Group #31913864 — subgroups

Subgroups — #31913864

Order 18176.

414 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 275 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×C₄⋊(C₇₁⋊C₁₆)	C₂	1	6	yes	maximal
2	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
4	C₄⋊(C₇₁⋊C₁₆)	—	2	12	no	—
5	C₂×C₄×C₈×C₇₁	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
7	C₄³×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₇₁⋊C₁₆	—	2	12	no	—
9	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₇₁	C₈	1	3	yes	—
12	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
13	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
14	C₄×C₈×C₇₁	—	2	6	no	—
15	C₄×C₈×C₇₁	D₄	1	6	yes	—
16	C₄×C₈×C₇₁	Q₈	1	6	yes	—
17	C₇₁⋊C₁₆	—	4	4	no	—
18	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
19	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
20	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	—	2	6	no	—
22	C₂×C₈×C₇₁	C₄⋊C₄	1	6	yes	—
23	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
24	C₂×C₈×C₇₁	C₂×Q₈	1	3	yes	—
25	C₄²×C₇₁	—	2	6	no	—
26	C₄²×C₇₁	C₁₆	1	4	yes	complemented
27	C₄²×C₇₁	C₄⋊C₄	1	6	yes	—
28	C₄²×C₇₁	M₁₆	1	6	yes	—
29	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	6	no	—
31	C₂×C₄×C₇₁	C₄⋊C₈	1	6	yes	—
32	C₂×C₄×C₇₁	C₄⋊C₈	1	6	yes	—
33	C₂×C₄×C₇₁	C₂×C₁₆	1	6	yes	—
34	C₂×C₄×C₇₁	M₃₂	1	6	yes	—
35	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	3	yes	—
36	C₂×C₄×C₇₁	C₂×M₁₆	1	3	yes	—
37	C₈×C₇₁	—	2	6	no	—
38	C₈×C₇₁	#127	1	1	yes	characteristic
39	C₈×C₇₁	C₄⋊Q₈	1	3	yes	—
40	C₂²×C₇₁	C₂×C₄⋊C₈	1	3	yes	—
41	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
42	C₂²×C₇₁	C₂×M₃₂	1	3	yes	—
43	C₄×C₇₁	—	2	6	no	—
44	C₄×C₇₁	C₄⋊C₁₆	1	12	yes	complemented
45	C₄×C₇₁	#384	1	1	yes	characteristic
46	C₄×C₇₁	#423	1	3	yes	—
47	#12898	—	71	1	no	maximal, Sylow
48	C₂×C₇₁	#1842	1	1	yes	characteristic
49	C₂×C₇₁	C₂×C₄⋊C₁₆	1	3	yes	—
50	C₂×C₇₁	#2150	1	3	yes	—
51	C₂×C₄⋊C₁₆	—	71	6	no	—
52	C₄²×C₈	D₇₁	1	1	yes	characteristic
53	C₇₁	#12898	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₁₆	—	71	4	no	—
55	C₂×C₄×C₈	Dic₇₁	1	3	yes	—
56	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
57	C₄³	Dic₇₁	1	1	yes	characteristic
58	C₄⋊C₁₆	—	142	12	no	—
59	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
60	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
61	C₂×C₄²	C₇₁⋊C₈	1	3	yes	—
62	C₂×C₄²	C₇₁⋊C₈	1	1	yes	characteristic
63	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
64	C₂×C₁₆	—	142	12	no	—
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	Dic₁₄₂	1	6	yes	—
67	C₄×C₈	D₂₈₄	1	6	yes	—
68	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
69	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
70	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	C₄⋊Dic₇₁	1	6	yes	—
73	C₂×C₈	C₂×D₂₈₄	1	3	yes	—
74	C₂×C₈	C₂×Dic₁₄₂	1	3	yes	—
75	C₄²	—	2	6	no	—
76	C₄²	C₇₁⋊C₁₆	1	4	yes	complemented
77	C₄²	C₄⋊Dic₇₁	1	6	yes	—
78	C₄²	#11765849	1	6	yes	—
79	C₁₆	—	284	4	no	—
80	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	6	yes	—
83	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	6	yes	—
84	C₂×C₄	C₂×C₇₁⋊C₁₆	1	6	yes	—
85	C₂×C₄	#14556068	1	6	yes	—
86	C₂×C₄	C₂×C₄⋊Dic₇₁	1	3	yes	—
87	C₂×C₄	C₂×#11765849	1	3	yes	—
88	C₈	—	2	6	no	—
89	C₈	#14556125	1	1	yes	characteristic
90	C₈	#14556128	1	3	yes	—
91	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	3	yes	—
92	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
93	C₂²	C₂×#14556068	1	3	yes	—
94	C₄	—	2	6	no	—
95	C₄	C₄⋊(C₇₁⋊C₁₆)	1	12	yes	complemented
96	C₄	#17985869	1	1	yes	characteristic
97	C₄	#17986023	1	3	yes	—
98	C₂	#22588492	1	1	yes	characteristic
99	C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	1	3	yes	—
100	C₂	#22590096	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal

