Group #34044760 — subgroups

Subgroups — #34044760

Order 19072.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#18011870	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11924746	C₂	1	1	yes	maximal, characteristic
3	#23749379	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924746	C₂²	1	2	yes	—
6	#18011870	—	2	2	no	—
7	#18011870	—	2	2	no	—
8	C₂×C₄⋊Dic₁₄₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₄₉	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₁₄₉	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
12	C₂²⋊#11924746	—	2	2	no	—
13	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₄₉	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₄₉	—	2	2	no	—
16	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
18	C₂×#11924746	—	2	2	no	—
19	C₂×#11924746	—	2	4	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₄⋊Dic₁₄₉	—	2	2	no	—
22	C₂²⋊Dic₁₄₉	—	2	4	no	—
23	C₄×Dic₁₄₉	—	2	2	no	—
24	C₄×Dic₁₄₉	—	2	2	no	—
25	C₂⁴×C₁₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₄₉	—	2	4	no	—
27	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
30	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₄₉	—	2	2	no	—
32	C₂×C₈×C₁₄₉	—	2	4	no	—
33	C₂×Dic₁₄₉	—	2	2	no	—
34	C₂×Dic₁₄₉	—	2	2	no	—
35	C₂×Dic₁₄₉	—	2	4	no	—
36	C₂×Dic₁₄₉	—	2	4	no	—
37	#11924746	—	4	2	no	—
38	C₂³×C₁₄₉	—	2	2	no	—
39	C₂³×C₁₄₉	—	2	2	no	—
40	C₂³×C₁₄₉	—	2	2	no	—
41	C₂³×C₁₄₉	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄×C₁₄₉	—	2	4	no	—
45	C₂×C₄×C₁₄₉	—	2	4	no	—
46	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₄₉	#41	1	4	yes	—
50	C₈×C₁₄₉	—	4	2	no	—
51	C₂²×C₁₄₉	—	2	2	no	—
52	C₂²×C₁₄₉	—	2	2	no	—
53	C₂²×C₁₄₉	—	2	2	no	—
54	C₂²×C₁₄₉	—	4	4	no	—
55	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₄₉	#117	1	2	yes	—
57	C₂²×C₁₄₉	#131	1	2	yes	—
58	C₂²×C₁₄₉	#131	1	2	yes	—
59	C₄×C₁₄₉	—	2	2	no	—
60	C₄×C₁₄₉	#123	1	2	yes	—
61	C₄×C₁₄₉	#123	1	2	yes	—
62	Dic₁₄₉	—	4	2	no	—
63	Dic₁₄₉	—	4	2	no	—
64	C₂×C₁₄₉	—	4	2	no	—
65	C₂×C₁₄₉	#381	1	1	yes	characteristic
66	C₂×C₁₄₉	#408	1	1	yes	characteristic
67	C₂×C₁₄₉	#408	1	1	yes	characteristic
68	C₂×C₁₄₉	#433	1	4	yes	—
69	C₁₄₉	#1818	1	1	yes	characteristic, complemented, Sylow
70	#1818	—	149	1	no	maximal, Sylow
71	C₂×#117	—	149	1	no	—
72	C₂×C₂²⋊C₈	—	149	1	no	—
73	C₂×C₂²⋊C₈	D₁₄₉	1	1	yes	characteristic
74	#336	—	149	4	no	—
75	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
76	C₂²×C₈	—	149	2	no	—
77	C₂²×C₈	C₂×D₁₄₉	1	2	yes	—
78	#117	—	298	2	no	—
79	#117	—	298	2	no	—
80	C₂×C₄⋊C₄	—	149	1	no	—
81	C₂×C₄⋊C₄	—	149	1	no	—
82	C₂×C₂²⋊C₄	—	149	2	no	—
83	C₂×C₄²	—	149	2	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	—	298	2	no	—
86	C₂⁴	C₄×D₁₄₉	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	149	2	no	—
89	C₂²×C₄	—	149	2	no	—
90	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
91	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
92	C₂²×C₄	D₅₉₆	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	298	2	no	—
97	C₂×C₈	—	298	4	no	—
98	C₄⋊C₄	—	298	2	no	—
99	C₄⋊C₄	—	298	2	no	—
100	C₂²⋊C₄	—	298	4	no	—
101	C₄²	—	298	2	no	—
102	C₄²	—	298	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	#14558713	1	2	yes	—
107	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	298	4	no	—
112	C₂×C₄	—	298	4	no	—
113	C₂×C₄	—	298	2	no	—
114	C₂×C₄	—	298	2	no	—
115	C₂×C₄	#14558713	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
117	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
118	C₂×C₄	#14558735	1	4	yes	—
119	C₈	—	4	2	no	—
120	C₈	—	596	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂×#14558713	1	1	yes	characteristic
126	C₂²	#18011867	1	2	yes	—
127	C₂²	#18011917	1	2	yes	—
128	C₂²	#18011919	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	596	2	no	—
131	C₄	—	596	2	no	—
132	C₄	#18011892	1	2	yes	—
133	C₄	#18011895	1	2	yes	—
134	C₂	—	4	2	no	—
135	C₂	#23749486	1	1	yes	characteristic
136	C₂	#23749591	1	1	yes	characteristic
137	C₂	#23749593	1	1	yes	characteristic
138	C₂	#23749683	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011870

