Group #13471 — subgroups

Subgroups — #13471

Order 256.

153 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 55 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×#368	C₂	1	1	yes	maximal, characteristic
2	#2151	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	1	yes	maximal, characteristic
4	C₂×#358	C₂	1	1	yes	maximal, characteristic
5	#1979	C₂	1	1	yes	maximal, characteristic
6	#1952	C₂	1	1	yes	maximal, characteristic
7	#1387	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#108	C₂²	1	1	yes	characteristic
12	C₂×#108	C₂²	1	1	yes	characteristic
13	#428	—	2	1	no	—
14	C₂×#107	C₂²	1	1	yes	characteristic
15	#419	—	2	1	no	—
16	C₂×#101	—	2	1	no	—
17	#405	C₂²	1	1	yes	characteristic
18	#368	—	2	2	no	—
19	C₄⋊C₁₆	—	2	2	no	—
20	#362	—	2	2	no	—
21	#358	—	2	2	no	—
22	#343	—	2	1	no	—
23	C₂²×Q₈	—	2	1	no	—
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂×Q₁₆	—	2	1	no	—
29	C₂×Q₁₆	Q₈	1	2	yes	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	D₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	1	yes	characteristic
34	#117	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄	—	2	1	no	—
36	M₃₂	—	2	1	no	—
37	M₃₂	—	2	1	no	—
38	C₂×C₁₆	—	2	1	no	—
39	C₂×C₁₆	—	2	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	D₄	1	2	yes	—
42	#108	—	2	1	no	—
43	#108	—	2	1	no	—
44	#108	—	2	1	no	—
45	#108	D₄	1	2	yes	—
46	#107	—	2	1	no	—
47	#107	—	2	1	no	—
48	#107	—	4	1	no	—
49	#106	—	4	1	no	—
50	#103	—	4	1	no	—
51	#103	—	4	1	no	—
52	#101	—	4	2	no	—
53	#97	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	4	1	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	1	no	—
61	C₂²×C₄	—	4	1	no	—
62	C₂²×C₄	—	4	1	no	—
63	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
64	Q₁₆	—	2	2	no	—
65	Q₁₆	—	2	2	no	—
66	Q₁₆	—	4	1	no	—
67	Q₁₆	—	4	1	no	—
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	1	no	—
70	M₁₆	—	2	1	no	—
71	M₁₆	—	4	1	no	—
72	M₁₆	—	4	1	no	—
73	M₁₆	—	4	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈	#41	1	1	yes	characteristic
81	C₂×C₈	#41	1	1	yes	characteristic
82	C₂×C₈	#41	1	1	yes	characteristic
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄⋊C₄	D₈	1	1	yes	characteristic
86	C₄⋊C₄	D₈	1	1	yes	characteristic
87	C₂²⋊C₄	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	4	1	no	—
91	C₂³	#135	1	1	yes	characteristic
92	Q₈	—	4	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	8	1	no	—
95	Q₈	—	8	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	—	8	1	no	—
101	C₂×C₄	—	8	1	no	—
102	C₂×C₄	—	16	1	no	—
103	C₂×C₄	#121	1	1	yes	characteristic
104	C₂×C₄	#121	1	1	yes	characteristic
105	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄	#137	1	1	yes	characteristic
109	C₈	—	4	2	no	—
110	C₈	—	4	1	no	—
111	C₈	—	8	1	no	—
112	C₈	#121	1	1	yes	characteristic
113	C₈	C₂²⋊Q₈	1	1	yes	characteristic
114	C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈	#126	1	1	yes	characteristic
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	#464	1	1	yes	characteristic
119	C₂²	#466	1	1	yes	characteristic
120	C₂²	#478	1	1	yes	characteristic
121	C₄	—	4	1	no	—
122	C₄	—	4	1	no	—
123	C₄	—	8	2	no	—
124	C₄	—	16	1	no	—
125	C₄	#396	1	1	yes	characteristic
126	C₄	#459	1	1	yes	characteristic
127	C₄	#468	1	1	yes	characteristic
128	C₄	#474	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#2067	1	1	yes	characteristic
132	C₂	#2222	1	1	yes	characteristic
133	C₂	#2223	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#368

C₂

yes

maximal, characteristic

#2151

C₂

yes

maximal, characteristic

C₂×#362

C₂

yes

maximal, characteristic

C₂×#358

C₂

yes

maximal, characteristic

#1979

C₂

yes

maximal, characteristic

#1952

C₂

yes

maximal, characteristic

#1387

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

C₂×#108

C₂²

yes

characteristic

#428

—

C₂×#107

C₂²

yes

—

—

C₂²

yes

characteristic

#368

—

C₄⋊C₁₆

—

#362

—

#358

—

#343

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Q₈

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

#117

D₄

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#135

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#121

yes

characteristic

104

C₂×C₄

#121

yes

characteristic

105

C₂×C₄

C₂²⋊Q₈

yes

characteristic

106

C₂×C₄

C₂×D₈

yes

characteristic

107

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

108

C₂×C₄

#137

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

#121

yes

characteristic

113

C₈

C₂²⋊Q₈

yes

characteristic

114

C₈

C₂²⋊Q₈

yes

characteristic

115

C₈

#126

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

#464

yes

characteristic

119

C₂²

#466

yes

characteristic

120

C₂²

#478

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#396

yes

characteristic

126

C₄

#459

yes

characteristic

127

C₄

#468

yes

characteristic

128

C₄

#474

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

#2067

yes

characteristic

132

C₂

#2222

yes

characteristic

133

C₂

#2223

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³

prime radical

C₂³

#135

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#368	2
#2151	2
C₂×#362	2
C₂×#358	2
#1979	2
#1952	2
#1387	2

subgroup

index

Subgroups — #13471

Conjugacy classes of subgroups

Maximal subgroups