Group #31792593 — subgroups

Subgroups — #31792593

Order 17536.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22578216	C₂	1	4	yes	maximal, complemented
2	#22577675	C₂	1	4	yes	maximal, complemented
3	#22577585	C₂	1	4	yes	maximal, complemented
4	#22577521	C₂	1	2	yes	maximal, complemented
5	#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984136	C₂²	1	4	yes	—
7	#17984120	—	2	8	no	—
8	#17984119	C₂²	1	8	yes	—
9	#17984111	—	2	4	no	—
10	#17984110	C₂²	1	2	yes	—
11	Q₁₆×D₁₃₇	—	2	4	no	—
12	#17984094	C₂²	1	4	yes	—
13	#17984024	C₂²	1	2	yes	—
14	#17984016	C₂²	1	4	yes	—
15	#17984007	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₃₇	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
18	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#14555511	—	2	8	no	—
20	#14555510	—	4	4	no	—
21	#14555510	C₂³	1	4	yes	—
22	Q₈×D₁₃₇	—	2	8	no	—
23	Dic₅₄₈	D₄	1	2	yes	—
24	#14555500	—	2	8	no	—
25	#14555499	D₄	1	4	yes	—
26	#14555497	—	2	8	no	—
27	D₁₀₉₆	D₄	1	2	yes	—
28	#14555494	C₂³	1	4	yes	—
29	#14555493	—	2	4	no	—
30	C₈×D₁₃₇	—	2	4	no	—
31	C₂×Q₈×C₁₃₇	C₂³	1	4	yes	—
32	Q₁₆×C₁₃₇	—	2	4	no	—
33	Q₁₆×C₁₃₇	—	2	4	no	—
34	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₃₇	—	4	2	no	—
36	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
37	Dic₂₇₄	—	4	4	no	—
38	Dic₂₇₄	C₂×D₄	1	4	yes	—
39	D₅₄₈	—	4	4	no	—
40	D₅₄₈	C₂×D₄	1	4	yes	—
41	C₂²⋊D₁₃₇	—	2	4	no	—
42	C₄×D₁₃₇	—	2	4	no	—
43	C₄×D₁₃₇	—	4	8	no	—
44	#11764040	—	2	4	no	—
45	Q₈×C₁₃₇	—	2	8	no	—
46	C₂×C₄×C₁₃₇	—	2	1	no	—
47	C₂×C₄×C₁₃₇	—	2	4	no	—
48	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
50	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
51	C₂²×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₃₇	—	2	4	no	—
53	C₄×C₁₃₇	—	2	2	no	—
54	C₄×C₁₃₇	—	4	4	no	—
55	C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
56	Dic₁₃₇	—	2	4	no	—
57	C₂×C₁₃₇	—	2	1	no	—
58	C₂×C₁₃₇	D₄²	1	1	yes	characteristic
59	D₁₃₇	—	4	4	no	—
60	C₁₃₇	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	137	1	no	maximal, Sylow
62	#578	—	137	4	no	—
63	#494	D₁₃₇	1	1	yes	characteristic, complemented
64	#473	—	137	4	no	—
65	#456	—	137	4	no	—
66	#443	—	137	2	no	—
67	2₋¹⁺⁴	—	137	4	no	—
68	#137	—	274	8	no	—
69	#137	—	137	8	no	—
70	#135	—	274	4	no	—
71	#135	—	137	2	no	—
72	C₂×Q₁₆	—	274	4	no	—
73	C₂×Q₁₆	C₂×D₁₃₇	1	4	yes	—
74	#131	—	137	4	no	—
75	C₄⋊Q₈	C₂×D₁₃₇	1	2	yes	—
76	#108	—	137	2	no	—
77	#104	—	137	4	no	—
78	#101	—	137	4	no	—
79	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
80	#41	—	274	8	no	—
81	#41	—	548	4	no	—
82	#41	—	137	4	no	—
83	C₂×Q₈	—	274	8	no	—
84	C₂×Q₈	C₂²×D₁₃₇	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	274	8	no	—
88	Q₁₆	—	137	2	no	—
89	SD₁₆	—	274	8	no	—
90	SD₁₆	—	137	4	no	—
91	D₈	—	137	2	no	—
92	M₁₆	—	274	4	no	—
93	M₁₆	—	137	4	no	—
94	C₂×C₈	—	274	4	no	—
95	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	548	4	no	—
100	Q₈	—	137	4	no	—
101	D₄	—	274	4	no	—
102	D₄	—	548	4	no	—
103	D₄	—	137	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	274	4	no	—
107	C₂×C₄	—	548	8	no	—
108	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
109	C₈	—	274	4	no	—
110	C₈	D₄×D₁₃₇	1	2	yes	—
111	C₈	D₄×D₁₃₇	1	2	yes	—
112	C₂²	—	274	4	no	—
113	C₂²	#17984133	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	274	4	no	—
117	C₄	C₂×D₄×D₁₃₇	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	548	4	no	—
120	C₂	#22578038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578216

