Group #31797950 — subgroups

Subgroups — #31797950

Order 17536.

184 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 95 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×#17984184	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555522	C₂	1	1	yes	maximal, characteristic
3	#22578497	C₂	1	2	yes	maximal, complemented
4	C₄×#14555522	C₂	1	2	yes	maximal, complemented
5	#22578367	C₂	1	1	yes	maximal, characteristic
6	#17984184	C₄	1	4	yes	complemented
7	C₂×#14555522	C₄	1	4	yes	complemented
8	C₂×#14555522	C₂²	1	1	yes	characteristic
9	C₂×#14555522	C₂²	1	2	yes	—
10	C₂×#14555522	C₂²	1	1	yes	characteristic
11	C₂×#14555519	C₄	1	1	yes	characteristic
12	C₂²×#11764047	C₂²	1	1	yes	characteristic
13	#17984143	C₄	1	2	yes	—
14	C₄×#11764047	C₂²	1	2	yes	—
15	#17984044	C₄	1	1	yes	characteristic
16	#14555522	C₂×C₄	1	2	yes	—
17	#14555522	C₂×C₄	1	2	yes	—
18	#14555522	C₂×C₄	1	4	yes	—
19	#14555519	C₈	1	4	yes	—
20	C₂×#11764047	C₂×C₄	1	2	yes	—
21	C₂×#11764047	C₂×C₄	1	2	yes	—
22	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
23	C₂×#11764047	C₂×C₄	1	2	yes	—
24	C₂×#11764047	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
26	#14555489	C₈	1	2	yes	—
27	C₂²⋊Dic₁₃₇	C₈	1	1	yes	characteristic
28	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
29	C₂²⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
30	#11764047	C₄²	1	2	yes	—
31	#11764047	C₂×C₈	1	4	yes	—
32	#11764047	C₂²×C₄	1	1	yes	characteristic
33	#11764047	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₃₇	C₄²	1	2	yes	—
35	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
36	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
38	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
41	C₂²×C₁₃₇	—	2	2	no	—
42	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
43	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₃₇	—	2	2	no	—
45	Dic₁₃₇	—	2	2	no	—
46	Dic₁₃₇	C₄×C₈	1	2	yes	—
47	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
48	Dic₁₃₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₁₃₇	—	2	2	no	—
50	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂×C₁₃₇	#504	1	2	yes	—
52	C₁₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
53	#2108	—	137	1	no	maximal, Sylow
54	C₂×M₃₂	—	137	1	no	—
55	C₂²×C₁₆	—	137	1	no	—
56	#405	—	137	1	no	—
57	#346	—	137	2	no	—
58	C₄×C₁₆	—	137	2	no	—
59	C₂×M₁₆	—	137	1	no	—
60	C₂²×C₈	—	137	1	no	—
61	#117	—	137	1	no	—
62	M₃₂	—	137	4	no	—
63	C₂×C₁₆	—	137	1	no	—
64	C₂×C₁₆	—	137	2	no	—
65	C₂×C₁₆	—	137	1	no	—
66	C₂×C₁₆	—	137	4	no	—
67	#97	—	137	2	no	—
68	C₄×C₈	—	137	2	no	—
69	C₂²×C₄	—	137	1	no	—
70	M₁₆	—	137	4	no	—
71	C₂×C₈	—	137	1	no	—
72	C₂×C₈	—	137	2	no	—
73	C₂×C₈	—	137	1	no	—
74	C₂×C₈	—	137	2	no	—
75	C₂×C₈	—	137	2	no	—
76	C₄⋊C₄	—	137	2	no	—
77	C₂²⋊C₄	—	137	1	no	—
78	C₂²⋊C₄	#11764049	1	1	yes	characteristic
79	C₄²	—	137	2	no	—
80	C₁₆	—	137	2	no	—
81	C₁₆	—	137	2	no	—
82	C₁₆	—	137	4	no	—
83	C₂³	C₂×#11764049	1	1	yes	characteristic
84	C₂×C₄	—	137	2	no	—
85	C₂×C₄	—	137	2	no	—
86	C₂×C₄	—	137	2	no	—
87	C₂×C₄	—	137	1	no	—
88	C₂×C₄	—	137	1	no	—
89	C₂×C₄	C₂×#11764049	1	2	yes	—
90	C₈	—	137	1	no	—
91	C₈	—	137	1	no	—
92	C₈	—	137	2	no	—
93	C₈	—	137	4	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄×#11764049	1	2	yes	—
96	C₂²	C₂²×#11764049	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	274	2	no	—
99	C₄	—	137	1	no	—
100	C₄	—	137	1	no	—
101	C₄	—	137	2	no	—
102	C₂	—	2	2	no	—
103	C₂	C₂×C₄×#11764049	1	1	yes	characteristic
104	C₂	#22578524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984184

