Group #34044198 — subgroups

Subgroups — #34044198

Order 19072.

146 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 55 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	#23750023	C₂	1	1	yes	maximal, characteristic
2	#23749921	C₂	1	1	yes	maximal, characteristic
3	#23749836	C₂	1	1	yes	maximal, characteristic
4	#23749755	C₂	1	1	yes	maximal, characteristic
5	#23749373	C₂	1	1	yes	maximal, characteristic
6	#23749372	C₂	1	1	yes	maximal, characteristic
7	#335×C₁₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₉₆	—	2	1	no	—
9	#18011911	C₂²	1	1	yes	characteristic
10	#18011911	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₉₈	C₂²	1	1	yes	characteristic
12	#18011845	—	2	1	no	—
13	#18011843	—	2	1	no	—
14	#18011843	—	2	1	no	—
15	C₄⋊#11924746	C₂²	1	1	yes	characteristic
16	#18011840	—	2	1	no	—
17	#18011836	—	2	1	no	—
18	#18011836	—	2	1	no	—
19	#18011821	—	2	1	no	—
20	C₄⋊C₈×C₁₄₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₄₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₉₈	—	2	1	no	—
24	C₂×Dic₂₉₈	—	2	1	no	—
25	C₂×Dic₂₉₈	D₄	1	1	yes	characteristic
26	Dic₅₉₆	—	4	2	no	—
27	C₂×#11924746	—	2	1	no	—
28	C₄⋊Dic₁₄₉	—	2	1	no	—
29	C₄⋊Dic₁₄₉	—	2	1	no	—
30	C₄⋊Dic₁₄₉	—	2	1	no	—
31	C₄⋊Dic₁₄₉	—	2	1	no	—
32	C₄⋊Dic₁₄₉	D₄	1	1	yes	characteristic
33	#14558715	—	2	1	no	—
34	C₄×Dic₁₄₉	—	2	1	no	—
35	C₂×C₈×C₁₄₉	—	2	1	no	—
36	C₂×C₈×C₁₄₉	—	2	1	no	—
37	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
39	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
40	Dic₂₉₈	—	2	1	no	—
41	Dic₂₉₈	—	2	1	no	—
42	Dic₂₉₈	—	4	1	no	—
43	Dic₂₉₈	—	4	1	no	—
44	Dic₂₉₈	—	8	1	no	—
45	C₂×Dic₁₄₉	—	2	1	no	—
46	C₂×Dic₁₄₉	—	2	1	no	—
47	C₂×Dic₁₄₉	—	4	1	no	—
48	C₂×Dic₁₄₉	—	4	1	no	—
49	#11924746	—	4	1	no	—
50	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
53	C₈×C₁₄₉	—	2	1	no	—
54	C₈×C₁₄₉	—	4	1	no	—
55	C₈×C₁₄₉	—	4	1	no	—
56	C₈×C₁₄₉	Q₁₆	1	2	yes	complemented
57	C₂²×C₁₄₉	#121	1	1	yes	characteristic
58	C₄×C₁₄₉	—	2	1	no	—
59	C₄×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₁₄₉	#137	1	1	yes	characteristic
62	C₄×C₁₄₉	#137	1	1	yes	characteristic
63	Dic₁₄₉	—	4	1	no	—
64	Dic₁₄₉	—	4	1	no	—
65	Dic₁₄₉	—	8	1	no	—
66	Dic₁₄₉	—	8	1	no	—
67	C₂×C₁₄₉	#462	1	1	yes	characteristic
68	C₂×C₁₄₉	#468	1	1	yes	characteristic
69	C₂×C₁₄₉	#471	1	1	yes	characteristic
70	C₁₄₉	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	149	1	no	maximal, Sylow
72	#498	—	149	1	no	—
73	#477	—	149	1	no	—
74	#462	—	149	1	no	—
75	#446	—	149	1	no	—
76	#335	D₁₄₉	1	1	yes	characteristic
77	#333	—	149	1	no	—
78	#332	—	149	1	no	—
79	C₂×Q₁₆	—	298	1	no	—
80	C₄⋊Q₈	—	149	1	no	—
81	C₄⋊Q₈	—	149	1	no	—
82	C₄×Q₈	—	149	1	no	—
83	#107	—	298	1	no	—
84	#106	—	298	1	no	—
85	#106	—	298	1	no	—
86	C₄⋊C₈	—	298	1	no	—
87	C₄⋊C₈	—	149	1	no	—
88	C₄⋊C₈	C₂×D₁₄₉	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₁₄₉	1	1	yes	characteristic
90	#103	—	298	1	no	—
91	#103	—	298	1	no	—
92	#97	—	298	1	no	—
93	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
94	C₂×Q₈	—	298	1	no	—
95	C₂×Q₈	—	298	1	no	—
96	C₂×Q₈	—	149	1	no	—
97	Q₁₆	—	596	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	298	1	no	—
101	C₂×C₈	D₅₉₆	1	1	yes	characteristic
102	C₂×C₈	D₅₉₆	1	1	yes	characteristic
103	C₄⋊C₄	—	298	1	no	—
104	C₄⋊C₄	—	298	1	no	—
105	C₄⋊C₄	—	298	1	no	—
106	C₄⋊C₄	—	298	1	no	—
107	C₄⋊C₄	—	298	1	no	—
108	C₄⋊C₄	—	149	1	no	—
109	C₄²	—	298	1	no	—
110	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
111	Q₈	—	298	1	no	—
112	Q₈	—	298	1	no	—
113	Q₈	—	596	1	no	—
114	Q₈	—	596	1	no	—
115	Q₈	—	1192	1	no	—
116	C₂×C₄	—	298	1	no	—
117	C₂×C₄	—	298	1	no	—
118	C₂×C₄	—	596	1	no	—
119	C₂×C₄	—	596	1	no	—
120	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
121	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
122	C₂×C₄	#14558737	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	596	1	no	—
127	C₈	Dic₅₉₆	1	2	yes	complemented
128	C₂²	#18011886	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	596	1	no	—
131	C₄	—	596	1	no	—
132	C₄	—	1192	1	no	—
133	C₄	—	1192	1	no	—
134	C₄	C₂×Dic₅₉₆	1	1	yes	characteristic
135	C₄	#18011935	1	1	yes	characteristic
136	C₄	#18011943	1	1	yes	characteristic
137	C₄	#18011944	1	1	yes	characteristic
138	C₂	#23749836	1	1	yes	characteristic
139	C₂	#23749872	1	1	yes	characteristic
140	C₂	#23749889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750023

