Group #31796510 — subgroups

Subgroups — #31796510

Order 17536.

200 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 78 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	#22578366	C₂	1	2	yes	maximal
2	#22578272	C₂	1	1	yes	maximal, characteristic
3	#22578271	C₂	1	1	yes	maximal, characteristic
4	#22578270	C₂	1	2	yes	maximal
5	#22577357	C₂	1	1	yes	maximal, characteristic
6	D₁₃₇⋊M₁₆	C₂²	1	2	yes	—
7	#17984167	C₂²	1	2	yes	—
8	#17984142	C₄	1	4	yes	complemented
9	C₈×#10587739	C₄	1	4	yes	complemented
10	C₂×#14555493	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
12	#17984045	C₂²	1	1	yes	characteristic
13	#17983999	C₄	1	1	yes	characteristic, complemented
14	#17983998	C₄	1	1	yes	characteristic, complemented
15	C₈×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
17	#14555520	—	2	2	no	—
18	#14555519	—	2	2	no	—
19	#14555518	C₂×C₄	1	4	yes	—
20	C₄⋊#10587739	—	2	2	no	—
21	C₂²⋊#10587739	—	2	2	no	—
22	C₄×#10587739	C₂×C₄	1	4	yes	—
23	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
24	#14555493	—	2	2	no	—
25	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
26	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
27	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
28	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
29	#14555489	—	2	1	no	—
30	#14555487	—	2	1	no	—
31	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×#10587739	—	2	4	no	—
36	#11764047	—	2	4	no	—
37	C₂²×D₁₃₇	#41	1	1	yes	characteristic
38	C₂×Dic₁₃₇	—	2	1	no	—
39	C₂×Dic₁₃₇	#41	1	1	yes	characteristic
40	C₄×D₁₃₇	C₂²×C₄	1	2	yes	—
41	C₄×D₁₃₇	#41	1	2	yes	—
42	#11764040	—	2	1	no	—
43	#11764040	C₄²	1	2	yes	—
44	C₂×C₄×C₁₃₇	—	2	1	no	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₃₇	—	2	1	no	—
47	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
48	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
49	C₂²×C₁₃₇	#117	1	1	yes	characteristic
50	C₂×D₁₃₇	—	4	1	no	—
51	C₂×D₁₃₇	#117	1	2	yes	—
52	#10587739	—	4	4	no	—
53	C₄×C₁₃₇	—	4	1	no	—
54	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₁₃₇	#117	1	1	yes	characteristic
56	Dic₁₃₇	—	4	1	no	—
57	Dic₁₃₇	#117	1	2	yes	—
58	C₂×C₁₃₇	—	2	1	no	—
59	C₂×C₁₃₇	#376	1	1	yes	characteristic
60	D₁₃₇	—	2	2	no	—
61	C₁₃₇	#1747	1	1	yes	characteristic, complemented, Sylow
62	#1747	—	137	1	no	maximal, Sylow
63	#405	—	137	2	no	—
64	#405	—	137	1	no	—
65	#338	—	137	1	no	—
66	H(3,ℤ/4)	—	137	2	no	—
67	H(3,ℤ/4)	—	137	1	no	—
68	C₂×M₁₆	—	137	2	no	—
69	C₂×M₁₆	—	137	1	no	—
70	C₂²×C₈	—	137	1	no	—
71	#117	—	137	2	no	—
72	#117	—	137	1	no	—
73	#97	—	137	4	no	—
74	#97	—	137	1	no	—
75	#97	—	137	1	no	—
76	C₄×C₈	—	137	4	no	—
77	C₄×C₈	—	137	1	no	—
78	C₄×C₈	#10587739	1	1	yes	characteristic, complemented
79	C₂²×C₄	—	137	1	no	—
80	M₁₆	—	274	2	no	—
81	M₁₆	—	274	2	no	—
82	M₁₆	—	274	2	no	—
83	C₂×C₈	—	137	4	no	—
84	C₂×C₈	—	137	1	no	—
85	C₂×C₈	—	137	1	no	—
86	C₂×C₈	—	137	2	no	—
87	C₂×C₈	—	137	2	no	—
88	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
89	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
90	C₄⋊C₄	—	274	1	no	—
91	C₄⋊C₄	—	274	2	no	—
92	C₂²⋊C₄	—	274	1	no	—
93	C₂²⋊C₄	—	274	2	no	—
94	C₄²	—	137	4	no	—
95	C₄²	—	137	1	no	—
96	C₄²	C₂×#10587739	1	1	yes	characteristic
97	C₂³	—	137	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	274	4	no	—
100	C₂×C₄	—	274	1	no	—
101	C₂×C₄	—	137	1	no	—
102	C₂×C₄	—	137	2	no	—
103	C₂×C₄	—	137	2	no	—
104	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	274	4	no	—
107	C₈	—	274	1	no	—
108	C₈	—	137	2	no	—
109	C₈	C₄×#10587739	1	1	yes	characteristic
110	C₈	C₄×#10587739	1	1	yes	characteristic
111	C₂²	—	548	1	no	—
112	C₂²	—	137	2	no	—
113	C₂²	#17984167	1	1	yes	characteristic
114	C₄	—	4	1	no	—
115	C₄	—	548	1	no	—
116	C₄	—	548	4	no	—
117	C₄	—	137	2	no	—
118	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
119	C₄	#17984167	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	274	2	no	—
122	C₂	#22578339	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578366

