Group #31780408 — subgroups

Subgroups — #31780408

Order 17536.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22577808	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577799	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577775	C₂	1	1	yes	maximal, characteristic
4	#22577755	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577749	C₂	1	1	yes	maximal, characteristic
6	#22577124	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555498	—	2	2	no	—
9	#17984088	C₂²	1	1	yes	characteristic
10	#17984088	C₂²	1	1	yes	characteristic
11	#17984022	—	2	2	no	—
12	#17984019	—	2	2	no	—
13	#17984011	—	2	2	no	—
14	#17984009	—	2	2	no	—
15	C₄×#11764040	C₂²	1	1	yes	characteristic
16	C₄×#11764040	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
18	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#127×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₇₄	—	2	1	no	—
22	C₂×Dic₂₇₄	—	2	1	no	—
23	#14555498	—	4	4	no	—
24	C₂×#11764040	D₄	1	2	yes	—
25	C₂×#11764040	D₄	1	2	yes	—
26	C₄⋊Dic₁₃₇	—	2	1	no	—
27	C₄⋊Dic₁₃₇	—	2	1	no	—
28	C₄⋊Dic₁₃₇	—	2	1	no	—
29	C₄⋊Dic₁₃₇	—	2	1	no	—
30	C₂×Q₈×C₁₃₇	—	2	1	no	—
31	C₂×D₄×C₁₃₇	—	2	1	no	—
32	C₂×D₄×C₁₃₇	—	2	1	no	—
33	C₂×D₄×C₁₃₇	—	2	1	no	—
34	SD₁₆×C₁₃₇	—	4	4	no	—
35	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
36	C₄⋊C₄×C₁₃₇	—	2	1	no	—
37	C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
39	Dic₂₇₄	—	4	2	no	—
40	Dic₂₇₄	—	4	2	no	—
41	C₂×Dic₁₃₇	—	4	1	no	—
42	C₂×Dic₁₃₇	—	4	1	no	—
43	#11764040	—	2	2	no	—
44	#11764040	—	2	2	no	—
45	C₂³×C₁₃₇	—	4	1	no	—
46	Q₈×C₁₃₇	—	4	2	no	—
47	D₄×C₁₃₇	—	4	2	no	—
48	D₄×C₁₃₇	—	4	2	no	—
49	D₄×C₁₃₇	—	4	2	no	—
50	C₂×C₄×C₁₃₇	—	4	1	no	—
51	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₃₇	SD₁₆	1	4	yes	complemented
55	C₂²×C₁₃₇	—	8	1	no	—
56	C₂²×C₁₃₇	—	8	1	no	—
57	C₂²×C₁₃₇	—	8	1	no	—
58	C₂²×C₁₃₇	#127	1	1	yes	characteristic
59	C₄×C₁₃₇	—	8	1	no	—
60	C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
61	C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₃₇	#137	1	2	yes	—
63	Dic₁₃₇	—	8	1	no	—
64	Dic₁₃₇	—	8	1	no	—
65	C₂×C₁₃₇	—	16	1	no	—
66	C₂×C₁₃₇	#492	1	1	yes	characteristic
67	C₂×C₁₃₇	#496	1	1	yes	characteristic
68	C₂×C₁₃₇	#497	1	1	yes	characteristic
69	C₁₃₇	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	137	1	no	maximal, Sylow
71	#500	—	137	1	no	—
72	#498	—	137	1	no	—
73	#492	—	137	1	no	—
74	#492	D₁₃₇	1	1	yes	characteristic
75	#487	—	137	1	no	—
76	#486	—	137	1	no	—
77	#322	—	137	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	274	2	no	—
80	C₄⋊Q₈	—	137	1	no	—
81	C₄⋊Q₈	—	137	1	no	—
82	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
83	#127	C₂×D₁₃₇	1	1	yes	characteristic
84	#107	—	274	2	no	—
85	#106	—	274	2	no	—
86	#103	—	274	2	no	—
87	#102	—	274	2	no	—
88	C₄×C₈	—	137	1	no	—
89	C₄×C₈	—	137	1	no	—
90	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	274	1	no	—
93	C₂×Q₈	—	274	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	548	4	no	—
99	C₂×C₈	—	137	2	no	—
100	C₂×C₈	—	137	2	no	—
101	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	274	1	no	—
105	C₄⋊C₄	—	274	1	no	—
106	C₄⋊C₄	—	274	1	no	—
107	C₄⋊C₄	—	274	1	no	—
108	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	548	2	no	—
112	Q₈	—	548	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	548	1	no	—
118	C₂×C₄	—	548	1	no	—
119	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
122	C₈	—	274	2	no	—
123	C₈	—	274	2	no	—
124	C₈	#14555498	1	4	yes	complemented
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#17984084	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	1096	1	no	—
131	C₄	—	1096	1	no	—
132	C₄	C₂×#14555498	1	2	yes	—
133	C₄	#17984115	1	2	yes	—
134	C₄	#17984117	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#22577775	1	1	yes	characteristic
137	C₂	#22577789	1	1	yes	characteristic
138	C₂	#22577794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577808

