Group #10613 — subgroups

Subgroups — #10613

Order 256.

138 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 49 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	#1574	C₂	1	1	yes	maximal, characteristic
2	#1574	C₂	1	1	yes	maximal, characteristic
3	#1574	C₂	1	1	yes	maximal, characteristic
4	#1549	C₂	1	1	yes	maximal, characteristic
5	#1548	C₂	1	1	yes	maximal, characteristic
6	#1546	C₂	1	1	yes	maximal, characteristic
7	#1309	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	#431	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	#423	C₂²	1	1	yes	characteristic
14	#423	C₂²	1	1	yes	characteristic
15	#335	—	2	1	no	—
16	#335	—	2	1	no	—
17	#335	—	2	1	no	—
18	#334	—	2	1	no	—
19	#334	—	2	1	no	—
20	#334	—	2	1	no	—
21	#333	—	2	1	no	—
22	#333	—	2	1	no	—
23	#332	—	2	1	no	—
24	#332	—	2	1	no	—
25	#332	—	2	1	no	—
26	#332	—	2	1	no	—
27	C₂²×Q₈	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₄⋊Q₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	D₄	1	1	yes	characteristic
45	C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₄⋊C₈	D₄	1	1	yes	characteristic
47	C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₂²⋊C₈	—	2	1	no	—
51	C₂²⋊C₈	—	2	1	no	—
52	C₂²⋊C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	8	1	no	—
59	C₂×Q₈	—	8	1	no	—
60	C₂²×C₄	—	4	1	no	—
61	C₂²×C₄	#41	1	1	yes	characteristic
62	C₂²×C₄	#41	1	1	yes	characteristic
63	C₂²×C₄	#41	1	1	yes	characteristic
64	M₁₆	—	8	1	no	—
65	M₁₆	—	8	1	no	—
66	M₁₆	—	8	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	4	1	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	C₂×D₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	C₂×Q₈	1	1	yes	characteristic
84	C₂³	#126	1	1	yes	characteristic
85	Q₈	—	8	2	no	—
86	Q₈	—	8	1	no	—
87	Q₈	—	8	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	#121	1	1	yes	characteristic
97	C₂×C₄	#121	1	1	yes	characteristic
98	C₂×C₄	#121	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₈	—	4	1	no	—
103	C₈	—	4	1	no	—
104	C₈	—	4	1	no	—
105	C₈	—	4	1	no	—
106	C₈	—	4	1	no	—
107	C₈	—	4	1	no	—
108	C₈	—	8	1	no	—
109	C₈	—	8	1	no	—
110	C₈	—	8	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	#396	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	8	2	no	—
119	C₄	#471	1	1	yes	characteristic
120	C₄	#471	1	1	yes	characteristic
121	C₄	#471	1	1	yes	characteristic
122	C₄	#472	1	1	yes	characteristic
123	C₄	#473	1	1	yes	characteristic
124	C₄	#473	1	1	yes	characteristic
125	C₂	—	4	1	no	—
126	C₂	#2063	1	1	yes	characteristic
127	C₂	#2064	1	1	yes	characteristic
128	C₂	#2064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1574

C₂

yes

maximal, characteristic

#1574

C₂

yes

maximal, characteristic

#1574

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal, characteristic

#1546

C₂

yes

maximal, characteristic

#1309

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₂³

#126

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Q₈

yes

characteristic

100

C₂×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄

C₂²⋊Q₈

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#396

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#471

yes

characteristic

120

C₄

#471

yes

characteristic

121

C₄

#471

yes

characteristic

122

C₄

#472

yes

characteristic

123

C₄

#473

yes

characteristic

124

C₄

#473

yes

characteristic

125

C₂

—

126

C₂

#2063

yes

characteristic

127

C₂

#2064

yes

characteristic

128

C₂

#2064

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#396

commutator, Frattini, cosocle

C₂×C₄²

C₂³

prime radical

C₂³

#126

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#1574	2
#1549	2
#1548	2
#1546	2
#1309	2

subgroup

index

Subgroups — #10613

Conjugacy classes of subgroups

Maximal subgroups