Group #31806129 — subgroups

Subgroups — #31806129

Order 17792.

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

normal subgroups only

This group has 372 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	C₂²×#14555768	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₃₉	C₂	1	2	yes	maximal
3	#22580718	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580717	C₂	1	2	yes	maximal, complemented
5	#341×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
7	C₂×#14555769	C₂²	1	2	yes	—
8	C₂×#14555768	—	2	2	no	—
9	C₂×#14555768	—	2	4	no	—
10	C₂×#14555768	C₄	1	4	yes	—
11	M₁₆×D₁₃₉	—	2	4	no	—
12	M₁₆×D₁₃₉	C₄	1	8	yes	—
13	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₃₉	—	2	2	no	—
15	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂³×D₁₃₉	Q₈	1	1	yes	characteristic
18	C₂²×Dic₁₃₉	Q₈	1	1	yes	characteristic
19	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
20	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
22	C₂×C₄×D₁₃₉	D₄	1	4	yes	—
23	#14555769	—	2	2	no	—
24	#14555769	C₂×C₄	1	4	yes	—
25	#14555768	—	2	8	no	—
26	#14555768	—	2	8	no	—
27	C₂×C₁₃₉⋊C₈	—	2	2	no	—
28	C₂×C₁₃₉⋊C₈	—	2	2	no	—
29	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₈×D₁₃₉	—	2	8	no	—
31	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₃₉	—	2	2	no	—
33	M₁₆×C₁₃₉	C₂×C₄	1	4	yes	—
34	C₂×C₈×C₁₃₉	—	2	2	no	—
35	C₂×C₈×C₁₃₉	—	2	2	no	—
36	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
37	C₂²×D₁₃₉	—	4	2	no	—
38	C₂²×D₁₃₉	C₄⋊C₄	1	2	yes	—
39	C₂²×D₁₃₉	C₄⋊C₄	1	4	yes	—
40	C₂×Dic₁₃₉	C₄⋊C₄	1	4	yes	—
41	C₂×Dic₁₃₉	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₃₉	C₄²	1	4	yes	—
43	C₄×D₁₃₉	C₂²⋊C₄	1	4	yes	—
44	C₄×D₁₃₉	C₂²⋊C₄	1	4	yes	—
45	C₄×D₁₃₉	C₂²⋊C₄	1	4	yes	—
46	C₁₃₉⋊C₈	—	2	4	no	—
47	C₁₃₉⋊C₈	—	2	2	no	—
48	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₈×C₁₃₉	—	2	2	no	—
54	C₈×C₁₃₉	—	2	4	no	—
55	C₂²×C₁₃₉	—	2	2	no	—
56	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×D₁₃₉	—	2	4	no	—
59	C₂×D₁₃₉	—	4	4	no	—
60	C₂×D₁₃₉	#95	1	4	yes	—
61	C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
64	Dic₁₃₉	#95	1	4	yes	—
65	C₂×C₁₃₉	—	2	2	no	—
66	C₂×C₁₃₉	C₂×#95	1	1	yes	characteristic
67	C₂×C₁₃₉	#430	1	2	yes	—
68	D₁₃₉	—	2	4	no	—
69	C₁₃₉	#1738	1	1	yes	characteristic, complemented, Sylow
70	#1738	—	139	1	no	maximal, Sylow
71	C₂²×M₁₆	—	139	2	no	—
72	C₂²×M₁₆	—	139	1	no	—
73	#341	—	139	2	no	—
74	#341	—	139	1	no	—
75	#341	D₁₃₉	1	1	yes	characteristic, complemented
76	C₂³×C₄	—	139	1	no	—
77	C₂×M₁₆	—	278	4	no	—
78	C₂×M₁₆	—	278	2	no	—
79	C₂×M₁₆	—	278	4	no	—
80	C₂×M₁₆	—	139	2	no	—
81	C₂×M₁₆	—	139	8	no	—
82	C₂×M₁₆	—	139	4	no	—
83	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
84	C₂²×C₈	—	278	2	no	—
85	C₂²×C₈	—	139	1	no	—
86	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
87	C₂⁴	—	139	1	no	—
88	C₂²×C₄	—	139	1	no	—
89	C₂²×C₄	—	139	4	no	—
90	C₂²×C₄	—	139	2	no	—
91	C₂²×C₄	—	139	4	no	—
92	C₂²×C₄	—	139	2	no	—
93	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
94	M₁₆	—	2	2	no	—
95	M₁₆	—	278	8	no	—
96	M₁₆	—	278	8	no	—
97	M₁₆	—	278	2	no	—
98	M₁₆	—	139	4	no	—
99	M₁₆	C₄×D₁₃₉	1	4	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	278	8	no	—
103	C₂×C₈	—	278	2	no	—
104	C₂×C₈	—	278	2	no	—
105	C₂×C₈	—	139	2	no	—
106	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
107	C₂³	—	556	2	no	—
108	C₂³	—	139	2	no	—
109	C₂³	—	139	4	no	—
110	C₂³	Q₈×D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	—	139	4	no	—
112	C₂×C₄	—	139	2	no	—
113	C₂×C₄	—	139	4	no	—
114	C₂×C₄	—	139	4	no	—
115	C₂×C₄	—	139	4	no	—
116	C₂×C₄	—	139	4	no	—
117	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
119	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
120	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	4	no	—
123	C₈	—	278	4	no	—
124	C₈	—	278	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	278	4	no	—
127	C₂²	—	556	4	no	—
128	C₂²	—	139	4	no	—
129	C₂²	C₄⋊C₄×D₁₃₉	1	2	yes	—
130	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
131	C₄	—	139	4	no	—
132	C₄	C₄²×D₁₃₉	1	1	yes	characteristic
133	C₄	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
134	C₄	C₂²⋊C₄×D₁₃₉	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	278	4	no	—
137	C₂	#95×D₁₃₉	1	1	yes	characteristic
138	C₂	#22581000	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14555768

