Group #31797919 — subgroups

Subgroups — #31797919

Order 17536.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	Q₈×#11764049	C₂	1	2	yes	maximal
2	#22578492	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11764049	C₂	1	2	yes	maximal
4	#22578487	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#10587739	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11764049	—	2	2	no	—
7	C₄⋊#11764049	C₂²	1	2	yes	—
8	C₄×#11764049	—	2	2	no	—
9	C₄×#11764049	C₂²	1	1	yes	characteristic
10	C₄×#11764049	C₂²	1	1	yes	characteristic
11	Q₈×#10587739	C₂²	1	2	yes	—
12	#17984157	C₄	1	1	yes	characteristic
13	#17984151	C₄	1	2	yes	—
14	C₈×#10587739	C₂²	1	1	yes	characteristic
15	Q₁₆×D₁₃₇	C₄	1	1	yes	characteristic
16	C₂×#11764049	—	2	2	no	—
17	C₂×#11764049	—	2	1	no	—
18	C₂×#11764049	D₄	1	2	yes	—
19	#14555518	C₂×C₄	1	1	yes	characteristic
20	C₄⋊#10587739	—	2	2	no	—
21	C₄⋊#10587739	C₂×C₄	1	2	yes	—
22	C₄×#10587739	—	2	2	no	—
23	C₄×#10587739	C₂³	1	1	yes	characteristic
24	Q₈×D₁₃₇	C₂×C₄	1	2	yes	—
25	Dic₅₄₈	C₈	1	1	yes	characteristic, complemented
26	#14555500	C₈	1	2	yes	complemented
27	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
28	Q₁₆×C₁₃₇	C₈	1	1	yes	characteristic, complemented
29	#11764049	—	2	2	no	—
30	#11764049	—	4	1	no	—
31	#11764049	—	4	2	no	—
32	C₂×#10587739	—	2	2	no	—
33	C₂×#10587739	C₂×D₄	1	1	yes	characteristic
34	C₂×#10587739	#41	1	1	yes	characteristic
35	#11764047	—	2	1	no	—
36	Dic₂₇₄	—	2	2	no	—
37	Dic₂₇₄	C₂×C₈	1	2	yes	—
38	C₄×D₁₃₇	—	2	2	no	—
39	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
40	#11764040	C₂×C₈	1	1	yes	characteristic
41	Q₈×C₁₃₇	C₂×C₈	1	2	yes	—
42	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
43	C₂×D₁₃₇	C₄×D₄	1	1	yes	characteristic
44	#10587739	—	2	1	no	—
45	#10587739	—	4	2	no	—
46	#10587739	#137	1	2	yes	—
47	C₄×C₁₃₇	—	2	2	no	—
48	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
49	Dic₁₃₇	—	2	2	no	—
50	Dic₁₃₇	#131	1	1	yes	characteristic
51	C₂×C₁₃₇	C₈×D₄	1	1	yes	characteristic
52	D₁₃₇	#441	1	1	yes	characteristic
53	D₁₃₇	#444	1	1	yes	characteristic
54	C₁₃₇	#1579	1	1	yes	characteristic, complemented, Sylow
55	#1579	—	137	1	no	maximal, Sylow
56	C₈×Q₈	—	137	2	no	—
57	C₄×Q₁₆	—	137	1	no	—
58	#334	—	137	1	no	—
59	Q₈⋊C₈	—	137	2	no	—
60	#322	—	137	1	no	—
61	C₂×Q₁₆	—	137	1	no	—
62	C₄×Q₈	—	137	2	no	—
63	#107	—	137	1	no	—
64	C₄⋊C₈	—	274	2	no	—
65	C₄⋊C₈	—	137	2	no	—
66	#103	—	137	2	no	—
67	C₄×C₈	—	274	2	no	—
68	C₄×C₈	—	137	1	no	—
69	C₄×C₈	—	137	1	no	—
70	C₄×C₈	—	137	1	no	—
71	C₂×Q₈	—	137	2	no	—
72	Q₁₆	—	137	2	no	—
73	Q₁₆	—	137	1	no	—
74	Q₁₆	#11764049	1	1	yes	characteristic, complemented
75	C₂×C₈	—	274	2	no	—
76	C₂×C₈	—	274	1	no	—
77	C₂×C₈	—	137	2	no	—
78	C₂×C₈	—	137	1	no	—
79	C₂×C₈	—	137	1	no	—
80	C₄⋊C₄	—	274	2	no	—
81	C₄⋊C₄	—	137	2	no	—
82	C₄²	—	274	2	no	—
83	C₄²	—	137	1	no	—
84	Q₈	—	274	2	no	—
85	Q₈	—	137	2	no	—
86	Q₈	C₂×#11764049	1	2	yes	—
87	C₂×C₄	—	274	2	no	—
88	C₂×C₄	—	274	2	no	—
89	C₂×C₄	—	137	1	no	—
90	C₂×C₄	—	137	1	no	—
91	C₂×C₄	—	137	1	no	—
92	C₈	—	274	1	no	—
93	C₈	—	274	2	no	—
94	C₈	—	548	1	no	—
95	C₈	—	548	2	no	—
96	C₈	—	137	1	no	—
97	C₈	C₂×#11764049	1	1	yes	characteristic
98	C₂²	—	137	1	no	—
99	C₄	—	2	2	no	—
100	C₄	—	274	1	no	—
101	C₄	—	274	2	no	—
102	C₄	—	548	2	no	—
103	C₄	—	137	2	no	—
104	C₄	—	137	1	no	—
105	C₄	C₂²×#11764049	1	1	yes	characteristic
106	C₂	—	137	1	no	—
107	C₂	—	137	1	no	—
108	C₂	D₄×#11764049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11764049

