Group #22580841 — subgroups

Subgroups — #22580841

Order 8896.

190 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 69 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×Dic₁₃₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃₉⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃₉⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₃₉	C₂²	1	1	yes	characteristic
12	Dic₁₃₉⋊C₄	—	2	2	no	—
13	C₂²⋊Dic₁₃₉	—	2	2	no	—
14	C₂²⋊Dic₁₃₉	C₄	1	4	yes	complemented
15	C₄×Dic₁₃₉	—	2	2	no	—
16	C₂⁴×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
20	C₂×Dic₁₃₉	—	2	1	no	—
21	C₂×Dic₁₃₉	—	2	1	no	—
22	C₂×Dic₁₃₉	—	2	1	no	—
23	C₂×Dic₁₃₉	—	2	1	no	—
24	C₂×Dic₁₃₉	—	2	1	no	—
25	C₂×Dic₁₃₉	—	2	1	no	—
26	C₂×Dic₁₃₉	—	2	1	no	—
27	C₂×Dic₁₃₉	—	2	1	no	—
28	C₂×Dic₁₃₉	—	2	1	no	—
29	C₂×Dic₁₃₉	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₃₉	D₄	1	2	yes	—
32	C₂³×C₁₃₉	—	2	1	no	—
33	C₂³×C₁₃₉	—	2	1	no	—
34	C₂³×C₁₃₉	—	2	1	no	—
35	C₂³×C₁₃₉	—	2	1	no	—
36	C₂³×C₁₃₉	—	2	1	no	—
37	C₂³×C₁₃₉	—	2	1	no	—
38	C₂³×C₁₃₉	C₂×C₄	1	2	yes	—
39	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	—	2	1	no	—
41	C₂×C₄×C₁₃₉	—	2	1	no	—
42	C₂×C₄×C₁₃₉	—	2	1	no	—
43	C₂×C₄×C₁₃₉	—	2	1	no	—
44	C₂×C₄×C₁₃₉	—	2	1	no	—
45	C₂×C₄×C₁₃₉	D₄	1	2	yes	—
46	C₂²×C₁₃₉	—	2	2	no	—
47	C₂²×C₁₃₉	—	2	2	no	—
48	C₂²×C₁₃₉	—	2	2	no	—
49	C₂²×C₁₃₉	—	4	1	no	—
50	C₂²×C₁₃₉	—	4	1	no	—
51	C₂²×C₁₃₉	—	4	1	no	—
52	C₂²×C₁₃₉	—	4	1	no	—
53	C₂²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₁₃₉	#41	1	1	yes	characteristic
57	C₂²×C₁₃₉	#41	1	1	yes	characteristic
58	C₂²×C₁₃₉	#41	1	1	yes	characteristic
59	C₂²×C₁₃₉	#41	1	1	yes	characteristic
60	C₄×C₁₃₉	—	2	2	no	—
61	C₄×C₁₃₉	—	4	1	no	—
62	Dic₁₃₉	—	2	2	no	—
63	Dic₁₃₉	—	2	2	no	—
64	Dic₁₃₉	—	2	2	no	—
65	Dic₁₃₉	—	4	1	no	—
66	C₂×C₁₃₉	—	4	2	no	—
67	C₂×C₁₃₉	#117	1	1	yes	characteristic
68	C₂×C₁₃₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₁₃₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₁₃₉	#121	1	1	yes	characteristic
71	C₂×C₁₃₉	#123	1	1	yes	characteristic
72	C₂×C₁₃₉	#124	1	1	yes	characteristic
73	C₂×C₁₃₉	#126	1	1	yes	characteristic
74	C₁₃₉	#388	1	1	yes	characteristic, complemented, Sylow
75	#388	—	139	1	no	maximal, Sylow
76	C₂×C₄⋊C₄	—	139	1	no	—
77	C₂×C₂²⋊C₄	—	139	1	no	—
78	C₂×C₂²⋊C₄	—	139	1	no	—
79	C₂×C₂²⋊C₄	D₁₃₉	1	1	yes	characteristic
80	C₂×C₄²	—	139	1	no	—
81	#95	—	139	1	no	—
82	#95	—	139	1	no	—
83	C₂⁴	C₂×D₁₃₉	1	1	yes	characteristic
84	C₂²×C₄	—	139	1	no	—
85	C₂²×C₄	—	139	1	no	—
86	C₂²×C₄	—	139	1	no	—
87	C₂²×C₄	—	139	1	no	—
88	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
90	C₄⋊C₄	—	278	2	no	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₂²⋊C₄	—	278	2	no	—
93	C₂²⋊C₄	—	139	4	no	—
94	C₄²	—	278	2	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	2	1	no	—
97	C₂³	—	2	1	no	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	1	no	—
101	C₂³	C₄×D₁₃₉	1	2	yes	—
102	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	278	1	no	—
109	C₂×C₄	—	278	1	no	—
110	C₂×C₄	—	278	1	no	—
111	C₂×C₄	—	278	1	no	—
112	C₂×C₄	—	278	1	no	—
113	C₂×C₄	—	278	1	no	—
114	C₂×C₄	—	278	1	no	—
115	C₂×C₄	—	278	1	no	—
116	C₂×C₄	—	278	1	no	—
117	C₂×C₄	—	139	2	no	—
118	C₂×C₄	—	139	2	no	—
119	C₂×C₄	—	139	2	no	—
120	C₂×C₄	C₂²⋊D₁₃₉	1	2	yes	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂×C₄×D₁₃₉	1	1	yes	characteristic
129	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
130	C₂²	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
131	C₂²	#14555784	1	1	yes	characteristic
132	C₂²	#14555784	1	1	yes	characteristic
133	C₂²	#14555784	1	1	yes	characteristic
134	C₂²	#14555785	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	4	1	no	—
137	C₄	—	278	2	no	—
138	C₄	—	278	2	no	—
139	C₄	—	278	2	no	—
140	C₄	—	556	1	no	—
141	C₂	—	4	2	no	—
142	C₂	#17984798	1	1	yes	characteristic
143	C₂	(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
144	C₂	C₄×C₂²⋊D₁₃₉	1	1	yes	characteristic
145	C₂	#17984813	1	1	yes	characteristic
146	C₂	#17984823	1	1	yes	characteristic
147	C₂	#17984828	1	1	yes	characteristic
148	C₂	#17984835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₃₉⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₃₉

