Group #31912145 — subgroups

Subgroups — #31912145

Order 18176.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	2	yes	maximal, complemented
2	#22590563	C₂	1	2	yes	maximal
3	#22587950	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1709	C₂	1	1	yes	maximal, characteristic
6	#17986404	C₂²	1	2	yes	—
7	#17986350	—	2	2	no	—
8	#17985803	—	2	4	no	—
9	#17985800	—	2	4	no	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	#494×C₇₁	C₂²	1	1	yes	characteristic
12	#493×C₇₁	C₂²	1	1	yes	characteristic
13	#486×C₇₁	—	2	2	no	—
14	C₈²×C₇₁	C₂²	1	1	yes	characteristic
15	#14556128	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
17	#14556062	—	2	2	no	—
18	#14556062	—	2	2	no	—
19	#14556053	—	4	4	no	—
20	C₂×Q₁₆×C₇₁	—	2	2	no	—
21	C₂×D₈×C₇₁	—	2	2	no	—
22	C₄⋊Q₈×C₇₁	—	2	1	no	—
23	#127×C₇₁	—	2	1	no	—
24	#102×C₇₁	—	4	4	no	—
25	C₄×C₈×C₇₁	D₄	1	2	yes	—
26	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₄₂	—	4	2	no	—
28	C₄⋊Dic₇₁	—	4	2	no	—
29	C₄⋊Dic₇₁	—	4	2	no	—
30	C₇₁⋊C₁₆	—	2	4	no	—
31	C₂×Q₈×C₇₁	—	4	1	no	—
32	C₂×D₄×C₇₁	—	4	1	no	—
33	C₂×D₄×C₇₁	—	4	2	no	—
34	Q₁₆×C₇₁	—	4	4	no	—
35	D₈×C₇₁	—	4	4	no	—
36	C₂×C₈×C₇₁	D₈	1	4	yes	—
37	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₇₁	—	4	2	no	—
39	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂³×C₇₁	—	8	1	no	—
41	Q₈×C₇₁	—	8	2	no	—
42	D₄×C₇₁	—	8	2	no	—
43	D₄×C₇₁	—	8	4	no	—
44	C₂×C₄×C₇₁	—	8	1	no	—
45	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
47	C₂×Dic₇₁	—	8	2	no	—
48	Dic₁₄₂	—	8	4	no	—
49	C₈×C₇₁	—	2	4	no	—
50	C₈×C₇₁	#135	1	4	yes	—
51	C₂²×C₇₁	—	16	1	no	—
52	C₂²×C₇₁	—	16	2	no	—
53	C₂²×C₇₁	#493	1	1	yes	characteristic
54	C₄×C₇₁	—	16	1	no	—
55	C₄×C₇₁	#495	1	2	yes	—
56	C₄×C₇₁	#508	1	4	yes	—
57	Dic₇₁	—	16	2	no	—
58	#12511	—	71	1	no	maximal, Sylow
59	C₂×C₇₁	—	32	1	no	—
60	C₂×C₇₁	#1709	1	1	yes	characteristic
61	C₂×C₇₁	#2249	1	2	yes	—
62	#2241	—	71	2	no	—
63	#2240	—	71	2	no	—
64	#1710	—	71	1	no	—
65	#1709	D₇₁	1	1	yes	characteristic
66	C₈×C₁₆	—	71	1	no	—
67	C₇₁	#12511	1	1	yes	characteristic, complemented, Sylow
68	#500	—	71	2	no	—
69	#494	C₂×D₇₁	1	1	yes	characteristic
70	#493	C₂×D₇₁	1	1	yes	characteristic
71	#487	—	142	2	no	—
72	#486	—	2	2	no	—
73	#358	—	142	4	no	—
74	#357	—	142	4	no	—
75	C₄×C₁₆	—	71	2	no	—
76	C₈²	C₂×D₇₁	1	1	yes	characteristic
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	142	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	—	71	4	no	—
83	#107	—	142	2	no	—
84	#107	—	142	2	no	—
85	#103	—	284	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
88	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	284	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	#11765850	1	4	yes	—
96	C₂×C₈	D₄×D₇₁	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	284	2	no	—
99	C₄⋊C₄	—	284	2	no	—
100	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
101	C₁₆	—	142	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	568	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	568	2	no	—
109	C₂×C₄	#14556124	1	1	yes	characteristic
110	C₂×C₄	C₂×#11765850	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14556147	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#17986375	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	1136	2	no	—
118	C₄	#17986382	1	2	yes	—
119	C₄	#17986429	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#22587945	1	1	yes	characteristic
122	C₂	#22590599	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590567

