Group #72225086 — subgroups

Subgroups — #72225086

Order 4096.

1137 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1137 conjugacy classes of subgroups — too many to draw. Only its 651 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	#72225076	C₂	1	3	yes	maximal
2	#72225071	C₄	1	12	yes	complemented
3	#72225067	C₂	1	1	yes	maximal, characteristic
4	#72225038	C₄	1	3	yes	complemented
5	#72216231	C₂	1	3	yes	maximal
6	#72216226	C₄	1	6	yes	—
7	C₈×#1398	C₄	1	6	yes	—
8	#72216220	C₂²	1	3	yes	—
9	C₁₆×M₆₄	C₂²	1	3	yes	—
10	#72216182	C₂²	1	1	yes	characteristic
11	C₃₂²	C₄	1	1	yes	characteristic, complemented
12	C₄×#1398	C₈	1	12	yes	—
13	#149495	C₈	1	12	yes	—
14	C₈×M₆₄	C₂×C₄	1	6	yes	—
15	C₈×M₆₄	C₂×C₄	1	6	yes	—
16	C₈×#346	C₂×C₄	1	3	yes	—
17	C₂×#7052	C₂×C₄	1	3	yes	—
18	C₂×C₁₆²	C₂³	1	1	yes	characteristic
19	#93666	C₈	1	24	yes	—
20	#93665	C₂×C₄	1	12	yes	—
21	#93664	C₂×C₄	1	6	yes	—
22	C₁₆×C₃₂	C₂×C₄	1	3	yes	—
23	C₁₆×C₃₂	C₂×C₄	1	3	yes	—
24	C₂×#1398	—	2	24	no	—
25	C₄×M₆₄	C₂×C₈	1	12	yes	—
26	C₄×M₆₄	C₂×C₈	1	12	yes	—
27	C₄×#346	C₂×C₈	1	6	yes	—
28	C₂×#1311	C₂×C₈	1	6	yes	—
29	C₂×C₈×C₁₆	C₄²	1	3	yes	—
30	C₂×C₈×C₁₆	C₂²×C₄	1	3	yes	—
31	C₄×C₈²	C₄²	1	1	yes	characteristic
32	#7330	—	2	24	no	—
33	#7329	C₂×C₈	1	12	yes	—
34	#7329	C₂×C₈	1	12	yes	—
35	C₈×C₃₂	C₄²	1	6	yes	—
36	C₈×C₃₂	C₄²	1	6	yes	—
37	C₈×C₃₂	C₄²	1	12	yes	—
38	#7052	C₂×C₈	1	12	yes	—
39	C₁₆²	C₂²×C₄	1	1	yes	characteristic
40	C₁₆²	C₂²×C₄	1	3	yes	—
41	C₂×M₆₄	—	2	12	no	—
42	C₂×M₆₄	M₃₂	1	24	yes	—
43	C₂×#346	—	2	12	no	—
44	C₂×C₄×C₁₆	C₄×C₈	1	6	yes	—
45	C₂×C₄×C₁₆	C₂²×C₈	1	6	yes	—
46	C₄²×C₈	C₄×C₈	1	3	yes	—
47	C₂×C₈²	C₄×C₈	1	3	yes	—
48	C₂×C₈²	C₂×C₄²	1	1	yes	characteristic
49	#1398	—	4	48	no	—
50	#1397	—	2	12	no	—
51	#1397	M₃₂	1	24	yes	—
52	C₄×C₃₂	C₄×C₈	1	24	yes	—
53	C₄×C₃₂	C₄×C₈	1	12	yes	—
54	C₄×C₃₂	C₄×C₈	1	12	yes	—
55	#1311	—	2	12	no	—