C₂

yes

maximal, characteristic

C₂×M₁₆×D₁₃₉

C₂

yes

maximal

#22580718

C₂

yes

maximal, characteristic, complemented

#22580717

C₂

yes

maximal, complemented

#341×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×#14555769

C₂²

yes

—

C₂×#14555768

—

C₂×#14555768

—

C₂×#14555768

C₄

yes

—

M₁₆×D₁₃₉

—

M₁₆×D₁₃₉

C₄

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₃₉

—

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

Q₈

yes

characteristic

C₂²×Dic₁₃₉

Q₈

yes

characteristic

C₂×C₄×D₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

D₄

yes

—

#14555769

—

#14555769

C₂×C₄

yes

—

#14555768

—

#14555768

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₈×D₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₄⋊C₄

yes

—

C₂²×D₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₄×D₁₃₉

C₄²

yes

—

C₄×D₁₃₉

C₂²⋊C₄

yes

—

C₄×D₁₃₉

C₂²⋊C₄

yes

—

C₄×D₁₃₉

C₂²⋊C₄

yes

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

#95

yes

—

C₄×C₁₃₉

C₂×C₄²

yes

characteristic

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

—

Dic₁₃₉

#95

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#95

yes

characteristic

C₂×C₁₃₉

#430

yes

—

D₁₃₉

—

C₁₃₉

#1738

yes

characteristic, complemented, Sylow

#1738

—

139

maximal, Sylow

C₂²×M₁₆

—

139

—

C₂²×M₁₆

—

139

—

#341

—

139

—

#341

—

139

—

#341

D₁₃₉

yes

characteristic, complemented

C₂³×C₄

—

139

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

278

—

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂⁴

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

M₁₆

—

278

—

M₁₆

—

139

—

M₁₆

C₄×D₁₃₉

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

278

—

103

C₂×C₈

—

278

—

104

C₂×C₈

—

278

—

105

C₂×C₈

—

139

—

106

C₂×C₈

C₄×D₁₃₉

yes

—

107

C₂³

—

556

—

108

C₂³

—

139

—

109

C₂³

—

139

—

110

C₂³

Q₈×D₁₃₉

yes

characteristic

111

C₂×C₄

—

139

—

112

C₂×C₄

—

139

—

113

C₂×C₄

—

139

—

114

C₂×C₄

—

139

—

115

C₂×C₄

—

139

—

116

C₂×C₄

—

139

—

117

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

118

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

119

C₂×C₄

D₄×D₁₃₉

yes

—

120

C₂×C₄

D₄×D₁₃₉

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

278

—

124

C₈

—

278

—

125

C₂²

—

126

C₂²

—

278

—

127

C₂²

—

556

—

128

C₂²

—

139

—

129

C₂²

C₄⋊C₄×D₁₃₉

yes

—

130

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

131

C₄

—

139

—

132

C₄

C₄²×D₁₃₉

yes

characteristic

133

C₄

C₂²⋊C₄×D₁₃₉

yes

characteristic

134

C₄

C₂²⋊C₄×D₁₃₉

yes

—

135

C₂

—

136

C₂

—

278

—

137

C₂

#95×D₁₃₉

yes

characteristic

138

C₂

#22581000

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#341×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃₉	Q₈
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#341	D₁₃₉
hypocenter	C₁₃₉	#1738
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₉

commutator

C₄×C₁₃₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#341×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₄⋊C₄

prime radical

C₂³×D₁₃₉

Q₈

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
3. #22580718	C₂	semidirect
4. #22580717	C₂	semidirect
5. #341×C₁₃₉	C₂	semidirect
69. C₁₃₉	#1738	semidirect
75. #341	D₁₃₉	semidirect

H ≅ G/N

type

3. #22580718

C₂

semidirect

4. #22580717

C₂

semidirect

5. #341×C₁₃₉

C₂

semidirect

69. C₁₃₉

#1738

semidirect

75. #341

D₁₃₉

semidirect

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

subgroup	index
C₂²×#14555768	2
C₂×M₁₆×D₁₃₉	2
#22580718	2
#22580717	2
#341×C₁₃₉	2
#1738	139

subgroup

index

C₂×M₁₆×D₁₃₉

#341×C₁₃₉

139

Subgroups — #31806129

Conjugacy classes of subgroups

Complements

Maximal subgroups