C₂

yes

maximal, complemented

#22590563

C₂

yes

maximal

#22587950

C₂

yes

maximal, characteristic, complemented

C₈×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₇₁×#1709

C₂

yes

maximal, characteristic

#17986404

C₂²

yes

—

#17986350

—

#17985803

—

#17985800

—

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#494×C₇₁

C₂²

yes

characteristic

#493×C₇₁

C₂²

yes

characteristic

#486×C₇₁

—

C₈²×C₇₁

C₂²

yes

characteristic

#14556128

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

#14556062

—

#14556053

—

C₂×Q₁₆×C₇₁

—

C₂×D₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

#127×C₇₁

—

#102×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

Q₁₆×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#495

yes

—

C₄×C₇₁

#508

yes

—

Dic₇₁

—

#12511

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1709

yes

characteristic

C₂×C₇₁

#2249

yes

—

—

—

—

D₇₁

yes

characteristic

C₈×C₁₆

—

C₇₁

#12511

yes

characteristic, complemented, Sylow

#500

—

#494

C₂×D₇₁

yes

characteristic

#493

C₂×D₇₁

yes

characteristic

#487

—

142

—

#486

—

#358

—

142

—

#357

—

142

—

C₄×C₁₆

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

#127

—

C₂×C₁₆

—

#107

—

142

—

#107

—

142

—

#103

—

284

—

#102

—

C₄×C₈

C₂²⋊D₇₁

yes

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#11765850

yes

—

C₂×C₈

D₄×D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

100

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

101

C₁₆

—

142

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

568

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

568

—

109

C₂×C₄

#14556124

yes

characteristic

110

C₂×C₄

C₂×#11765850

yes

—

111

C₈

—

112

C₈

#14556147

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#17986375

yes

characteristic

116

C₄

—

117

C₄

—

1136

—

118

C₄

#17986382

yes

—

119

C₄

#17986429

yes

—

120

C₂

—

121

C₂

#22587945

yes

characteristic

122

C₂

#22590599

yes

—

	subgroup	G/H
center	C₂²	#17986375
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1709	C₂
socle	C₂²×C₇₁	#493
prime radical	#493×C₇₁	C₂²
hypercenter	#1709	D₇₁
hypocenter	C₇₁	#12511
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#17986375

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1709

C₂

socle

C₂²×C₇₁

#493

prime radical

#493×C₇₁

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #22590567	C₂	semidirect
3. #22587950	C₂	semidirect
4. C₈×C₇₁⋊C₁₆	C₂	semidirect
67. C₇₁	#12511	semidirect

H ≅ G/N

type

1. #22590567

C₂

semidirect

3. #22587950

C₂

semidirect

4. C₈×C₇₁⋊C₁₆

C₂

semidirect

67. C₇₁

#12511

semidirect

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

subgroup	index
#22590567	2
#22590563	2
#22587950	2
C₈×C₇₁⋊C₁₆	2
C₇₁×#1709	2
#12511	71

subgroup

index

#22590567

#22590563

#22587950

C₈×C₇₁⋊C₁₆

C₇₁×#1709

#12511

Subgroups — #31912145

Conjugacy classes of subgroups

Complements

Maximal subgroups