Group #31805761 — subgroups

Subgroups — #31805761

Order 17792.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22581150	C₂	1	2	yes	maximal, complemented
2	C₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580706	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580697	C₂	1	2	yes	maximal, complemented
5	#493×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555773	—	2	2	no	—
7	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
8	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
10	Dic₁₃₉⋊C₈	—	2	1	no	—
11	#17984763	—	2	2	no	—
12	C₈×Dic₁₃₉	—	2	1	no	—
13	C₂×D₈×C₁₃₉	—	2	1	no	—
14	C₂×D₈×C₁₃₉	—	2	1	no	—
15	#127×C₁₃₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
18	#14555773	—	4	4	no	—
19	C₂×C₁₃₉⋊C₈	—	2	1	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
22	Dic₁₃₉⋊C₄	—	2	1	no	—
23	C₄×Dic₁₃₉	—	2	1	no	—
24	C₂×D₄×C₁₃₉	—	2	2	no	—
25	C₂×D₄×C₁₃₉	—	2	2	no	—
26	C₂×D₄×C₁₃₉	—	2	2	no	—
27	D₈×C₁₃₉	—	4	2	no	—
28	D₈×C₁₃₉	—	4	2	no	—
29	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	Dic₂₇₈	—	2	1	no	—
33	Dic₂₇₈	D₈	1	2	yes	complemented
34	C₂×Dic₁₃₉	—	2	1	no	—
35	C₂×Dic₁₃₉	—	2	1	no	—
36	C₁₃₉⋊C₈	—	4	2	no	—
37	C₁₃₉⋊C₈	—	4	1	no	—
38	C₂³×C₁₃₉	—	4	2	no	—
39	D₄×C₁₃₉	—	4	2	no	—
40	D₄×C₁₃₉	—	4	2	no	—
41	D₄×C₁₃₉	—	4	2	no	—
42	D₄×C₁₃₉	—	4	2	no	—
43	D₄×C₁₃₉	—	8	2	no	—
44	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
47	C₈×C₁₃₉	—	2	1	no	—
48	C₈×C₁₃₉	SD₁₆	1	2	yes	complemented
49	C₂²×C₁₃₉	—	8	2	no	—
50	C₂²×C₁₃₉	—	8	2	no	—
51	C₂²×C₁₃₉	—	8	2	no	—
52	C₂²×C₁₃₉	#121	1	1	yes	characteristic
53	C₄×C₁₃₉	—	2	1	no	—
54	C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
55	C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₁₃₉	#135	1	1	yes	characteristic
57	C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₁₃₉	—	2	2	no	—
59	Dic₁₃₉	—	4	1	no	—
60	C₂×C₁₃₉	—	16	2	no	—
61	C₂×C₁₃₉	#460	1	1	yes	characteristic
62	C₂×C₁₃₉	#466	1	1	yes	characteristic
63	C₂×C₁₃₉	#472	1	1	yes	characteristic
64	C₁₃₉	#1668	1	1	yes	characteristic, complemented, Sylow
65	#1668	—	139	1	no	maximal, Sylow
66	#493	D₁₃₉	1	1	yes	characteristic
67	#460	—	139	2	no	—
68	C₈×Q₈	—	139	1	no	—
69	#335	—	139	1	no	—
70	#331	—	139	2	no	—
71	C₂×SD₁₆	—	278	2	no	—
72	C₂×D₈	—	2	1	no	—
73	C₂×D₈	—	2	1	no	—
74	#127	C₂×D₁₃₉	1	2	yes	—
75	C₄×Q₈	—	139	1	no	—
76	C₄⋊C₈	—	278	1	no	—
77	C₄⋊C₈	—	139	1	no	—
78	C₄⋊C₈	—	139	2	no	—
79	#102	—	278	2	no	—
80	C₄×C₈	—	278	1	no	—
81	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
82	C₂×Q₈	—	139	1	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	SD₁₆	—	556	4	no	—
87	D₈	—	4	2	no	—
88	D₈	—	4	2	no	—
89	C₂×C₈	—	278	2	no	—
90	C₂×C₈	—	278	1	no	—
91	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
93	C₄⋊C₄	—	278	1	no	—
94	C₄⋊C₄	—	139	1	no	—
95	C₄²	—	278	1	no	—
96	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
97	C₂³	—	4	2	no	—
98	Q₈	—	278	1	no	—
99	Q₈	—	139	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	8	2	no	—
105	C₂×C₄	—	278	1	no	—
106	C₂×C₄	—	278	1	no	—
107	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
109	C₂×C₄	#14555784	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	556	2	no	—
112	C₈	—	556	1	no	—
113	C₈	#14555773	1	2	yes	complemented
114	C₂²	—	8	2	no	—
115	C₂²	—	8	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	#17984813	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	278	2	no	—
120	C₄	—	556	1	no	—
121	C₄	D₈×D₁₃₉	1	1	yes	characteristic
122	C₄	C₂×#14555773	1	1	yes	characteristic
123	C₄	#17984861	1	1	yes	characteristic
124	C₄	#17984868	1	1	yes	characteristic
125	C₂	—	16	2	no	—
126	C₂	#22581150	1	1	yes	characteristic
127	C₂	#22581186	1	1	yes	characteristic
128	C₂	#22581220	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581150

