Group #70698 — subgroups

Subgroups — #70698

Order 384.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×C₃×#108	C₂	1	1	yes	maximal, characteristic
2	#5106	C₂	1	1	yes	maximal, characteristic
3	#5101	C₂	1	1	yes	maximal, characteristic
4	#4559	C₂	1	2	yes	maximal
5	#4557	C₂	1	2	yes	maximal
6	#1833	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	#926	C₂²	1	2	yes	—
10	#926	C₂²	1	1	yes	characteristic
11	C₂×#208	C₂²	1	1	yes	characteristic
12	C₃×#108	—	2	1	no	—
13	C₃×#108	—	2	1	no	—
14	#818	—	2	1	no	—
15	#817	—	2	1	no	—
16	#815	C₄	1	2	yes	complemented
17	C₈×Dic₃	C₄	1	2	yes	complemented
18	C₂×#108	D₃	1	1	yes	characteristic
19	#427	—	3	1	no	—
20	#405	—	3	1	no	—
21	#344	—	3	2	no	—
22	H(3,ℤ/4)	—	3	2	no	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₃×M₁₆	—	2	2	no	—
25	C₃×M₁₆	—	2	2	no	—
26	C₂×C₃×C₈	—	2	2	no	—
27	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
33	#217	—	2	1	no	—
34	#217	—	2	2	no	—
35	C₄⋊Dic₃	—	2	1	no	—
36	C₄⋊Dic₃	—	2	2	no	—
37	C₄×Dic₃	—	2	2	no	—
38	C₄×Dic₃	C₂×C₄	1	2	yes	—
39	#208	—	2	1	no	—
40	#208	—	2	1	no	—
41	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×M₁₆	—	3	1	no	—
43	C₂×M₁₆	C₂×D₃	1	2	yes	—
44	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
45	#117	—	3	1	no	—
46	#117	—	3	2	no	—
47	#108	—	2	1	no	—
48	#108	—	2	1	no	—
49	#107	—	6	1	no	—
50	#106	—	6	1	no	—
51	#97	—	3	2	no	—
52	C₄×C₈	—	3	2	no	—
53	C₂³×C₃	#41	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃×C₄	#41	1	1	yes	characteristic
58	C₂×C₃×C₄	#41	1	1	yes	characteristic
59	C₂×C₃×C₄	#41	1	1	yes	characteristic
60	C₂×Dic₃	—	2	2	no	—
61	C₂×Dic₃	—	4	2	no	—
62	C₃×C₈	—	4	2	no	—
63	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₃⋊C₈	—	2	2	no	—
68	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	6	1	no	—
72	M₁₆	—	6	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	3	2	no	—
75	C₂×C₈	Dic₆	1	1	yes	characteristic
76	C₂×C₈	Dic₆	1	1	yes	characteristic
77	C₂×C₈	C₄×D₃	1	1	yes	characteristic
78	C₂×C₈	C₄×D₃	1	1	yes	characteristic
79	C₂×C₈	#67	1	1	yes	characteristic
80	C₂×C₈	#67	1	1	yes	characteristic
81	C₄⋊C₄	—	6	1	no	—
82	C₄⋊C₄	—	6	2	no	—
83	C₂²⋊C₄	—	6	1	no	—
84	C₂²⋊C₄	—	6	2	no	—
85	C₄²	—	3	2	no	—
86	C₄²	—	6	2	no	—
87	C₂²×C₃	—	4	1	no	—
88	C₂²×C₃	#117	1	1	yes	characteristic
89	C₂²×C₃	#124	1	1	yes	characteristic
90	C₂²×C₃	#125	1	1	yes	characteristic
91	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₃×C₄	#117	1	1	yes	characteristic
93	C₃×C₄	#124	1	1	yes	characteristic
94	C₃×C₄	#125	1	1	yes	characteristic
95	Dic₃	—	4	2	no	—
96	Dic₃	—	8	2	no	—
97	C₂³	#239	1	1	yes	characteristic
98	C₂×C₄	—	6	2	no	—
99	C₂×C₄	—	12	2	no	—
100	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
102	C₂×C₄	#237	1	1	yes	characteristic
103	C₂×C₄	#237	1	1	yes	characteristic
104	C₂×C₄	#239	1	1	yes	characteristic
105	C₂×C₄	C₂×#67	1	1	yes	characteristic
106	C₈	—	4	2	no	—
107	C₈	—	6	2	no	—
108	C₈	Dic₃⋊C₄	1	1	yes	characteristic
109	C₈	Dic₃⋊C₄	1	1	yes	characteristic
110	C₈	Dic₃⋊C₄	1	1	yes	characteristic
111	C₈	Dic₃⋊C₄	1	1	yes	characteristic
112	C₂×C₃	—	2	1	no	—
113	C₂×C₃	—	2	1	no	—
114	C₂×C₃	—	2	1	no	—
115	C₂×C₃	#382	1	1	yes	characteristic
116	C₂²	—	4	1	no	—
117	C₂²	#890	1	1	yes	characteristic
118	C₂²	#892	1	1	yes	characteristic
119	C₂²	#947	1	1	yes	characteristic
120	C₄	—	12	2	no	—
121	C₄	—	24	2	no	—
122	C₄	#890	1	1	yes	characteristic
123	C₄	#892	1	1	yes	characteristic
124	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
125	C₄	#936	1	1	yes	characteristic
126	C₃	#1833	1	1	yes	characteristic, complemented, Sylow
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#4977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#108

