Group #70001 — subgroups

Subgroups — #70001

Order 384.

252 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 85 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	#5819	C₂	1	1	yes	maximal, characteristic
2	C₃×#408	C₂	1	1	yes	maximal, characteristic
3	#5237	C₂	1	1	yes	maximal, characteristic
4	#5124	C₂	1	2	yes	maximal, complemented
5	#4552	C₂	1	2	yes	maximal, complemented
6	#1858	—	3	1	no	maximal, Sylow
7	C₂×C₃×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	#948	—	2	4	no	—
11	C₂×#208	—	2	2	no	—
12	C₂×#208	C₂²	1	2	yes	—
13	C₂×#208	C₂²	1	2	yes	—
14	C₃×C₂²⋊C₈	—	2	2	no	—
15	#836	C₄	1	2	yes	—
16	#833	C₄	1	2	yes	—
17	#822	—	2	4	no	—
18	#568	—	3	1	no	—
19	#430	—	3	2	no	—
20	#413	—	3	1	no	—
21	#408	D₃	1	1	yes	characteristic
22	#341	—	3	2	no	—
23	C₃×#41	—	2	2	no	—
24	C₃×#41	—	2	2	no	—
25	C₂×C₃×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×D₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×D₄	Q₈	1	2	yes	—
28	C₂²×C₃×C₄	D₄	1	2	yes	—
29	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×C₈	—	2	1	no	—
32	C₂×C₃×C₈	—	2	1	no	—
33	C₂×C₃×C₈	—	2	2	no	—
34	#208	—	2	4	no	—
35	#208	—	2	2	no	—
36	#208	—	2	2	no	—
37	#208	C₂×C₄	1	4	yes	—
38	#208	D₄	1	4	yes	—
39	C₂×C₃⋊C₈	—	2	4	no	—
40	C₂×C₃⋊C₈	—	2	2	no	—
41	C₂×C₃⋊C₈	—	2	2	no	—
42	C₂×#41	C₂×D₃	1	1	yes	characteristic
43	#131	—	6	4	no	—
44	C₂×M₁₆	—	3	2	no	—
45	C₂×M₁₆	—	3	2	no	—
46	C₂×M₁₆	—	6	2	no	—
47	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
49	#108	—	6	4	no	—
50	#101	—	3	2	no	—
51	#100	—	3	2	no	—
52	C₂²⋊C₈	—	2	2	no	—
53	C₂³×C₃	C₄⋊C₄	1	2	yes	—
54	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
55	C₃×Q₈	—	2	2	no	—
56	C₃×D₄	—	2	4	no	—
57	C₃×D₄	—	2	2	no	—
58	C₂×C₃×C₄	—	2	2	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
61	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄	#41	1	2	yes	—
65	C₃×C₈	—	4	1	no	—
66	C₃×C₈	—	4	1	no	—
67	C₃⋊C₈	—	2	4	no	—
68	C₃⋊C₈	—	2	4	no	—
69	#41	—	2	2	no	—
70	#41	—	2	2	no	—
71	C₂×Q₈	C₄×D₃	1	1	yes	characteristic
72	C₂×D₄	Dic₆	1	2	yes	—
73	C₂×D₄	C₄×D₃	1	1	yes	characteristic
74	C₂²×C₄	#67	1	2	yes	—
75	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
76	M₁₆	—	2	2	no	—
77	M₁₆	—	3	4	no	—
78	M₁₆	—	3	4	no	—
79	M₁₆	—	6	4	no	—
80	M₁₆	—	6	2	no	—
81	M₁₆	—	6	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	6	4	no	—
86	C₂×C₈	—	6	2	no	—
87	C₂×C₈	—	6	2	no	—
88	C₂²×C₃	—	2	2	no	—
89	C₂²×C₃	—	4	1	no	—
90	C₂²×C₃	—	4	2	no	—
91	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
93	C₃×C₄	—	2	2	no	—
94	C₃×C₄	C₄×D₄	1	2	yes	—
95	C₃×C₄	#120	1	1	yes	characteristic
96	C₃×C₄	#123	1	1	yes	characteristic
97	C₂³	Dic₃⋊C₄	1	2	yes	—
98	C₂³	C₂×Dic₆	1	1	yes	characteristic
99	Q₈	—	2	2	no	—
100	D₄	—	2	4	no	—
101	D₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
105	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
106	C₂×C₄	D₃×D₄	1	2	yes	—
107	C₂×C₄	#237	1	2	yes	—
108	C₂×C₄	C₂×#67	1	1	yes	characteristic
109	C₈	—	4	1	no	—
110	C₈	—	4	1	no	—
111	C₈	—	6	4	no	—
112	C₈	—	6	4	no	—
113	C₂×C₃	—	2	1	no	—
114	C₂×C₃	—	2	2	no	—
115	C₂×C₃	—	4	2	no	—
116	C₂×C₃	#385	1	1	yes	characteristic
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	#878	1	2	yes	—
121	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	#67⋊C₄	1	2	yes	—
124	C₄	#935	1	1	yes	characteristic
125	C₄	#937	1	1	yes	characteristic
126	C₃	#1858	1	1	yes	characteristic, complemented, Sylow
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#4946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5819

