Group #31806525 — subgroups

Subgroups — #31806525

Order 17792.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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×#17984777	C₂	1	1	yes	maximal, characteristic
2	#22580989	C₂	1	1	yes	maximal, characteristic
3	#22580911	C₂	1	1	yes	maximal, characteristic
4	#22580724	C₂	1	1	yes	maximal, characteristic
5	#22580710	C₂	1	1	yes	maximal, characteristic
6	#22580709	C₂	1	1	yes	maximal, characteristic
7	#344×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555769	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	#17984801	C₂²	1	1	yes	characteristic
11	#17984801	C₂²	1	1	yes	characteristic
12	#17984777	—	2	1	no	—
13	#17984777	—	2	1	no	—
14	#17984775	—	2	1	no	—
15	#17984773	—	2	1	no	—
16	#17984752	C₄	1	2	yes	complemented
17	C₈×Dic₁₃₉	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
20	#117×C₁₃₉	C₂²	1	1	yes	characteristic
21	#14555769	—	2	1	no	—
22	#14555769	—	2	1	no	—
23	C₂×C₁₃₉⋊C₈	—	2	1	no	—
24	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₁₃₉⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₁₃₉⋊C₈	Q₈	1	1	yes	characteristic
29	C₂×C₁₃₉⋊C₈	Q₈	1	1	yes	characteristic
30	C₄⋊Dic₁₃₉	—	2	1	no	—
31	C₄⋊Dic₁₃₉	—	2	1	no	—
32	C₂²⋊Dic₁₃₉	—	2	1	no	—
33	C₂²⋊Dic₁₃₉	—	2	1	no	—
34	C₄×Dic₁₃₉	—	2	1	no	—
35	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₃₉	—	2	1	no	—
38	M₁₆×C₁₃₉	—	2	1	no	—
39	M₁₆×C₁₃₉	—	2	1	no	—
40	M₁₆×C₁₃₉	—	2	1	no	—
41	C₂×C₈×C₁₃₉	—	2	1	no	—
42	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₃₉	—	2	1	no	—
44	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
45	C₄²×C₁₃₉	—	2	1	no	—
46	C₂×Dic₁₃₉	—	2	2	no	—
47	C₂×Dic₁₃₉	—	4	1	no	—
48	C₁₃₉⋊C₈	—	4	1	no	—
49	C₁₃₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₁₃₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₁₃₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₁₃₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂³×C₁₃₉	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	—	4	1	no	—
55	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
61	C₈×C₁₃₉	—	2	2	no	—
62	C₈×C₁₃₉	—	4	1	no	—
63	C₂²×C₁₃₉	—	4	1	no	—
64	C₂²×C₁₃₉	#117	1	1	yes	characteristic
65	C₂²×C₁₃₉	#124	1	1	yes	characteristic
66	C₂²×C₁₃₉	#125	1	1	yes	characteristic
67	C₄×C₁₃₉	—	8	1	no	—
68	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₁₃₉	#117	1	1	yes	characteristic
70	C₄×C₁₃₉	#124	1	1	yes	characteristic
71	C₄×C₁₃₉	#125	1	1	yes	characteristic
72	Dic₁₃₉	—	4	2	no	—
73	Dic₁₃₉	—	8	1	no	—
74	C₂×C₁₃₉	—	2	1	no	—
75	C₂×C₁₃₉	—	2	1	no	—
76	C₂×C₁₃₉	—	2	1	no	—
77	C₂×C₁₃₉	#382	1	1	yes	characteristic
78	C₁₃₉	#1833	1	1	yes	characteristic, complemented, Sylow
79	#1833	—	139	1	no	maximal, Sylow
80	C₂×#108	—	139	1	no	—
81	#427	—	139	1	no	—
82	#405	—	139	1	no	—
83	#344	—	139	1	no	—
84	#344	D₁₃₉	1	1	yes	characteristic
85	H(3,ℤ/4)	—	139	1	no	—
86	H(3,ℤ/4)	—	139	1	no	—
87	C₂×M₁₆	—	139	1	no	—
88	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
90	C₂²×C₈	—	139	1	no	—
91	#117	—	139	1	no	—
92	#117	—	139	1	no	—
93	#117	C₂×D₁₃₉	1	1	yes	characteristic
94	#108	—	278	1	no	—
95	#108	—	278	1	no	—
96	#107	—	278	1	no	—
97	#106	—	278	1	no	—
98	#97	—	139	2	no	—
99	C₄×C₈	—	139	2	no	—
100	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	278	1	no	—
106	M₁₆	—	278	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	278	1	no	—
109	C₂×C₈	—	139	1	no	—
110	C₂×C₈	—	139	1	no	—
111	C₂×C₈	—	139	1	no	—
112	C₂×C₈	—	139	1	no	—
113	C₂×C₈	—	139	1	no	—
114	C₂×C₈	—	139	1	no	—
115	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	278	1	no	—
118	C₄⋊C₄	—	278	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	278	1	no	—
121	C₂²⋊C₄	—	278	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	278	1	no	—
124	C₄²	—	139	2	no	—
125	C₂³	#14555785	1	1	yes	characteristic
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	278	2	no	—
128	C₂×C₄	—	556	1	no	—
129	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
130	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
131	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
132	C₂×C₄	#14555784	1	1	yes	characteristic
133	C₂×C₄	#14555784	1	1	yes	characteristic
134	C₂×C₄	#14555785	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	4	1	no	—
137	C₈	—	556	1	no	—
138	C₈	—	139	1	no	—
139	C₈	—	139	1	no	—
140	C₈	—	139	1	no	—
141	C₈	—	139	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#17984799	1	1	yes	characteristic
144	C₂²	#17984828	1	1	yes	characteristic
145	C₂²	#17984832	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	556	2	no	—
148	C₄	—	1112	1	no	—
149	C₄	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
150	C₄	#17984798	1	1	yes	characteristic
151	C₄	#17984828	1	1	yes	characteristic
152	C₄	#17984832	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#22580817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984777