56	C₈×C₁₆	C₄×C₈	1	12	yes	—
57	C₈×C₁₆	C₂×C₄²	1	3	yes	—
58	C₈×C₁₆	C₂×C₄²	1	3	yes	—
59	C₈×C₁₆	C₂²×C₈	1	6	yes	—
60	C₂²×C₁₆	—	2	6	no	—
61	C₂²×C₁₆	C₂×M₃₂	1	12	yes	—
62	C₂×C₄×C₈	—	2	6	no	—
63	C₂×C₄×C₈	C₈²	1	3	yes	—
64	C₂×C₄×C₈	C₂×C₄×C₈	1	3	yes	—
65	C₄³	C₈²	1	1	yes	characteristic
66	M₆₄	—	2	48	no	—
67	M₆₄	—	4	24	no	—
68	C₂×C₃₂	—	2	24	no	—
69	C₂×C₃₂	#346	1	24	yes	—
70	C₂×C₃₂	#346	1	24	yes	—
71	#346	—	4	24	no	—
72	C₄×C₁₆	—	2	6	no	—
73	C₄×C₁₆	C₈²	1	12	yes	—
74	C₄×C₁₆	C₂×C₄×C₈	1	6	yes	—
75	C₄×C₁₆	C₂×C₄×C₈	1	6	yes	—
76	C₄×C₁₆	C₂×M₃₂	1	12	yes	—
77	C₈²	—	2	6	no	—
78	C₈²	C₄³	1	1	yes	characteristic
79	C₈²	C₂×C₄×C₈	1	3	yes	—
80	C₂²×C₈	—	2	3	no	—
81	C₂²×C₈	C₄×M₃₂	1	6	yes	—
82	C₂×C₄²	—	2	3	no	—
83	C₂×C₄²	C₂×C₈²	1	1	yes	characteristic
84	C₂×C₁₆	—	2	24	no	—
85	C₂×C₁₆	—	2	12	no	—
86	C₂×C₁₆	—	4	12	no	—
87	C₂×C₁₆	C₂×#346	1	12	yes	—
88	C₂×C₁₆	C₄×M₃₂	1	12	yes	—
89	C₄×C₈	—	2	6	no	—
90	C₄×C₈	—	4	12	no	—
91	C₄×C₈	C₂×C₈²	1	3	yes	—
92	C₄×C₈	C₄²×C₈	1	3	yes	—
93	C₄×C₈	C₄×M₃₂	1	6	yes	—
94	C₃₂	—	2	24	no	—
95	C₃₂	—	4	24	no	—
96	C₃₂	#1398	1	48	yes	complemented
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	C₈×M₃₂	1	3	yes	—
99	C₂×C₈	—	2	12	no	—
100	C₂×C₈	—	4	6	no	—
101	C₂×C₈	—	4	3	no	—
102	C₂×C₈	C₈×M₃₂	1	6	yes	—
103	C₂×C₈	C₄×#346	1	6	yes	—
104	C₄²	—	4	6	no	—
105	C₄²	C₄×C₈²	1	1	yes	characteristic
106	C₄²	C₈×M₃₂	1	3	yes	—
107	C₁₆	—	2	12	no	—
108	C₁₆	—	4	12	no	—
109	C₁₆	C₂×#1398	1	24	yes	—
110	C₂³	#122211	1	1	yes	characteristic
111	C₂×C₄	—	2	6	no	—
112	C₂×C₄	—	8	3	no	—
113	C₂×C₄	#122211	1	3	yes	—
114	C₂×C₄	C₈×#346	1	3	yes	—
115	C₈	—	2	6	no	—
116	C₈	—	8	3	no	—
117	C₈	C₄×#1398	1	12	yes	—
118	C₂²	—	2	3	no	—
119	C₂²	#72216182	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	16	1	no	—
122	C₄	C₈×#1398	1	6	yes	—
123	C₂	—	4	1	no	—
124	C₂	#72216231	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#72225076

