Group #31776554 — subgroups

Subgroups — #1543×C₁₃₇ #31776554

Order 17536.

246 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 96 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	#532×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#431×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#333×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#331×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₁₃₇	C₄	1	2	yes	—
6	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	#127×C₁₃₇	C₂²	1	1	yes	characteristic
8	#124×C₁₃₇	C₄	1	2	yes	—
9	#121×C₁₃₇	—	2	2	no	—
10	C₄×D₄×C₁₃₇	—	2	2	no	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
13	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
14	C₄×C₈×C₁₃₇	—	2	2	no	—
15	#41×C₁₃₇	—	4	4	no	—
16	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₁₃₇	—	2	2	no	—
18	C₂×D₄×C₁₃₇	—	2	2	no	—
19	C₂×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃₇	—	2	2	no	—
21	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
23	C₂×C₈×C₁₃₇	—	2	4	no	—
24	C₄⋊C₄×C₁₃₇	—	2	2	no	—
25	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
26	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃₇	D₄	1	2	yes	—
28	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂³×C₁₃₇	—	2	2	no	—
30	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
31	Q₈×C₁₃₇	—	4	2	no	—
32	D₄×C₁₃₇	—	4	2	no	—
33	D₄×C₁₃₇	—	4	4	no	—
34	D₄×C₁₃₇	—	4	4	no	—
35	C₂×C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄×C₁₃₇	—	4	4	no	—
39	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
43	C₈×C₁₃₇	—	4	4	no	—
44	C₂²×C₁₃₇	—	2	1	no	—
45	C₂²×C₁₃₇	—	2	2	no	—
46	C₂²×C₁₃₇	—	4	2	no	—
47	C₂²×C₁₃₇	—	4	4	no	—
48	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₄×C₁₃₇	—	2	1	no	—
50	C₄×C₁₃₇	—	2	2	no	—
51	C₄×C₁₃₇	—	4	2	no	—
52	C₄×C₁₃₇	#100	1	2	yes	—
53	C₄×C₁₃₇	#135	1	4	yes	—
54	C₂×C₁₃₇	—	4	1	no	—
55	C₂×C₁₃₇	—	8	2	no	—
56	C₂×C₁₃₇	C₂×#100	1	1	yes	characteristic
57	C₂×C₁₃₇	#416	1	2	yes	—
58	C₁₃₇	#1543	1	1	yes	characteristic, direct product factor, Sylow
59	#1543	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	#532	C₂×C₁₃₇	1	1	yes	characteristic
61	#431	C₂×C₁₃₇	1	2	yes	complemented
62	#333	C₂×C₁₃₇	1	2	yes	complemented
63	#331	C₂×C₁₃₇	1	2	yes	complemented
64	C₂×#41	C₄×C₁₃₇	1	2	yes	—
65	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
66	#127	C₂²×C₁₃₇	1	1	yes	characteristic
67	#124	C₄×C₁₃₇	1	2	yes	—
68	#121	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
71	C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
72	C₂²⋊C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	#41	—	4	4	no	—
75	C₂×Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
81	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
82	C₂×C₈	—	2	4	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₂²⋊C₄	—	2	4	no	—
85	C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
86	C₄²	D₄×C₁₃₇	1	2	yes	—
87	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	C₂³	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
99	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
102	C₈	—	4	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#100×C₁₃₇	1	2	yes	—
112	C₄	#135×C₁₃₇	1	4	yes	—
113	C₂	—	4	1	no	—
114	C₂	—	8	2	no	—
115	C₂	C₂×#100×C₁₃₇	1	1	yes	characteristic
116	C₂	#416×C₁₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#532×C₁₃₇

C₂

yes

maximal, characteristic

#431×C₁₃₇

C₂

yes

maximal, complemented

#333×C₁₃₇

C₂

yes

maximal, complemented

#331×C₁₃₇

C₂

yes

maximal, complemented

C₂×#41×C₁₃₇

C₄

yes

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

characteristic

#127×C₁₃₇

C₂²

yes

characteristic

#124×C₁₃₇

C₄

yes

—

#121×C₁₃₇

—

C₄×D₄×C₁₃₇

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

—

C₂²⋊C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

—

C₂×D₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#100

yes

—

C₄×C₁₃₇

#135

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#100

yes

characteristic

C₂×C₁₃₇

#416

yes

—

C₁₃₇

#1543

yes

characteristic, direct product factor, Sylow

#1543

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

#532

C₂×C₁₃₇

yes

characteristic

#431

C₂×C₁₃₇

yes

complemented

#333

C₂×C₁₃₇

yes

complemented

#331

C₂×C₁₃₇

yes

complemented

C₂×#41

C₄×C₁₃₇

yes

—

C₄⋊Q₈

C₂²×C₁₃₇

yes

characteristic

#127

C₂²×C₁₃₇

yes

characteristic

#124

C₄×C₁₃₇

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂²×C₁₃₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₃₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×C₄×C₁₃₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₁₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₃₇

yes

characteristic

C₂²×C₄

D₄×C₁₃₇

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₄×C₁₃₇

yes

characteristic

C₄²

D₄×C₁₃₇

yes

—

C₄²

C₂³×C₁₃₇

yes

characteristic

C₂³

—

C₂³

C₂²⋊C₄×C₁₃₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₁₃₇

yes

—

C₂×C₄

C₂²⋊C₄×C₁₃₇

yes

characteristic

100

C₂×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

101

C₂×C₄

C₂×D₄×C₁₃₇

yes

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊C₄×C₁₃₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#100×C₁₃₇

yes

—

112

C₄

#135×C₁₃₇

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#100×C₁₃₇

yes

characteristic

116

C₂

#416×C₁₃₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	#532×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₃₇

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₄²

C₂³×C₁₃₇

socle

C₂²×C₁₃₇

C₂×C₂²⋊C₄

prime radical

#532×C₁₃₇

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

N	H ≅ G/N	type
2. #431×C₁₃₇	C₂	semidirect
3. #333×C₁₃₇	C₂	semidirect
4. #331×C₁₃₇	C₂	semidirect
58. C₁₃₇	#1543	direct (N × H)
59. #1543	C₁₃₇	direct (N × H)
61. #431	C₂×C₁₃₇	semidirect
62. #333	C₂×C₁₃₇	semidirect
63. #331	C₂×C₁₃₇	semidirect

H ≅ G/N

type

2. #431×C₁₃₇

C₂

semidirect

3. #333×C₁₃₇

C₂

semidirect

4. #331×C₁₃₇

C₂

semidirect

58. C₁₃₇

#1543

direct (N × H)

59. #1543

C₁₃₇

direct (N × H)

61. #431

C₂×C₁₃₇

semidirect

62. #333

C₂×C₁₃₇

semidirect

63. #331

C₂×C₁₃₇

semidirect

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

subgroup	index
#532×C₁₃₇	2
#431×C₁₃₇	2
#333×C₁₃₇	2
#331×C₁₃₇	2
#1543	137

subgroup

index

#532×C₁₃₇

#431×C₁₃₇

#333×C₁₃₇

#331×C₁₃₇

#1543

137

Subgroups — #1543×C₁₃₇ #31776554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1543×C137 #31776554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1543×C₁₃₇ #31776554