Group #72231932 — subgroups

Subgroups — #72231932

Order 1024.

198 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 105 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	#149495	C₂	1	1	yes	maximal, characteristic
2	#122922	C₂	1	2	yes	maximal
3	#93877	C₂	1	4	yes	maximal, complemented
4	C₄×M₆₄	C₄	1	1	yes	characteristic
5	C₄×M₆₄	C₄	1	1	yes	characteristic
6	#12013	—	2	2	no	—
7	C₂×#1311	C₂²	1	1	yes	characteristic
8	#7330	C₄	1	2	yes	—
9	#7330	C₂²	1	2	yes	—
10	#7099	—	2	2	no	—
11	#7099	C₂²	1	4	yes	—
12	C₂×M₆₄	—	2	1	no	—
13	C₂×M₆₄	C₈	1	2	yes	—
14	#2163	—	4	2	no	—
15	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₆	Q₈	1	1	yes	characteristic
17	C₂×C₈²	D₄	1	1	yes	characteristic
18	#1397	C₂×C₄	1	2	yes	—
19	#1397	C₂×C₄	1	2	yes	—
20	C₄×C₃₂	—	2	1	no	—
21	C₄×C₃₂	C₈	1	1	yes	characteristic
22	C₄×C₃₂	C₈	1	1	yes	characteristic
23	#1370	—	4	2	no	—
24	#1367	—	2	4	no	—
25	#1311	C₂×C₄	1	1	yes	characteristic
26	#1311	D₄	1	1	yes	characteristic
27	#1311	Q₈	1	1	yes	characteristic
28	#1311	C₂³	1	1	yes	characteristic
29	C₂²×C₁₆	—	2	1	no	—
30	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₁₆	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	SD₁₆	1	2	yes	—
34	M₆₄	—	2	4	no	—
35	M₆₄	—	2	4	no	—
36	C₂×C₃₂	—	2	2	no	—
37	C₂×C₃₂	C₂×C₈	1	2	yes	—
38	C₂×C₃₂	C₂×C₈	1	2	yes	—
39	C₄⋊C₁₆	—	4	4	no	—
40	C₂²⋊C₁₆	—	8	2	no	—
41	#346	—	8	2	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₁₆	C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
46	C₄×C₁₆	M₁₆	1	1	yes	characteristic
47	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
48	C₄×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₈²	C₄⋊C₄	1	1	yes	characteristic
50	C₈²	SD₁₆	1	2	yes	—
51	C₈²	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
54	C₂²×C₈	#108	1	1	yes	characteristic
55	C₂×C₄²	#106	1	1	yes	characteristic
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	4	2	no	—
61	C₂×C₁₆	—	8	4	no	—
62	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	C₄⋊C₈	1	1	yes	characteristic
68	C₄×C₈	#106	1	2	yes	—
69	C₄×C₈	#106	1	1	yes	characteristic
70	C₄×C₈	#108	1	1	yes	characteristic
71	C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₈	C₂×SD₁₆	1	2	yes	—
73	C₃₂	—	2	4	no	—
74	C₃₂	—	2	2	no	—
75	C₃₂	M₃₂	1	4	yes	—
76	C₂²×C₄	#334	1	1	yes	characteristic
77	C₂²×C₄	#365	1	2	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	#334	1	1	yes	characteristic
85	C₂×C₈	#334	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₈	C₂×#108	1	1	yes	characteristic
88	C₄²	#334	1	1	yes	characteristic
89	C₄²	#365	1	2	yes	—
90	C₄²	C₂×#106	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	4	1	no	—
94	C₁₆	—	4	1	no	—
95	C₁₆	—	16	4	no	—
96	C₁₆	#364	1	2	yes	—
97	C₁₆	C₂×M₃₂	1	2	yes	—
98	C₂³	#1368	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	#1368	1	2	yes	—
103	C₂×C₄	#1368	1	1	yes	characteristic
104	C₂×C₄	C₂×#334	1	1	yes	characteristic
105	C₂×C₄	C₂×#365	1	2	yes	—
106	C₈	—	2	1	no	—
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	4	2	no	—
110	C₈	#1369	1	2	yes	—
111	C₈	#2150	1	1	yes	characteristic
112	C₈	C₂×#364	1	1	yes	characteristic
113	C₂²	—	2	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	C₂×#1368	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	#11996	1	1	yes	characteristic
120	C₄	C₂×#1369	1	1	yes	characteristic
121	C₂	—	4	1	no	—
122	C₂	#122921	1	1	yes	characteristic
123	C₂	#151024	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#149495

