Group #31796008 — subgroups

Subgroups — #31796008

Order 17536.

190 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22578420	C₂	1	2	yes	maximal
2	#22578245	C₂	1	2	yes	maximal, complemented
3	#22578244	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#17984155	C₄	1	4	yes	—
7	C₂²⋊#11764047	C₄	1	4	yes	—
8	C₄×#11764047	C₂²	1	2	yes	—
9	C₄×#11764047	C₂²	1	2	yes	—
10	C₂×#14555494	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	#17983998	C₄	1	1	yes	characteristic
13	#17983998	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
17	C₂×#11764047	C₂×C₄	1	4	yes	—
18	C₂×#11764047	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
20	#14555494	—	2	2	no	—
21	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
22	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	M₁₆×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	#11764047	—	2	4	no	—
31	#11764047	—	2	4	no	—
32	C₂×Dic₁₃₇	—	2	4	no	—
33	C₂×Dic₁₃₇	C₄²	1	2	yes	—
34	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
35	#11764040	—	2	1	no	—
36	#11764040	M₁₆	1	2	yes	—
37	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	—	2	1	no	—
39	C₂×C₄×C₁₃₇	—	2	1	no	—
40	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₈×C₁₃₇	—	2	1	no	—
43	C₈×C₁₃₇	M₁₆	1	2	yes	—
44	C₂²×C₁₃₇	—	2	1	no	—
45	C₂²×C₁₃₇	—	2	2	no	—
46	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₁₃₇	—	2	1	no	—
48	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
49	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
50	Dic₁₃₇	—	2	2	no	—
51	Dic₁₃₇	#131	1	4	yes	—
52	C₂×C₁₃₇	—	4	1	no	—
53	C₂×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
54	C₂×C₁₃₇	#405	1	2	yes	—
55	C₁₃₇	#1453	1	1	yes	characteristic, complemented, Sylow
56	#1453	—	137	1	no	maximal, Sylow
57	#431	—	137	2	no	—
58	C₄×M₁₆	—	137	1	no	—
59	#322	—	137	1	no	—
60	#322	—	137	2	no	—
61	C₈²	—	137	1	no	—
62	C₂×M₁₆	—	137	1	no	—
63	C₂×M₁₆	#10587739	1	1	yes	characteristic
64	C₂×C₄²	—	137	1	no	—
65	C₄⋊C₈	—	137	4	no	—
66	C₂²⋊C₈	—	137	4	no	—
67	#97	—	137	1	no	—
68	#97	—	137	1	no	—
69	C₄×C₈	—	137	2	no	—
70	C₄×C₈	—	137	2	no	—
71	C₄×C₈	—	137	1	no	—
72	C₄×C₈	—	137	1	no	—
73	C₂²×C₄	—	137	2	no	—
74	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	274	2	no	—
77	C₂×C₈	—	137	4	no	—
78	C₂×C₈	—	137	4	no	—
79	C₂×C₈	—	137	1	no	—
80	C₂×C₈	—	137	1	no	—
81	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
82	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
83	C₄²	—	137	2	no	—
84	C₄²	—	137	1	no	—
85	C₄²	—	137	1	no	—
86	C₂³	C₄×#10587739	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	274	4	no	—
90	C₂×C₄	—	137	2	no	—
91	C₂×C₄	—	137	2	no	—
92	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
93	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	274	4	no	—
96	C₈	—	274	4	no	—
97	C₈	—	274	1	no	—
98	C₈	—	137	2	no	—
99	C₈	#14555519	1	2	yes	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	274	2	no	—
105	C₄	—	137	4	no	—
106	C₄	C₂×#14555519	1	1	yes	characteristic
107	C₄	D₁₃₇⋊M₁₆	1	1	yes	characteristic
108	C₂	—	4	1	no	—
109	C₂	C₄×#14555519	1	1	yes	characteristic
110	C₂	#22578369	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578420

C₂

yes

maximal

#22578245

C₂

yes

maximal, complemented

#22578244

C₂

yes

maximal, characteristic, complemented

C₈×#11764047

C₂

yes

maximal, characteristic, complemented

M₁₆×Dic₁₃₇

C₂

yes

maximal, characteristic

#17984155

C₄

yes

—

C₂²⋊#11764047

C₄

yes

—

C₄×#11764047

C₂²

yes

—

C₄×#11764047

C₂²

yes

—

C₂×#14555494

C₄

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17983998

C₄

yes

characteristic

#17983998

C₄

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₇

C₄

yes

characteristic

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂×C₄

yes

—

C₂²×Dic₁₃₇

C₂×C₄

yes

—

#14555494

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

#11764047

—

#11764047

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄²

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

#11764040

—

#11764040

M₁₆

yes

—

C₂³×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

M₁₆

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄²

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

#131

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₄×M₁₆

yes

characteristic

C₂×C₁₃₇

#405

yes

—

C₁₃₇

#1453

yes

characteristic, complemented, Sylow

#1453

—

137

maximal, Sylow

#431

—

137

—

C₄×M₁₆

—

137

—

#322

—

137

—

#322

—

137

—

C₈²

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

#10587739

yes

characteristic

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

#97

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×#10587739

yes

characteristic

M₁₆

—

M₁₆

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₂³

C₄×#10587739

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

274

—

C₈

—

274

—

C₈

—

274

—

C₈

—

137

—

C₈

#14555519

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₄×#10587739

yes

characteristic

103

C₄

—

104

C₄

—

274

—

105

C₄

—

137

—

106

C₄

C₂×#14555519

yes

characteristic

107

C₄

D₁₃₇⋊M₁₆

yes

characteristic

108

C₂

—

109

C₂

C₄×#14555519

yes

characteristic

110

C₂

#22578369

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	C₂×M₁₆×C₁₃₇	C₄
prime radical	C₂³×C₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#1453
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂×C₄²

Fitting

C₂×M₁₆×C₁₃₇

C₄

prime radical

C₂³×C₁₃₇

C₄²

cosocle

C₄×Dic₁₃₇

C₂³

C₂×M₁₆

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #22578245	C₂	semidirect
3. #22578244	C₂	semidirect
4. C₈×#11764047	C₂	semidirect
55. C₁₃₇	#1453	semidirect

H ≅ G/N

type

2. #22578245

C₂

semidirect

3. #22578244

C₂

semidirect

4. C₈×#11764047

C₂

semidirect

55. C₁₃₇

#1453

semidirect

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

subgroup	index
#22578420	2
#22578245	2
#22578244	2
C₈×#11764047	2
M₁₆×Dic₁₃₇	2
#1453	137

subgroup

index

M₁₆×Dic₁₃₇

137

Subgroups — #31796008

Conjugacy classes of subgroups

Complements

Maximal subgroups