Group #31912141 — subgroups

Subgroups — #31912141

Order 18176.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22590567	C₂	1	4	yes	maximal
2	#22587958	C₂	1	1	yes	maximal, characteristic
3	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1713	C₂	1	1	yes	maximal, characteristic
5	#17986404	—	2	1	no	—
6	#17986404	C₂²	1	2	yes	—
7	#17986379	—	2	1	no	—
8	#17985803	—	2	8	no	—
9	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	#500×C₇₁	—	2	1	no	—
11	#500×C₇₁	—	2	1	no	—
12	#494×C₇₁	C₂²	1	2	yes	—
13	C₈²×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₈₄	—	4	2	no	—
15	#14556128	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
17	#14556062	—	2	4	no	—
18	#14556062	—	4	2	no	—
19	C₂×Q₁₆×C₇₁	—	2	4	no	—
20	C₄⋊Q₈×C₇₁	—	2	2	no	—
21	#107×C₇₁	—	4	2	no	—
22	#107×C₇₁	—	4	2	no	—
23	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₄₂	—	4	2	no	—
27	Dic₂₈₄	—	8	4	no	—
28	C₄⋊Dic₇₁	—	4	2	no	—
29	C₄⋊Dic₇₁	—	4	2	no	—
30	C₇₁⋊C₁₆	—	2	4	no	—
31	C₂×Q₈×C₇₁	—	4	2	no	—
32	Q₁₆×C₇₁	—	4	8	no	—
33	C₂×C₈×C₇₁	D₈	1	2	yes	—
34	C₂×C₈×C₇₁	D₈	1	2	yes	—
35	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
36	C₄⋊C₄×C₇₁	—	4	2	no	—
37	C₄⋊C₄×C₇₁	—	4	2	no	—
38	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₇₁	—	8	4	no	—
40	C₂×C₄×C₇₁	—	8	2	no	—
41	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₇₁	—	8	2	no	—
45	Dic₁₄₂	—	8	4	no	—
46	C₈×C₇₁	SD₃₂	1	4	yes	—
47	C₈×C₇₁	Q₃₂	1	4	yes	—
48	C₈×C₇₁	#135	1	4	yes	—
49	C₂²×C₇₁	#493	1	1	yes	characteristic
50	C₄×C₇₁	—	16	2	no	—
51	C₄×C₇₁	#495	1	2	yes	—
52	C₄×C₇₁	C₂×SD₃₂	1	2	yes	—
53	C₄×C₇₁	C₂×Q₃₂	1	2	yes	—
54	Dic₇₁	—	16	2	no	—
55	#12510	—	71	1	no	maximal, Sylow
56	C₂×C₇₁	#1709	1	1	yes	characteristic
57	C₂×C₇₁	#2247	1	1	yes	characteristic
58	C₂×C₇₁	#2248	1	1	yes	characteristic
59	#2241	—	71	4	no	—
60	#1713	—	71	1	no	—
61	#1713	D₇₁	1	1	yes	characteristic
62	C₈×C₁₆	—	71	1	no	—
63	C₇₁	#12510	1	1	yes	characteristic, complemented, Sylow
64	#500	—	2	1	no	—
65	#500	—	2	1	no	—
66	#500	—	71	2	no	—
67	#500	—	142	1	no	—
68	#494	—	142	1	no	—
69	#494	C₂×D₇₁	1	2	yes	—
70	#358	—	142	8	no	—
71	C₄×C₁₆	—	71	2	no	—
72	C₈²	C₂×D₇₁	1	1	yes	characteristic
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	—	284	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	142	2	no	—
77	C₂×C₁₆	—	71	4	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	#107	—	284	2	no	—
81	#107	—	142	4	no	—
82	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	284	2	no	—
87	Q₁₆	—	4	8	no	—
88	Q₁₆	—	568	4	no	—
89	C₂×C₈	#11765850	1	2	yes	—
90	C₂×C₈	#11765850	1	2	yes	—
91	C₂×C₈	D₄×D₇₁	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	284	2	no	—
95	C₄⋊C₄	—	284	2	no	—
96	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
97	C₁₆	—	142	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	568	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	568	2	no	—
102	C₂×C₄	#14556124	1	1	yes	characteristic
103	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
104	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
105	C₈	#14556071	1	4	yes	—
106	C₈	#14556074	1	4	yes	—
107	C₈	#14556147	1	4	yes	—
108	C₂²	#17986375	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	1136	2	no	—
111	C₄	#17986382	1	2	yes	—
112	C₄	C₂×#14556071	1	2	yes	—
113	C₄	C₂×#14556074	1	2	yes	—
114	C₂	#22587945	1	1	yes	characteristic
115	C₂	#22590592	1	1	yes	characteristic
116	C₂	#22590597	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590567

C₂

yes

maximal

#22587958

C₂

yes

maximal, characteristic

C₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic

C₇₁×#1713

C₂

yes

maximal, characteristic

#17986404

—

#17986404

C₂²

yes

—

#17986379

—

#17985803

—

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#500×C₇₁

—

#500×C₇₁

—

#494×C₇₁

C₂²

yes

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

#14556128

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

#14556062

—

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

—

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

SD₃₂

yes

—

C₈×C₇₁

Q₃₂

yes

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#495

yes

—

C₄×C₇₁

C₂×SD₃₂

yes

—

C₄×C₇₁

C₂×Q₃₂

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

C₈×C₁₆

—

C₇₁

#12510

yes

characteristic, complemented, Sylow

—

—

—

—

142

—

#494

—

142

—

#494

C₂×D₇₁

yes

—

#358

—

142

—

C₄×C₁₆

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

284

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

284

—

#107

—

142

—

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

Q₁₆

—

Q₁₆

—

568

—

C₂×C₈

#11765850

yes

—

C₂×C₈

#11765850

yes

—

C₂×C₈

D₄×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

C₁₆

—

142

—

Q₈

—

Q₈

—

568

—

100

C₂×C₄

—

101

C₂×C₄

—

568

—

102

C₂×C₄

#14556124

yes

characteristic

103

C₂×C₄

C₂×#11765850

yes

characteristic

104

C₂×C₄

C₂×#11765850

yes

characteristic

105

C₈

#14556071

yes

—

106

C₈

#14556074

yes

—

107

C₈

#14556147

yes

—

108

C₂²

#17986375

yes

characteristic

109

C₄

—

110

C₄

—

1136

—

111

C₄

#17986382

yes

—

112

C₄

C₂×#14556071

yes

—

113

C₄

C₂×#14556074

yes

—

114

C₂

#22587945

yes

characteristic

115

C₂

#22590592

yes

characteristic

116

C₂

#22590597

yes

characteristic

	subgroup	G/H
center	C₂²	#17986375
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1713	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	#1713	D₇₁
hypocenter	C₇₁	#12510
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#17986375

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1713

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
63. C₇₁	#12510	semidirect

H ≅ G/N

type

63. C₇₁

#12510

semidirect

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

subgroup	index
#22590567	2
#22587958	2
C₈×C₇₁⋊C₁₆	2
C₇₁×#1713	2
#12510	71

subgroup

index

#22590567

#22587958

C₈×C₇₁⋊C₁₆

C₇₁×#1713

#12510

Subgroups — #31912141

Conjugacy classes of subgroups

Complements

Maximal subgroups