Group #31912170 — subgroups

Subgroups — #31912170

Order 18176.

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

normal subgroups only

This group has 354 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	#22590596	C₂	1	4	yes	maximal, complemented
2	#22587958	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587955	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556073	—	2	8	no	—
6	#17986404	—	2	2	no	—
7	#17986379	C₂²	1	2	yes	—
8	#17986376	—	2	2	no	—
9	#17986376	C₂²	1	2	yes	—
10	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
11	C₈²×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₈₄	—	2	4	no	—
13	C₂×D₅₆₈	—	2	4	no	—
14	C₂×D₅₆₈	—	4	4	no	—
15	#14556128	—	2	2	no	—
16	#14556125	—	2	2	no	—
17	#14556073	—	4	16	no	—
18	#14556062	—	4	4	no	—
19	C₂×C₁₆×C₇₁	D₄	1	4	yes	—
20	C₄×C₈×C₇₁	D₄	1	2	yes	—
21	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₄₂	—	4	2	no	—
23	C₂×D₂₈₄	—	4	2	no	—
24	C₂×D₂₈₄	—	4	4	no	—
25	Dic₂₈₄	—	4	8	no	—
26	D₅₆₈	—	4	8	no	—
27	D₅₆₈	—	8	8	no	—
28	C₄⋊Dic₇₁	—	4	4	no	—
29	C₂×C₈×C₇₁	D₈	1	4	yes	—
30	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
31	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₁₆×C₇₁	D₈	1	8	yes	complemented
33	C₂²×D₇₁	—	8	2	no	—
34	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
36	C₂×Dic₇₁	—	8	2	no	—
37	D₂₈₄	—	8	4	no	—
38	D₂₈₄	—	8	8	no	—
39	Dic₁₄₂	—	8	4	no	—
40	C₈×C₇₁	SD₃₂	1	8	yes	complemented
41	C₈×C₇₁	C₂×D₈	1	4	yes	—
42	C₂²×C₇₁	#493	1	1	yes	characteristic
43	C₂×D₇₁	—	16	2	no	—
44	C₂×D₇₁	—	16	4	no	—
45	C₄×C₇₁	#493	1	2	yes	—
46	C₄×C₇₁	C₂×SD₃₂	1	4	yes	—
47	Dic₇₁	—	16	2	no	—
48	#12517	—	71	1	no	maximal, Sylow
49	C₂×C₇₁	#1712	1	1	yes	characteristic
50	C₂×C₇₁	#2248	1	2	yes	—
51	D₇₁	—	32	2	no	—
52	#2248	—	71	4	no	—
53	#1713	—	71	1	no	—
54	#1712	—	71	1	no	—
55	C₈×C₁₆	D₇₁	1	1	yes	characteristic, complemented
56	C₇₁	#12517	1	1	yes	characteristic, complemented, Sylow
57	C₂×SD₃₂	—	142	8	no	—
58	#500	—	142	2	no	—
59	#494	—	71	2	no	—
60	#493	—	71	2	no	—
61	#493	—	142	2	no	—
62	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
63	C₈²	C₂×D₇₁	1	1	yes	characteristic
64	C₂×Q₁₆	—	142	4	no	—
65	C₂×D₈	—	284	4	no	—
66	C₂×D₈	—	142	4	no	—
67	C₄⋊Q₈	—	142	2	no	—
68	#127	—	142	2	no	—
69	SD₃₂	—	284	16	no	—
70	C₂×C₁₆	D₂₈₄	1	4	yes	—
71	#107	—	284	4	no	—
72	C₄×C₈	D₂₈₄	1	2	yes	—
73	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
74	C₂×Q₈	—	284	2	no	—
75	C₂×D₄	—	284	2	no	—
76	C₂×D₄	—	284	4	no	—
77	Q₁₆	—	284	8	no	—
78	D₈	—	284	8	no	—
79	D₈	—	568	8	no	—
80	C₂×C₈	D₅₆₈	1	4	yes	—
81	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
82	C₄⋊C₄	—	284	4	no	—
83	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
84	C₁₆	D₅₆₈	1	8	yes	complemented
85	C₂³	—	568	2	no	—
86	Q₈	—	568	4	no	—
87	D₄	—	568	4	no	—
88	D₄	—	568	8	no	—
89	C₂×C₄	—	568	2	no	—
90	C₂×C₄	#14556125	1	1	yes	characteristic
91	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
92	C₈	#14556073	1	8	yes	complemented
93	C₈	C₂×D₅₆₈	1	4	yes	—
94	C₂²	—	1136	2	no	—
95	C₂²	—	1136	4	no	—
96	C₂²	#17986376	1	1	yes	characteristic
97	C₄	—	1136	2	no	—
98	C₄	#17986376	1	2	yes	—
99	C₄	C₂×#14556073	1	4	yes	—
100	C₂	—	2272	2	no	—
101	C₂	#22587955	1	1	yes	characteristic
102	C₂	#22590596	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590596

