Group #31809124 — subgroups

Subgroups — #31809124

Order 17792.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₁₃₉	C₂	1	2	yes	maximal
2	#22580860	C₂	1	2	yes	maximal
3	#22580839	C₂	1	8	yes	maximal
4	#22580819	C₂	1	1	yes	maximal, characteristic
5	#22580799	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₁₃₉	—	2	8	no	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
10	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
13	C₁₃₉⋊#95	C₂²	1	4	yes	—
14	C₁₃₉⋊#95	C₂²	1	4	yes	—
15	C₁₃₉⋊#95	C₂²	1	8	yes	—
16	C₂²×Q₈×C₁₃₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₃₉	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
19	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
21	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
22	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
23	C₄⋊Dic₁₃₉	—	2	8	no	—
24	C₄⋊Dic₁₃₉	—	2	4	no	—
25	C₄×Dic₁₃₉	—	2	8	no	—
26	C₄×Dic₁₃₉	—	2	2	no	—
27	C₄×Dic₁₃₉	D₄	1	4	yes	—
28	C₂×Q₈×C₁₃₉	—	2	4	no	—
29	C₂×Q₈×C₁₃₉	—	2	4	no	—
30	C₂×Q₈×C₁₃₉	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₃₉	C₂×C₄	1	16	yes	—
35	C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
36	C₂×Dic₁₃₉	—	2	4	no	—
37	C₂×Dic₁₃₉	—	2	4	no	—
38	C₂×Dic₁₃₉	—	2	2	no	—
39	C₂×Dic₁₃₉	—	2	8	no	—
40	C₂×Dic₁₃₉	—	2	4	no	—
41	C₂×Dic₁₃₉	C₂×D₄	1	4	yes	—
42	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
43	Q₈×C₁₃₉	—	2	16	no	—
44	C₂×C₄×C₁₃₉	—	2	4	no	—
45	C₂×C₄×C₁₃₉	—	2	4	no	—
46	C₂×C₄×C₁₃₉	—	2	4	no	—
47	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃₉	#41	1	2	yes	—
52	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
55	C₂²×C₁₃₉	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₁₃₉	—	2	8	no	—
57	C₄×C₁₃₉	—	2	4	no	—
58	C₄×C₁₃₉	C₄×D₄	1	4	yes	—
59	Dic₁₃₉	—	2	4	no	—
60	Dic₁₃₉	—	4	4	no	—
61	Dic₁₃₉	—	4	2	no	—
62	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₁₃₉	#519	1	2	yes	—
64	C₂×C₁₃₉	#536	1	2	yes	—
65	C₂×C₁₃₉	#541	1	2	yes	—
66	C₁₃₉	#2327	1	1	yes	characteristic, complemented, Sylow
67	#2327	—	139	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	139	2	no	—
70	#391	—	139	2	no	—
71	#387	—	139	8	no	—
72	#382	—	139	1	no	—
73	#376	—	139	1	no	—
74	C₂²×Q₈	C₂×D₁₃₉	1	2	yes	—
75	C₄⋊Q₈	Dic₁₃₉	1	8	yes	complemented
76	C₄×Q₈	—	278	8	no	—
77	C₂×C₄⋊C₄	—	139	2	no	—
78	C₂×C₄⋊C₄	—	139	4	no	—
79	C₂×C₄⋊C₄	C₂×D₁₃₉	1	4	yes	—
80	C₂×C₄²	—	139	4	no	—
81	C₂×C₄²	—	139	1	no	—
82	C₂×C₄²	—	139	1	no	—
83	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
84	#95	—	139	4	no	—
85	#95	—	139	4	no	—
86	#95	—	139	8	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₁₃₉	1	8	yes	—
90	C₂²×C₄	—	139	2	no	—
91	C₂²×C₄	—	139	4	no	—
92	C₂²×C₄	—	139	2	no	—
93	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
96	C₄⋊C₄	—	278	8	no	—
97	C₄⋊C₄	—	278	4	no	—
98	C₄⋊C₄	C₂×Dic₁₃₉	1	16	yes	—
99	C₄²	—	278	8	no	—
100	C₄²	—	278	2	no	—
101	C₄²	—	139	4	no	—
102	C₄²	C₂×Dic₁₃₉	1	4	yes	—
103	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	278	4	no	—
109	C₂×C₄	—	278	4	no	—
110	C₂×C₄	—	278	2	no	—
111	C₂×C₄	—	278	8	no	—
112	C₂×C₄	—	278	4	no	—
113	C₂×C₄	—	139	4	no	—
114	C₂×C₄	C₂²×Dic₁₃₉	1	4	yes	—
115	C₂×C₄	C₂²×Dic₁₃₉	1	8	yes	—
116	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
117	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
118	C₂×C₄	#14555784	1	2	yes	—
119	C₂²	C₂³×Dic₁₃₉	1	1	yes	characteristic
120	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
121	C₂²	C₂×#14555784	1	1	yes	characteristic
122	C₂²	#17984890	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	278	4	no	—
126	C₄	—	556	4	no	—
127	C₄	—	556	2	no	—
128	C₄	D₄×Dic₁₃₉	1	4	yes	—
129	C₂	C₂×D₄×Dic₁₃₉	1	1	yes	characteristic
130	C₂	#22581431	1	2	yes	—
131	C₂	#22581528	1	2	yes	—
132	C₂	#22581564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₃₉

