Group #72070 — subgroups

Subgroups — #72070

Order 384.

256 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂²×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#836	C₂	1	2	yes	maximal
3	#5225	C₂	1	2	yes	maximal
4	#5102	C₂	1	1	yes	maximal, characteristic
5	#4552	C₂	1	1	yes	maximal, characteristic
6	#2047	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×Q₁₆	—	2	4	no	—
9	C₂×C₃×Q₁₆	—	2	2	no	—
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	#926	C₂²	1	1	yes	characteristic
12	C₂×#208	C₂²	1	2	yes	—
13	C₂×#208	C₂²	1	1	yes	characteristic
14	#836	—	2	4	no	—
15	#835	—	2	4	no	—
16	Dic₃⋊C₈	—	2	2	no	—
17	C₂²×Q₁₆	D₃	1	1	yes	characteristic
18	#424	—	3	1	no	—
19	#419	—	3	2	no	—
20	C₂×#101	—	3	2	no	—
21	#341	—	3	1	no	—
22	C₂×C₃×Q₈	—	2	2	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×C₃×Q₈	—	2	2	no	—
25	C₂×C₃×Q₈	—	2	2	no	—
26	C₂×C₃×Q₈	—	2	2	no	—
27	C₂×C₃×Q₈	D₄	1	4	yes	—
28	C₂²×C₃×C₄	—	2	2	no	—
29	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₃×Q₁₆	—	2	8	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₂×C₃×C₈	D₄	1	2	yes	—
33	#217	—	2	1	no	—
34	C₄⋊Dic₃	D₄	1	2	yes	—
35	C₄×Dic₃	—	2	1	no	—
36	#208	—	2	2	no	—
37	#208	—	2	2	no	—
38	#208	—	2	2	no	—
39	C₂×C₃⋊C₈	—	2	1	no	—
40	C₂×C₃⋊C₈	—	2	2	no	—
41	C₂²×Q₈	C₂×D₃	1	2	yes	—
42	C₂×Q₁₆	—	2	4	no	—
43	C₂×Q₁₆	—	2	2	no	—
44	C₂×M₁₆	—	3	2	no	—
45	C₂×M₁₆	—	3	1	no	—
46	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
47	#117	—	3	1	no	—
48	C₄⋊C₈	—	6	2	no	—
49	#103	—	6	4	no	—
50	#101	—	6	4	no	—
51	C₂³×C₃	#41	1	1	yes	characteristic
52	C₃×Q₈	—	2	4	no	—
53	C₃×Q₈	—	2	4	no	—
54	C₃×Q₈	—	4	2	no	—
55	C₃×Q₈	—	4	2	no	—
56	C₂×C₃×C₄	—	2	4	no	—
57	C₂×C₃×C₄	—	4	2	no	—
58	C₂×C₃×C₄	—	4	2	no	—
59	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄	#41	1	2	yes	—
63	C₂×Dic₃	—	2	2	no	—
64	C₃×C₈	—	2	2	no	—
65	C₃⋊C₈	—	4	1	no	—
66	C₃⋊C₈	—	4	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×Q₈	#67	1	4	yes	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
75	Q₁₆	—	2	8	no	—
76	M₁₆	—	6	2	no	—
77	M₁₆	—	6	2	no	—
78	M₁₆	—	6	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	6	2	no	—
81	C₂×C₈	—	6	1	no	—
82	C₂×C₈	#67	1	2	yes	—
83	C₄⋊C₄	—	3	2	no	—
84	C₂²⋊C₄	—	6	1	no	—
85	C₄²	—	6	1	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	#121	1	1	yes	characteristic
88	C₂²×C₃	#123	1	2	yes	—
89	C₃×C₄	—	4	4	no	—
90	C₃×C₄	#120	1	1	yes	characteristic
91	C₃×C₄	#121	1	2	yes	—
92	C₃×C₄	#124	1	1	yes	characteristic
93	Dic₃	—	4	2	no	—
94	C₂³	#237	1	1	yes	characteristic
95	Q₈	—	2	4	no	—
96	Q₈	—	2	4	no	—
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	6	2	no	—
103	C₂×C₄	D₃×D₄	1	1	yes	characteristic
104	C₂×C₄	#237	1	2	yes	—
105	C₂×C₄	C₂×#67	1	1	yes	characteristic
106	C₂×C₄	C₂×#67	1	2	yes	—
107	C₈	—	2	2	no	—
108	C₈	—	12	2	no	—
109	C₈	—	12	1	no	—
110	C₂×C₃	—	2	2	no	—
111	C₂×C₃	#394	1	1	yes	characteristic
112	C₂×C₃	#473	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	#935	1	2	yes	—
115	C₂²	#938	1	1	yes	characteristic
116	C₄	—	4	4	no	—
117	C₄	—	12	2	no	—
118	C₄	#936	1	1	yes	characteristic
119	C₄	#939	1	2	yes	—
120	C₄	#953	1	1	yes	characteristic
121	C₃	#2047	1	1	yes	characteristic, complemented, Sylow
122	C₂	—	2	2	no	—
123	C₂	#5192	1	2	yes	—
124	C₂	#5222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×Q₁₆

C₂

yes

maximal, characteristic

C₂×#836

C₂

yes

maximal

#5225

C₂

yes

maximal

#5102

C₂

yes

maximal, characteristic

#4552

C₂

yes

maximal, characteristic

#2047

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₂×C₃×Q₁₆

—

C₂×C₃×Q₁₆

—

C₂²×C₃×C₈

C₂²

yes

characteristic

#926

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₂×#208

C₂²

yes

characteristic

#836

—

#835

—

Dic₃⋊C₈

—

C₂²×Q₁₆

D₃

yes

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

#217

—

C₄⋊Dic₃

D₄

yes

—

C₄×Dic₃

—

#208

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

C₂×D₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃

yes

characteristic

#117

—

C₄⋊C₈

—

#103

—

#101

—

C₂³×C₃

#41

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#67

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

#123

yes

—

C₃×C₄

—

C₃×C₄

#120

yes

characteristic

C₃×C₄

#121

yes

—

C₃×C₄

#124

yes

characteristic

Dic₃

—

C₂³

#237

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

D₃×D₄

yes

characteristic

104

C₂×C₄

#237

yes

—

105

C₂×C₄

C₂×#67

yes

characteristic

106

C₂×C₄

C₂×#67

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂×C₃

—

111

C₂×C₃

#394

yes

characteristic

112

C₂×C₃

#473

yes

—

113

C₂²

—

114

C₂²

#935

yes

—

115

C₂²

#938

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#936

yes

characteristic

119

C₄

#939

yes

—

120

C₄

#953

yes

characteristic

121

C₃

#2047

yes

characteristic, complemented, Sylow

122

C₂

—

123

C₂

#5192

yes

—

124

C₂

#5222

yes

characteristic

	subgroup	G/H
center	C₂²	#938
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×Q₁₆	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
hypercenter	C₂²×Q₁₆	D₃
hypocenter	C₃	#2047
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#938

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂²×C₃×Q₁₆

C₂

C₂²×C₃

C₂³×C₃

C₂²×Q₁₆

D₃

hypocenter

C₃

#2047

LCS

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
121. C₃	#2047	semidirect

H ≅ G/N

type

121. C₃

#2047

semidirect

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

subgroup	index
C₂²×C₃×Q₁₆	2
C₂×#836	2
#5225	2
#5102	2
#4552	2
#2047	3

subgroup

index

C₂²×C₃×Q₁₆

Subgroups — #72070

Conjugacy classes of subgroups

Complements

Maximal subgroups