C₄²×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

—

C₂×C₄×C₈×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₈

yes

—

C₂×C₄²×C₇₁

C₈

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₁₆

yes

complemented

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

M₁₆

yes

—

C₂³×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

#127

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

C₂×C₄⋊C₈

yes

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₂²×C₇₁

C₂×M₃₂

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₄⋊C₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₂×C₄⋊C₁₆

yes

—

C₂×C₇₁

#2150

yes

—

C₂×C₄⋊C₁₆

—

C₄²×C₈

D₇₁

yes

characteristic

C₇₁

#12898

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₄³

Dic₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

C₇₁⋊C₈

yes

—

C₂×C₄²

C₇₁⋊C₈

yes

characteristic

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₁₆

—

142

—

C₄×C₈

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

D₂₈₄

yes

—

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

C₂×D₂₈₄

yes

—

C₂×C₈

C₂×Dic₁₄₂

yes

—

C₄²

—

C₄²

C₇₁⋊C₁₆

yes

complemented

C₄²

C₄⋊Dic₇₁

yes

—

C₄²

#11765849

yes

—

C₁₆

—

284

—

C₂³

C₂²×C₇₁⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

—

C₂×C₄

#14556068

yes

—

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂×#11765849

yes

—

C₈

—

C₈

#14556125

yes

characteristic

C₈

#14556128

yes

—

C₂²

C₂×C₄⋊(C₇₁⋊C₈)

yes

—

C₂²

C₂²×C₇₁⋊C₁₆

yes

characteristic

C₂²

C₂×#14556068

yes

—

C₄

—

C₄

C₄⋊(C₇₁⋊C₁₆)

yes

C₄

yes

C₄

yes

—

C₂

#22588492

yes

characteristic

C₂

C₂×C₄⋊(C₇₁⋊C₁₆)

yes

—

100

C₂

#22590096

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#12898
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₄²×C₈×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂²×C₈

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₄²×C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
26. C₄²×C₇₁	C₁₆	semidirect
44. C₄×C₇₁	C₄⋊C₁₆	semidirect
53. C₇₁	#12898	semidirect
76. C₄²	C₇₁⋊C₁₆	semidirect
95. C₄	C₄⋊(C₇₁⋊C₁₆)	semidirect

H ≅ G/N

type

26. C₄²×C₇₁

C₁₆

semidirect

44. C₄×C₇₁

C₄⋊C₁₆

semidirect

53. C₇₁

#12898

semidirect

76. C₄²

C₇₁⋊C₁₆

semidirect

95. C₄

C₄⋊(C₇₁⋊C₁₆)

semidirect

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₄²×C₈×C₇₁	2
#12898	71

subgroup

index

C₂×C₄⋊(C₇₁⋊C₁₆)

C₄²×C₈×C₇₁

#12898

Subgroups — #31913864

Conjugacy classes of subgroups

Complements

Maximal subgroups