C₂

yes

maximal

#22580860

C₂

yes

maximal

#22580839

C₂

yes

maximal

#22580819

C₂

yes

maximal, characteristic

#22580799

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic

Q₈×Dic₁₃₉

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₉

C₂²

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₁₃₉⋊#95

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

—

C₂²×Q₈×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₂²×Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂³×C₄

yes

characteristic

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×#41

yes

characteristic

C₂²×C₁₃₉

2₋¹⁺⁴

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃₉

#519

yes

—

C₂×C₁₃₉

#536

yes

—

C₂×C₁₃₉

#541

yes

—

C₁₃₉

#2327

yes

characteristic, complemented, Sylow

#2327

—

139

maximal, Sylow

C₂×C₄⋊Q₈

D₁₃₉

yes

characteristic

C₂×C₄×Q₈

—

139

—

#391

—

139

—

#387

—

139

—

#382

—

139

—

#376

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

—

C₄⋊Q₈

Dic₁₃₉

yes

complemented

C₄×Q₈

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#95

—

139

—

#95

—

139

—

#95

—

139

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃₉

yes

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

C₂×Dic₁₃₉

yes

—

C₄²

—

278

—

100

C₄²

—

278

—

101

C₄²

—

139

—

102

C₄²

C₂×Dic₁₃₉

yes

—

103

C₂³

C₂³×D₁₃₉

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

278

—

109

C₂×C₄

—

278

—

110

C₂×C₄

—

278

—

111

C₂×C₄

—

278

—

112

C₂×C₄

—

278

—

113

C₂×C₄

—

139

—

114

C₂×C₄

C₂²×Dic₁₃₉

yes

—

115

C₂×C₄

C₂²×Dic₁₃₉

yes

—

116

C₂×C₄

C₂²×Dic₁₃₉

yes

—

117

C₂×C₄

D₄×D₁₃₉

yes

—

118

C₂×C₄

#14555784

yes

—

119

C₂²

C₂³×Dic₁₃₉

yes

characteristic

120

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

121

C₂²

C₂×#14555784

yes

characteristic

122

C₂²

#17984890

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

278

—

126

C₄

—

556

—

127

C₄

—

556

—

128

C₄

D₄×Dic₁₃₉

yes

—

129

C₂

C₂×D₄×Dic₁₃₉

yes

characteristic

130

C₂

#22581431

yes

—

131

C₂

#22581528

yes

—

132

C₂

#22581564

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₁₃₉	C₂
socle, prime radical, cosocle	C₂³×C₁₃₉	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	#2327
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₉

commutator

C₂²×C₁₃₉

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₁₃₉

C₂

socle, prime radical, cosocle

C₂³×C₁₃₉

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
17. C₄⋊Q₈×C₁₃₉	C₄	semidirect
66. C₁₃₉	#2327	semidirect
75. C₄⋊Q₈	Dic₁₃₉	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₁₃₉

C₄

semidirect

66. C₁₃₉

#2327

semidirect

75. C₄⋊Q₈

Dic₁₃₉

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₃₉	2
#22580860	2
#22580839	2
#22580819	2
#22580799	2
C₂×C₄⋊Q₈×C₁₃₉	2
#2327	139

subgroup

index

C₂×Q₈×Dic₁₃₉

C₂×C₄⋊Q₈×C₁₃₉

139

Subgroups — #31809124

Conjugacy classes of subgroups

Complements

Maximal subgroups