Group #31918444 — subgroups

Subgroups — #31918444

Order 18176.

174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22590593	C₂	1	1	yes	maximal, characteristic
2	#22586520	C₂	1	4	yes	maximal
3	#22586474	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#2253	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₆₈	—	2	1	no	—
6	C₂×Dic₅₆₈	C₂²	1	2	yes	—
7	#17986379	—	2	1	no	—
8	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
9	#499×C₇₁	—	2	1	no	—
10	#367×C₇₁	—	2	1	no	—
11	#366×C₇₁	C₂²	1	2	yes	—
12	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₈₄	—	2	2	no	—
14	C₂×Dic₂₈₄	—	4	1	no	—
15	#14556128	—	4	1	no	—
16	Dic₅₆₈	—	2	4	no	—
17	Dic₅₆₈	—	4	2	no	—
18	C₇₁⋊C₃₂	—	2	2	no	—
19	#125×C₇₁	—	4	1	no	—
20	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
22	#107×C₇₁	—	2	2	no	—
23	#106×C₇₁	—	4	1	no	—
24	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₄₂	—	4	2	no	—
26	Dic₂₈₄	—	4	2	no	—
27	Dic₂₈₄	—	4	2	no	—
28	Dic₂₈₄	—	8	2	no	—
29	C₄⋊Dic₇₁	—	8	2	no	—
30	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₇₁	—	4	2	no	—
33	C₄⋊C₄×C₇₁	—	8	2	no	—
34	C₄²×C₇₁	D₈	1	1	yes	characteristic
35	C₁₆×C₇₁	SD₁₆	1	2	yes	—
36	C₁₆×C₇₁	#41	1	2	yes	—
37	C₂×C₄×C₇₁	—	8	2	no	—
38	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
39	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₇₁	—	8	2	no	—
41	Dic₁₄₂	—	8	2	no	—
42	Dic₁₄₂	—	8	2	no	—
43	C₈×C₇₁	SD₃₂	1	2	yes	—
44	C₈×C₇₁	#124	1	1	yes	characteristic
45	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
46	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
47	C₄×C₇₁	—	2	2	no	—
48	C₄×C₇₁	—	16	2	no	—
49	C₄×C₇₁	#486	1	1	yes	characteristic
50	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
51	Dic₇₁	—	16	2	no	—
52	#13724	—	71	1	no	maximal, Sylow
53	C₂×C₇₁	#2240	1	1	yes	characteristic
54	C₂×C₇₁	#2264	1	2	yes	—
55	#2253	D₇₁	1	1	yes	characteristic
56	#2247	—	71	1	no	—
57	#1416	—	71	4	no	—
58	#1397	—	71	1	no	—
59	C₇₁	#13724	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₃₂	—	71	2	no	—
61	C₂×Q₃₂	—	142	1	no	—
62	#499	—	2	1	no	—
63	#494	—	142	1	no	—
64	C₂×C₃₂	—	71	2	no	—
65	#367	—	2	1	no	—
66	#366	C₂×D₇₁	1	2	yes	—
67	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
68	C₂×Q₁₆	—	284	1	no	—
69	C₂×Q₁₆	—	142	2	no	—
70	C₄⋊Q₈	—	284	1	no	—
71	#125	—	4	1	no	—
72	Q₃₂	—	284	2	no	—
73	Q₃₂	—	142	4	no	—
74	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#106	—	4	1	no	—
78	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
79	C₃₂	—	142	2	no	—
80	C₂×Q₈	—	284	2	no	—
81	Q₁₆	—	284	2	no	—
82	Q₁₆	—	284	2	no	—
83	Q₁₆	—	568	2	no	—
84	C₂×C₈	#11765850	1	1	yes	characteristic
85	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	568	2	no	—
89	C₄²	#11765850	1	1	yes	characteristic
90	C₁₆	#11765852	1	2	yes	—
91	C₁₆	#11765866	1	2	yes	—
92	Q₈	—	568	2	no	—
93	Q₈	—	568	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	568	2	no	—
96	C₂×C₄	#14556069	1	2	yes	—
97	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
98	C₈	#14556071	1	2	yes	—
99	C₈	#14556117	1	1	yes	characteristic
100	C₈	C₂×#11765852	1	1	yes	characteristic
101	C₂²	C₂×#14556069	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	1136	2	no	—
105	C₄	#17986347	1	1	yes	characteristic
106	C₄	C₂×#14556071	1	1	yes	characteristic
107	C₂	#22590565	1	1	yes	characteristic
108	C₂	#22590657	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590593

C₂

yes

maximal, characteristic

#22586520

C₂

yes

maximal

#22586474

C₂

yes

maximal, characteristic

C₇₁×#2253

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

C₂²

yes

—

#17986379

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

#499×C₇₁

—

#367×C₇₁

—

#366×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

Dic₅₆₈

—

Dic₅₆₈

—

C₇₁⋊C₃₂

—

#125×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

#41

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

SD₃₂

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#486

yes

characteristic

C₄×C₇₁

C₂×SD₃₂

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#2253

D₇₁

yes

—

—

—

C₇₁

#13724

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

142

—

#499

—

#494

—

142

—

C₂×C₃₂

—

#367

—

#366

C₂×D₇₁

yes

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

284

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

284

—

#125

—

Q₃₂

—

284

—

Q₃₂

—

142

—

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₃₂

—

142

—

C₂×Q₈

—

284

—

Q₁₆

—

284

—

Q₁₆

—

284

—

Q₁₆

—

568

—

C₂×C₈

#11765850

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

568

—

C₄²

#11765850

yes

characteristic

C₁₆

#11765852

yes

—

C₁₆

#11765866

yes

—

Q₈

—

568

—

Q₈

—

568

—

C₂×C₄

—

C₂×C₄

—

568

—

C₂×C₄

#14556069

yes

—

C₂×C₄

C₂×#11765850

yes

characteristic

C₈

#14556071

yes

—

C₈

#14556117

yes

characteristic

100

C₈

C₂×#11765852

yes

characteristic

101

C₂²

C₂×#14556069

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

1136

—

105

C₄

#17986347

yes

characteristic

106

C₄

C₂×#14556071

yes

characteristic

107

C₂

#22590565

yes

characteristic

108

C₂

#22590657

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556069
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#2253	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#2253	D₇₁
hypocenter	C₇₁	#13724
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

subgroup

G/H

center

C₂²

C₂×#14556069

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#2253

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

N	H ≅ G/N	type
59. C₇₁	#13724	semidirect

H ≅ G/N

type

59. C₇₁

#13724

semidirect

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

subgroup	index
#22590593	2
#22586520	2
#22586474	2
C₇₁×#2253	2
#13724	71

subgroup

index

C₇₁×#2253

Subgroups — #31918444

Conjugacy classes of subgroups

Complements

Maximal subgroups