C₂

yes

maximal

#72225071

C₄

yes

complemented

#72225067

C₂

yes

maximal, characteristic

#72225038

C₄

yes

complemented

#72216231

C₂

yes

maximal

#72216226

C₄

yes

—

C₈×#1398

C₄

yes

—

#72216220

C₂²

yes

—

C₁₆×M₆₄

C₂²

yes

—

#72216182

C₂²

yes

characteristic

C₃₂²

C₄

yes

characteristic, complemented

C₄×#1398

C₈

yes

—

#149495

C₈

yes

—

C₈×M₆₄

C₂×C₄

yes

—

C₈×M₆₄

C₂×C₄

yes

—

C₈×#346

C₂×C₄

yes

—

C₂×#7052

C₂×C₄

yes

—

C₂×C₁₆²

C₂³

yes

characteristic

#93666

C₈

yes

—

#93665

C₂×C₄

yes

—

#93664

C₂×C₄

yes

—

C₁₆×C₃₂

C₂×C₄

yes

—

C₁₆×C₃₂

C₂×C₄

yes

—

C₂×#1398

—

C₄×M₆₄

C₂×C₈

yes

—

C₄×M₆₄

C₂×C₈

yes

—

C₄×#346

C₂×C₈

yes

—

C₂×#1311

C₂×C₈

yes

—

C₂×C₈×C₁₆

C₄²

yes

—

C₂×C₈×C₁₆

C₂²×C₄

yes

—

C₄×C₈²

C₄²

yes

characteristic

#7330

—

#7329

C₂×C₈

yes

—

#7329

C₂×C₈

yes

—

C₈×C₃₂

C₄²

yes

—

C₈×C₃₂

C₄²

yes

—

C₈×C₃₂

C₄²

yes

—

#7052

C₂×C₈

yes

—

C₁₆²

C₂²×C₄

yes

characteristic

C₁₆²

C₂²×C₄

yes

—

C₂×M₆₄

—

C₂×M₆₄

M₃₂

yes

—

C₂×#346

—

C₂×C₄×C₁₆

C₄×C₈

yes

—

C₂×C₄×C₁₆

C₂²×C₈

yes

—

C₄²×C₈

C₄×C₈

yes

—

C₂×C₈²

C₄×C₈

yes

—

C₂×C₈²

C₂×C₄²

yes

—

—

M₃₂

yes

—

C₄×C₃₂

C₄×C₈

yes

—

C₄×C₃₂

C₄×C₈

yes

—

C₄×C₃₂

C₄×C₈

yes

—

#1311

—

C₈×C₁₆

C₄×C₈

yes

—

C₈×C₁₆

C₂×C₄²

yes

—

C₈×C₁₆

C₂×C₄²

yes

—

C₈×C₁₆

C₂²×C₈

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×M₃₂

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₈²

yes

—

C₂×C₄×C₈

yes

—

C₄³

C₈²

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

#346

yes

—

C₂×C₃₂

#346

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₈²

yes

—

C₄×C₁₆

C₂×C₄×C₈

yes

—

C₄×C₁₆

C₂×C₄×C₈

yes

—

C₄×C₁₆

C₂×M₃₂

yes

—

C₈²

—

C₈²

C₄³

yes

characteristic

C₈²

C₂×C₄×C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₄×M₃₂

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₈²

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#346

yes

—

C₂×C₁₆

C₄×M₃₂

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈²

yes

—

C₄×C₈

C₄²×C₈

yes

—

C₄×C₈

C₄×M₃₂

yes

—

C₃₂

—

C₃₂

—

C₃₂

#1398

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₈×M₃₂

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₈×M₃₂

yes

—

103

C₂×C₈

C₄×#346

yes

—

104

C₄²

—

105

C₄²

C₄×C₈²

yes

characteristic

106

C₄²

C₈×M₃₂

yes

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

C₂×#1398

yes

—

110

C₂³

#122211

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#122211

yes

—

114

C₂×C₄

C₈×#346

yes

—

115

C₈

—

116

C₈

—

117

C₈

C₄×#1398

yes

—

118

C₂²

—

119

C₂²

#72216182

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

C₈×#1398

yes

—

123

C₂

—

124

C₂

#72216231

yes

—

	subgroup	G/H
center	C₈²	C₄³
commutator	C₄²	C₄×C₈²
Frattini, cosocle	C₂×C₁₆²	C₂³
socle	C₂²	#72216182
prime radical	C₂³	#122211
LCS	G ▷ C₄² ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₈²

C₄³

commutator

C₄²

C₄×C₈²

Frattini, cosocle

C₂×C₁₆²

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₁

C₁ ◁ C₈² ◁ G

G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #72225071	C₄	semidirect
4. #72225038	C₄	semidirect
11. C₃₂²	C₄	semidirect
96. C₃₂	#1398	semidirect

H ≅ G/N

type

2. #72225071

C₄

semidirect

4. #72225038

C₄

semidirect

11. C₃₂²

C₄

semidirect

96. C₃₂

#1398

semidirect

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

subgroup	index
#72225076	2
#72225067	2
#72216231	2

subgroup

index

#72225076

#72225067

#72216231

Subgroups — #72225086

Conjugacy classes of subgroups

Complements

Maximal subgroups