Group #31797951 — subgroups

Subgroups — #31797951

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×#17984185	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984183	C₂	1	1	yes	maximal, characteristic
3	#22578497	C₂	1	2	yes	maximal, complemented
4	C₄×#14555522	C₂	1	2	yes	maximal, complemented
5	#22578368	C₂	1	1	yes	maximal, characteristic
6	#17984185	C₄	1	4	yes	complemented
7	#17984183	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×#14555520	C₄	1	1	yes	characteristic
12	C₂×#14555518	C₂²	1	1	yes	characteristic
13	#17984143	C₄	1	2	yes	—
14	C₄×#11764047	C₂²	1	2	yes	—
15	#17984046	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	#14555520	C₈	1	4	yes	—
20	#14555518	C₂×C₄	1	2	yes	—
21	#14555518	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	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄⋊Dic₁₃₇	C₈	1	1	yes	characteristic
27	#14555487	C₈	1	2	yes	—
28	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
29	C₄⋊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	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
36	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×D₁₃₇	C₄²	1	2	yes	—
38	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
42	C₂×D₁₃₇	—	2	1	no	—
43	C₂×D₁₃₇	—	2	1	no	—
44	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
45	C₄×C₁₃₇	—	2	2	no	—
46	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
47	Dic₁₃₇	—	2	2	no	—
48	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
49	Dic₁₃₇	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂×C₁₃₇	#504	1	1	yes	characteristic
52	C₂×C₁₃₇	#504	1	1	yes	characteristic
53	D₁₃₇	—	2	2	no	—
54	C₁₃₇	#2108	1	1	yes	characteristic, complemented, Sylow
55	#2108	—	137	1	no	maximal, Sylow
56	C₂×M₃₂	—	137	1	no	—
57	C₂²×C₁₆	—	137	1	no	—
58	#405	—	137	1	no	—
59	#346	—	137	2	no	—
60	C₄×C₁₆	—	137	2	no	—
61	C₂×M₁₆	—	137	1	no	—
62	C₂²×C₈	—	137	1	no	—
63	#117	—	137	1	no	—
64	M₃₂	—	137	4	no	—
65	C₂×C₁₆	—	137	1	no	—
66	C₂×C₁₆	—	137	2	no	—
67	C₂×C₁₆	—	137	1	no	—
68	C₂×C₁₆	—	137	4	no	—
69	#97	—	137	2	no	—
70	C₄×C₈	—	137	2	no	—
71	C₂²×C₄	—	137	1	no	—
72	M₁₆	—	137	4	no	—
73	C₂×C₈	—	137	1	no	—
74	C₂×C₈	—	137	2	no	—
75	C₂×C₈	—	137	1	no	—
76	C₂×C₈	—	137	2	no	—
77	C₂×C₈	—	137	2	no	—
78	C₄⋊C₄	—	137	1	no	—
79	C₄⋊C₄	#11764049	1	1	yes	characteristic
80	C₂²⋊C₄	—	137	2	no	—
81	C₄²	—	137	2	no	—
82	C₁₆	—	137	2	no	—
83	C₁₆	—	137	2	no	—
84	C₁₆	—	137	4	no	—
85	C₂³	—	137	1	no	—
86	C₂×C₄	—	137	2	no	—
87	C₂×C₄	—	137	1	no	—
88	C₂×C₄	—	137	2	no	—
89	C₂×C₄	—	137	2	no	—
90	C₂×C₄	C₂×#11764049	1	2	yes	—
91	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
92	C₈	—	137	1	no	—
93	C₈	—	137	1	no	—
94	C₈	—	137	2	no	—
95	C₈	—	137	4	no	—
96	C₂²	—	274	1	no	—
97	C₂²	—	274	1	no	—
98	C₂²	—	137	2	no	—
99	C₂²	C₂²×#11764049	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	274	2	no	—
102	C₄	—	137	1	no	—
103	C₄	—	137	1	no	—
104	C₄	C₄×#11764049	1	2	yes	—
105	C₂	—	274	2	no	—
106	C₂	C₂×C₄×#11764049	1	1	yes	characteristic
107	C₂	#22578524	1	1	yes	characteristic
108	C₂	#22578525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984185