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₃₉

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₁₃₉⋊#95

C₂

yes

maximal, characteristic, complemented

C₁₃₉⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

C₄

yes

complemented

C₄×Dic₁₃₉

—

C₂⁴×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

D₄

yes

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂×C₄

yes

—

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×C₄

yes

characteristic

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×D₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#117

yes

characteristic

C₂×C₁₃₉

C₄×D₄

yes

characteristic

C₂×C₁₃₉

C₄×D₄

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#388

—

139

maximal, Sylow

C₂×C₄⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₂²⋊C₄

D₁₃₉

yes

characteristic

C₂×C₄²

—

139

—

#95

—

139

—

#95

—

139

—

C₂⁴

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₂²⋊C₄

—

C₂²⋊C₄

—

278

—

C₂²⋊C₄

—

139

—

C₄²

—

278

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₄×D₁₃₉

yes

—

102

C₂³

C₂²×D₁₃₉

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

278

—

109

C₂×C₄

—

278

—

110

C₂×C₄

—

278

—

111

C₂×C₄

—

278

—

112

C₂×C₄

—

278

—

113

C₂×C₄

—

278

—

114

C₂×C₄

—

278

—

115

C₂×C₄

—

278

—

116

C₂×C₄

—

278

—

117

C₂×C₄

—

139

—

118

C₂×C₄

—

139

—

119

C₂×C₄

—

139

—

120

C₂×C₄

C₂²⋊D₁₃₉

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×C₄×D₁₃₉

yes

characteristic

129

C₂²

D₄×D₁₃₉

yes

characteristic

130

C₂²

C₂×C₂²⋊D₁₃₉

yes

characteristic

131

C₂²

#14555784

yes

characteristic

132

C₂²

#14555784

yes

characteristic

133

C₂²

#14555784

yes

characteristic

134

C₂²

#14555785

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

278

—

138

C₄

—

278

—

139

C₄

—

278

—

140

C₄

—

556

—

141

C₂

—

142

C₂

#17984798

yes

characteristic

143

C₂

(C₂²⋊D₁₃₉)⋊C₄

yes

characteristic

144

C₂

C₄×C₂²⋊D₁₃₉

yes

characteristic

145

C₂

#17984813

yes

characteristic

146

C₂

#17984823

yes

characteristic

147

C₂

#17984828

yes

characteristic

148

C₂

#17984835

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₄×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂³
hypercenter	C₂×C₂²⋊C₄	D₁₃₉
hypocenter	C₁₃₉	#388
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₄×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×Dic₁₃₉⋊C₄	C₂	semidirect
4. C₂×C₄×Dic₁₃₉	C₂	semidirect
5. C₁₃₉⋊#95	C₂	semidirect
6. C₁₃₉⋊#95	C₂	semidirect
14. C₂²⋊Dic₁₃₉	C₄	semidirect
74. C₁₃₉	#388	semidirect

H ≅ G/N

type

1. C₂×Dic₁₃₉⋊C₄

C₂

semidirect

4. C₂×C₄×Dic₁₃₉

C₂

semidirect

5. C₁₃₉⋊#95

C₂

semidirect

6. C₁₃₉⋊#95

C₂

semidirect

14. C₂²⋊Dic₁₃₉

C₄

semidirect

74. C₁₃₉

#388

semidirect

Subgroups — #22580841

Conjugacy classes of subgroups

Complements