #23749649	C₂	semidirect
5. #23749591	C₂	semidirect
6. C₂×#97×C₁₄₉	C₂	semidirect
39. C₂×C₈×C₁₄₉	D₄	semidirect
79. C₁₄₉	#2932	semidirect
86. C₂×#97	D₁₄₉	semidirect
117. C₂×C₈	D₅₉₆	semidirect

H ≅ G/N

type

1. C₂×#18011917

C₂

semidirect

3. #23749698

C₂

semidirect

4. #23749649

C₂

semidirect

5. #23749591

C₂

semidirect

6. C₂×#97×C₁₄₉

C₂

semidirect

39. C₂×C₈×C₁₄₉

D₄

semidirect

79. C₁₄₉

#2932

semidirect

86. C₂×#97

D₁₄₉

semidirect

117. C₂×C₈

D₅₉₆

semidirect

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

subgroup	index
C₂×#18011917	2
#23750175	2
#23749698	2
#23749649	2
#23749591	2
C₂×#97×C₁₄₉	2
#2932	149

subgroup

index

C₂×#97×C₁₄₉

149

Subgroups — #34053410

Conjugacy classes of subgroups

Complements

Maximal subgroups