C₂

yes

maximal, characteristic

C₂×#17984183

C₂

yes

maximal, characteristic

#22578497

C₂

yes

maximal, complemented

C₄×#14555522

C₂

yes

maximal, complemented

#22578368

C₂

yes

maximal, characteristic

#17984185

C₄

yes

complemented

#17984183

C₄

yes

complemented

C₂×#14555522

C₂²

yes

characteristic

C₂×#14555522

C₂²

yes

—

C₂×#14555522

C₂²

yes

characteristic

C₂×#14555520

C₄

yes

characteristic

C₂×#14555518

C₂²

yes

characteristic

#17984143

C₄

yes

—

C₄×#11764047

C₂²

yes

—

#17984046

C₄

yes

characteristic

#14555522

C₂×C₄

yes

—

#14555522

C₂×C₄

yes

—

#14555522

C₂×C₄

yes

—

#14555520

C₈

yes

—

#14555518

C₂×C₄

yes

—

#14555518

C₂×C₄

yes

—

C₂×#11764047

C₂×C₄

yes

characteristic

C₂×#11764047

C₂×C₄

yes

—

C₂×#11764047

C₂³

yes

characteristic

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₃₇

C₈

yes

characteristic

#14555487

C₈

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₇

C₈

yes

characteristic

#11764047

C₄²

yes

—

#11764047

C₂×C₈

yes

—

#11764047

C₂²×C₄

yes

characteristic

#11764047

C₂²×C₄

yes

characteristic

C₂²×D₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₄×D₁₃₇

C₄²

yes

—

C₄×D₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

characteristic

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄×C₈

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×C₈

yes

—

Dic₁₃₇

—

Dic₁₃₇

C₂×C₄²

yes

characteristic

Dic₁₃₇

C₂²×C₈

yes

characteristic

C₂×C₁₃₇

C₂×C₄×C₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

—

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

—

C₄⋊C₄

#11764049

yes

characteristic

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₁₆

—

137

—

C₁₆

—

137

—

C₁₆

—

137

—

C₂³

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

C₂×#11764049

yes

—

C₂×C₄

C₂×#11764049

yes

characteristic

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₈

—

137

—

C₂²

—

274

—

C₂²

—

274

—

C₂²

—

137

—

C₂²

C₂²×#11764049

yes

characteristic

100

C₄

—

101

C₄

—

274

—

102

C₄

—

137

—

103

C₄

—

137

—

104

C₄

C₄×#11764049

yes

—

105

C₂

—

274

—

106

C₂

C₂×C₄×#11764049

yes

characteristic

107

C₂

#22578524

yes

characteristic

108

C₂

#22578525

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Fitting	C₄⋊C₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764047	C₂³
hypercenter	C₄⋊C₄	#11764049
hypocenter	C₁₃₇	#2108
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#11764049

commutator

C₂×C₁₃₇

C₂×C₄×C₈

Fitting

C₄⋊C₄×C₁₃₇

C₈

socle

C₂²×C₁₃₇

C₂²×C₈

prime radical

C₂²×D₁₃₇

C₂×C₈

C₂³

C₄⋊C₄

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄⋊C₄

N	H ≅ G/N	type
3. #22578497	C₂	semidirect
4. C₄×#14555522	C₂	semidirect
6. #17984185	C₄	semidirect
7. #17984183	C₄	semidirect
54. C₁₃₇	#2108	semidirect

H ≅ G/N

type

3. #22578497

C₂

semidirect

4. C₄×#14555522

C₂

semidirect

6. #17984185

C₄

semidirect

7. #17984183

C₄

semidirect

54. C₁₃₇

#2108

semidirect

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

subgroup	index
C₂×#17984185	2
C₂×#17984183	2
#22578497	2
C₄×#14555522	2
#22578368	2
#2108	137

subgroup

index

137

Subgroups — #31797951

Conjugacy classes of subgroups

Complements

Maximal subgroups