C₂

yes

maximal

#22578272

C₂

yes

maximal, characteristic

#22578271

C₂

yes

maximal, characteristic

#22578270

C₂

yes

maximal

#22577357

C₂

yes

maximal, characteristic

D₁₃₇⋊M₁₆

C₂²

yes

—

#17984167

C₂²

yes

—

#17984142

C₄

yes

complemented

C₈×#10587739

C₄

yes

complemented

C₂×#14555493

C₂²

yes

characteristic

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

#17984045

C₂²

yes

characteristic

#17983999

C₄

yes

characteristic, complemented

#17983998

C₄

yes

characteristic, complemented

C₈×Dic₁₃₇

C₄

yes

characteristic, complemented

C₄×C₈×C₁₃₇

C₄

yes

characteristic, complemented

#14555520

—

#14555519

—

#14555518

C₂×C₄

yes

—

C₄⋊#10587739

—

C₂²⋊#10587739

—

C₄×#10587739

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

#14555493

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₈×D₁₃₇

C₂×C₄

yes

—

C₈×D₁₃₇

C₂×C₄

yes

—

#14555489

—

#14555487

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₂×#10587739

—

#11764047

—

C₂²×D₁₃₇

#41

yes

characteristic

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

#41

yes

characteristic

C₄×D₁₃₇

C₂²×C₄

yes

—

C₄×D₁₃₇

#41

yes

—

#11764040

—

#11764040

C₄²

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₄²

yes

characteristic

C₈×C₁₃₇

C₄²

yes

characteristic

C₂²×C₁₃₇

#117

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

#117

yes

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

#117

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

#117

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#376

yes

characteristic

D₁₃₇

—

C₁₃₇

#1747

yes

characteristic, complemented, Sylow

#1747

—

137

maximal, Sylow

#405

—

137

—

#405

—

137

—

#338

—

137

—

H(3,ℤ/4)

—

137

—

H(3,ℤ/4)

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

#117

—

137

—

#97

—

137

—

#97

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

#10587739

yes

characteristic, complemented

C₂²×C₄

—

137

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₂²⋊C₄

—

274

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₂³

—

137

—

C₂×C₄

—

C₂×C₄

—

274

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

137

—

102

C₂×C₄

—

137

—

103

C₂×C₄

—

137

—

104

C₂×C₄

C₂²×#10587739

yes

characteristic

105

C₈

—

106

C₈

—

274

—

107

C₈

—

274

—

108

C₈

—

137

—

109

C₈

C₄×#10587739

yes

characteristic

110

C₈

C₄×#10587739

yes

characteristic

111

C₂²

—

548

—

112

C₂²

—

137

—

113

C₂²

#17984167

yes

characteristic

114

C₄

—

115

C₄

—

548

—

116

C₄

—

548

—

117

C₄

—

137

—

118

C₄

C₂×C₄×#10587739

yes

characteristic

119

C₄

#17984167

yes

characteristic

120

C₂

—

121

C₂

—

274

—

122

C₂

#22578339

yes

characteristic

	subgroup	G/H
center	C₈	C₄×#10587739
commutator	C₄×C₁₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#376
prime radical	C₂²×D₁₃₇	#41
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1747
LCS	G ▷ C₄×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄×#10587739

commutator

C₄×C₁₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄×C₈×C₁₃₇

C₄

C₂×C₁₃₇

C₂²×D₁₃₇

C₂×C₄×D₁₃₇

C₂³

C₄×C₈

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₈ ◁ C₄×C₈

N	H ≅ G/N	type
8. #17984142	C₄	semidirect
9. C₈×#10587739	C₄	semidirect
13. #17983999	C₄	semidirect
14. #17983998	C₄	semidirect
15. C₈×Dic₁₃₇	C₄	semidirect
16. C₄×C₈×C₁₃₇	C₄	semidirect
61. C₁₃₇	#1747	semidirect
78. C₄×C₈	#10587739	semidirect

H ≅ G/N

type

8. #17984142

C₄

semidirect

9. C₈×#10587739

C₄

semidirect

13. #17983999

C₄

semidirect

14. #17983998

C₄

semidirect

15. C₈×Dic₁₃₇

C₄

semidirect

16. C₄×C₈×C₁₃₇

C₄

semidirect

61. C₁₃₇

#1747

semidirect

78. C₄×C₈

#10587739

semidirect

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

subgroup	index
#22578366	2
#22578272	2
#22578271	2
#22578270	2
#22577357	2
#1747	137

subgroup

index

137

Subgroups — #31796510

Conjugacy classes of subgroups

Complements

Maximal subgroups