Group #31796731 — subgroups

Subgroups — #31796731

Order 17536.

212 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#17984159	C₂	1	2	yes	maximal
2	C₂×#17984158	C₂	1	2	yes	maximal
3	#22578408	C₂	1	1	yes	maximal, characteristic
4	#22578407	C₂	1	1	yes	maximal, characteristic
5	#97×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555520	C₂²	1	2	yes	—
7	C₂×#14555520	C₂²	1	1	yes	characteristic
8	C₂×#14555520	C₂²	1	1	yes	characteristic
9	#17984159	—	2	4	no	—
10	#17984158	—	2	4	no	—
11	#17984155	—	2	2	no	—
12	C₄⋊#11764047	—	2	2	no	—
13	C₂×C₈×D₁₃₇	C₂²	1	2	yes	—
14	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
15	#17983999	C₄	1	1	yes	characteristic
16	#17983998	C₄	1	2	yes	—
17	#97×C₁₃₇	C₄	1	1	yes	characteristic
18	#14555520	—	2	2	no	—
19	#14555520	—	2	2	no	—
20	#14555520	—	2	2	no	—
21	#14555520	—	2	2	no	—
22	C₂×#11764047	—	2	2	no	—
23	C₂×#11764047	—	2	1	no	—
24	C₂×#11764047	—	2	1	no	—
25	C₂×C₄×D₁₃₇	Q₈	1	2	yes	—
26	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×#11764040	C₂×C₄	1	2	yes	—
28	C₈×D₁₃₇	D₄	1	4	yes	—
29	C₈×D₁₃₇	Q₈	1	4	yes	—
30	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₇	D₄	1	2	yes	—
32	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	#11764047	—	4	2	no	—
35	#11764047	—	4	1	no	—
36	#11764047	—	4	1	no	—
37	C₂²×D₁₃₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
39	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₄×D₁₃₇	—	4	2	no	—
41	C₄×D₁₃₇	C₂×D₄	1	2	yes	—
42	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
43	#11764040	C₄⋊C₄	1	4	yes	—
44	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₁₃₇	—	2	2	no	—
49	C₂×D₁₃₇	C₄⋊Q₈	1	2	yes	—
50	C₄×C₁₃₇	—	2	2	no	—
51	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
52	Dic₁₃₇	—	2	2	no	—
53	Dic₁₃₇	#127	1	1	yes	characteristic
54	Dic₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₁₃₇	#384	1	1	yes	characteristic
56	C₂×C₁₃₇	#430	1	2	yes	—
57	D₁₃₇	—	2	2	no	—
58	C₁₃₇	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	137	1	no	maximal, Sylow
60	C₂×#108	—	137	2	no	—
61	C₂×#108	—	137	2	no	—
62	#423	—	137	1	no	—
63	#423	—	137	1	no	—
64	C₂×#97	—	137	1	no	—
65	C₂×M₁₆	—	137	2	no	—
66	C₂×M₁₆	—	137	1	no	—
67	C₂×M₁₆	—	137	1	no	—
68	C₂²×C₈	—	137	2	no	—
69	C₂×C₄²	—	137	1	no	—
70	#108	—	274	4	no	—
71	#108	—	274	4	no	—
72	C₄⋊C₈	—	274	2	no	—
73	C₄⋊C₈	—	274	2	no	—
74	#97	—	137	2	no	—
75	#97	—	137	1	no	—
76	#97	#10587739	1	1	yes	characteristic
77	C₂²×C₄	—	137	2	no	—
78	C₂²×C₄	—	137	1	no	—
79	M₁₆	—	274	2	no	—
80	M₁₆	—	274	2	no	—
81	M₁₆	—	274	2	no	—
82	M₁₆	—	274	2	no	—
83	C₂×C₈	—	274	2	no	—
84	C₂×C₈	—	274	1	no	—
85	C₂×C₈	—	274	1	no	—
86	C₂×C₈	—	137	2	no	—
87	C₂×C₈	—	137	4	no	—
88	C₂×C₈	—	137	4	no	—
89	C₂×C₈	C₂×#10587739	1	2	yes	—
90	C₄²	—	137	2	no	—
91	C₄²	—	137	1	no	—
92	C₄²	C₂×#10587739	1	1	yes	characteristic
93	C₂³	—	137	1	no	—
94	C₂×C₄	—	548	2	no	—
95	C₂×C₄	—	137	2	no	—
96	C₂×C₄	—	137	1	no	—
97	C₂×C₄	—	137	2	no	—
98	C₂×C₄	—	137	2	no	—
99	C₂×C₄	C₄⋊#10587739	1	2	yes	—
100	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
101	C₈	—	548	2	no	—
102	C₈	—	548	1	no	—
103	C₈	—	548	1	no	—
104	C₈	—	137	4	no	—
105	C₈	C₄⋊#10587739	1	4	yes	—
106	C₂²	—	274	2	no	—
107	C₂²	—	137	2	no	—
108	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	274	2	no	—
111	C₄	—	137	1	no	—
112	C₄	—	137	1	no	—
113	C₄	C₂×C₄⋊#10587739	1	2	yes	—
114	C₂	—	274	2	no	—
115	C₂	#22578349	1	1	yes	characteristic
116	C₂	#22578418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984159

