Group #31912868 — subgroups

Subgroups — #31912868

Order 18176.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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×#17985829	C₂	1	2	yes	maximal
2	C₄×#14556064	C₂	1	2	yes	maximal
3	#22586434	C₂	1	2	yes	maximal
4	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556064	C₂²	1	4	yes	—
6	C₄×#11765849	—	2	2	no	—
7	#17985829	C₄	1	8	yes	complemented
8	#17985764	—	2	2	no	—
9	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#346×C₇₁	C₄	1	4	yes	—
13	C₂×#11765849	—	2	4	no	—
14	#14556064	—	2	4	no	—
15	#14556064	C₂×C₄	1	8	yes	—
16	#14556039	—	4	2	no	—
17	C₄×C₇₁⋊C₈	—	4	2	no	—
18	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	—	2	2	no	—
22	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
24	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₁	Q₈	1	2	yes	—
27	#11765849	—	2	8	no	—
28	#11765849	—	4	4	no	—
29	C₂×C₇₁⋊C₈	—	4	4	no	—
30	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
31	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₇₁	—	2	2	no	—
33	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	#41	1	2	yes	—
38	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
39	C₄²×C₇₁	D₈	1	2	yes	—
40	C₁₆×C₇₁	—	2	2	no	—
41	C₁₆×C₇₁	C₄²	1	4	yes	—
42	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₁	—	4	2	no	—
44	C₂×C₄×C₇₁	#107	1	4	yes	—
45	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
47	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#135	1	2	yes	—
49	C₈×C₇₁	—	2	2	no	—
50	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
51	C₈×C₇₁	#117	1	1	yes	characteristic
52	C₈×C₇₁	C₄×Q₈	1	2	yes	—
53	C₇₁⋊C₈	—	4	8	no	—
54	C₂²×C₇₁	—	2	2	no	—
55	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
56	C₂²×C₇₁	C₄×Q₁₆	1	2	yes	—
57	C₄×C₇₁	—	2	4	no	—
58	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₇₁	#427	1	1	yes	characteristic
60	C₄×C₇₁	C₄×D₈	1	2	yes	—
61	#12676	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
64	C₂×C₇₁	#2161	1	2	yes	—
65	C₂×#368	—	71	2	no	—
66	C₂×#346	D₇₁	1	1	yes	characteristic
67	C₄×#108	—	71	2	no	—
68	#1381	—	71	2	no	—
69	C₇₁	#12676	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
71	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
72	C₂×#108	—	71	4	no	—
73	C₄×M₁₆	—	142	2	no	—
74	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
75	#368	—	71	8	no	—
76	#346	Dic₇₁	1	4	yes	—
77	#341	—	142	2	no	—
78	C₂×M₁₆	—	142	4	no	—
79	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
81	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
84	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
85	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
86	#108	—	71	8	no	—
87	#108	—	142	4	no	—
88	#97	—	284	2	no	—
89	C₄×C₈	—	284	2	no	—
90	C₄×C₈	Dic₁₄₂	1	2	yes	—
91	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
92	C₂²×C₄	Dic₂₈₄	1	2	yes	—
93	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
94	M₁₆	—	284	4	no	—
95	M₁₆	—	142	8	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	284	4	no	—
98	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
99	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
100	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
101	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
102	C₂×C₈	#11765865	1	2	yes	—
103	C₄²	C₄⋊Dic₇₁	1	2	yes	—
104	C₄²	D₅₆₈	1	2	yes	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄×Dic₇₁	1	4	yes	—
107	C₂³	C₂×Dic₂₈₄	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#14556062	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
111	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
112	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
113	C₂×C₄	#14556150	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	284	8	no	—
116	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
117	C₈	#14556087	1	1	yes	characteristic
118	C₈	C₄×Dic₁₄₂	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂×#14556062	1	1	yes	characteristic
121	C₂²	C₄×Dic₂₈₄	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
124	C₄	#17986037	1	1	yes	characteristic
125	C₄	C₄×D₅₆₈	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₄×#14556062	1	1	yes	characteristic
128	C₂	#22590135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985829

C₂

yes

maximal

C₄×#14556064

C₂

yes

maximal

#22586434

C₂

yes

maximal

C₂×#346×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556064

C₂²

yes

—

C₄×#11765849

—

#17985829

C₄

yes

complemented

#17985764

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

C₄

yes

—

C₂×#11765849

—

#14556064

—

#14556064

C₂×C₄

yes

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

Q₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₄²

yes

—

C₂³×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#107

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×#107

yes

characteristic

C₂²×C₇₁

C₄×Q₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₁

#427

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

—

#12676

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₄×#107

yes

characteristic

C₂×C₇₁

#2161

yes

—

C₂×#368

—

C₂×#346

D₇₁

yes

characteristic

C₄×#108

—

#1381

—

C₇₁

#12676

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×#108

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#368

—

#346

Dic₇₁

yes

—

#341

—

142

—

C₂×M₁₆

—

142

—

C₂²×C₈

D₂₈₄

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

D₂₈₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

#108

—

#108

—

142

—

#97

—

284

—

C₄×C₈

—

284

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₂²×C₄

Dic₂₈₄

yes

—

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

M₁₆

—

284

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₂×C₈

C₂×C₄×D₇₁

yes

—

100

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

101

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

102

C₂×C₈

#11765865

yes

—

103

C₄²

C₄⋊Dic₇₁

yes

—

104

C₄²

D₅₆₈

yes

—

105

C₁₆

—

106

C₁₆

C₄×Dic₇₁

yes

—

107

C₂³

C₂×Dic₂₈₄

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

#14556062

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

111

C₂×C₄

C₄×D₂₈₄

yes

—

112

C₂×C₄

C₂×D₅₆₈

yes

characteristic

113

C₂×C₄

#14556150

yes

—

114

C₈

—

115

C₈

—

284

—

116

C₈

C₂×C₄×Dic₇₁

yes

characteristic

117

C₈

#14556087

yes

characteristic

118

C₈

C₄×Dic₁₄₂

yes

—

119

C₂²

—

120

C₂²

C₂×#14556062

yes

characteristic

121

C₂²

C₄×Dic₂₈₄

yes

—

122

C₄

—

123

C₄

C₄×C₄⋊Dic₇₁

yes

characteristic

124

C₄

#17986037

yes

characteristic

125

C₄

C₄×D₅₆₈

yes

—

126

C₂

—

127

C₂

C₄×#14556062

yes

characteristic

128

C₂

#22590135

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#346×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	#12676
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₅₆₈

commutator

C₈×C₇₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#346×C₇₁

C₂

socle

C₂²×C₇₁

C₂×#107

prime radical

C₂³×C₇₁

C₂×Q₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
7. #17985829	C₄	semidirect
69. C₇₁	#12676	semidirect

H ≅ G/N

type

7. #17985829

C₄

semidirect

69. C₇₁

#12676

semidirect

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

subgroup	index
C₂×#17985829	2
C₄×#14556064	2
#22586434	2
C₂×#346×C₇₁	2
#12676	71

subgroup

index

C₂×#346×C₇₁

Subgroups — #31912868

Conjugacy classes of subgroups

Complements

Maximal subgroups