Group #7683 — subgroups

Subgroups — #7683

Order 256.

216 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 77 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	#2859	C₂	1	1	yes	maximal, characteristic
2	#1767	C₂	1	1	yes	maximal, characteristic
3	#1744	C₂	1	1	yes	maximal, characteristic
4	#1271	C₂	1	4	yes	maximal
5	C₂×#125	—	2	1	no	—
6	C₂×C₄⋊C₈	—	2	1	no	—
7	C₂×#97	—	2	1	no	—
8	C₂×C₄×C₈	—	2	1	no	—
9	#389	—	2	1	no	—
10	#389	—	2	1	no	—
11	#389	C₄	1	2	yes	—
12	#389	C₂²	1	2	yes	—
13	#384	C₄	1	1	yes	characteristic
14	#384	C₄	1	1	yes	characteristic
15	#382	—	2	1	no	—
16	C₄³	C₂²	1	1	yes	characteristic
17	#336	—	2	1	no	—
18	#336	C₂²	1	2	yes	—
19	#336	C₂²	1	2	yes	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	—	2	2	no	—
22	#125	—	4	4	no	—
23	C₂×C₄⋊C₄	—	2	1	no	—
24	C₂×C₄⋊C₄	—	2	1	no	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	—	2	1	no	—
27	C₂×C₄⋊C₄	—	2	1	no	—
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₄²	D₄	1	1	yes	characteristic
35	C₂×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	C₄⋊C₈	—	2	4	no	—
39	#97	—	4	2	no	—
40	C₄×C₈	—	2	4	no	—
41	#95	—	2	2	no	—
42	#95	—	2	2	no	—
43	C₂²×C₄	—	2	2	no	—
44	C₂²×C₄	—	2	2	no	—
45	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	4	2	no	—
57	C₂×C₈	—	4	2	no	—
58	C₄⋊C₄	—	2	4	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	4	no	—
66	C₄²	—	2	2	no	—
67	C₄²	—	2	2	no	—
68	C₄²	—	2	2	no	—
69	C₄²	—	2	2	no	—
70	C₄²	—	2	2	no	—
71	C₄²	—	4	1	no	—
72	C₄²	SD₁₆	1	2	yes	—
73	C₄²	Q₁₆	1	2	yes	—
74	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	#101	1	2	yes	—
95	C₂×C₄	#103	1	2	yes	—
96	C₂×C₄	#103	1	2	yes	—
97	C₂×C₄	#104	1	2	yes	—
98	C₂×C₄	#104	1	2	yes	—
99	C₂×C₄	#131	1	2	yes	—
100	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
103	C₈	—	4	4	no	—
104	C₈	—	4	4	no	—
105	C₂²	#408	1	1	yes	characteristic
106	C₂²	#410	1	1	yes	characteristic
107	C₂²	C₂×#101	1	1	yes	characteristic
108	C₂²	C₂×#103	1	1	yes	characteristic
109	C₂²	#418	1	1	yes	characteristic
110	C₂²	C₂×#104	1	1	yes	characteristic
111	C₂²	C₂×#104	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	—	2	2	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	2	no	—
118	C₄	—	4	1	no	—
119	C₄	—	8	2	no	—
120	C₄	—	8	2	no	—
121	C₄	#330	1	4	yes	—
122	C₂	#1470	1	1	yes	characteristic
123	C₂	#1484	1	1	yes	characteristic
124	C₂	#1493	1	1	yes	characteristic
125	C₂	#1503	1	1	yes	characteristic
126	C₂	#1505	1	1	yes	characteristic
127	C₂	C₂×#330	1	1	yes	characteristic
128	C₂	#1549	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2859

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

#1744

C₂

yes

maximal, characteristic

#1271

C₂

yes

maximal

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#389

—

#389

—

#389

C₄

yes

—

#389

C₂²

yes

—

#384

C₄

yes

characteristic

#384

C₄

yes

characteristic

#382

—

C₄³

C₂²

yes

characteristic

#336

—

#336

C₂²

yes

—

#336

C₂²

yes

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#101

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

#103

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

#104

yes

—

C₂×C₄

#131

yes

—

100

C₂×C₄

C₂×SD₁₆

yes

characteristic

101

C₂×C₄

C₂×Q₁₆

yes

characteristic

102

C₂×C₄

AGL(1,ℤ/8)

yes

—

103

C₈

—

104

C₈

—

105

C₂²

#408

yes

characteristic

106

C₂²

#410

yes

characteristic

107

C₂²

C₂×#101

yes

characteristic

108

C₂²

C₂×#103

yes

characteristic

109

C₂²

#418

yes

characteristic

110

C₂²

C₂×#104

yes

characteristic

111

C₂²

C₂×#104

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#330

yes

—

122

C₂

#1470

yes

characteristic

123

C₂

#1484

yes

characteristic

124

C₂

#1493

yes

characteristic

125

C₂

#1503

yes

characteristic

126

C₂

#1505

yes

characteristic

127

C₂

C₂×#330

yes

characteristic

128

C₂

#1549

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×C₂²⋊C₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

subgroup	index
#2859	2
#1767	2
#1744	2
#1271	2

subgroup

index

Subgroups — #7683

Conjugacy classes of subgroups

Maximal subgroups