Group #31779870 — subgroups

Subgroups — #31779870

Order 17536.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22578119	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577595	C₂	1	2	yes	maximal, complemented
3	#22577152	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577132	C₂	1	2	yes	maximal, complemented
5	#492×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555495	—	2	2	no	—
7	#17984077	—	2	2	no	—
8	#17984063	—	2	2	no	—
9	C₄×D₅₄₈	C₂²	1	2	yes	—
10	#17984046	—	2	2	no	—
11	C₄⋊#11764040	C₂²	1	2	yes	—
12	#17984009	—	2	2	no	—
13	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	#127×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×D₅₄₈	—	2	2	no	—
18	C₂×D₅₄₈	D₄	1	2	yes	—
19	C₂×C₄×D₁₃₇	—	2	2	no	—
20	#14555495	—	4	4	no	—
21	C₂×#11764040	—	2	2	no	—
22	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
23	#14555487	—	2	4	no	—
24	#14555487	—	2	2	no	—
25	C₄×Dic₁₃₇	—	2	2	no	—
26	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
27	C₂×D₄×C₁₃₇	—	2	2	no	—
28	C₂×D₄×C₁₃₇	—	2	1	no	—
29	SD₁₆×C₁₃₇	—	4	4	no	—
30	C₂×C₈×C₁₃₇	—	2	1	no	—
31	C₄⋊C₄×C₁₃₇	—	2	2	no	—
32	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₃₇	—	2	2	no	—
34	D₅₄₈	—	2	2	no	—
35	D₅₄₈	—	2	2	no	—
36	D₅₄₈	—	8	2	no	—
37	C₂×Dic₁₃₇	—	2	2	no	—
38	C₄×D₁₃₇	—	4	2	no	—
39	C₄×D₁₃₇	—	4	2	no	—
40	#11764040	—	4	2	no	—
41	C₂³×C₁₃₇	—	4	1	no	—
42	Q₈×C₁₃₇	—	4	2	no	—
43	D₄×C₁₃₇	—	4	2	no	—
44	D₄×C₁₃₇	—	4	2	no	—
45	D₄×C₁₃₇	—	4	2	no	—
46	C₂×C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₇	—	2	2	no	—
50	C₂²×C₁₃₇	—	8	2	no	—
51	C₂²×C₁₃₇	—	8	1	no	—
52	C₂²×C₁₃₇	#120	1	1	yes	characteristic
53	C₂×D₁₃₇	—	4	2	no	—
54	C₂×D₁₃₇	—	4	2	no	—
55	C₂×D₁₃₇	—	4	2	no	—
56	C₄×C₁₃₇	—	2	1	no	—
57	C₄×C₁₃₇	—	4	2	no	—
58	C₄×C₁₃₇	#135	1	2	yes	—
59	C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₁₃₇	—	4	2	no	—
61	C₂×C₁₃₇	—	16	1	no	—
62	C₂×C₁₃₇	#449	1	2	yes	—
63	C₂×C₁₃₇	#453	1	1	yes	characteristic
64	D₁₃₇	—	8	2	no	—
65	C₁₃₇	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	137	1	no	maximal, Sylow
67	#555	—	137	1	no	—
68	#492	D₁₃₇	1	1	yes	characteristic, complemented
69	#459	—	137	2	no	—
70	#333	—	137	1	no	—
71	D₄⋊C₈	—	137	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	274	2	no	—
74	C₄⋊Q₈	C₂×D₁₃₇	1	1	yes	characteristic
75	#127	C₂×D₁₃₇	1	1	yes	characteristic
76	#124	—	274	2	no	—
77	#121	—	274	2	no	—
78	C₄×D₄	—	137	2	no	—
79	#117	—	274	2	no	—
80	C₄⋊C₈	—	137	2	no	—
81	#102	—	274	2	no	—
82	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₁₃₇	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	274	2	no	—
87	C₂×D₄	—	137	2	no	—
88	C₂²×C₄	—	274	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	548	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	274	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	137	2	no	—
95	C₂²⋊C₄	—	274	4	no	—
96	C₂²⋊C₄	—	274	2	no	—
97	C₄²	—	274	2	no	—
98	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	274	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	274	2	no	—
106	D₄	—	274	2	no	—
107	D₄	—	1096	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	274	2	no	—
110	C₂×C₄	—	548	2	no	—
111	C₂×C₄	—	548	2	no	—
112	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	548	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	548	2	no	—
119	C₂²	—	548	2	no	—
120	C₂²	—	548	2	no	—
121	C₂²	#17984056	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	548	2	no	—
125	C₄	#17984109	1	2	yes	—
126	C₄	#17984115	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	1096	2	no	—
129	C₂	#22577547	1	2	yes	—
130	C₂	#22577570	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578119

