Group #31806452 — subgroups

Subgroups — #31806452

Order 17792.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#17984849	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984777	C₂	1	2	yes	maximal, complemented
3	#22580937	C₂	1	1	yes	maximal, characteristic
4	C₁₃₉⋊#338	C₂	1	2	yes	maximal, complemented
5	#413×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984849	—	2	4	no	—
7	C₂×#14555769	C₄	1	2	yes	—
8	C₂×#14555769	C₂²	1	2	yes	—
9	C₂×#14555769	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃₉⋊C₈	C₄	1	2	yes	—
11	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	#17984777	—	2	4	no	—
13	#17984761	—	2	1	no	—
14	#17984759	—	2	1	no	—
15	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
17	#101×C₁₃₉	—	2	1	no	—
18	#100×C₁₃₉	—	2	1	no	—
19	#14555769	—	2	4	no	—
20	#14555769	—	2	2	no	—
21	#14555769	—	2	2	no	—
22	#14555769	—	2	2	no	—
23	C₂×C₁₃₉⋊C₈	—	2	4	no	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	C₂×C₁₃₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
29	#41×C₁₃₉	—	2	2	no	—
30	#41×C₁₃₉	—	2	2	no	—
31	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
32	C₂×D₄×C₁₃₉	D₄	1	1	yes	characteristic
33	C₂×D₄×C₁₃₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₃₉	—	2	2	no	—
37	M₁₆×C₁₃₉	—	2	2	no	—
38	C₂×C₈×C₁₃₉	—	2	2	no	—
39	C₁₃₉⋊C₈	—	2	2	no	—
40	C₁₃₉⋊C₈	—	4	2	no	—
41	C₁₃₉⋊C₈	C₂²⋊C₄	1	2	yes	—
42	C₁₃₉⋊C₈	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₁₃₉	C₄⋊C₄	1	2	yes	—
44	C₂³×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
45	Q₈×C₁₃₉	—	2	2	no	—
46	D₄×C₁₃₉	—	2	4	no	—
47	D₄×C₁₃₉	—	2	2	no	—
48	C₂×C₄×C₁₃₉	—	2	2	no	—
49	C₂×C₄×C₁₃₉	—	2	2	no	—
50	C₂×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₉	#41	1	2	yes	—
55	C₈×C₁₃₉	—	4	2	no	—
56	C₂²×C₁₃₉	—	2	2	no	—
57	C₂²×C₁₃₉	—	4	1	no	—
58	C₂²×C₁₃₉	—	4	2	no	—
59	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₃₉	—	2	2	no	—
62	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₃₉	#117	1	1	yes	characteristic
64	C₄×C₁₃₉	#121	1	2	yes	—
65	C₂×C₁₃₉	—	2	1	no	—
66	C₂×C₁₃₉	—	2	2	no	—
67	C₂×C₁₃₉	—	4	2	no	—
68	C₂×C₁₃₉	#380	1	1	yes	characteristic
69	C₁₃₉	#1813	1	1	yes	characteristic, complemented, Sylow
70	#1813	—	139	1	no	maximal, Sylow
71	C₂×#131	—	139	1	no	—
72	C₂×#108	—	139	2	no	—
73	#413	—	139	1	no	—
74	#413	D₁₃₉	1	1	yes	characteristic
75	#338	—	139	2	no	—
76	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
77	#131	—	278	4	no	—
78	C₂×M₁₆	—	139	2	no	—
79	C₂×M₁₆	—	139	2	no	—
80	C₂×M₁₆	—	139	1	no	—
81	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
82	C₂²×C₈	—	139	2	no	—
83	C₂²×C₈	—	139	1	no	—
84	#108	—	278	4	no	—
85	#101	—	2	1	no	—
86	#101	—	278	1	no	—
87	#100	—	2	1	no	—
88	#100	—	278	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
92	C₂×D₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
93	C₂×D₄	Dic₂₇₈	1	2	yes	—
94	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	278	2	no	—
99	M₁₆	—	278	4	no	—
100	M₁₆	—	278	2	no	—
101	M₁₆	—	278	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	278	4	no	—
104	C₂×C₈	—	278	2	no	—
105	C₂×C₈	—	139	2	no	—
106	C₂×C₈	—	139	1	no	—
107	C₂×C₈	—	139	1	no	—
108	C₂×C₈	—	139	4	no	—
109	C₂³	Dic₁₃₉⋊C₄	1	2	yes	—
110	C₂³	C₂×Dic₂₇₈	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	Dic₁₃₉⋊C₄	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
120	C₂×C₄	#14555784	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	278	2	no	—
123	C₈	—	556	2	no	—
124	C₈	—	139	2	no	—
125	C₈	—	139	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
130	C₂²	#17984819	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
133	C₄	#17984798	1	1	yes	characteristic
134	C₄	#17984814	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#22580811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984849

