Group #31917955 — subgroups

Subgroups — #31917955

Order 18176.

190 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22590124	C₂	1	2	yes	maximal
2	#22586542	C₂	1	2	yes	maximal, complemented
3	#22586540	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986037	—	2	2	no	—
6	#17985827	—	2	2	no	—
7	#17985825	C₂²	1	4	yes	—
8	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₇₁	C₄	1	4	yes	complemented
10	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
12	#14556087	—	4	2	no	—
13	#14556062	—	2	4	no	—
14	#14556060	—	4	2	no	—
15	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
19	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
20	C₃₂×C₇₁	C₂×C₄	1	2	yes	—
21	C₃₂×C₇₁	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₇₁	—	4	4	no	—
23	C₂²⋊Dic₇₁	—	8	2	no	—
24	C₄×Dic₇₁	—	8	2	no	—
25	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
26	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
28	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
29	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
32	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₂³×C₇₁	D₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₁	#107	1	2	yes	—
35	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
36	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
38	C₂×Dic₇₁	—	8	4	no	—
39	C₈×C₇₁	#107	1	2	yes	—
40	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
42	C₂²×C₇₁	—	2	2	no	—
43	C₂²×C₇₁	#366	1	2	yes	—
44	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
45	C₄×C₇₁	#366	1	2	yes	—
46	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
47	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
48	Dic₇₁	—	16	4	no	—
49	#13633	—	71	1	no	maximal, Sylow
50	C₂×C₇₁	—	2	2	no	—
51	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
52	C₂×C₇₁	#2262	1	2	yes	—
53	C₂²×C₃₂	D₇₁	1	1	yes	characteristic
54	#2158	—	71	2	no	—
55	#1424	—	71	2	no	—
56	#1423	—	71	2	no	—
57	C₇₁	#13633	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
59	#427	—	142	2	no	—
60	C₂×C₃₂	Dic₇₁	1	4	yes	complemented
61	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
62	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
63	#367	—	142	2	no	—
64	#366	—	71	4	no	—
65	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
66	#117	—	284	2	no	—
67	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
68	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
70	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
71	#107	—	142	4	no	—
72	#106	—	284	2	no	—
73	C₃₂	C₂×Dic₇₁	1	2	yes	—
74	C₃₂	C₂×Dic₇₁	1	2	yes	—
75	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
76	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
77	C₂×C₈	D₅₆₈	1	1	yes	characteristic
78	C₂×C₈	Dic₂₈₄	1	2	yes	—
79	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
80	C₄⋊C₄	—	284	4	no	—
81	C₂²⋊C₄	—	568	2	no	—
82	C₄²	—	568	2	no	—
83	C₁₆	C₄⋊Dic₇₁	1	2	yes	—
84	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
85	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
86	C₂³	D₁₁₃₆	1	1	yes	characteristic
87	C₂×C₄	—	568	4	no	—
88	C₂×C₄	#14556062	1	2	yes	—
89	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
90	C₂×C₄	Dic₅₆₈	1	2	yes	—
91	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
92	C₈	#14556062	1	2	yes	—
93	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
94	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#17985825	1	2	yes	—
97	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
98	C₄	—	1136	4	no	—
99	C₄	#17985825	1	2	yes	—
100	C₄	C₂×#14556062	1	1	yes	characteristic
101	C₄	C₂×Dic₅₆₈	1	1	yes	characteristic
102	C₂	—	2	2	no	—
103	C₂	C₂×#17985825	1	1	yes	characteristic
104	C₂	#22590649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590124