C₂

yes

maximal, characteristic

#23749921

C₂

yes

maximal, characteristic

#23749836

C₂

yes

maximal, characteristic

#23749755

C₂

yes

maximal, characteristic

#23749373

C₂

yes

maximal, characteristic

#23749372

C₂

yes

maximal, characteristic

#335×C₁₄₉

C₂

yes

maximal, characteristic

C₂×Dic₅₉₆

—

#18011911

C₂²

yes

characteristic

#18011911

C₂²

yes

characteristic

C₄×Dic₂₉₈

C₂²

yes

—

—

—

C₂²

yes

—

—

—

—

C₄⋊C₈×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

C₂²

yes

characteristic

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

D₄

yes

characteristic

Dic₅₉₆

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

D₄

yes

characteristic

#14558715

—

C₄×Dic₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

characteristic

C₂×C₈×C₁₄₉

D₄

yes

characteristic

C₄²×C₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₈×C₁₄₉

—

C₈×C₁₄₉

—

C₈×C₁₄₉

—

C₈×C₁₄₉

Q₁₆

yes

complemented

C₂²×C₁₄₉

#121

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂×Q₁₆

yes

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#1692

—

149

maximal, Sylow

#498

—

149

—

#477

—

149

—

#462

—

149

—

#446

—

149

—

#335

D₁₄₉

yes

characteristic

#333

—

149

—

#332

—

149

—

C₂×Q₁₆

—

298

—

C₄⋊Q₈

—

149

—

C₄⋊Q₈

—

149

—

C₄×Q₈

—

149

—

#107

—

298

—

#106

—

298

—

#106

—

298

—

C₄⋊C₈

—

298

—

C₄⋊C₈

—

149

—

C₄⋊C₈

C₂×D₁₄₉

yes

characteristic

C₄⋊C₈

C₂×D₁₄₉

yes

characteristic

#103

—

298

—

#103

—

298

—

#97

—

298

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

Q₁₆

—

596

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

298

—

101

C₂×C₈

D₅₉₆

yes

characteristic

102

C₂×C₈

D₅₉₆

yes

characteristic

103

C₄⋊C₄

—

298

—

104

C₄⋊C₄

—

298

—

105

C₄⋊C₄

—

298

—

106

C₄⋊C₄

—

298

—

107

C₄⋊C₄

—

298

—

108

C₄⋊C₄

—

149

—

109

C₄²

—

298

—

110

C₄²

C₂²×D₁₄₉

yes

characteristic

111

Q₈

—

298

—

112

Q₈

—

298

—

113

Q₈

—

596

—

114

Q₈

—

596

—

115

Q₈

—

1192

—

116

C₂×C₄

—

298

—

117

C₂×C₄

—

298

—

118

C₂×C₄

—

596

—

119

C₂×C₄

—

596

—

120

C₂×C₄

D₄×D₁₄₉

yes

characteristic

121

C₂×C₄

C₂×D₅₉₆

yes

characteristic

122

C₂×C₄

#14558737

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

596

—

127

C₈

Dic₅₉₆

yes

complemented

128

C₂²

#18011886

yes

characteristic

129

C₄

—

130

C₄

—

596

—

131

C₄

—

596

—

132

C₄

—

1192

—

133

C₄

—

1192

—

134

C₄

C₂×Dic₅₉₆

yes

characteristic

135

C₄

#18011935

yes

characteristic

136

C₄

#18011943

yes

characteristic

137

C₄

#18011944

yes

characteristic

138

C₂

#23749836

yes

characteristic

139

C₂

#23749872

yes

characteristic

140

C₂

#23749889

yes

characteristic

	subgroup	G/H
center	C₂²	#18011886
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	#335×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	#121
hypercenter	#335	D₁₄₉
hypocenter	C₁₄₉	#1692
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#18011886

commutator, cosocle

C₄²×C₁₄₉

C₂³

Frattini

C₄²

C₂²×D₁₄₉

Fitting

#335×C₁₄₉

C₂

C₂²×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
56. C₈×C₁₄₉	Q₁₆	semidirect
70. C₁₄₉	#1692	semidirect
127. C₈	Dic₅₉₆	semidirect

H ≅ G/N

type

56. C₈×C₁₄₉

Q₁₆

semidirect

70. C₁₄₉

#1692

semidirect

127. C₈

Dic₅₉₆

semidirect

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

subgroup	index
#23750023	2
#23749921	2
#23749836	2
#23749755	2
#23749373	2
#23749372	2
#335×C₁₄₉	2
#1692	149

subgroup

index

#335×C₁₄₉

149

Subgroups — #34044198

Conjugacy classes of subgroups

Complements

Maximal subgroups