C₂

yes

maximal, characteristic

#122922

C₂

yes

maximal

#93877

C₂

yes

maximal, complemented

C₄×M₆₄

C₄

yes

characteristic

C₄×M₆₄

C₄

yes

characteristic

#12013

—

C₂×#1311

C₂²

yes

characteristic

#7330

C₄

yes

—

#7330

C₂²

yes

—

#7099

—

#7099

C₂²

yes

—

C₂×M₆₄

—

C₂×M₆₄

C₈

yes

—

#2163

—

C₂×C₄×C₁₆

C₂×C₄

yes

characteristic

C₂×C₄×C₁₆

Q₈

yes

characteristic

C₂×C₈²

D₄

yes

characteristic

#1397

C₂×C₄

yes

—

#1397

C₂×C₄

yes

—

C₄×C₃₂

—

C₄×C₃₂

C₈

yes

characteristic

C₄×C₃₂

C₈

yes

—

—

C₂×C₄

yes

characteristic

#1311

D₄

yes

characteristic

#1311

Q₈

yes

characteristic

#1311

C₂³

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₈

yes

characteristic

C₂²×C₁₆

M₁₆

yes

characteristic

C₂×C₄×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈

SD₁₆

yes

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₈

yes

—

C₂×C₃₂

C₂×C₈

yes

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₄⋊C₄

yes

characteristic

C₄×C₁₆

C₄⋊C₄

yes

characteristic

C₄×C₁₆

C₂×C₈

yes

characteristic

C₄×C₁₆

M₁₆

yes

characteristic

C₄×C₁₆

C₂²×C₄

yes

characteristic

C₄×C₁₆

C₂×Q₈

yes

characteristic

C₈²

C₄⋊C₄

yes

characteristic

C₈²

SD₁₆

yes

—

C₈²

C₂×D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₄⋊C₈

yes

C₂²×C₈

yes

C₂×C₄²

yes

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₈

yes

characteristic

C₂×C₁₆

C₄⋊C₈

yes

characteristic

C₂×C₁₆

C₂²×C₈

yes

characteristic

C₂×C₁₆

C₂×M₁₆

yes

characteristic

C₄×C₈

—

C₄×C₈

C₄⋊C₈

yes

characteristic

C₄×C₈

#106

yes

—

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×SD₁₆

yes

—

C₃₂

—

C₃₂

—

C₃₂

M₃₂

yes

—

C₂²×C₄

#334

yes

characteristic

C₂²×C₄

#365

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×C₄⋊C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

yes

—

C₄²

C₂×#106

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#364

yes

—

C₁₆

C₂×M₃₂

yes

—

C₂³

#1368

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#1368

yes

—

103

C₂×C₄

#1368

yes

characteristic

104

C₂×C₄

C₂×#334

yes

characteristic

105

C₂×C₄

C₂×#365

yes

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#1369

yes

—

111

C₈

#2150

yes

characteristic

112

C₈

C₂×#364

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×#1368

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#11996

yes

characteristic

120

C₄

C₂×#1369

yes

characteristic

121

C₂

—

122

C₂

#122921

yes

characteristic

123

C₂

#151024

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#1368
commutator	C₂×C₁₆	C₂²×C₈
Frattini, cosocle	#1311	C₂³
prime radical	C₂³	#1368
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#1368

commutator

C₂×C₁₆

C₂²×C₈

C₂³

C₂³

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G

LECS

G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #93877	C₂	semidirect

H ≅ G/N

type

3. #93877

C₂

semidirect

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

subgroup	index
#149495	2
#122922	2
#93877	2

subgroup

index

#149495

#122922

#93877

Subgroups — #72231932

Conjugacy classes of subgroups

Complements

Maximal subgroups