C₂

yes

maximal

#22578492

C₂

yes

maximal, characteristic

Q₈⋊#11764049

C₂

yes

maximal

#22578487

C₂

yes

maximal, characteristic

Q₁₆×#10587739

C₂

yes

maximal, characteristic

C₄⋊#11764049

—

C₄⋊#11764049

C₂²

yes

—

C₄×#11764049

—

C₄×#11764049

C₂²

yes

characteristic

C₄×#11764049

C₂²

yes

characteristic

Q₈×#10587739

C₂²

yes

—

#17984157

C₄

yes

characteristic

#17984151

C₄

yes

—

C₈×#10587739

C₂²

yes

characteristic

Q₁₆×D₁₃₇

C₄

yes

—

—

D₄

yes

—

#14555518

C₂×C₄

yes

characteristic

C₄⋊#10587739

—

C₄⋊#10587739

C₂×C₄

yes

—

C₄×#10587739

—

C₄×#10587739

C₂³

yes

characteristic

Q₈×D₁₃₇

C₂×C₄

yes

—

Dic₅₄₈

C₈

yes

characteristic, complemented

#14555500

C₈

yes

complemented

C₈×D₁₃₇

C₂×C₄

yes

characteristic

Q₁₆×C₁₃₇

C₈

yes

characteristic, complemented

—

—

—

—

C₂×D₄

yes

yes

—

Dic₂₇₄

—

Dic₂₇₄

C₂×C₈

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

C₂²×C₄

yes

characteristic

#11764040

C₂×C₈

yes

characteristic

Q₈×C₁₃₇

C₂×C₈

yes

—

C₈×C₁₃₇

C₂×C₈

yes

characteristic

C₂×D₁₃₇

C₄×D₄

yes

—

—

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×C₈

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

#131

yes

characteristic

C₂×C₁₃₇

C₈×D₄

yes

D₁₃₇

yes

D₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1579

—

137

maximal, Sylow

C₈×Q₈

—

137

—

C₄×Q₁₆

—

137

—

#334

—

137

—

Q₈⋊C₈

—

137

—

#322

—

137

—

C₂×Q₁₆

—

137

—

C₄×Q₈

—

137

—

#107

—

137

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

#103

—

137

—

C₄×C₈

—

274

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

137

—

Q₁₆

—

137

—

Q₁₆

—

137

—

Q₁₆

#11764049

yes

characteristic, complemented

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄²

—

274

—

C₄²

—

137

—

Q₈

—

274

—

Q₈

—

137

—

Q₈

C₂×#11764049

yes

—

C₂×C₄

—

274

—

C₂×C₄

—

274

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₂×C₄

—

137

—

C₈

—

274

—

C₈

—

274

—

C₈

—

548

—

C₈

—

548

—

C₈

—

137

—

C₈

C₂×#11764049

yes

characteristic

C₂²

—

137

—

C₄

—

100

C₄

—

274

—

101

C₄

—

274

—

102

C₄

—

548

—

103

C₄

—

137

—

104

C₄

—

137

—

105

C₄

C₂²×#11764049

yes

characteristic

106

C₂

—

137

—

107

C₂

—

137

—

108

C₂

D₄×#11764049

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#11764049
commutator	C₄×C₁₃₇	C₂²×C₈
Frattini	C₄	C₂²×#11764049
Fitting	Q₁₆×C₁₃₇	C₈
socle	C₂×C₁₃₇	C₈×D₄
prime radical	C₂×D₁₃₇	C₄×D₄
cosocle	C₄×#10587739	C₂³
hypercenter	Q₁₆	#11764049
hypocenter	C₁₃₇	#1579
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

subgroup

G/H

center

C₂

D₄×#11764049

commutator

C₄×C₁₃₇

C₂²×C₈

Frattini

C₄

C₂²×#11764049

Fitting

Q₁₆×C₁₃₇

C₈

socle

C₂×C₁₃₇

C₈×D₄

prime radical

C₂×D₁₃₇

C₄×D₄

C₂³

Q₁₆

C₁₃₇

G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

N	H ≅ G/N	type
25. Dic₅₄₈	C₈	semidirect
26. #14555500	C₈	semidirect
28. Q₁₆×C₁₃₇	C₈	semidirect
54. C₁₃₇	#1579	semidirect
74. Q₁₆	#11764049	semidirect

H ≅ G/N

type

25. Dic₅₄₈

C₈

semidirect

26. #14555500

C₈

semidirect

28. Q₁₆×C₁₃₇

C₈

semidirect

54. C₁₃₇

#1579

semidirect

74. Q₁₆

#11764049

semidirect

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

subgroup	index
Q₈×#11764049	2
#22578492	2
Q₈⋊#11764049	2
#22578487	2
Q₁₆×#10587739	2
#1579	137

subgroup

index

Q₁₆×#10587739

137

Subgroups — #31797919

Conjugacy classes of subgroups

Complements

Maximal subgroups