Group #31806256 — subgroups

Subgroups — #31806256

Order 17792.

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

normal subgroups only

This group has 292 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	C₂×Dic₁₃₉⋊C₈	C₂	1	2	yes	maximal
2	C₂×#17984752	C₂	1	2	yes	maximal
3	C₄²×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#22580708	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₃₉	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
11	Dic₁₃₉⋊C₈	C₄	1	8	yes	complemented
12	#17984752	—	2	4	no	—
13	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₃₉⋊C₈	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	—	2	2	no	—
20	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₃₉	—	2	4	no	—
22	C₄×Dic₁₃₉	—	2	4	no	—
23	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₃₉	D₄	1	2	yes	—
25	C₄×Dic₁₃₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
31	C₄²×C₁₃₉	—	2	2	no	—
32	C₂×Dic₁₃₉	—	2	2	no	—
33	C₂×Dic₁₃₉	—	2	2	no	—
34	C₂×Dic₁₃₉	—	2	2	no	—
35	C₂×Dic₁₃₉	C₄²	1	2	yes	—
36	C₂×Dic₁₃₉	C₄²	1	2	yes	—
37	C₂×Dic₁₃₉	C₄⋊C₄	1	4	yes	—
38	C₂×Dic₁₃₉	M₁₆	1	4	yes	—
39	C₁₃₉⋊C₈	—	2	4	no	—
40	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄×C₁₃₉	M₁₆	1	4	yes	—
44	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃₉	#41	1	2	yes	—
48	C₈×C₁₃₉	—	2	4	no	—
49	C₂²×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₃₉	#117	1	1	yes	characteristic
52	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₃₉	—	2	4	no	—
55	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
56	C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
57	Dic₁₃₉	—	2	2	no	—
58	Dic₁₃₉	—	2	4	no	—
59	Dic₁₃₉	#97	1	4	yes	complemented
60	C₂×C₁₃₉	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₃₉	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₃₉	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₃₉	#423	1	2	yes	—
64	C₂×C₁₃₉	#432	1	2	yes	—
65	C₁₃₉	#1767	1	1	yes	characteristic, complemented, Sylow
66	#1767	—	139	1	no	maximal, Sylow
67	C₂×C₄⋊C₈	—	139	2	no	—
68	C₂×#97	—	139	2	no	—
69	C₄³	—	139	1	no	—
70	#336	—	139	1	no	—
71	#336	D₁₃₉	1	1	yes	characteristic
72	C₂²×C₈	—	139	2	no	—
73	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
74	C₂×C₄²	—	139	2	no	—
75	C₂×C₄²	—	139	2	no	—
76	C₂×C₄²	—	139	1	no	—
77	C₂×C₄²	—	139	1	no	—
78	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
79	C₄⋊C₈	—	139	8	no	—
80	#97	—	278	4	no	—
81	C₂²×C₄	—	139	2	no	—
82	C₂²×C₄	—	139	1	no	—
83	C₂²×C₄	—	139	1	no	—
84	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	278	2	no	—
89	C₂×C₈	—	278	2	no	—
90	C₂×C₈	—	139	4	no	—
91	C₂×C₈	C₄×D₁₃₉	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	278	4	no	—
94	C₄²	—	278	4	no	—
95	C₄²	—	139	4	no	—
96	C₄²	—	139	2	no	—
97	C₄²	—	139	2	no	—
98	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	278	2	no	—
102	C₂×C₄	—	278	2	no	—
103	C₂×C₄	—	278	2	no	—
104	C₂×C₄	—	139	2	no	—
105	C₂×C₄	—	139	4	no	—
106	C₂×C₄	—	139	2	no	—
107	C₂×C₄	—	139	4	no	—
108	C₂×C₄	#14555768	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
110	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
112	C₂×C₄	#14555784	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	278	4	no	—
115	C₂²	C₄²×D₁₃₉	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
117	C₂²	#17984798	1	1	yes	characteristic
118	C₂²	M₁₆×D₁₃₉	1	2	yes	—
119	C₂²	C₂×#14555768	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	278	2	no	—
122	C₄	—	278	4	no	—
123	C₄	—	139	4	no	—
124	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
125	C₄	#17984807	1	2	yes	—
126	C₂	#22580804	1	1	yes	characteristic
127	C₂	#97×D₁₃₉	1	1	yes	characteristic
128	C₂	C₄×#14555768	1	1	yes	characteristic
129	C₂	#22580975	1	2	yes	—
130	C₂	#22581007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₃₉⋊C₈

