Group #72225323 — subgroups

Subgroups — #72225323

Order 1024.

298 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₄×#1398	C₂	1	1	yes	maximal, characteristic
2	#129088	C₂	1	2	yes	maximal
3	C₂×#1398	—	2	2	no	—
4	C₄×M₆₄	C₄	1	1	yes	characteristic
5	C₄×M₆₄	C₄	1	1	yes	characteristic
6	C₄×#346	C₂²	1	1	yes	characteristic
7	#11999	C₄	1	4	yes	—
8	#8266	—	2	2	no	—
9	#7157	—	2	2	no	—
10	C₂×M₆₄	—	2	1	no	—
11	C₂×M₆₄	—	2	1	no	—
12	C₂×#346	C₂×C₄	1	2	yes	—
13	C₂×#346	D₄	1	2	yes	—
14	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆	Q₈	1	1	yes	characteristic
16	C₄²×C₈	D₄	1	1	yes	characteristic
17	#1564	—	2	4	no	—
18	#1450	—	4	2	no	—
19	#1398	—	2	8	no	—
20	#1397	—	2	1	no	—
21	#1397	C₈	1	2	yes	complemented
22	C₄×C₃₂	—	2	1	no	—
23	C₄×C₃₂	C₈	1	2	yes	complemented
24	#1368	—	2	4	no	—
25	#1368	C₈	1	8	yes	complemented
26	#1276	—	4	2	no	—
27	C₂²×C₁₆	C₄²	1	1	yes	characteristic
28	C₂²×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₆	C₄⋊C₄	1	2	yes	—
30	#431	—	4	4	no	—
31	C₂×C₄×C₈	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	SD₁₆	1	2	yes	—
34	C₂×C₄×C₈	Q₁₆	1	1	yes	characteristic
35	C₄³	D₈	1	1	yes	characteristic
36	M₆₄	—	2	4	no	—
37	M₆₄	—	2	4	no	—
38	C₂×C₃₂	—	2	2	no	—
39	C₂×C₃₂	—	2	2	no	—
40	#346	—	2	4	no	—
41	#346	C₂×C₈	1	4	yes	—
42	#346	M₁₆	1	4	yes	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
46	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
47	C₄×C₁₆	M₁₆	1	1	yes	characteristic
48	C₄×C₁₆	M₁₆	1	1	yes	characteristic
49	#335	—	4	4	no	—
50	#334	—	2	8	no	—
51	#322	—	8	2	no	—
52	C₈²	—	8	2	no	—
53	C₂²×C₈	#95	1	1	yes	characteristic
54	C₂²×C₈	#103	1	2	yes	—
55	C₂²×C₈	#106	1	1	yes	characteristic
56	C₂×C₄²	—	4	1	no	—
57	C₂×C₄²	#102	1	2	yes	—
58	C₂×C₄²	#107	1	1	yes	characteristic
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
63	C₂×C₁₆	C₄×C₈	1	2	yes	—
64	C₂×C₁₆	#97	1	1	yes	characteristic
65	C₂×C₁₆	#97	1	2	yes	—
66	C₂×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₁₆	C₄⋊C₈	1	4	yes	—
69	C₄⋊C₈	—	4	8	no	—
70	C₂²⋊C₈	—	8	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	4	1	no	—
74	C₄×C₈	—	8	4	no	—
75	C₄×C₈	C₂²⋊C₈	1	4	yes	—
76	C₄×C₈	#104	1	4	yes	—
77	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₈	#108	1	1	yes	characteristic
80	C₄×C₈	#108	1	1	yes	characteristic
81	C₃₂	—	2	4	no	—
82	C₃₂	—	2	4	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	#340	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	8	8	no	—
89	C₂×C₈	Q₈⋊C₈	1	4	yes	—
90	C₂×C₈	#334	1	1	yes	characteristic
91	C₂×C₈	#334	1	1	yes	characteristic
92	C₂×C₈	#336	1	1	yes	characteristic
93	C₂×C₈	#336	1	2	yes	—
94	C₂×C₈	#339	1	2	yes	—
95	C₂×C₈	#341	1	1	yes	characteristic
96	C₂×C₈	#365	1	2	yes	—
97	C₂×C₈	#365	1	2	yes	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	1	no	—
100	C₄²	—	8	2	no	—
101	C₄²	D₄⋊C₈	1	4	yes	—
102	C₄²	#335	1	1	yes	characteristic
103	C₄²	#335	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	2	4	no	—
107	C₁₆	C₈²	1	1	yes	characteristic
108	C₁₆	#322	1	1	yes	characteristic
109	C₁₆	#322	1	2	yes	—
110	C₂³	#1384	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#1277	1	1	yes	characteristic
117	C₂×C₄	#1279	1	2	yes	—
118	C₂×C₄	#1383	1	2	yes	—
119	C₂×C₄	#1385	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	16	8	no	—
122	C₈	#1273	1	1	yes	characteristic
123	C₈	#1276	1	1	yes	characteristic
124	C₈	#1278	1	2	yes	—
125	C₈	#1368	1	2	yes	—
126	C₈	#1368	1	1	yes	characteristic
127	C₈	#1368	1	1	yes	characteristic
128	C₂²	—	4	1	no	—
129	C₂²	#7157	1	1	yes	characteristic
130	C₂²	#7174	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	4	1	no	—
134	C₄	#7015	1	1	yes	characteristic
135	C₄	#7156	1	1	yes	characteristic
136	C₄	#7173	1	2	yes	—
137	C₂	—	2	1	no	—
138	C₂	—	2	2	no	—
139	C₂	#93269	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#1398