C₂

yes

maximal, characteristic, complemented

#22577595

C₂

yes

maximal, complemented

#22577152

C₂

yes

maximal, characteristic, complemented

#22577132

C₂

yes

maximal, complemented

#492×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#14555495

—

#17984077

—

#17984063

—

C₄×D₅₄₈

C₂²

yes

—

#17984046

—

C₄⋊#11764040

C₂²

yes

—

#17984009

—

C₂×SD₁₆×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

#127×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×D₅₄₈

—

C₂×D₅₄₈

D₄

yes

—

C₂×C₄×D₁₃₇

—

#14555495

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

D₄

yes

—

#14555487

—

#14555487

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

SD₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂²×D₁₃₇

—

D₅₄₈

—

D₅₄₈

—

D₅₄₈

—

C₂×Dic₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#120

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#135

yes

—

C₄×C₁₃₇

AGL(1,ℤ/8)

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#449

yes

—

C₂×C₁₃₇

#453

yes

characteristic

D₁₃₇

—

C₁₃₇

#1624

yes

characteristic, complemented, Sylow

#1624

—

137

maximal, Sylow

#555

—

137

—

#492

D₁₃₇

yes

characteristic, complemented

#459

—

137

—

#333

—

137

—

D₄⋊C₈

—

137

—

C₂×SD₁₆

—

C₂×D₈

—

274

—

C₄⋊Q₈

C₂×D₁₃₇

yes

characteristic

#127

C₂×D₁₃₇

yes

characteristic

#124

—

274

—

#121

—

274

—

C₄×D₄

—

137

—

#117

—

274

—

C₄⋊C₈

—

137

—

#102

—

274

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

C₂×Q₈

C₂²⋊D₁₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

274

—

C₂×D₄

—

137

—

C₂²×C₄

—

274

—

SD₁₆

—

D₈

—

548

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₄²

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

100

C₂³

—

274

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

274

—

106

D₄

—

274

—

107

D₄

—

1096

—

108

C₂×C₄

—

109

C₂×C₄

—

274

—

110

C₂×C₄

—

548

—

111

C₂×C₄

—

548

—

112

C₂×C₄

D₄×D₁₃₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

114

C₈

—

115

C₈

—

548

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

548

—

119

C₂²

—

548

—

120

C₂²

—

548

—

121

C₂²

#17984056

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

548

—

125

C₄

#17984109

yes

—

126

C₄

#17984115

yes

—

127

C₂

—

128

C₂

—

1096

—

129

C₂

#22577547

yes

—

130

C₂

#22577570

yes

characteristic

	subgroup	G/H
center	C₂²	#17984056
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#492×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#120
hypercenter	#492	D₁₃₇
hypocenter	C₁₃₇	#1624
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#17984056

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#492×C₁₃₇

C₂

C₂²×C₁₃₇

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #22578119	C₂	semidirect
2. #22577595	C₂	semidirect
3. #22577152	C₂	semidirect
4. #22577132	C₂	semidirect
5. #492×C₁₃₇	C₂	semidirect
65. C₁₃₇	#1624	semidirect
68. #492	D₁₃₇	semidirect

H ≅ G/N

type

1. #22578119

C₂

semidirect

2. #22577595

C₂

semidirect

3. #22577152

C₂

semidirect

4. #22577132

C₂

semidirect

5. #492×C₁₃₇

C₂

semidirect

65. C₁₃₇

#1624

semidirect

68. #492

D₁₃₇

semidirect

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

subgroup	index
#22578119	2
#22577595	2
#22577152	2
#22577132	2
#492×C₁₃₇	2
#1624	137

subgroup

index

#492×C₁₃₇

137

Subgroups — #31779870

Conjugacy classes of subgroups

Complements

Maximal subgroups