C₂

yes

maximal

#22586542

C₂

yes

maximal, complemented

#22586540

C₂

yes

maximal, complemented

C₂²×C₃₂×C₇₁

C₂

yes

maximal, characteristic

#17986037

—

#17985827

—

#17985825

C₂²

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₄

yes

complemented

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

—

—

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

Q₈

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

C₂×C₄

yes

—

C₃₂×C₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

Q₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₃₂

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

#107

yes

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₂×Q₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#366

yes

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

#366

yes

—

C₄×C₇₁

C₂×#107

yes

characteristic

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

Dic₇₁

—

#13633

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×#366

yes

characteristic

C₂×C₇₁

#2262

yes

—

C₂²×C₃₂

D₇₁

yes

—

—

—

C₇₁

#13633

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

#427

—

142

—

C₂×C₃₂

Dic₇₁

yes

complemented

C₂×C₃₂

C₂×D₇₁

yes

characteristic

C₂×C₃₂

C₂×D₇₁

yes

characteristic

#367

—

142

—

#366

—

C₂²×C₈

D₂₈₄

yes

characteristic

#117

—

284

—

C₂×C₁₆

Dic₁₄₂

yes

—

C₂×C₁₆

D₂₈₄

yes

characteristic

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#107

—

142

—

#106

—

284

—

C₃₂

C₂×Dic₇₁

yes

—

C₃₂

C₂×Dic₇₁

yes

—

C₂²×C₄

D₅₆₈

yes

characteristic

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

D₅₆₈

yes

characteristic

C₂×C₈

Dic₂₈₄

yes

—

C₂×C₈

C₂×D₂₈₄

yes

characteristic

C₄⋊C₄

—

284

—

C₂²⋊C₄

—

568

—

C₄²

—

568

—

C₁₆

C₄⋊Dic₇₁

yes

—

C₁₆

C₂²×Dic₇₁

yes

characteristic

C₁₆

C₂×Dic₁₄₂

yes

characteristic

C₂³

D₁₁₃₆

yes

characteristic

C₂×C₄

—

568

—

C₂×C₄

#14556062

yes

—

C₂×C₄

D₁₁₃₆

yes

characteristic

C₂×C₄

Dic₅₆₈

yes

—

C₂×C₄

C₂×D₅₆₈

yes

characteristic

C₈

#14556062

yes

—

C₈

C₂×C₄⋊Dic₇₁

yes

characteristic

C₈

C₂×Dic₂₈₄

yes

characteristic

C₂²

—

C₂²

#17985825

yes

—

C₂²

C₂×D₁₁₃₆

yes

characteristic

C₄

—

1136

—

C₄

#17985825

yes

—

100

C₄

C₂×#14556062

yes

characteristic

101

C₄

C₂×Dic₅₆₈

yes

characteristic

102

C₂

—

103

C₂

C₂×#17985825

yes

characteristic

104

C₂

#22590649

yes

—

	subgroup	G/H
center	C₂×C₄	D₁₁₃₆
commutator	C₁₆×C₇₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂²×C₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₁₆
prime radical	C₂³×C₇₁	D₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂²×C₃₂	D₇₁
hypocenter	C₇₁	#13633
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂×C₄

D₁₁₃₆

commutator

C₁₆×C₇₁

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₂²×C₃₂×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₁₆

prime radical

C₂³×C₇₁

D₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₂²×C₃₂

D₇₁

hypocenter

C₇₁

#13633

LCS

G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
2. #22586542	C₂	semidirect
3. #22586540	C₂	semidirect
9. C₂×C₃₂×C₇₁	C₄	semidirect
57. C₇₁	#13633	semidirect
60. C₂×C₃₂	Dic₇₁	semidirect

H ≅ G/N

type

2. #22586542

C₂

semidirect

3. #22586540

C₂

semidirect

9. C₂×C₃₂×C₇₁

C₄

semidirect

57. C₇₁

#13633

semidirect

60. C₂×C₃₂

Dic₇₁

semidirect

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

subgroup	index
#22590124	2
#22586542	2
#22586540	2
C₂²×C₃₂×C₇₁	2
#13633	71

subgroup

index

C₂²×C₃₂×C₇₁

Subgroups — #31917955

Conjugacy classes of subgroups

Complements

Maximal subgroups