Group #31797125 — subgroups

Subgroups — #31797125

Order 17536.

160 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 76 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×#17984163	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984161	C₂	1	1	yes	maximal, characteristic
3	#22578324	C₂	1	2	yes	maximal, complemented
4	C₄⋊#14555513	C₂	1	2	yes	maximal, complemented
5	#22577360	C₂	1	1	yes	maximal, characteristic
6	#17984163	—	2	2	no	—
7	#17984161	—	2	2	no	—
8	C₂×#14555513	C₂²	1	2	yes	—
9	C₂×#14555513	C₂²	1	2	yes	—
10	C₂×#14555493	C₄	1	1	yes	characteristic
11	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
12	#17984045	C₄	1	1	yes	characteristic
13	#17983998	C₄	1	1	yes	characteristic
14	C₄×#11764040	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	#97×C₁₃₇	C₄	1	1	yes	characteristic
17	#14555513	—	2	2	no	—
18	#14555513	—	2	2	no	—
19	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
20	#14555493	—	2	2	no	—
21	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
22	C₂×#11764040	C₂³	1	1	yes	characteristic
23	C₈×D₁₃₇	D₄	1	2	yes	—
24	C₈×D₁₃₇	Q₈	1	2	yes	—
25	#14555489	C₈	1	2	yes	—
26	#14555487	C₈	1	2	yes	—
27	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
33	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₄×D₁₃₇	C₄⋊C₄	1	2	yes	—
35	C₄×D₁₃₇	M₁₆	1	2	yes	—
36	#11764040	—	2	1	no	—
37	#11764040	C₂×D₄	1	1	yes	characteristic
38	#11764040	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₁₃₇	—	2	1	no	—
42	C₈×C₁₃₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
44	C₂×D₁₃₇	—	2	2	no	—
45	C₂×D₁₃₇	C₄⋊C₈	1	2	yes	—
46	C₄×C₁₃₇	—	2	2	no	—
47	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
49	Dic₁₃₇	—	2	2	no	—
50	Dic₁₃₇	C₄⋊C₈	1	2	yes	—
51	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₁₃₇	#504	1	2	yes	—
53	D₁₃₇	—	2	2	no	—
54	C₁₃₇	#2151	1	1	yes	characteristic, complemented, Sylow
55	#2151	—	137	1	no	maximal, Sylow
56	C₂×M₃₂	—	137	1	no	—
57	C₂²×C₁₆	—	137	1	no	—
58	#405	—	137	1	no	—
59	C₄⋊C₁₆	—	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	—	274	2	no	—
65	C₂×C₁₆	—	274	2	no	—
66	C₂×C₁₆	—	137	2	no	—
67	C₂×C₁₆	—	137	2	no	—
68	#97	—	137	1	no	—
69	#97	#10587739	1	1	yes	characteristic
70	C₄×C₈	—	137	1	no	—
71	C₄×C₈	—	137	1	no	—
72	C₂²×C₄	—	137	1	no	—
73	M₁₆	—	274	2	no	—
74	C₂×C₈	—	137	1	no	—
75	C₂×C₈	—	137	1	no	—
76	C₂×C₈	—	137	2	no	—
77	C₂×C₈	—	137	2	no	—
78	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
79	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
80	C₄⋊C₄	—	137	2	no	—
81	C₂²⋊C₄	—	137	2	no	—
82	C₄²	—	137	1	no	—
83	C₄²	C₂×#10587739	1	1	yes	characteristic
84	C₁₆	—	274	2	no	—
85	C₁₆	—	274	2	no	—
86	C₂³	—	137	1	no	—
87	C₂×C₄	—	137	2	no	—
88	C₂×C₄	—	137	1	no	—
89	C₂×C₄	—	137	2	no	—
90	C₂×C₄	—	137	2	no	—
91	C₂×C₄	#14555518	1	2	yes	—
92	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	274	1	no	—
95	C₈	—	137	1	no	—
96	C₈	—	137	1	no	—
97	C₈	C₄⋊#10587739	1	2	yes	—
98	C₂²	—	274	2	no	—
99	C₂²	—	137	2	no	—
100	C₂²	C₂×#14555518	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	—	274	2	no	—
103	C₄	—	137	2	no	—
104	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
105	C₄	D₁₃₇⋊M₁₆	1	1	yes	characteristic
106	C₂	—	274	2	no	—
107	C₂	#22578404	1	1	yes	characteristic
108	C₂	#22578462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984163

