Group #31807132 — subgroups

Subgroups — #31807132

Order 17792.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#17984774	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984774	C₂	1	1	yes	maximal, characteristic
3	#22580851	C₂	1	2	yes	maximal
4	#22580716	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
8	C₂×Dic₁₃₉⋊C₄	—	2	2	no	—
9	#17984774	—	2	2	no	—
10	#17984774	C₄	1	4	yes	complemented
11	C₁₃₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃₉	—	2	2	no	—
16	C₂²×Dic₁₃₉	—	2	2	no	—
17	C₂²×Dic₁₃₉	—	2	2	no	—
18	C₄⋊Dic₁₃₉	—	2	2	no	—
19	C₄⋊Dic₁₃₉	—	2	2	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
22	Dic₁₃₉⋊C₄	—	4	4	no	—
23	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
24	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₃₉	—	2	1	no	—
26	C₂×C₈×C₁₃₉	—	2	1	no	—
27	C₂×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₃₉	Q₈	1	2	yes	—
30	C₂×C₈×C₁₃₉	Q₈	1	2	yes	—
31	C₄²×C₁₃₉	—	2	1	no	—
32	C₄²×C₁₃₉	—	2	1	no	—
33	C₂×Dic₁₃₉	—	2	4	no	—
34	C₂×Dic₁₃₉	—	4	2	no	—
35	C₂×Dic₁₃₉	—	4	2	no	—
36	C₂×Dic₁₃₉	—	4	2	no	—
37	C₂×Dic₁₃₉	—	4	2	no	—
38	C₂×Dic₁₃₉	—	4	2	no	—
39	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄×C₁₃₉	SD₁₆	1	4	yes	—
43	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
49	C₈×C₁₃₉	—	2	2	no	—
50	C₈×C₁₃₉	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₁₃₉	#121	1	1	yes	characteristic
53	C₂²×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₃₉	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₁₃₉	#135	1	2	yes	—
56	C₄×C₁₃₉	—	2	4	no	—
57	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₁₃₉	#125	1	1	yes	characteristic
61	Dic₁₃₉	—	4	4	no	—
62	Dic₁₃₉	—	8	2	no	—
63	C₂×C₁₃₉	#389	1	1	yes	characteristic
64	C₂×C₁₃₉	C₄×SD₁₆	1	2	yes	—
65	C₂×C₁₃₉	#465	1	2	yes	—
66	C₂×C₁₃₉	#498	1	1	yes	characteristic
67	C₂×C₁₃₉	#499	1	1	yes	characteristic
68	C₁₃₉	#1941	1	1	yes	characteristic, complemented, Sylow
69	#1941	—	139	1	no	maximal, Sylow
70	C₂×#106	—	139	1	no	—
71	C₂×#106	—	139	1	no	—
72	C₂×C₄×C₈	D₁₃₉	1	1	yes	characteristic
73	#389	—	139	2	no	—
74	#340	—	139	2	no	—
75	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	278	2	no	—
78	C₂×C₄⋊C₄	—	139	2	no	—
79	C₂×C₄⋊C₄	—	139	2	no	—
80	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
81	#106	—	278	2	no	—
82	#106	—	139	4	no	—
83	C₄×C₈	—	2	2	no	—
84	#95	—	278	2	no	—
85	C₂²×C₄	—	278	2	no	—
86	C₂²×C₄	—	278	2	no	—
87	C₂²×C₄	D₅₅₆	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₄×D₁₃₉	1	2	yes	—
92	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
93	C₂×C₈	Dic₂₇₈	1	2	yes	—
94	C₂×C₈	Dic₂₇₈	1	2	yes	—
95	C₄⋊C₄	—	278	2	no	—
96	C₄⋊C₄	—	278	2	no	—
97	C₄⋊C₄	—	278	2	no	—
98	C₄⋊C₄	—	556	4	no	—
99	C₄⋊C₄	—	139	4	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	278	4	no	—
106	C₂×C₄	—	556	2	no	—
107	C₂×C₄	—	556	2	no	—
108	C₂×C₄	—	556	2	no	—
109	C₂×C₄	—	556	2	no	—
110	C₂×C₄	—	556	2	no	—
111	C₂×C₄	#14555774	1	4	yes	—
112	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
116	C₂×C₄	#14555785	1	1	yes	characteristic
117	C₂×C₄	#14555785	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	Dic₁₃₉⋊C₄	1	4	yes	complemented
120	C₂²	C₄×D₅₅₆	1	1	yes	characteristic
121	C₂²	#17984816	1	1	yes	characteristic
122	C₂²	#17984842	1	1	yes	characteristic
123	C₂²	C₂×#14555774	1	2	yes	—
124	C₂²	#17984864	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	556	4	no	—
127	C₄	—	1112	2	no	—
128	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
129	C₄	C₄×Dic₂₇₈	1	1	yes	characteristic
130	C₄	#17984821	1	1	yes	characteristic
131	C₄	#17984834	1	1	yes	characteristic
132	C₂	#22580851	1	1	yes	characteristic
133	C₂	C₄×#14555774	1	2	yes	—
134	C₂	#22581182	1	2	yes	—
135	C₂	#22581351	1	1	yes	characteristic
136	C₂	#22581355	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984774