C₂

yes

maximal

C₂×#17984158

C₂

yes

maximal

#22578408

C₂

yes

maximal, characteristic

#22578407

C₂

yes

maximal, characteristic

#97×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555520

C₂²

yes

—

C₂×#14555520

C₂²

yes

characteristic

C₂×#14555520

C₂²

yes

—

—

—

—

C₂×C₈×D₁₃₇

C₂²

yes

—

C₄²×D₁₃₇

C₂²

yes

characteristic

#17983999

C₄

yes

characteristic

#17983998

C₄

yes

—

#97×C₁₃₇

C₄

yes

—

—

—

—

—

—

—

C₂×C₄×D₁₃₇

Q₈

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₂×#11764040

C₂×C₄

yes

—

C₈×D₁₃₇

D₄

yes

—

C₈×D₁₃₇

Q₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

—

—

C₂²×D₁₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂×D₄

yes

—

C₄×D₁₃₇

C₂×Q₈

yes

—

#11764040

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄⋊Q₈

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

Dic₁₃₇

—

Dic₁₃₇

#127

yes

characteristic

Dic₁₃₇

C₄⋊Q₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

D₁₃₇

—

C₁₃₇

#1849

yes

characteristic, complemented, Sylow

#1849

—

137

maximal, Sylow

C₂×#108

—

137

—

C₂×#108

—

137

—

#423

—

137

—

#423

—

137

—

C₂×#97

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

#108

—

274

—

#108

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

274

—

#97

—

137

—

#97

—

137

—

#97

#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

C₂×#10587739

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

548

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₄⋊#10587739

yes

—

100

C₂×C₄

C₂²×#10587739

yes

characteristic

101

C₈

—

548

—

102

C₈

—

548

—

103

C₈

—

548

—

104

C₈

—

137

—

105

C₈

C₄⋊#10587739

yes

—

106

C₂²

—

274

—

107

C₂²

—

137

—

108

C₂²

C₂×C₄⋊#10587739

yes

characteristic

109

C₄

—

110

C₄

—

274

—

111

C₄

—

137

—

112

C₄

—

137

—

113

C₄

C₂×C₄⋊#10587739

yes

—

114

C₂

—

274

—

115

C₂

#22578349

yes

characteristic

116

C₂

#22578418

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#97×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₃₇	C₂×Q₈
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#1849
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

#97×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₃₇

C₂×Q₈

cosocle

C₂×C₄×D₁₃₇

C₂³

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
58. C₁₃₇	#1849	semidirect

H ≅ G/N

type

58. C₁₃₇

#1849

semidirect

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

subgroup	index
C₂×#17984159	2
C₂×#17984158	2
#22578408	2
#22578407	2
#97×D₁₃₇	2
#1849	137

subgroup

index

#97×D₁₃₇

137

Subgroups — #31796731

Conjugacy classes of subgroups

Complements

Maximal subgroups