Group #31912169 — subgroups

Subgroups — #31912169

Order 18176.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22590595	C₂	1	4	yes	maximal, complemented
2	#22587958	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#1712	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556072	—	2	8	no	—
6	#17986404	—	2	2	no	—
7	#17986379	C₂²	1	2	yes	—
8	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
9	#493×C₇₁	—	2	2	no	—
10	#493×C₇₁	C₂²	1	2	yes	—
11	C₈²×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₈₄	—	2	4	no	—
13	#14556128	—	2	2	no	—
14	#14556072	—	4	16	no	—
15	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
16	#14556062	—	4	4	no	—
17	C₂×D₈×C₇₁	—	2	4	no	—
18	C₂×D₈×C₇₁	—	4	4	no	—
19	#127×C₇₁	—	2	2	no	—
20	C₄×C₈×C₇₁	D₄	1	2	yes	—
21	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₄₂	—	4	2	no	—
23	Dic₂₈₄	—	4	8	no	—
24	C₄⋊Dic₇₁	—	4	4	no	—
25	C₇₁⋊C₁₆	D₈	1	8	yes	complemented
26	C₂×D₄×C₇₁	—	4	2	no	—
27	C₂×D₄×C₇₁	—	4	4	no	—
28	D₈×C₇₁	—	4	8	no	—
29	D₈×C₇₁	—	8	8	no	—
30	C₂×C₈×C₇₁	D₈	1	4	yes	—
31	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
32	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂³×C₇₁	—	8	2	no	—
34	D₄×C₇₁	—	8	4	no	—
35	D₄×C₇₁	—	8	8	no	—
36	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
38	C₂×Dic₇₁	—	8	2	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	—	16	2	no	—
43	C₂²×C₇₁	—	16	4	no	—
44	C₂²×C₇₁	#493	1	1	yes	characteristic
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	—	32	2	no	—
50	C₂×C₇₁	#1712	1	1	yes	characteristic
51	C₂×C₇₁	#2248	1	2	yes	—
52	#2248	—	71	4	no	—
53	#1713	—	71	1	no	—
54	#1712	D₇₁	1	1	yes	characteristic
55	C₈×C₁₆	—	71	1	no	—
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	—	2	2	no	—
61	#493	C₂×D₇₁	1	2	yes	—
62	C₄×C₁₆	—	71	2	no	—
63	C₈²	C₂×D₇₁	1	1	yes	characteristic
64	C₂×Q₁₆	—	142	4	no	—
65	C₂×D₈	—	2	4	no	—
66	C₂×D₈	—	4	4	no	—
67	C₄⋊Q₈	—	142	2	no	—
68	#127	—	2	2	no	—
69	SD₃₂	—	284	16	no	—
70	C₂×C₁₆	—	71	4	no	—
71	#107	—	284	4	no	—
72	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
73	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
74	C₂×Q₈	—	284	2	no	—
75	C₂×D₄	—	4	2	no	—
76	C₂×D₄	—	4	4	no	—
77	Q₁₆	—	284	8	no	—
78	D₈	—	4	8	no	—
79	D₈	—	8	8	no	—
80	C₂×C₈	#11765850	1	4	yes	—
81	C₂×C₈	D₄×D₇₁	1	2	yes	—
82	C₄⋊C₄	—	284	4	no	—
83	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
84	C₁₆	—	71	8	no	—
85	C₂³	—	8	2	no	—
86	Q₈	—	568	4	no	—
87	D₄	—	8	4	no	—
88	D₄	—	8	8	no	—
89	C₂×C₄	—	568	2	no	—
90	C₂×C₄	#14556124	1	1	yes	characteristic
91	C₂×C₄	C₂×#11765850	1	2	yes	—
92	C₈	#14556072	1	8	yes	complemented
93	C₈	D₈×D₇₁	1	4	yes	—
94	C₂²	—	16	2	no	—
95	C₂²	—	16	4	no	—
96	C₂²	#17986375	1	1	yes	characteristic
97	C₄	—	1136	2	no	—
98	C₄	#17986374	1	2	yes	—
99	C₄	C₂×#14556072	1	4	yes	—
100	C₂	—	32	2	no	—
101	C₂	#22587954	1	1	yes	characteristic
102	C₂	#22590595	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590595

C₂

yes

maximal, complemented

#22587958

C₂

yes

maximal, characteristic, complemented

C₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₇₁×#1712

C₂

yes

maximal, characteristic

C₂×#14556072

—

#17986404

—

#17986379

C₂²

yes

—

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#493×C₇₁

—

#493×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

#14556128

—

#14556072

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

#127×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

D₈

yes

complemented

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

SD₃₂

yes

complemented

C₈×C₇₁

C₂×D₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

#493

yes

—

C₄×C₇₁

C₂×SD₃₂

yes

—

Dic₇₁

—

#12517

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1712

yes

characteristic

C₂×C₇₁

#2248

yes

—

#2248

—

#1713

—

#1712

D₇₁

yes

characteristic

C₈×C₁₆

—

C₇₁

#12517

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

142

—

#500

—

142

—

#494

—

#493

—

#493

C₂×D₇₁

yes

—

C₄×C₁₆

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

142

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

142

—

#127

—

SD₃₂

—

284

—

C₂×C₁₆

—

#107

—

284

—

C₄×C₈

C₂²⋊D₇₁

yes

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

284

—

D₈

—

D₈

—

C₂×C₈

#11765850

yes

—

C₂×C₈

D₄×D₇₁

yes

—

C₄⋊C₄

—

284

—

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

568

—

D₄

—

D₄

—

C₂×C₄

—

568

—

C₂×C₄

#14556124

yes

characteristic

C₂×C₄

C₂×#11765850

yes

—

C₈

#14556072

yes

complemented

C₈

D₈×D₇₁

yes

—

C₂²

—

C₂²

—

C₂²

#17986375

yes

characteristic

C₄

—

1136

—

C₄

#17986374

yes

—

C₄

C₂×#14556072

yes

—

100

C₂

—

101

C₂

#22587954

yes

characteristic

102

C₂

#22590595

yes

—

	subgroup	G/H
center	C₂²	#17986375
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting, prime radical	C₇₁×#1712	C₂
socle	C₂²×C₇₁	#493
hypercenter	#1712	D₇₁
hypocenter	C₇₁	#12517
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#17986375

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting, prime radical

C₇₁×#1712

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #22590595	C₂	semidirect
2. #22587958	C₂	semidirect
3. C₈×C₇₁⋊C₁₆	C₂	semidirect
25. C₇₁⋊C₁₆	D₈	semidirect
40. C₈×C₇₁	SD₃₂	semidirect
56. C₇₁	#12517	semidirect
92. C₈	#14556072	semidirect

H ≅ G/N

type

1. #22590595

C₂

semidirect

2. #22587958

C₂

semidirect

3. C₈×C₇₁⋊C₁₆

C₂

semidirect

25. C₇₁⋊C₁₆

D₈

semidirect

40. C₈×C₇₁

SD₃₂

semidirect

56. C₇₁

#12517

semidirect

92. C₈

#14556072

semidirect

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

subgroup	index
#22590595	2
#22587958	2
C₈×C₇₁⋊C₁₆	2
C₇₁×#1712	2
#12517	71

subgroup

index

#22590595

#22587958

C₈×C₇₁⋊C₁₆

C₇₁×#1712

#12517

Subgroups — #31912169

Conjugacy classes of subgroups

Complements

Maximal subgroups