C₂

yes

maximal, characteristic

C₂×#17984774

C₂

yes

maximal, characteristic

#22580851

C₂

yes

maximal

#22580716

C₂

yes

maximal

C₂×C₄×C₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

—

#17984774

—

#17984774

C₄

yes

complemented

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

Dic₁₃₉⋊C₄

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

Q₈

yes

—

C₂×C₈×C₁₃₉

Q₈

yes

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

SD₁₆

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄⋊C₄

yes

complemented

C₂²×C₁₃₉

C₄×D₄

yes

characteristic

C₂²×C₁₃₉

#121

yes

characteristic

C₂²×C₁₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₃₉

C₂×SD₁₆

yes

—

C₂²×C₁₃₉

#135

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

C₄×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₉

#125

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₂×C₁₃₉

C₄×SD₁₆

yes

—

C₂×C₁₃₉

#465

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1941

—

139

maximal, Sylow

C₂×#106

—

139

—

C₂×#106

—

139

—

C₂×C₄×C₈

D₁₃₉

yes

characteristic

#389

—

139

—

#340

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#106

—

278

—

#106

—

139

—

C₄×C₈

—

#95

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₂×C₈

Dic₂₇₈

yes

—

C₂×C₈

Dic₂₇₈

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

556

—

C₄⋊C₄

—

139

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₅₅₆

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

556

—

107

C₂×C₄

—

556

—

108

C₂×C₄

—

556

—

109

C₂×C₄

—

556

—

110

C₂×C₄

—

556

—

111

C₂×C₄

#14555774

yes

—

112

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

114

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

115

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

116

C₂×C₄

#14555785

yes

characteristic

117

C₂×C₄

#14555785

yes

characteristic

118

C₈

—

119

C₈

Dic₁₃₉⋊C₄

yes

complemented

120

C₂²

C₄×D₅₅₆

yes

characteristic

121

C₂²

#17984816

yes

characteristic

122

C₂²

#17984842

yes

characteristic

123

C₂²

C₂×#14555774

yes

—

124

C₂²

#17984864

yes

—

125

C₄

—

126

C₄

—

556

—

127

C₄

—

1112

—

128

C₄

C₂×Dic₁₃₉⋊C₄

yes

characteristic

129

C₄

C₄×Dic₂₇₈

yes

characteristic

130

C₄

#17984821

yes

characteristic

131

C₄

#17984834

yes

characteristic

132

C₂

#22580851

yes

characteristic

133

C₂

C₄×#14555774

yes

—

134

C₂

#22581182

yes

—

135

C₂

#22581351

yes

characteristic

136

C₂

#22581355

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×C₄×C₈×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₉
hypocenter	C₁₃₉	#1941
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×C₄×C₈×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₂×C₄×C₈

D₁₃₉

hypocenter

C₁₃₉

#1941

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
10. #17984774	C₄	semidirect
50. C₈×C₁₃₉	C₄⋊C₄	semidirect
68. C₁₃₉	#1941	semidirect
119. C₈	Dic₁₃₉⋊C₄	semidirect

H ≅ G/N

type

10. #17984774

C₄

semidirect

50. C₈×C₁₃₉

C₄⋊C₄

semidirect

68. C₁₃₉

#1941

semidirect

119. C₈

Dic₁₃₉⋊C₄

semidirect

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

subgroup	index
C₂×#17984774	2
#22580851	2
#22580716	2
C₂×C₄×C₈×C₁₃₉	2
#1941	139

subgroup

index

C₂×C₄×C₈×C₁₃₉

139

Subgroups — #31807132

Conjugacy classes of subgroups

Complements

Maximal subgroups