C₂

yes

maximal

C₂×#17984752

C₂

yes

maximal

C₄²×Dic₁₃₉

C₂

yes

maximal, characteristic

#22580708

C₂

yes

maximal, characteristic

#336×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₈

C₄

yes

complemented

#17984752

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

Q₈

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄²

yes

—

C₂×Dic₁₃₉

C₄²

yes

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

M₁₆

yes

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

M₁₆

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄²

yes

characteristic

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃₉

#117

yes

characteristic

C₂²×C₁₃₉

C₂×M₁₆

yes

—

C₂²×C₁₃₉

C₂×M₁₆

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

C₄×Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

#97

yes

complemented

C₂×C₁₃₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₃₉

C₂×#97

yes

characteristic

C₂×C₁₃₉

C₄×M₁₆

yes

characteristic

C₂×C₁₃₉

#423

yes

—

C₂×C₁₃₉

#432

yes

—

C₁₃₉

#1767

yes

characteristic, complemented, Sylow

#1767

—

139

maximal, Sylow

C₂×C₄⋊C₈

—

139

—

C₂×#97

—

139

—

C₄³

—

139

—

#336

—

139

—

#336

D₁₃₉

yes

characteristic

C₂²×C₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

139

—

#97

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₄×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₄×D₁₃₉

yes

—

C₄²

—

C₄²

—

278

—

C₄²

—

278

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

—

139

—

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

278

—

102

C₂×C₄

—

278

—

103

C₂×C₄

—

278

—

104

C₂×C₄

—

139

—

105

C₂×C₄

—

139

—

106

C₂×C₄

—

139

—

107

C₂×C₄

—

139

—

108

C₂×C₄

#14555768

yes

—

109

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

110

C₂×C₄

D₄×D₁₃₉

yes

characteristic

111

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

112

C₂×C₄

#14555784

yes

—

113

C₈

—

114

C₈

—

278

—

115

C₂²

C₄²×D₁₃₉

yes

characteristic

116

C₂²

C₄⋊C₄×D₁₃₉

yes

characteristic

117

C₂²

#17984798

yes

characteristic

118

C₂²

M₁₆×D₁₃₉

yes

—

119

C₂²

C₂×#14555768

yes

—

120

C₄

—

121

C₄

—

278

—

122

C₄

—

278

—

123

C₄

—

139

—

124

C₄

(C₂²⋊D₁₃₉)⋊C₄

yes

—

125

C₄

#17984807

yes

—

126

C₂

#22580804

yes

characteristic

127

C₂

#97×D₁₃₉

yes

characteristic

128

C₂

C₄×#14555768

yes

characteristic

129

C₂

#22580975

yes

—

130

C₂

#22581007

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂×C₄²
Fitting	#336×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#336	D₁₃₉
hypocenter	C₁₃₉	#1767
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂×C₄²

Fitting

#336×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. Dic₁₃₉⋊C₈	C₄	semidirect
59. Dic₁₃₉	#97	semidirect
65. C₁₃₉	#1767	semidirect

H ≅ G/N

type

11. Dic₁₃₉⋊C₈

C₄

semidirect

59. Dic₁₃₉

#97

semidirect

65. C₁₃₉

#1767

semidirect

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

subgroup	index
C₂×Dic₁₃₉⋊C₈	2
C₂×#17984752	2
C₄²×Dic₁₃₉	2
#22580708	2
#336×C₁₃₉	2
#1767	139

subgroup

index

C₂×Dic₁₃₉⋊C₈

C₂×#17984752

C₄²×Dic₁₃₉

#22580708

#336×C₁₃₉

#1767

139

Subgroups — #31806256

Conjugacy classes of subgroups

Complements

Maximal subgroups