Group #34056036 — subgroups

Subgroups — #34056036

Order 19072.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#23750182	C₂	1	2	yes	maximal
2	#23749921	C₂	1	4	yes	maximal, complemented
3	#23749866	C₂	1	4	yes	maximal
4	#23749836	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₉₆	C₂²	1	4	yes	—
7	#18011911	C₂²	1	2	yes	—
8	#18011911	C₂²	1	2	yes	—
9	#18011903	—	2	2	no	—
10	#18011890	—	2	2	no	—
11	#18011890	C₂²	1	4	yes	—
12	C₄×Dic₂₉₈	—	2	2	no	—
13	C₄×Dic₂₉₈	C₂²	1	4	yes	—
14	#18011870	—	2	2	no	—
15	#18011845	C₂²	1	4	yes	—
16	#18011836	C₂²	1	8	yes	—
17	C₂²×C₈×C₁₄₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₄₉	C₂²	1	4	yes	—
20	C₂×Dic₂₉₈	—	2	2	no	—
21	C₂×Dic₂₉₈	C₂³	1	4	yes	—
22	Dic₅₉₆	—	4	4	no	—
23	C₄⋊Dic₁₄₉	—	2	2	no	—
24	C₄⋊Dic₁₄₉	—	2	2	no	—
25	C₄⋊Dic₁₄₉	—	2	2	no	—
26	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
27	#14558715	—	2	4	no	—
28	#14558715	—	2	4	no	—
29	C₂²⋊Dic₁₄₉	—	2	4	no	—
30	C₄×Dic₁₄₉	—	2	4	no	—
31	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₄₉	—	4	2	no	—
35	C₂×C₈×C₁₄₉	C₂³	1	4	yes	—
36	C₄²×C₁₄₉	D₄	1	2	yes	—
37	C₄²×C₁₄₉	C₂³	1	2	yes	—
38	Dic₂₉₈	—	2	4	no	—
39	Dic₂₉₈	—	2	4	no	—
40	Dic₂₉₈	—	4	4	no	—
41	C₂×Dic₁₄₉	—	2	4	no	—
42	C₂×Dic₁₄₉	—	2	4	no	—
43	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₄₉	—	2	1	no	—
45	C₂×C₄×C₁₄₉	—	2	1	no	—
46	C₂×C₄×C₁₄₉	—	4	1	no	—
47	C₂×C₄×C₁₄₉	Q₁₆	1	4	yes	complemented
48	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
52	C₈×C₁₄₉	—	2	4	no	—
53	C₂²×C₁₄₉	—	2	1	no	—
54	C₂²×C₁₄₉	—	2	1	no	—
55	C₂²×C₁₄₉	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₁₄₉	—	2	1	no	—
58	C₄×C₁₄₉	—	2	2	no	—
59	C₄×C₁₄₉	C₂×Q₁₆	1	4	yes	—
60	C₄×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₁₄₉	—	4	4	no	—
63	Dic₁₄₉	—	4	4	no	—
64	C₂×C₁₄₉	—	2	2	no	—
65	C₂×C₁₄₉	#537	1	1	yes	characteristic
66	C₂×C₁₄₉	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂×C₁₄₉	#575	1	1	yes	characteristic
68	C₁₄₉	#3249	1	1	yes	characteristic, complemented, Sylow
69	#3249	—	149	1	no	maximal, Sylow
70	#533	—	149	2	no	—
71	#477	—	149	4	no	—
72	#467	—	149	4	no	—
73	#462	—	149	4	no	—
74	C₂×C₄⋊C₈	D₁₄₉	1	1	yes	characteristic
75	C₂×Q₁₆	—	149	4	no	—
76	C₂²×C₈	C₂×D₁₄₉	1	2	yes	—
77	C₄⋊Q₈	—	149	2	no	—
78	C₄⋊Q₈	—	149	2	no	—
79	#125	—	298	2	no	—
80	C₂²⋊Q₈	—	298	2	no	—
81	C₂²⋊Q₈	—	149	4	no	—
82	C₄×Q₈	—	298	2	no	—
83	C₄×Q₈	—	149	4	no	—
84	#117	—	298	2	no	—
85	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
86	#107	—	149	4	no	—
87	C₄⋊C₈	C₂×D₁₄₉	1	4	yes	—
88	#103	—	149	8	no	—
89	C₂×Q₈	—	298	2	no	—
90	C₂×Q₈	—	149	4	no	—
91	C₂²×C₄	D₅₉₆	1	1	yes	characteristic
92	C₂²×C₄	D₅₉₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
94	Q₁₆	—	596	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₁₄₉	1	4	yes	—
97	C₄⋊C₄	—	298	2	no	—
98	C₄⋊C₄	—	298	2	no	—
99	C₄⋊C₄	—	298	2	no	—
100	C₄⋊C₄	—	298	4	no	—
101	C₄⋊C₄	—	298	4	no	—
102	C₄⋊C₄	—	149	4	no	—
103	C₂²⋊C₄	—	298	4	no	—
104	C₄²	—	298	4	no	—
105	C₄²	D₅₉₆	1	2	yes	—
106	C₄²	C₂²×D₁₄₉	1	2	yes	—
107	C₂³	C₂×D₅₉₆	1	1	yes	characteristic
108	Q₈	—	298	4	no	—
109	Q₈	—	298	4	no	—
110	Q₈	—	596	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	298	4	no	—
115	C₂×C₄	—	298	4	no	—
116	C₂×C₄	Dic₅₉₆	1	4	yes	complemented
117	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
118	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
119	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
120	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₅₉₆	1	2	yes	—
125	C₂²	C₂²×D₅₉₆	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	596	4	no	—
129	C₄	—	596	4	no	—
130	C₄	C₂×Dic₅₉₆	1	4	yes	—
131	C₄	#18011956	1	1	yes	characteristic
132	C₄	#18011959	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#23750206	1	1	yes	characteristic
135	C₂	C₂²×Dic₅₉₆	1	1	yes	characteristic
136	C₂	#23750425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750182

