Group #34055065 — subgroups

Subgroups — #34055065

Order 19072.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	1	yes	maximal, characteristic, complemented
2	#23749989	C₂	1	4	yes	maximal, complemented
3	#23749971	C₂	1	4	yes	maximal, complemented
4	#23749630	C₂	1	4	yes	maximal, complemented
5	C₄×#14558720	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#127×C₁₄₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#14558720	C₂²	1	2	yes	—
8	#18011911	C₂²	1	2	yes	complemented
9	#18011903	C₂²	1	2	yes	complemented
10	#18011890	—	2	2	no	—
11	C₄×Dic₂₉₈	—	2	2	no	—
12	#18011870	C₂²	1	2	yes	—
13	#18011832	C₂²	1	16	yes	complemented
14	#18011822	C₂²	1	2	yes	complemented
15	C₄×#11924746	C₂²	1	2	yes	complemented
16	C₂²×D₄×C₁₄₉	—	2	2	no	—
17	C₂²×D₄×C₁₄₉	C₂²	1	2	yes	—
18	#127×C₁₄₉	—	2	2	no	—
19	#127×C₁₄₉	C₂²	1	2	yes	—
20	#127×C₁₄₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₉₈	—	2	2	no	—
23	#14558720	—	2	4	no	—
24	C₂×#11924746	C₂³	1	4	yes	—
25	C₄⋊Dic₁₄₉	—	2	2	no	—
26	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
27	#14558715	—	2	4	no	—
28	C₂²⋊Dic₁₄₉	—	2	2	no	—
29	C₄×Dic₁₄₉	—	2	2	no	—
30	C₂⁴×C₁₄₉	—	2	2	no	—
31	C₂×D₄×C₁₄₉	—	2	2	no	—
32	C₂×D₄×C₁₄₉	—	2	4	no	—
33	C₂×D₄×C₁₄₉	—	2	4	no	—
34	C₂×D₄×C₁₄₉	—	2	8	no	—
35	C₂×D₄×C₁₄₉	—	4	8	no	—
36	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
38	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
39	C₄²×C₁₄₉	D₄	1	2	yes	—
40	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
42	Dic₂₉₈	—	4	4	no	—
43	C₂×Dic₁₄₉	—	2	4	no	—
44	#11924746	—	2	4	no	—
45	C₂³×C₁₄₉	—	2	4	no	—
46	C₂³×C₁₄₉	—	4	8	no	—
47	C₂³×C₁₄₉	—	4	4	no	—
48	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	D₄×C₁₄₉	—	2	8	no	—
50	D₄×C₁₄₉	—	4	8	no	—
51	D₄×C₁₄₉	—	4	4	no	—
52	D₄×C₁₄₉	—	4	8	no	—
53	C₂×C₄×C₁₄₉	—	2	4	no	—
54	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₄₉	#41	1	4	yes	—
58	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₄₉	—	2	2	no	—
60	C₂²×C₁₄₉	—	4	4	no	—
61	C₂²×C₁₄₉	—	4	8	no	—
62	C₂²×C₁₄₉	—	8	8	no	—
63	C₂²×C₁₄₉	#124	1	2	yes	—
64	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₄₉	—	2	2	no	—
66	C₄×C₁₄₉	#124	1	2	yes	—
67	C₄×C₁₄₉	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₁₄₉	C₂×#41	1	2	yes	—
69	Dic₁₄₉	—	4	4	no	—
70	C₂×C₁₄₉	—	2	2	no	—
71	C₂×C₁₄₉	—	8	4	no	—
72	C₂×C₁₄₉	C₂×#124	1	1	yes	characteristic
73	C₂×C₁₄₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	C₁₄₉	#3138	1	1	yes	characteristic, complemented, Sylow
75	#3138	—	149	1	no	maximal, Sylow
76	#533	—	149	1	no	—
77	C₂×#127	D₁₄₉	1	1	yes	characteristic
78	#490	—	149	4	no	—
79	#486	—	149	4	no	—
80	#418	—	149	4	no	—
81	C₄×M₁₆	—	149	1	no	—
82	C₂²×D₄	—	2	2	no	—
83	C₂²×D₄	C₂×D₁₄₉	1	2	yes	—
84	C₂×M₁₆	—	149	2	no	—
85	C₄⋊Q₈	—	149	2	no	—
86	#127	—	2	2	no	—
87	#127	C₂×D₁₄₉	1	2	yes	—
88	#127	C₂×D₁₄₉	1	2	yes	—
89	#125	—	149	2	no	—
90	C₂²⋊Q₈	—	298	2	no	—
91	C₄×Q₈	—	298	2	no	—
92	#117	—	149	2	no	—
93	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
94	#102	—	149	16	no	—
95	#97	—	149	2	no	—
96	C₄×C₈	—	149	2	no	—
97	C₂⁴	—	2	2	no	—
98	C₂×Q₈	—	298	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	2	8	no	—
103	C₂×D₄	—	4	8	no	—
104	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
105	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
107	M₁₆	—	298	4	no	—
108	C₂×C₈	—	149	4	no	—
109	C₄⋊C₄	—	298	4	no	—
110	C₄⋊C₄	—	298	2	no	—
111	C₄⋊C₄	—	149	4	no	—
112	C₂²⋊C₄	—	298	2	no	—
113	C₄²	—	298	2	no	—
114	C₄²	C₂²⋊D₁₄₉	1	2	yes	—
115	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
116	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
117	C₂³	—	2	4	no	—
118	C₂³	—	4	8	no	—
119	C₂³	—	4	4	no	—
120	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
121	Q₈	—	596	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	8	no	—
124	D₄	—	4	8	no	—
125	D₄	—	4	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	298	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
131	C₂×C₄	#14558735	1	4	yes	—
132	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
133	C₈	—	298	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	8	no	—
136	C₂²	—	4	4	no	—
137	C₂²	—	8	8	no	—
138	C₂²	#18011898	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₄₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	596	4	no	—
142	C₄	#18011898	1	2	yes	—
143	C₄	#18011937	1	4	yes	—
144	C₄	C₂×#14558735	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	8	4	no	—
147	C₂	C₂×#18011898	1	1	yes	characteristic
148	C₂	C₂×#18011937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23750182

