Group #31807319 — subgroups

Subgroups — #31807319

Order 17792.

276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 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×Dic₅₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984778	C₂	1	1	yes	maximal, characteristic
3	#22580961	C₂	1	2	yes	maximal
4	C₂×#17984760	C₂	1	2	yes	maximal
5	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
7	C₂×Dic₅₅₆	—	2	1	no	—
8	C₂×Dic₅₅₆	—	2	1	no	—
9	C₂×Dic₅₅₆	—	2	2	no	—
10	C₂×Dic₅₅₆	C₄	1	4	yes	complemented
11	C₂×#14555769	C₂²	1	2	yes	—
12	#17984778	—	2	2	no	—
13	#17984765	—	2	4	no	—
14	#17984760	—	2	4	no	—
15	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	#117×C₁₃₉	C₂²	1	1	yes	characteristic
18	#97×C₁₃₉	—	2	1	no	—
19	C₄×C₈×C₁₃₉	—	2	1	no	—
20	C₂×Dic₂₇₈	—	2	2	no	—
21	C₂×Dic₂₇₈	—	2	2	no	—
22	C₂×Dic₂₇₈	—	2	2	no	—
23	C₂×Dic₂₇₈	—	2	2	no	—
24	C₂×Dic₂₇₈	—	2	2	no	—
25	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	—
26	C₂²×Dic₁₃₉	—	2	2	no	—
27	Dic₅₅₆	—	2	4	no	—
28	Dic₅₅₆	—	2	4	no	—
29	#14555769	—	2	2	no	—
30	#14555769	—	2	2	no	—
31	C₂×C₁₃₉⋊C₈	—	2	2	no	—
32	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₃₉	—	2	1	no	—
34	M₁₆×C₁₃₉	D₄	1	2	yes	—
35	C₂×C₈×C₁₃₉	—	2	1	no	—
36	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
40	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
41	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
42	C₄²×C₁₃₉	—	2	1	no	—
43	Dic₂₇₈	—	2	4	no	—
44	Dic₂₇₈	—	2	4	no	—
45	Dic₂₇₈	—	4	2	no	—
46	Dic₂₇₈	—	4	2	no	—
47	C₂×Dic₁₃₉	—	2	4	no	—
48	C₂×Dic₁₃₉	—	4	2	no	—
49	C₂×Dic₁₃₉	—	4	2	no	—
50	C₁₃₉⋊C₈	—	4	2	no	—
51	C₂³×C₁₃₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	—	2	2	no	—
53	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
59	C₈×C₁₃₉	—	2	2	no	—
60	C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
61	C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₁₃₉	—	2	2	no	—
63	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₁₃₉	#121	1	1	yes	characteristic
65	C₂²×C₁₃₉	#124	1	1	yes	characteristic
66	C₄×C₁₃₉	—	4	2	no	—
67	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
69	C₄×C₁₃₉	#121	1	1	yes	characteristic
70	C₄×C₁₃₉	#127	1	1	yes	characteristic
71	Dic₁₃₉	—	4	4	no	—
72	C₂×C₁₃₉	—	2	1	no	—
73	C₂×C₁₃₉	—	2	1	no	—
74	C₂×C₁₃₉	#390	1	1	yes	characteristic
75	C₂×C₁₃₉	#473	1	2	yes	—
76	C₁₃₉	#1979	1	1	yes	characteristic, complemented, Sylow
77	#1979	—	139	1	no	maximal, Sylow
78	C₂²×Q₁₆	—	139	1	no	—
79	C₂×#108	—	139	1	no	—
80	#419	—	139	2	no	—
81	C₂×#101	—	139	2	no	—
82	#405	D₁₃₉	1	1	yes	characteristic
83	C₂²×Q₈	—	139	2	no	—
84	C₂×Q₁₆	—	278	1	no	—
85	C₂×Q₁₆	—	278	1	no	—
86	C₂×Q₁₆	—	278	2	no	—
87	C₂×Q₁₆	—	139	4	no	—
88	C₂×M₁₆	—	139	2	no	—
89	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
91	#117	C₂×D₁₃₉	1	1	yes	characteristic
92	#108	—	278	2	no	—
93	#103	—	278	4	no	—
94	#101	—	278	4	no	—
95	#97	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₂×Q₈	—	278	2	no	—
98	C₂×Q₈	—	278	2	no	—
99	C₂×Q₈	—	278	2	no	—
100	C₂×Q₈	—	278	2	no	—
101	C₂×Q₈	—	278	2	no	—
102	C₂×Q₈	—	139	4	no	—
103	C₂²×C₄	—	278	2	no	—
104	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
105	Q₁₆	—	278	4	no	—
106	Q₁₆	—	278	4	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	278	2	no	—
109	M₁₆	—	278	2	no	—
110	M₁₆	D₅₅₆	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	278	2	no	—
113	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
114	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
116	C₂×C₈	D₅₅₆	1	2	yes	—
117	C₄⋊C₄	D₅₅₆	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#14555785	1	1	yes	characteristic
121	Q₈	—	278	4	no	—
122	Q₈	—	278	4	no	—
123	Q₈	—	556	2	no	—
124	Q₈	—	556	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	278	4	no	—
127	C₂×C₄	—	556	2	no	—
128	C₂×C₄	—	556	2	no	—
129	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
131	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
134	C₂×C₄	#14555785	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	556	2	no	—
137	C₈	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
138	C₈	(C₂×D₁₃₉)⋊C₄	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×D₅₅₆	1	1	yes	characteristic
141	C₂²	#17984816	1	1	yes	characteristic
142	C₂²	#17984830	1	1	yes	characteristic
143	C₄	—	4	2	no	—
144	C₄	—	556	4	no	—
145	C₄	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
146	C₄	C₄×D₅₅₆	1	1	yes	characteristic
147	C₄	#17984816	1	1	yes	characteristic
148	C₄	#17984839	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#22580855	1	1	yes	characteristic
152	C₂	#22581228	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₅₆