C₂

yes

maximal, characteristic

#22580989

C₂

yes

maximal, characteristic

#22580911

C₂

yes

maximal, characteristic

#22580724

C₂

yes

maximal, characteristic

#22580710

C₂

yes

maximal, characteristic

#22580709

C₂

yes

maximal, characteristic

#344×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

characteristic

#17984801

C₂²

yes

characteristic

#17984801

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₈×Dic₁₃₉

C₄

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

D₄

yes

characteristic

C₂×C₁₃₉⋊C₈

Q₈

yes

characteristic

C₂×C₁₃₉⋊C₈

Q₈

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₄⋊C₄

yes

characteristic

C₁₃₉⋊C₈

C₄⋊C₄

yes

characteristic

C₁₃₉⋊C₈

C₄⋊C₄

yes

characteristic

C₁₃₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#382

yes

characteristic

C₁₃₉

#1833

yes

characteristic, complemented, Sylow

#1833

—

139

maximal, Sylow

C₂×#108

—

139

—

#427

—

139

—

#405

—

139

—

#344

—

139

—

#344

D₁₃₉

yes

characteristic

H(3,ℤ/4)

—

139

—

H(3,ℤ/4)

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

#117

—

139

—

#117

—

139

—

#117

C₂×D₁₃₉

yes

characteristic

#108

—

278

—

#108

—

278

—

#107

—

278

—

#106

—

278

—

#97

—

139

—

C₄×C₈

—

139

—

100

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

278

—

106

M₁₆

—

278

—

107

C₂×C₈

—

108

C₂×C₈

—

278

—

109

C₂×C₈

—

139

—

110

C₂×C₈

—

139

—

111

C₂×C₈

—

139

—

112

C₂×C₈

—

139

—

113

C₂×C₈

—

139

—

114

C₂×C₈

—

139

—

115

C₂×C₈

C₄×D₁₃₉

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

278

—

118

C₄⋊C₄

—

278

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

278

—

121

C₂²⋊C₄

—

278

—

122

C₄²

—

123

C₄²

—

278

—

124

C₄²

—

139

—

125

C₂³

#14555785

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

278

—

128

C₂×C₄

—

556

—

129

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

130

C₂×C₄

D₄×D₁₃₉

yes

characteristic

131

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

132

C₂×C₄

#14555784

yes

characteristic

133

C₂×C₄

#14555784

yes

characteristic

134

C₂×C₄

#14555785

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

556

—

138

C₈

—

139

—

139

C₈

—

139

—

140

C₈

—

139

—

141

C₈

—

139

—

142

C₂²

—

143

C₂²

#17984799

yes

characteristic

144

C₂²

#17984828

yes

characteristic

145

C₂²

#17984832

yes

characteristic

146

C₄

—

147

C₄

—

556

—

148

C₄

—

1112

—

149

C₄

C₄⋊C₄×D₁₃₉

yes

characteristic

150

C₄

#17984798

yes

characteristic

151

C₄

#17984828

yes

characteristic

152

C₄

#17984832

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#22580817

yes

characteristic

	subgroup	G/H
center	C₄	#17984798
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#344×C₁₃₉	C₂
socle	C₂×C₁₃₉	#382
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#344	D₁₃₉
hypocenter	C₁₃₉	#1833
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#17984798

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#344×C₁₃₉

C₂

C₂×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #17984752	C₄	semidirect
17. C₈×Dic₁₃₉	C₄	semidirect
78. C₁₃₉	#1833	semidirect

H ≅ G/N

type

16. #17984752

C₄

semidirect

17. C₈×Dic₁₃₉

C₄

semidirect

78. C₁₃₉

#1833

semidirect

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

subgroup	index
C₂×#17984777	2
#22580989	2
#22580911	2
#22580724	2
#22580710	2
#22580709	2
#344×C₁₃₉	2
#1833	139

subgroup

index

#344×C₁₃₉

139

Subgroups — #31806525

Conjugacy classes of subgroups

Complements

Maximal subgroups