C₂

yes

maximal, characteristic

C₂×#17984161

C₂

yes

maximal, characteristic

#22578324

C₂

yes

maximal, complemented

C₄⋊#14555513

C₂

yes

maximal, complemented

#22577360

C₂

yes

maximal, characteristic

#17984163

—

#17984161

—

C₂×#14555513

C₂²

yes

—

C₂×#14555513

C₂²

yes

—

C₂×#14555493

C₄

yes

characteristic

C₂×C₈×D₁₃₇

C₂²

yes

characteristic

#17984045

C₄

yes

characteristic

#17983998

C₄

yes

characteristic

C₄×#11764040

C₂²

yes

characteristic

C₈×Dic₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₄

yes

characteristic

#14555513

—

#14555513

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

characteristic

#14555493

—

C₂×#11764040

C₂×C₄

yes

characteristic

C₂×#11764040

C₂³

yes

characteristic

C₈×D₁₃₇

D₄

yes

—

C₈×D₁₃₇

Q₈

yes

—

#14555489

C₈

yes

—

#14555487

C₈

yes

—

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

C₂²×D₁₃₇

C₂×C₈

yes

characteristic

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

characteristic

C₄×D₁₃₇

C₄⋊C₄

yes

—

C₄×D₁₃₇

M₁₆

yes

—

#11764040

—

#11764040

C₂×D₄

yes

characteristic

#11764040

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

C₄⋊C₈

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₇

C₂×M₁₆

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

C₄⋊C₈

yes

—

C₂×C₁₃₇

C₂×C₄⋊C₈

yes

characteristic

C₂×C₁₃₇

#504

yes

—

D₁₃₇

—

C₁₃₇

#2151

yes

characteristic, complemented, Sylow

#2151

—

137

maximal, Sylow

C₂×M₃₂

—

137

—

C₂²×C₁₆

—

137

—

#405

—

137

—

C₄⋊C₁₆

—

137

—

C₄⋊C₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

#117

—

137

—

M₃₂

—

274

—

C₂×C₁₆

—

274

—

C₂×C₁₆

—

137

—

C₂×C₁₆

—

137

—

#97

—

137

—

#97

#10587739

yes

characteristic

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

137

—

M₁₆

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

characteristic

C₁₆

—

274

—

C₁₆

—

274

—

C₂³

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

#14555518

yes

—

C₂×C₄

C₂²×#10587739

yes

characteristic

C₈

—

C₈

—

274

—

C₈

—

137

—

C₈

—

137

—

C₈

C₄⋊#10587739

yes

—

C₂²

—

274

—

C₂²

—

137

—

100

C₂²

C₂×#14555518

yes

characteristic

101

C₄

—

102

C₄

—

274

—

103

C₄

—

137

—

104

C₄

C₂×C₄⋊#10587739

yes

characteristic

105

C₄

D₁₃₇⋊M₁₆

yes

characteristic

106

C₂

—

274

—

107

C₂

#22578404

yes

characteristic

108

C₂

#22578462

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	#97×C₁₃₇	C₄
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	#97	#10587739
hypocenter	C₁₃₇	#2151
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10587739

commutator, socle

C₂²×C₁₃₇

C₂²×C₈

Fitting

#97×C₁₃₇

C₄

prime radical

C₂²×D₁₃₇

C₂×C₈

C₂³

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
3. #22578324	C₂	semidirect
4. C₄⋊#14555513	C₂	semidirect
54. C₁₃₇	#2151	semidirect

H ≅ G/N

type

3. #22578324

C₂

semidirect

4. C₄⋊#14555513

C₂

semidirect

54. C₁₃₇

#2151

semidirect

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

subgroup	index
C₂×#17984163	2
C₂×#17984161	2
#22578324	2
C₄⋊#14555513	2
#22577360	2
#2151	137

subgroup

index

137

Subgroups — #31797125

Conjugacy classes of subgroups

Complements

Maximal subgroups