C₂

yes

maximal, characteristic

C₃×#408

C₂

yes

maximal, characteristic

#5237

C₂

yes

maximal, characteristic

#5124

C₂

yes

maximal, complemented

#4552

C₂

yes

maximal, complemented

#1858

—

maximal, Sylow

C₂×C₃×#41

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

—

C₂²

yes

—

C₂×#208

C₂²

yes

—

C₃×C₂²⋊C₈

—

#836

C₄

yes

—

#833

C₄

yes

—

—

—

—

—

D₃

yes

characteristic

#341

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

C₂×C₄

yes

characteristic

C₂×C₃×D₄

C₂×C₄

yes

characteristic

C₂×C₃×D₄

Q₈

yes

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

—

—

—

C₂×C₄

yes

—

#208

D₄

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×#41

C₂×D₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂²×C₈

C₂×D₃

yes

—

—

—

C₂²⋊C₈

—

C₂³×C₃

C₄⋊C₄

yes

—

C₂³×C₃

C₂×Q₈

yes

characteristic

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₃

yes

characteristic

C₂×D₄

Dic₆

yes

—

C₂×D₄

C₄×D₃

yes

characteristic

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₃×C₄

—

C₃×C₄

C₄×D₄

yes

—

C₃×C₄

yes

C₃×C₄

yes

C₂³

Dic₃⋊C₄

yes

—

C₂³

C₂×Dic₆

yes

characteristic

Q₈

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

Dic₃⋊C₄

yes

—

105

C₂×C₄

C₂×C₄×D₃

yes

characteristic

106

C₂×C₄

D₃×D₄

yes

—

107

C₂×C₄

#237

yes

—

108

C₂×C₄

C₂×#67

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

—

116

C₂×C₃

#385

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#878

yes

—

121

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

122

C₄

—

123

C₄

#67⋊C₄

yes

—

124

C₄

#935

yes

characteristic

125

C₄

#937

yes

characteristic

126

C₃

#1858

yes

characteristic, complemented, Sylow

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#4946

yes

characteristic

	subgroup	G/H
center	C₄	#937
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#408	C₂
socle	C₂×C₃	#385
prime radical	C₂×C₃×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#408	D₃
hypocenter	C₃	#1858
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#937

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂×C₃

C₂×C₃×#41

C₂²

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
4. #5124	C₂	semidirect
5. #4552	C₂	semidirect
126. C₃	#1858	semidirect

H ≅ G/N

type

4. #5124

C₂

semidirect

5. #4552

C₂

semidirect

126. C₃

#1858

semidirect

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

subgroup	index
#5819	2
C₃×#408	2
#5237	2
#5124	2
#4552	2
#1858	3

subgroup

index

Subgroups — #70001

Conjugacy classes of subgroups

Complements

Maximal subgroups