C₂

yes

maximal, complemented

C₈×Dic₂₇₈

C₂

yes

maximal, characteristic, complemented

#22580706

C₂

yes

maximal, characteristic, complemented

#22580697

C₂

yes

maximal, complemented

#493×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555773

—

C₄×Dic₂₇₈

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

Dic₁₃₉⋊C₈

—

#17984763

—

C₈×Dic₁₃₉

—

C₂×D₈×C₁₃₉

—

C₂×D₈×C₁₃₉

—

#127×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

D₄

yes

characteristic

#14555773

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

D₄

yes

characteristic

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

D₈×C₁₃₉

—

D₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

D₈

yes

complemented

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

SD₁₆

yes

complemented

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₈

yes

characteristic

C₄×C₁₃₉

C₂×SD₁₆

yes

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#1668

—

139

maximal, Sylow

#493

D₁₃₉

yes

characteristic

#460

—

139

—

C₈×Q₈

—

139

—

#335

—

139

—

#331

—

139

—

C₂×SD₁₆

—

278

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₁₃₉

yes

—

C₄×Q₈

—

139

—

C₄⋊C₈

—

278

—

C₄⋊C₈

—

139

—

C₄⋊C₈

—

139

—

#102

—

278

—

C₄×C₈

—

278

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

139

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

556

—

D₈

—

D₈

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

Q₈

—

278

—

Q₈

—

139

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

278

—

106

C₂×C₄

—

278

—

107

C₂×C₄

D₄×D₁₃₉

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

109

C₂×C₄

#14555784

yes

characteristic

110

C₈

—

111

C₈

—

556

—

112

C₈

—

556

—

113

C₈

#14555773

yes

complemented

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#17984813

yes

characteristic

118

C₄

—

119

C₄

—

278

—

120

C₄

—

556

—

121

C₄

D₈×D₁₃₉

yes

characteristic

122

C₄

C₂×#14555773

yes

characteristic

123

C₄

#17984861

yes

characteristic

124

C₄

#17984868

yes

characteristic

125

C₂

—

126

C₂

#22581150

yes

characteristic

127

C₂

#22581186

yes

characteristic

128

C₂

#22581220

yes

characteristic

	subgroup	G/H
center	C₂²	#17984813
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting, prime radical	#493×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#121
hypercenter	#493	D₁₃₉
hypocenter	C₁₃₉	#1668
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#17984813

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting, prime radical

#493×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #22581150	C₂	semidirect
2. C₈×Dic₂₇₈	C₂	semidirect
3. #22580706	C₂	semidirect
4. #22580697	C₂	semidirect
33. Dic₂₇₈	D₈	semidirect
48. C₈×C₁₃₉	SD₁₆	semidirect
64. C₁₃₉	#1668	semidirect
113. C₈	#14555773	semidirect

H ≅ G/N

type

1. #22581150

C₂

semidirect

2. C₈×Dic₂₇₈

C₂

semidirect

3. #22580706

C₂

semidirect

4. #22580697

C₂

semidirect

33. Dic₂₇₈

D₈

semidirect

48. C₈×C₁₃₉

SD₁₆

semidirect

64. C₁₃₉

#1668

semidirect

113. C₈

#14555773

semidirect

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

subgroup	index
#22581150	2
C₈×Dic₂₇₈	2
#22580706	2
#22580697	2
#493×C₁₃₉	2
#1668	139

subgroup

index

C₈×Dic₂₇₈

#493×C₁₃₉

139

Subgroups — #31805761

Conjugacy classes of subgroups

Complements

Maximal subgroups