C₂

yes

maximal, characteristic, complemented

#23749989

C₂

yes

maximal, complemented

#23749971

C₂

yes

maximal, complemented

#23749630

C₂

yes

maximal, complemented

C₄×#14558720

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558720

C₂²

yes

—

#18011911

C₂²

yes

complemented

#18011903

C₂²

yes

complemented

#18011890

—

C₄×Dic₂₉₈

—

#18011870

C₂²

yes

—

#18011832

C₂²

yes

complemented

#18011822

C₂²

yes

complemented

C₄×#11924746

C₂²

yes

complemented

C₂²×D₄×C₁₄₉

—

C₂²×D₄×C₁₄₉

C₂²

yes

—

#127×C₁₄₉

—

#127×C₁₄₉

C₂²

yes

—

#127×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

#14558720

—

C₂×#11924746

C₂³

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂³

yes

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂⁴×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#124

yes

—

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

#124

yes

—

C₄×C₁₄₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₄₉

C₂×#41

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×#124

yes

characteristic

C₂×C₁₄₉

C₂×AGL(1,ℤ/8)

yes

—

C₁₄₉

#3138

yes

characteristic, complemented, Sylow

#3138

—

149

maximal, Sylow

#533

—

149

—

C₂×#127

D₁₄₉

yes

characteristic

#490

—

149

—

#486

—

149

—

#418

—

149

—

C₄×M₁₆

—

149

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₄₉

yes

—

C₂×M₁₆

—

149

—

C₄⋊Q₈

—

149

—

#127

—

#127

C₂×D₁₄₉

yes

—

#127

C₂×D₁₄₉

yes

—

#125

—

149

—

C₂²⋊Q₈

—

298

—

C₄×Q₈

—

298

—

#117

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#102

—

149

—

#97

—

149

—

C₄×C₈

—

149

—

C₂⁴

—

C₂×Q₈

—

298

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂²×D₁₄₉

yes

—

105

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

106

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

107

M₁₆

—

298

—

108

C₂×C₈

—

149

—

109

C₄⋊C₄

—

298

—

110

C₄⋊C₄

—

298

—

111

C₄⋊C₄

—

149

—

112

C₂²⋊C₄

—

298

—

113

C₄²

—

298

—

114

C₄²

C₂²⋊D₁₄₉

yes

—

115

C₄²

C₂²×D₁₄₉

yes

characteristic

116

C₄²

C₂²×D₁₄₉

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

121

Q₈

—

596

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

298

—

128

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

131

C₂×C₄

#14558735

yes

—

132

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

133

C₈

—

298

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#18011898

yes

—

139

C₂²

C₂²×C₂²⋊D₁₄₉

yes

characteristic

140

C₄

—

141

C₄

—

596

—

142

C₄

#18011898

yes

—

143

C₄

#18011937

yes

—

144

C₄

C₂×#14558735

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#18011898

yes

characteristic

148

C₂

C₂×#18011937

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₄₉
commutator, cosocle	C₂×C₄×C₁₄₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting, prime radical	C₂×#127×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂²×D₄
hypercenter	C₂×#127	D₁₄₉
hypocenter	C₁₄₉	#3138
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₄₉

commutator, cosocle

C₂×C₄×C₁₄₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting, prime radical

C₂×#127×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂²×D₄

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #23750182	C₂	semidirect
2. #23749989	C₂	semidirect
3. #23749971	C₂	semidirect
4. #23749630	C₂	semidirect
5. C₄×#14558720	C₂	semidirect
8. #18011911	C₂²	semidirect
9. #18011903	C₂²	semidirect
13. #18011832	C₂²	semidirect
14. #18011822	C₂²	semidirect
15. C₄×#11924746	C₂²	semidirect
74. C₁₄₉	#3138	semidirect

H ≅ G/N

type

1. #23750182

C₂

semidirect

2. #23749989

C₂

semidirect

3. #23749971

C₂

semidirect

4. #23749630

C₂

semidirect

5. C₄×#14558720

C₂

semidirect

8. #18011911

C₂²

semidirect

9. #18011903

C₂²

semidirect

13. #18011832

C₂²

semidirect

14. #18011822

C₂²

semidirect

15. C₄×#11924746

C₂²

semidirect

74. C₁₄₉

#3138

semidirect

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

subgroup	index
#23750182	2
#23749989	2
#23749971	2
#23749630	2
C₄×#14558720	2
C₂×#127×C₁₄₉	2
#3138	149

subgroup

index

C₂×#127×C₁₄₉

149

Subgroups — #34055065

Conjugacy classes of subgroups

Complements

Maximal subgroups