C₂

yes

maximal, complemented

#22577675

C₂

yes

maximal, complemented

#22577585

C₂

yes

maximal, complemented

#22577521

C₂

yes

maximal, complemented

#494×C₁₃₇

C₂

yes

maximal, characteristic, complemented

#17984136

C₂²

yes

—

#17984120

—

#17984119

C₂²

yes

—

#17984111

—

#17984110

C₂²

yes

—

Q₁₆×D₁₃₇

—

#17984094

C₂²

yes

—

#17984024

C₂²

yes

—

#17984016

C₂²

yes

—

#17984007

C₂²

yes

—

C₂×Q₁₆×C₁₃₇

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₃₇

—

Dic₅₄₈

D₄

yes

—

#14555500

—

#14555499

D₄

yes

—

#14555497

—

D₁₀₉₆

D₄

yes

—

#14555494

C₂³

yes

—

#14555493

—

C₈×D₁₃₇

—

C₂×Q₈×C₁₃₇

C₂³

yes

—

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

C₂×D₄

yes

—

D₅₄₈

—

D₅₄₈

C₂×D₄

yes

—

C₂²⋊D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

C₂×D₄

yes

—

C₈×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

H(5,𝔽₂)

yes

characteristic

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×D₄

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

D₄²

yes

characteristic

D₁₃₇

—

C₁₃₇

#3293

yes

characteristic, complemented, Sylow

#3293

—

137

maximal, Sylow

#578

—

137

—

#494

D₁₃₇

yes

characteristic, complemented

#473

—

137

—

#456

—

137

—

#443

—

137

—

2₋¹⁺⁴

—

137

—

#137

—

274

—

#137

—

137

—

#135

—

274

—

#135

—

137

—

C₂×Q₁₆

—

274

—

C₂×Q₁₆

C₂×D₁₃₇

yes

—

#131

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

—

#108

—

137

—

#104

—

137

—

#101

—

137

—

C₄×C₈

C₂×D₁₃₇

yes

characteristic

#41

—

274

—

#41

—

548

—

#41

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

C₂²×D₁₃₇

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

274

—

Q₁₆

—

137

—

SD₁₆

—

274

—

SD₁₆

—

137

—

D₈

—

137

—

M₁₆

—

274

—

M₁₆

—

137

—

C₂×C₈

—

274

—

C₂×C₈

C₂²×D₁₃₇

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₃₇

yes

characteristic

Q₈

—

Q₈

—

548

—

100

Q₈

—

137

—

101

D₄

—

274

—

102

D₄

—

548

—

103

D₄

—

137

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

274

—

107

C₂×C₄

—

548

—

108

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

109

C₈

—

274

—

110

C₈

D₄×D₁₃₇

yes

—

111

C₈

D₄×D₁₃₇

yes

—

112

C₂²

—

274

—

113

C₂²

#17984133

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

274

—

117

C₄

C₂×D₄×D₁₃₇

yes

—

118

C₂

—

119

C₂

—

548

—

120

C₂

#22578038

yes

characteristic

	subgroup	G/H
center	C₂	#22578038
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#494×C₁₃₇	C₂
socle	C₂×C₁₃₇	D₄²
hypercenter	#494	D₁₃₇
hypocenter	C₁₃₇	#3293
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#22578038

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#494×C₁₃₇

C₂

socle

C₂×C₁₃₇

D₄²

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #22578216	C₂	semidirect
2. #22577675	C₂	semidirect
3. #22577585	C₂	semidirect
4. #22577521	C₂	semidirect
5. #494×C₁₃₇	C₂	semidirect
60. C₁₃₇	#3293	semidirect
63. #494	D₁₃₇	semidirect

H ≅ G/N

type

1. #22578216

C₂

semidirect

2. #22577675

C₂

semidirect

3. #22577585

C₂

semidirect

4. #22577521

C₂

semidirect

5. #494×C₁₃₇

C₂

semidirect

60. C₁₃₇

#3293

semidirect

63. #494

D₁₃₇

semidirect

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

subgroup	index
#22578216	2
#22577675	2
#22577585	2
#22577521	2
#494×C₁₃₇	2
#3293	137

subgroup

index

#494×C₁₃₇

137

Subgroups — #31792593

Conjugacy classes of subgroups

Complements

Maximal subgroups