C₂

yes

maximal, characteristic, complemented

#22577799

C₂

yes

maximal, characteristic, complemented

#22577775

C₂

yes

maximal, characteristic

#22577755

C₂

yes

maximal, characteristic, complemented

#22577749

C₂

yes

maximal, characteristic

#22577124

C₂

yes

maximal, characteristic, complemented

#492×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555498

—

#17984088

C₂²

yes

characteristic

#17984088

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11764040

C₂²

yes

characteristic

C₂×SD₁₆×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

#127×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

#14555498

—

C₂×#11764040

D₄

yes

—

C₂×#11764040

D₄

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

SD₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

SD₁₆

yes

complemented

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#127

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×SD₁₆

yes

—

C₄×C₁₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1714

—

137

maximal, Sylow

#500

—

137

—

#498

—

137

—

#492

—

137

—

#492

D₁₃₇

yes

characteristic

#487

—

137

—

#486

—

137

—

#322

—

137

—

C₂×SD₁₆

—

C₂×SD₁₆

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

#127

C₂×D₁₃₇

yes

characteristic

#107

—

274

—

#106

—

274

—

#103

—

274

—

#102

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

548

—

C₂×C₈

—

137

—

100

C₂×C₈

—

137

—

101

C₂×C₈

C₂²⋊D₁₃₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

274

—

105

C₄⋊C₄

—

274

—

106

C₄⋊C₄

—

274

—

107

C₄⋊C₄

—

274

—

108

C₄²

C₂²×D₁₃₇

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

548

—

112

Q₈

—

548

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

548

—

118

C₂×C₄

—

548

—

119

C₂×C₄

D₄×D₁₃₇

yes

characteristic

120

C₂×C₄

D₄×D₁₃₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

122

C₈

—

274

—

123

C₈

—

274

—

124

C₈

#14555498

yes

complemented

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#17984084

yes

characteristic

129

C₄

—

130

C₄

—

1096

—

131

C₄

—

1096

—

132

C₄

C₂×#14555498

yes

—

133

C₄

#17984115

yes

—

134

C₄

#17984117

yes

—

135

C₂

—

136

C₂

#22577775

yes

characteristic

137

C₂

#22577789

yes

characteristic

138

C₂

#22577794

yes

characteristic

	subgroup	G/H
center	C₂²	#17984084
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#492×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#127
prime radical	#127×C₁₃₇	C₂²
hypercenter	#492	D₁₃₇
hypocenter	C₁₃₇	#1714
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#17984084

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#492×C₁₃₇

C₂

socle

C₂²×C₁₃₇

#127

prime radical

#127×C₁₃₇

C₂²

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #22577808	C₂	semidirect
2. #22577799	C₂	semidirect
4. #22577755	C₂	semidirect
6. #22577124	C₂	semidirect
54. C₈×C₁₃₇	SD₁₆	semidirect
69. C₁₃₇	#1714	semidirect
124. C₈	#14555498	semidirect

H ≅ G/N

type

1. #22577808

C₂

semidirect

2. #22577799

C₂

semidirect

4. #22577755

C₂

semidirect

6. #22577124

C₂

semidirect

54. C₈×C₁₃₇

SD₁₆

semidirect

69. C₁₃₇

#1714

semidirect

124. C₈

#14555498

semidirect

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

subgroup	index
#22577808	2
#22577799	2
#22577775	2
#22577755	2
#22577749	2
#22577124	2
#492×C₁₃₇	2
#1714	137

subgroup

index

#492×C₁₃₇

137

Subgroups — #31780408

Conjugacy classes of subgroups

Complements

Maximal subgroups