C₂

yes

maximal, characteristic

C₂²×#14555522

C₂

yes

maximal, characteristic

#22578497

C₂

yes

maximal, complemented

C₄×#14555522

C₂

yes

maximal, complemented

#22578367

C₂

yes

maximal, characteristic

#17984184

C₄

yes

complemented

C₂×#14555522

C₄

yes

complemented

C₂×#14555522

C₂²

yes

characteristic

C₂×#14555522

C₂²

yes

—

C₂×#14555522

C₂²

yes

characteristic

C₂×#14555519

C₄

yes

characteristic

C₂²×#11764047

C₂²

yes

characteristic

#17984143

C₄

yes

—

C₄×#11764047

C₂²

yes

—

#17984044

C₄

yes

characteristic

#14555522

C₂×C₄

yes

—

#14555522

C₂×C₄

yes

—

#14555522

C₂×C₄

yes

—

#14555519

C₈

yes

—

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂×C₄

yes

characteristic

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂³

yes

characteristic

C₂²×Dic₁₃₇

C₂×C₄

yes

characteristic

#14555489

C₈

yes

—

C₂²⋊Dic₁₃₇

C₈

yes

characteristic

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₃₇

C₈

yes

characteristic

#11764047

C₄²

yes

—

#11764047

C₂×C₈

yes

—

#11764047

C₂²×C₄

yes

characteristic

#11764047

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₇

C₄²

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₂³×C₁₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×C₈

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₄×C₈

yes

—

Dic₁₃₇

C₂×C₄²

yes

characteristic

Dic₁₃₇

C₂²×C₈

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₇

#504

yes

—

C₁₃₇

#2108

yes

characteristic, complemented, Sylow

#2108

—

137

maximal, Sylow

C₂×M₃₂

—

137

—

C₂²×C₁₆

—

137

—

#405

—

137

—

#346

—

137

—

C₄×C₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

M₃₂

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

#11764049

yes

characteristic

C₄²

—

137

—

C₁₆

—

137

—

C₁₆

—

137

—

C₁₆

—

137

—

C₂³

C₂×#11764049

yes

characteristic

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#11764049

yes

—

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₂²

—

C₂²

C₄×#11764049

yes

—

C₂²

C₂²×#11764049

yes

characteristic

C₄

—

C₄

—

274

—

C₄

—

137

—

100

C₄

—

137

—

101

C₄

—

137

—

102

C₂

—

103

C₂

C₂×C₄×#11764049

yes

characteristic

104

C₂

#22578524

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Fitting	C₂²⋊C₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×#11764047	C₂³
hypercenter	C₂²⋊C₄	#11764049
hypocenter	C₁₃₇	#2108
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#11764049

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Fitting

C₂²⋊C₄×C₁₃₇

C₈

socle

C₂²×C₁₃₇

C₂²×C₈

prime radical

C₂³×C₁₃₇

C₂×C₈

C₂³

C₂²⋊C₄

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂²⋊C₄

N	H ≅ G/N	type
3. #22578497	C₂	semidirect
4. C₄×#14555522	C₂	semidirect
6. #17984184	C₄	semidirect
7. C₂×#14555522	C₄	semidirect
52. C₁₃₇	#2108	semidirect

H ≅ G/N

type

3. #22578497

C₂

semidirect

4. C₄×#14555522

C₂

semidirect

6. #17984184

C₄

semidirect

7. C₂×#14555522

C₄

semidirect

52. C₁₃₇

#2108

semidirect

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

subgroup	index
C₂×#17984184	2
C₂²×#14555522	2
#22578497	2
C₄×#14555522	2
#22578367	2
#2108	137

subgroup

index

137

Subgroups — #31797950

Conjugacy classes of subgroups

Complements

Maximal subgroups