C₂

yes

maximal, characteristic

C₂×C₂²⋊#11924746

C₂

yes

maximal, characteristic

#23749379

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

—

#18011870

—

#18011870

—

C₂×C₄⋊Dic₁₄₉

C₄

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₄

yes

characteristic

C₂×C₂²⋊Dic₁₄₉

C₄

yes

—

C₂×C₄×Dic₁₄₉

C₂²

yes

—

C₂²⋊#11924746

—

C₂³×C₄×C₁₄₉

C₂²

yes

characteristic

C₂²×C₈×C₁₄₉

C₂²

yes

—

C₂²⋊C₈×C₁₄₉

—

C₂²×Dic₁₄₉

C₂×C₄

yes

—

C₂²×Dic₁₄₉

C₂×C₄

yes

—

C₂×#11924746

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂⁴×C₁₄₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂²⋊C₄

yes

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

#117

yes

—

C₂²×C₁₄₉

#131

yes

—

C₂²×C₁₄₉

#131

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#123

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

C₁₄₉

#1818

yes

characteristic, complemented, Sylow

#1818

—

149

maximal, Sylow

C₂×#117

—

149

—

C₂×C₂²⋊C₈

—

149

—

C₂×C₂²⋊C₈

D₁₄₉

yes

characteristic

#336

—

149

—

C₂³×C₄

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

—

#117

—

298

—

#117

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₂²⋊C₄

—

149

—

C₂×C₄²

—

149

—

C₂²⋊C₈

—

C₂²⋊C₈

—

298

—

C₂⁴

C₄×D₁₄₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₄×D₁₄₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

D₅₉₆

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

100

C₂²⋊C₄

—

298

—

101

C₄²

—

298

—

102

C₄²

—

298

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

#14558713

yes

—

107

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

298

—

112

C₂×C₄

—

298

—

113

C₂×C₄

—

298

—

114

C₂×C₄

—

298

—

115

C₂×C₄

#14558713

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

117

C₂×C₄

C₂×D₅₉₆

yes

characteristic

118

C₂×C₄

#14558735

yes

—

119

C₈

—

120

C₈

—

596

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×#14558713

yes

characteristic

126

C₂²

#18011867

yes

—

127

C₂²

#18011917

yes

—

128

C₂²

#18011919

yes

—

129

C₄

—

130

C₄

—

596

—

131

C₄

—

596

—

132

C₄

#18011892

yes

—

133

C₄

#18011895

yes

—

134

C₂

—

135

C₂

#23749486

yes

characteristic

136

C₂

#23749591

yes

characteristic

137

C₂

#23749593

yes

characteristic

138

C₂

#23749683

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₄₉
commutator, socle	C₂³×C₁₄₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₁₄₉	C₂
prime radical	C₂⁴×C₁₄₉	C₂×C₄
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₄₉
hypocenter	C₁₄₉	#1818
LCS	G ▷ C₂³×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₄₉

commutator, socle

C₂³×C₁₄₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₁₄₉

C₂

prime radical

C₂⁴×C₁₄₉

C₂×C₄

cosocle

C₂²×C₄×C₁₄₉

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #23749379	C₂	semidirect
69. C₁₄₉	#1818	semidirect

H ≅ G/N

type

3. #23749379

C₂

semidirect

69. C₁₄₉

#1818

semidirect

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

subgroup	index
C₂×#18011870	2
C₂×C₂²⋊#11924746	2
#23749379	2
C₂×C₂²⋊C₈×C₁₄₉	2
#1818	149

subgroup

index

C₂×#18011870

C₂×C₂²⋊#11924746

#23749379

C₂×C₂²⋊C₈×C₁₄₉

#1818

149

Subgroups — #34044760

Conjugacy classes of subgroups

Complements

Maximal subgroups