C₂

yes

maximal, characteristic

#129088

C₂

yes

maximal

C₂×#1398

—

C₄×M₆₄

C₄

yes

characteristic

C₄×M₆₄

C₄

yes

characteristic

C₄×#346

C₂²

yes

characteristic

#11999

C₄

yes

—

#8266

—

#7157

—

C₂×M₆₄

—

C₂×M₆₄

—

C₂×#346

C₂×C₄

yes

—

C₂×#346

D₄

yes

—

C₂×C₄×C₁₆

C₂×C₄

yes

characteristic

C₂×C₄×C₁₆

Q₈

yes

characteristic

C₄²×C₈

D₄

yes

—

—

—

—

C₈

yes

complemented

C₄×C₃₂

—

C₄×C₃₂

C₈

yes

complemented

#1368

—

#1368

C₈

yes

complemented

#1276

—

C₂²×C₁₆

C₄²

yes

characteristic

C₂²×C₁₆

C₂²⋊C₄

yes

characteristic

C₂²×C₁₆

C₄⋊C₄

yes

—

#431

—

C₂×C₄×C₈

C₂²⋊C₄

yes

—

C₂×C₄×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈

SD₁₆

yes

—

C₂×C₄×C₈

Q₁₆

yes

characteristic

C₄³

D₈

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

#346

C₂×C₈

yes

—

#346

M₁₆

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₈

yes

characteristic

C₄×C₁₆

C₂×C₈

yes

characteristic

C₄×C₁₆

M₁₆

yes

characteristic

C₄×C₁₆

M₁₆

yes

—

—

—

C₈²

—

C₂²×C₈

#95

yes

characteristic

C₂²×C₈

#103

yes

—

C₂²×C₈

#106

yes

characteristic

C₂×C₄²

—

C₂×C₄²

#102

yes

—

C₂×C₄²

#107

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×C₈

yes

characteristic

C₂×C₁₆

C₄×C₈

yes

—

C₂×C₁₆

#97

yes

characteristic

C₂×C₁₆

#97

yes

—

C₂×C₁₆

C₂²⋊C₈

yes

characteristic

C₂×C₁₆

C₂²⋊C₈

yes

characteristic

C₂×C₁₆

C₄⋊C₈

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₈

yes

—

C₄×C₈

#104

yes

—

C₄×C₈

C₄⋊C₈

yes

characteristic

C₄×C₈

C₄⋊C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₃₂

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#340

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₈⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#339

yes

—

C₂×C₈

#341

yes

characteristic

C₂×C₈

#365

yes

—

C₂×C₈

#365

yes

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

D₄⋊C₈

yes

—

102

C₄²

#335

yes

characteristic

103

C₄²

#335

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

C₈²

yes

characteristic

108

C₁₆

#322

yes

characteristic

109

C₁₆

#322

yes

—

110

C₂³

#1384

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#1277

yes

characteristic

117

C₂×C₄

#1279

yes

—

118

C₂×C₄

#1383

yes

—

119

C₂×C₄

#1385

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

#1273

yes

characteristic

123

C₈

#1276

yes

characteristic

124

C₈

#1278

yes

—

125

C₈

#1368

yes

—

126

C₈

#1368

yes

characteristic

127

C₈

#1368

yes

characteristic

128

C₂²

—

129

C₂²

#7157

yes

characteristic

130

C₂²

#7174

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#7015

yes

characteristic

135

C₄

#7156

yes

characteristic

136

C₄

#7173

yes

—

137

C₂

—

138

C₂

—

139

C₂

#93269

yes

characteristic

	subgroup	G/H
center	C₈	#1368
commutator	C₁₆	C₈²
Frattini, cosocle	C₄×#346	C₂²
socle	C₂	#93269
prime radical	C₂³	#1384
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#1368

commutator

C₁₆

C₈²

C₂²

C₂

C₂³

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#346 ◁ G

LECS

G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
21. #1397	C₈	semidirect
23. C₄×C₃₂	C₈	semidirect
25. #1368	C₈	semidirect

H ≅ G/N

type

21. #1397

C₈

semidirect

23. C₄×C₃₂

C₈

semidirect

25. #1368

C₈

semidirect

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

subgroup	index
C₄×#1398	2
#129088	2

subgroup

index

C₄×#1398

#129088

Subgroups — #72225323

Conjugacy classes of subgroups

Complements

Maximal subgroups