C₂

yes

maximal

#23749921

C₂

yes

maximal, complemented

#23749866

C₂

yes

maximal

#23749836

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₁₄₉

C₂

yes

maximal, characteristic

C₂×Dic₅₉₆

C₂²

yes

—

#18011911

C₂²

yes

—

#18011911

C₂²

yes

—

#18011903

—

#18011890

—

#18011890

C₂²

yes

—

C₄×Dic₂₉₈

—

C₄×Dic₂₉₈

C₂²

yes

—

#18011870

—

#18011845

C₂²

yes

—

#18011836

C₂²

yes

—

C₂²×C₈×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₂³

yes

—

Dic₅₉₆

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂³

yes

—

#14558715

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

C₂³

yes

—

C₄²×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

C₂³

yes

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

Q₁₆

yes

complemented

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×Q₁₆

yes

—

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂×Q₁₆

yes

—

C₄×C₁₄₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₄₉

2₋¹⁺⁴

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#537

yes

characteristic

C₂×C₁₄₉

C₂²×Q₁₆

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#3249

—

149

maximal, Sylow

#533

—

149

—

#477

—

149

—

#467

—

149

—

#462

—

149

—

C₂×C₄⋊C₈

D₁₄₉

yes

characteristic

C₂×Q₁₆

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

—

C₄⋊Q₈

—

149

—

C₄⋊Q₈

—

149

—

#125

—

298

—

C₂²⋊Q₈

—

298

—

C₂²⋊Q₈

—

149

—

C₄×Q₈

—

298

—

C₄×Q₈

—

149

—

#117

—

298

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#107

—

149

—

C₄⋊C₈

C₂×D₁₄₉

yes

—

#103

—

149

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂²×C₄

D₅₉₆

yes

characteristic

C₂²×C₄

D₅₉₆

yes

characteristic

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

Q₁₆

—

596

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₄₉

yes

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

100

C₄⋊C₄

—

298

—

101

C₄⋊C₄

—

298

—

102

C₄⋊C₄

—

149

—

103

C₂²⋊C₄

—

298

—

104

C₄²

—

298

—

105

C₄²

D₅₉₆

yes

—

106

C₄²

C₂²×D₁₄₉

yes

—

107

C₂³

C₂×D₅₉₆

yes

characteristic

108

Q₈

—

298

—

109

Q₈

—

298

—

110

Q₈

—

596

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

298

—

115

C₂×C₄

—

298

—

116

C₂×C₄

Dic₅₉₆

yes

complemented

117

C₂×C₄

C₂×D₅₉₆

yes

characteristic

118

C₂×C₄

C₂×D₅₉₆

yes

—

119

C₂×C₄

C₂×D₅₉₆

yes

—

120

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₅₉₆

yes

—

125

C₂²

C₂²×D₅₉₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

596

—

129

C₄

—

596

—

130

C₄

C₂×Dic₅₉₆

yes

—

131

C₄

#18011956

yes

characteristic

132

C₄

#18011959

yes

characteristic

133

C₂

—

134

C₂

#23750206

yes

characteristic

135

C₂

C₂²×Dic₅₉₆

yes

characteristic

136

C₂

#23750425

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₉₆
commutator, cosocle	C₂×C₄×C₁₄₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	C₂×C₄⋊C₈×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂²×D₄
prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₄₉
hypocenter	C₁₄₉	#3249
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₅₉₆

commutator, cosocle

C₂×C₄×C₁₄₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting

C₂×C₄⋊C₈×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂²×D₄

prime radical

C₂³×C₁₄₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₄₉

hypocenter

C₁₄₉

#3249

LCS

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #23749921	C₂	semidirect
4. #23749836	C₂	semidirect
47. C₂×C₄×C₁₄₉	Q₁₆	semidirect
68. C₁₄₉	#3249	semidirect
116. C₂×C₄	Dic₅₉₆	semidirect

H ≅ G/N

type

2. #23749921

C₂

semidirect

4. #23749836

C₂

semidirect

47. C₂×C₄×C₁₄₉

Q₁₆

semidirect

68. C₁₄₉

#3249

semidirect

116. C₂×C₄

Dic₅₉₆

semidirect

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

subgroup	index
#23750182	2
#23749921	2
#23749866	2
#23749836	2
C₂×C₄⋊C₈×C₁₄₉	2
#3249	149

subgroup

index

C₂×C₄⋊C₈×C₁₄₉

149

Subgroups — #34056036

Conjugacy classes of subgroups

Complements

Maximal subgroups