Group #31805463 — subgroups

Subgroups — #31805463

Order 17792.

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

normal subgroups only

This group has 240 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	#22581329	C₂	1	1	yes	maximal, characteristic
2	#22581153	C₂	1	2	yes	maximal, complemented
3	#22580697	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₁₃₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555772	—	2	2	no	—
7	C₂×#14555770	—	2	2	no	—
8	#17984839	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
10	#17984762	—	2	2	no	—
11	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	#127×C₁₃₉	C₂²	1	1	yes	characteristic
13	#124×C₁₃₉	—	2	2	no	—
14	#123×C₁₃₉	—	2	2	no	—
15	C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
16	C₄×D₄×C₁₃₉	—	2	2	no	—
17	C₂×D₅₅₆	—	2	1	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	#14555772	—	4	4	no	—
20	#14555770	—	4	4	no	—
21	C₂×C₁₃₉⋊C₈	—	2	1	no	—
22	C₂×C₁₃₉⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₁₃₉	D₄	1	2	yes	—
24	C₂×D₄×C₁₃₉	—	2	2	no	—
25	C₂×D₄×C₁₃₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₉	—	2	2	no	—
27	C₄⋊C₄×C₁₃₉	—	2	2	no	—
28	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
30	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
31	C₄²×C₁₃₉	—	2	2	no	—
32	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₃₉	—	4	1	no	—
34	D₅₅₆	—	4	2	no	—
35	D₅₅₆	—	4	2	no	—
36	D₅₅₆	—	4	2	no	—
37	C₁₃₉⋊C₈	—	2	2	no	—
38	C₁₃₉⋊C₈	—	4	2	no	—
39	C₂³×C₁₃₉	—	2	2	no	—
40	Q₈×C₁₃₉	—	2	2	no	—
41	Q₈×C₁₃₉	—	2	2	no	—
42	D₄×C₁₃₉	—	4	2	no	—
43	D₄×C₁₃₉	—	4	2	no	—
44	D₄×C₁₃₉	—	4	2	no	—
45	C₂×C₄×C₁₃₉	—	2	2	no	—
46	C₂×C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄×C₁₃₉	—	8	2	no	—
48	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
50	C₂²×C₁₃₉	—	4	4	no	—
51	C₂²×C₁₃₉	—	4	2	no	—
52	C₂²×C₁₃₉	#120	1	1	yes	characteristic
53	C₂×D₁₃₉	—	8	2	no	—
54	C₂×D₁₃₉	—	8	1	no	—
55	C₄×C₁₃₉	—	2	1	no	—
56	C₄×C₁₃₉	—	4	2	no	—
57	C₄×C₁₃₉	—	4	2	no	—
58	C₄×C₁₃₉	#135	1	2	yes	—
59	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₃₉	—	8	2	no	—
61	C₂×C₁₃₉	#449	1	2	yes	—
62	C₂×C₁₃₉	#453	1	1	yes	characteristic
63	D₁₃₉	—	16	1	no	—
64	C₁₃₉	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	139	1	no	maximal, Sylow
66	#559	D₁₃₉	1	1	yes	characteristic, complemented
67	#493	—	139	1	no	—
68	#460	—	139	2	no	—
69	#331	—	139	1	no	—
70	Q₈⋊C₈	—	139	2	no	—
71	C₂×SD₁₆	—	278	2	no	—
72	C₂×D₈	—	278	2	no	—
73	#127	—	139	1	no	—
74	#127	C₂×D₁₃₉	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₃₉	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	139	2	no	—
80	#102	—	278	2	no	—
81	C₄×C₈	—	139	1	no	—
82	C₂×Q₈	C₂²⋊D₁₃₉	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	278	2	no	—
85	C₂×D₄	—	278	1	no	—
86	C₂×D₄	C₂²⋊D₁₃₉	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	556	4	no	—
89	D₈	—	556	4	no	—
90	C₂×C₈	—	278	1	no	—
91	C₂×C₈	—	278	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₃₉	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	556	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	556	2	no	—
106	D₄	—	556	2	no	—
107	D₄	—	556	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
113	C₈	—	278	2	no	—
114	C₈	—	556	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	1112	2	no	—
118	C₂²	—	1112	1	no	—
119	C₂²	#17984811	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#17984862	1	2	yes	—
124	C₄	#17984867	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	2224	1	no	—
127	C₂	#22581098	1	2	yes	—
128	C₂	#22581121	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22581329

C₂

yes

maximal, characteristic

#22581153

C₂

yes

maximal, complemented

#22580697

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, complemented

#559×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555772

—

C₂×#14555770

—

#17984839

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984762

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

#127×C₁₃₉

C₂²

yes

characteristic

#124×C₁₃₉

—

#123×C₁₃₉

—

C₄×Q₈×C₁₃₉

C₂²

yes

—

C₄×D₄×C₁₃₉

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

#14555772

—

#14555770

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×Q₈×C₁₃₉

D₄

yes

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₂²⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#449

yes

—

C₂×C₁₃₉

#453

yes

characteristic

D₁₃₉

—

C₁₃₉

#1620

yes

characteristic, complemented, Sylow

#1620

—

139

maximal, Sylow

#559

D₁₃₉

yes

characteristic, complemented

#493

—

139

—

#460

—

139

—

#331

—

139

—

Q₈⋊C₈

—

139

—

C₂×SD₁₆

—

278

—

C₂×D₈

—

278

—

#127

—

139

—

#127

C₂×D₁₃₉

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₃₉

yes

—

C₄×D₄

—

C₄⋊C₈

—

139

—

#102

—

278

—

C₄×C₈

—

139

—

C₂×Q₈

C₂²⋊D₁₃₉

yes

—

C₂×D₄

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

—

SD₁₆

—

556

—

D₈

—

556

—

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

C₂³

—

556

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

556

—

106

D₄

—

556

—

107

D₄

—

556

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

113

C₈

—

278

—

114

C₈

—

556

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

1112

—

118

C₂²

—

1112

—

119

C₂²

#17984811

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#17984862

yes

—

124

C₄

#17984867

yes

—

125

C₂

—

126

C₂

—

2224

—

127

C₂

#22581098

yes

—

128

C₂

#22581121

yes

characteristic

	subgroup	G/H
center	C₂²	#17984811
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#559×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#120
prime radical	#22581329	C₂
hypercenter	#559	D₁₃₉
hypocenter	C₁₃₉	#1620
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#17984811

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#559×C₁₃₉

C₂

C₂²×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #22581153	C₂	semidirect
3. #22580697	C₂	semidirect
4. Q₈⋊(C₁₃₉⋊C₈)	C₂	semidirect
5. #559×C₁₃₉	C₂	semidirect
64. C₁₃₉	#1620	semidirect
66. #559	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581153

C₂

semidirect

3. #22580697

C₂

semidirect

4. Q₈⋊(C₁₃₉⋊C₈)

C₂

semidirect

5. #559×C₁₃₉

C₂

semidirect

64. C₁₃₉

#1620

semidirect

66. #559

D₁₃₉

semidirect

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

subgroup	index
#22581329	2
#22581153	2
#22580697	2
Q₈⋊(C₁₃₉⋊C₈)	2
#559×C₁₃₉	2
#1620	139

subgroup

index

#22581329

#22581153

#22580697

Q₈⋊(C₁₃₉⋊C₈)

#559×C₁₃₉

#1620

139

Subgroups — #31805463

Conjugacy classes of subgroups

Complements

Maximal subgroups