C₂

yes

maximal, characteristic

C₂×#17984778

C₂

yes

maximal, characteristic

#22580961

C₂

yes

maximal

C₂×#17984760

C₂

yes

maximal

#405×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

—

C₂×Dic₅₅₆

C₄

yes

complemented

C₂×#14555769

C₂²

yes

—

#17984778

—

#17984765

—

#17984760

—

C₂×M₁₆×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

—

C₄×C₈×C₁₃₉

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂×C₄

yes

—

C₂²×Dic₁₃₉

—

Dic₅₅₆

—

Dic₅₅₆

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

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×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₂²⋊C₄

yes

—

C₈×C₁₃₉

C₂²⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×D₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×D₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#390

yes

characteristic

C₂×C₁₃₉

#473

yes

—

C₁₃₉

#1979

yes

characteristic, complemented, Sylow

#1979

—

139

maximal, Sylow

C₂²×Q₁₆

—

139

—

C₂×#108

—

139

—

#419

—

139

—

C₂×#101

—

139

—

#405

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

278

—

C₂×Q₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

C₂×D₁₃₉

yes

characteristic

#108

—

278

—

#103

—

278

—

#101

—

278

—

#97

—

C₄×C₈

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

100

C₂×Q₈

—

278

—

101

C₂×Q₈

—

278

—

102

C₂×Q₈

—

139

—

103

C₂²×C₄

—

278

—

104

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

105

Q₁₆

—

278

—

106

Q₁₆

—

278

—

107

M₁₆

—

108

M₁₆

—

278

—

109

M₁₆

—

278

—

110

M₁₆

D₅₅₆

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

278

—

113

C₂×C₈

C₄×D₁₃₉

yes

—

114

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

115

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

116

C₂×C₈

D₅₅₆

yes

—

117

C₄⋊C₄

D₅₅₆

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#14555785

yes

characteristic

121

Q₈

—

278

—

122

Q₈

—

278

—

123

Q₈

—

556

—

124

Q₈

—

556

—

125

C₂×C₄

—

126

C₂×C₄

—

278

—

127

C₂×C₄

—

556

—

128

C₂×C₄

—

556

—

129

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

131

C₂×C₄

C₂×D₅₅₆

yes

characteristic

132

C₂×C₄

C₂×D₅₅₆

yes

characteristic

133

C₂×C₄

C₂×D₅₅₆

yes

characteristic

134

C₂×C₄

#14555785

yes

characteristic

135

C₈

—

136

C₈

—

556

—

137

C₈

(C₂×D₁₃₉)⋊C₄

yes

—

138

C₈

(C₂×D₁₃₉)⋊C₄

yes

—

139

C₂²

—

140

C₂²

C₄×D₅₅₆

yes

characteristic

141

C₂²

#17984816

yes

characteristic

142

C₂²

#17984830

yes

characteristic

143

C₄

—

144

C₄

—

556

—

145

C₄

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

146

C₄

C₄×D₅₅₆

yes

characteristic

147

C₄

#17984816

yes

characteristic

148

C₄

#17984839

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#22580855

yes

characteristic

152

C₂

#22581228

yes

—

	subgroup	G/H
center	C₂²	#17984816
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#405×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#121
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#1979
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#17984816

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#405×C₁₃₉

C₂

C₂²×C₁₃₉

C₂³×C₁₃₉

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
10. C₂×Dic₅₅₆	C₄	semidirect
76. C₁₃₉	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₅₅₆

C₄

semidirect

76. C₁₃₉

#1979

semidirect

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

subgroup	index
C₂²×Dic₅₅₆	2
C₂×#17984778	2
#22580961	2
C₂×#17984760	2
#405×C₁₃₉	2
#1979	139

subgroup

index

C₂²×Dic₅₅₆

#405×C₁₃₉

139

Subgroups — #31807319

Conjugacy classes of subgroups

Complements

Maximal subgroups