C₂

yes

maximal, complemented

#22587958

C₂

yes

maximal, characteristic, complemented

#22587955

C₂

yes

maximal, characteristic

C₈×C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#14556073

—

#17986404

—

#17986379

C₂²

yes

—

#17986376

—

#17986376

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

—

—

—

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

Dic₂₈₄

—

D₅₆₈

—

D₅₆₈

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

D₈

yes

complemented

C₂²×D₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

SD₃₂

yes

complemented

C₈×C₇₁

C₂×D₈

yes

—

C₂²×C₇₁

#493

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

#493

yes

—

C₄×C₇₁

C₂×SD₃₂

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

#2248

—

#1713

—

#1712

—

C₈×C₁₆

D₇₁

yes

characteristic, complemented

C₇₁

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

142

—

#500

—

142

—

#494

—

#493

—

#493

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

142

—

C₂×D₈

—

284

—

C₂×D₈

—

142

—

C₄⋊Q₈

—

142

—

#127

—

142

—

SD₃₂

—

284

—

C₂×C₁₆

D₂₈₄

yes

—

#107

—

284

—

C₄×C₈

D₂₈₄

yes

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

Q₁₆

—

284

—

D₈

—

284

—

D₈

—

568

—

C₂×C₈

D₅₆₈

yes

—

C₂×C₈

C₂×D₂₈₄

yes

—

C₄⋊C₄

—

284

—

C₄²

C₂×D₂₈₄

yes

characteristic

C₁₆

D₅₆₈

yes

complemented

C₂³

—

568

—

Q₈

—

568

—

D₄

—

568

—

D₄

—

568

—

C₂×C₄

—

568

—

C₂×C₄

#14556125

yes

characteristic

C₂×C₄

C₂×D₅₆₈

yes

—

C₈

#14556073

yes

complemented

C₈

C₂×D₅₆₈

yes

—

C₂²

—

1136

—

C₂²

—

1136

—

C₂²

#17986376

yes

characteristic

C₄

—

1136

—

C₄

#17986376

yes

—

C₄

C₂×#14556073

yes

—

100

C₂

—

2272

—

101

C₂

#22587955

yes

characteristic

102

C₂

#22590596

yes

—

	subgroup	G/H
center	C₂²	#17986376
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₈×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#493
prime radical	#22587955	C₂
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#12517
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#17986376

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₈×C₁₆×C₇₁

C₂

C₂²×C₇₁

C₂

C₈×C₁₆

D₇₁

hypocenter

C₇₁

#12517

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #22590596	C₂	semidirect
2. #22587958	C₂	semidirect
4. C₈×C₁₆×C₇₁	C₂	semidirect
32. C₁₆×C₇₁	D₈	semidirect
40. C₈×C₇₁	SD₃₂	semidirect
55. C₈×C₁₆	D₇₁	semidirect
56. C₇₁	#12517	semidirect
84. C₁₆	D₅₆₈	semidirect
92. C₈	#14556073	semidirect

H ≅ G/N

type

1. #22590596

C₂

semidirect

2. #22587958

C₂

semidirect

4. C₈×C₁₆×C₇₁

C₂

semidirect

32. C₁₆×C₇₁

D₈

semidirect

40. C₈×C₇₁

SD₃₂

semidirect

55. C₈×C₁₆

D₇₁

semidirect

56. C₇₁

#12517

semidirect

84. C₁₆

D₅₆₈

semidirect

92. C₈

#14556073

semidirect

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

subgroup	index
#22590596	2
#22587958	2
#22587955	2
C₈×C₁₆×C₇₁	2
#12517	71

subgroup

index

C₈×C₁₆×C₇₁

Subgroups — #31912170

Conjugacy classes of subgroups

Complements

Maximal subgroups