C₂

yes

maximal, characteristic

C₂×#17984777

C₂

yes

maximal, complemented

#22580937

C₂

yes

maximal, characteristic

C₁₃₉⋊#338

C₂

yes

maximal, complemented

#413×C₁₃₉

C₂

yes

maximal, characteristic

#17984849

—

C₂×#14555769

C₄

yes

—

C₂×#14555769

C₂²

yes

—

C₂×#14555769

C₂²

yes

characteristic

C₂²×C₁₃₉⋊C₈

C₄

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

—

—

C₂×#41×C₁₃₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₉

C₂²

yes

—

#101×C₁₃₉

—

#100×C₁₃₉

—

—

—

—

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

D₄

yes

—

#41×C₁₃₉

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×D₄×C₁₃₉

D₄

yes

characteristic

C₂×D₄×C₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₂²⋊C₄

yes

—

C₁₃₉⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₃₉

C₄⋊C₄

yes

—

C₂³×C₁₃₉

C₂×Q₈

yes

characteristic

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₂²⋊C₄

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#380

yes

characteristic

C₁₃₉

#1813

yes

characteristic, complemented, Sylow

#1813

—

139

maximal, Sylow

C₂×#131

—

139

—

C₂×#108

—

139

—

#413

—

139

—

#413

D₁₃₉

yes

characteristic

#338

—

139

—

C₂×#41

C₂×D₁₃₉

yes

characteristic

#131

—

278

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

139

—

C₂²×C₈

—

139

—

#108

—

278

—

#101

—

#101

—

278

—

#100

—

#100

—

278

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×D₄

C₂²⋊D₁₃₉

yes

characteristic

C₂×D₄

Dic₂₇₈

yes

—

C₂²×C₄

C₄×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

278

—

M₁₆

—

278

—

100

M₁₆

—

278

—

101

M₁₆

—

278

—

102

C₂×C₈

—

103

C₂×C₈

—

278

—

104

C₂×C₈

—

278

—

105

C₂×C₈

—

139

—

106

C₂×C₈

—

139

—

107

C₂×C₈

—

139

—

108

C₂×C₈

—

139

—

109

C₂³

Dic₁₃₉⋊C₄

yes

—

110

C₂³

C₂×Dic₂₇₈

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Dic₁₃₉⋊C₄

yes

—

117

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

118

C₂×C₄

D₄×D₁₃₉

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

120

C₂×C₄

#14555784

yes

—

121

C₈

—

122

C₈

—

278

—

123

C₈

—

556

—

124

C₈

—

139

—

125

C₈

—

139

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₁₃₉⋊C₄

yes

characteristic

130

C₂²

#17984819

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₁₃₉

yes

characteristic

133

C₄

#17984798

yes

characteristic

134

C₄

#17984814

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#22580811

yes

characteristic

	subgroup	G/H
center	C₄	#17984798
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#413×C₁₃₉	C₂
socle	C₂×C₁₃₉	#380
prime radical	C₂×#41×C₁₃₉	C₂²
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#413	D₁₃₉
hypocenter	C₁₃₉	#1813
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#17984798

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#413×C₁₃₉

C₂

socle

C₂×C₁₃₉

#380

prime radical

C₂×#41×C₁₃₉

C₂²

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#17984777	C₂	semidirect
4. C₁₃₉⋊#338	C₂	semidirect
69. C₁₃₉	#1813	semidirect

H ≅ G/N

type

2. C₂×#17984777

C₂

semidirect

4. C₁₃₉⋊#338

C₂

semidirect

69. C₁₃₉

#1813

semidirect

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

subgroup	index
C₂×#17984849	2
C₂×#17984777	2
#22580937	2
C₁₃₉⋊#338	2
#413×C₁₃₉	2
#1813	139

subgroup

index

C₂×#17984849

C₂×#17984777

#22580937

C₁₃₉⋊#338

#413×C₁₃₉

#1813

139

Subgroups — #31806452

Conjugacy classes of subgroups

Complements

Maximal subgroups