Group #31779911 — subgroups

Subgroups — #31779911

Order 17536.

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

normal subgroups only

This group has 218 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	#22578139	C₂	1	1	yes	maximal, characteristic
2	#22577775	C₂	1	1	yes	maximal, characteristic
3	#22577699	C₂	1	2	yes	maximal, complemented
4	#22577133	C₂	1	2	yes	maximal, complemented
5	#331×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555498	—	2	2	no	—
7	#17984088	C₂²	1	1	yes	characteristic
8	#17984075	—	2	2	no	—
9	#17984069	—	2	2	no	—
10	C₄×Dic₂₇₄	C₂²	1	2	yes	—
11	D₄×Dic₁₃₇	—	2	2	no	—
12	#17984022	—	2	2	no	—
13	#17984013	—	2	2	no	—
14	C₄×#11764040	C₂²	1	1	yes	characteristic
15	#127×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×Dic₂₇₄	—	2	1	no	—
18	C₂×Dic₂₇₄	D₄	1	2	yes	—
19	C₂²×Dic₁₃₇	—	2	2	no	—
20	#14555498	—	4	4	no	—
21	C₂×#11764040	—	2	1	no	—
22	C₄⋊Dic₁₃₇	—	2	2	no	—
23	C₄⋊Dic₁₃₇	D₄	1	2	yes	—
24	#14555489	—	2	2	no	—
25	C₂²⋊Dic₁₃₇	—	2	4	no	—
26	C₂²⋊Dic₁₃₇	—	2	2	no	—
27	C₄×Dic₁₃₇	—	2	2	no	—
28	C₂×D₄×C₁₃₇	—	2	2	no	—
29	C₂×D₄×C₁₃₇	D₄	1	2	yes	—
30	C₂×C₈×C₁₃₇	—	2	2	no	—
31	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	Dic₂₇₄	—	2	2	no	—
33	Dic₂₇₄	—	2	2	no	—
34	Dic₂₇₄	—	4	2	no	—
35	C₂×Dic₁₃₇	—	2	2	no	—
36	C₂×Dic₁₃₇	—	2	2	no	—
37	C₂×Dic₁₃₇	—	4	1	no	—
38	C₂×Dic₁₃₇	—	8	2	no	—
39	#11764040	—	2	2	no	—
40	C₂³×C₁₃₇	—	2	2	no	—
41	D₄×C₁₃₇	—	4	2	no	—
42	D₄×C₁₃₇	—	4	2	no	—
43	D₄×C₁₃₇	—	4	2	no	—
44	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₃₇	—	4	2	no	—
47	C₂²×C₁₃₇	—	4	4	no	—
48	C₂²×C₁₃₇	—	4	2	no	—
49	C₂²×C₁₃₇	#120	1	1	yes	characteristic
50	C₄×C₁₃₇	—	2	1	no	—
51	C₄×C₁₃₇	#135	1	2	yes	—
52	C₄×C₁₃₇	#137	1	2	yes	—
53	Dic₁₃₇	—	4	2	no	—
54	Dic₁₃₇	—	4	2	no	—
55	Dic₁₃₇	—	8	1	no	—
56	C₂×C₁₃₇	—	8	2	no	—
57	C₂×C₁₃₇	#452	1	2	yes	—
58	C₂×C₁₃₇	#453	1	1	yes	characteristic
59	C₁₃₇	#1630	1	1	yes	characteristic, complemented, Sylow
60	#1630	—	137	1	no	maximal, Sylow
61	#559	—	137	1	no	—
62	#492	—	137	1	no	—
63	#477	—	137	2	no	—
64	#331	D₁₃₇	1	1	yes	characteristic
65	Q₈⋊C₈	—	137	2	no	—
66	C₂×SD₁₆	—	274	2	no	—
67	C₄⋊Q₈	—	137	1	no	—
68	#127	C₂×D₁₃₇	1	1	yes	characteristic
69	#124	—	274	2	no	—
70	#123	—	274	2	no	—
71	C₄×Q₈	—	137	2	no	—
72	C₄×D₄	—	274	2	no	—
73	#107	—	274	2	no	—
74	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	—
75	#103	—	274	2	no	—
76	C₄×C₈	—	137	1	no	—
77	C₂×Q₈	—	274	1	no	—
78	C₂×Q₈	—	137	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	D₅₄₈	1	2	yes	—
81	C₂²×C₄	—	274	2	no	—
82	SD₁₆	—	548	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	274	1	no	—
85	C₄⋊C₄	—	274	2	no	—
86	C₄⋊C₄	—	274	2	no	—
87	C₄⋊C₄	—	137	2	no	—
88	C₂²⋊C₄	—	274	4	no	—
89	C₂²⋊C₄	—	274	2	no	—
90	C₄²	—	274	2	no	—
91	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	274	2	no	—
94	Q₈	—	274	2	no	—
95	Q₈	—	548	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	274	2	no	—
100	C₂×C₄	—	274	2	no	—
101	C₂×C₄	—	548	1	no	—
102	C₂×C₄	—	1096	2	no	—
103	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
104	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	274	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#17984058	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	548	2	no	—
112	C₄	—	548	2	no	—
113	C₄	—	1096	1	no	—
114	C₄	#17984107	1	2	yes	—
115	C₄	#17984117	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#22577566	1	2	yes	—
118	C₂	#22577573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578139