C₂

yes

maximal, characteristic

#5106

C₂

yes

maximal, characteristic

#5101

C₂

yes

maximal, characteristic

#4559

C₂

yes

maximal

#4557

C₂

yes

maximal

#1833

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

#926

C₂²

yes

—

#926

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₈×Dic₃

C₄

yes

complemented

C₂×#108

D₃

yes

—

—

—

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

characteristic

C₂×C₃×C₈

Q₈

yes

characteristic

C₂×C₃×C₈

Q₈

yes

characteristic

#217

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

C₂×C₄

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₂²×C₈

C₂×D₃

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

C₄⋊C₄

yes

characteristic

C₃×C₈

C₄⋊C₄

yes

characteristic

C₃×C₈

C₄⋊C₄

yes

characteristic

C₃×C₈

C₄⋊C₄

yes

characteristic

C₃⋊C₈

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆

yes

characteristic

C₂×C₈

Dic₆

yes

characteristic

C₂×C₈

C₄×D₃

yes

characteristic

C₂×C₈

C₄×D₃

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₃×C₄

C₂×C₄⋊C₄

yes

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

Dic₃

—

Dic₃

—

C₂³

#239

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×Dic₆

yes

characteristic

101

C₂×C₄

C₂×C₄×D₃

yes

characteristic

102

C₂×C₄

#237

yes

characteristic

103

C₂×C₄

#237

yes

characteristic

104

C₂×C₄

#239

yes

characteristic

105

C₂×C₄

C₂×#67

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

Dic₃⋊C₄

yes

characteristic

109

C₈

Dic₃⋊C₄

yes

characteristic

110

C₈

Dic₃⋊C₄

yes

characteristic

111

C₈

Dic₃⋊C₄

yes

characteristic

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂×C₃

—

115

C₂×C₃

#382

yes

characteristic

116

C₂²

—

117

C₂²

#890

yes

characteristic

118

C₂²

#892

yes

characteristic

119

C₂²

#947

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

#890

yes

characteristic

123

C₄

#892

yes

characteristic

124

C₄

C₂×Dic₃⋊C₄

yes

characteristic

125

C₄

#936

yes

characteristic

126

C₃

#1833

yes

characteristic, complemented, Sylow

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#4977

yes

characteristic

	subgroup	G/H
center	C₄	#892
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#108	C₂
socle	C₂×C₃	#382
prime radical	C₂³×C₃	#41
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#108	D₃
hypocenter	C₃	#1833
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#892

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#108

C₂

C₂×C₃

C₂³×C₃

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
16. #815	C₄	semidirect
17. C₈×Dic₃	C₄	semidirect
126. C₃	#1833	semidirect

H ≅ G/N

type

16. #815

C₄

semidirect

17. C₈×Dic₃

C₄

semidirect

126. C₃

#1833

semidirect

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

subgroup	index
C₂×C₃×#108	2
#5106	2
#5101	2
#4559	2
#4557	2
#1833	3

subgroup

index

C₂×C₃×#108

Subgroups — #70698

Conjugacy classes of subgroups

Complements

Maximal subgroups