C₂

yes

maximal, characteristic

#22577775

C₂

yes

maximal, characteristic

#22577699

C₂

yes

maximal, complemented

#22577133

C₂

yes

maximal, complemented

#331×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555498

—

#17984088

C₂²

yes

characteristic

#17984075

—

#17984069

—

C₄×Dic₂₇₄

C₂²

yes

—

D₄×Dic₁₃₇

—

#17984022

—

#17984013

—

C₄×#11764040

C₂²

yes

characteristic

#127×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

D₄

yes

—

C₂²×Dic₁₃₇

—

#14555498

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

D₄

yes

—

#14555489

—

C₂²⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×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

C₄×C₁₃₇

—

C₄×C₁₃₇

#135

yes

—

C₄×C₁₃₇

#137

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#452

yes

—

C₂×C₁₃₇

#453

yes

characteristic

C₁₃₇

#1630

yes

characteristic, complemented, Sylow

#1630

—

137

maximal, Sylow

#559

—

137

—

#492

—

137

—

#477

—

137

—

#331

D₁₃₇

yes

characteristic

Q₈⋊C₈

—

137

—

C₂×SD₁₆

—

274

—

C₄⋊Q₈

—

137

—

#127

C₂×D₁₃₇

yes

characteristic

#124

—

274

—

#123

—

274

—

C₄×Q₈

—

137

—

C₄×D₄

—

274

—

#107

—

274

—

C₄⋊C₈

C₂×D₁₃₇

yes

—

#103

—

274

—

C₄×C₈

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×D₄

—

C₂×D₄

D₅₄₈

yes

—

C₂²×C₄

—

274

—

SD₁₆

—

548

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₄²

—

274

—

C₄²

C₂²×D₁₃₇

yes

characteristic

C₂³

—

Q₈

—

274

—

Q₈

—

274

—

Q₈

—

548

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

274

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

548

—

102

C₂×C₄

—

1096

—

103

C₂×C₄

D₄×D₁₃₇

yes

—

104

C₂×C₄

C₂×D₅₄₈

yes

characteristic

105

C₈

—

106

C₈

—

274

—

107

C₂²

—

108

C₂²

—

109

C₂²

#17984058

yes

characteristic

110

C₄

—

111

C₄

—

548

—

112

C₄

—

548

—

113

C₄

—

1096

—

114

C₄

#17984107

yes

—

115

C₄

#17984117

yes

—

116

C₂

—

117

C₂

#22577566

yes

—

118

C₂

#22577573

yes

characteristic

	subgroup	G/H
center	C₂²	#17984058
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#331×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#120
prime radical	#127×C₁₃₇	C₂²
hypercenter	#331	D₁₃₇
hypocenter	C₁₃₇	#1630
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#17984058

commutator, cosocle

C₄²×C₁₃₇

C₂³

Frattini

C₄²

C₂²×D₁₃₇

Fitting

#331×C₁₃₇

C₂

socle

C₂²×C₁₃₇

#120

prime radical

#127×C₁₃₇

C₂²

D₁₃₇

C₁₃₇

G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
3. #22577699	C₂	semidirect
4. #22577133	C₂	semidirect
59. C₁₃₇	#1630	semidirect

H ≅ G/N

type

3. #22577699

C₂

semidirect

4. #22577133

C₂

semidirect

59. C₁₃₇

#1630

semidirect

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

subgroup	index
#22578139	2
#22577775	2
#22577699	2
#22577133	2
#331×C₁₃₇	2
#1630	137

subgroup

index

#331×C₁₃₇

137

Subgroups — #31779911

Conjugacy classes of subgroups

Complements

Maximal subgroups