Group #69982 — subgroups

Subgroups — #69982

Order 384.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#407	C₂	1	1	yes	maximal, characteristic
3	#5207	C₂	1	1	yes	maximal, characteristic
4	C₂×#822	C₂	1	2	yes	maximal, complemented
5	#4558	C₂	1	2	yes	maximal, complemented
6	#1810	—	3	1	no	maximal, Sylow
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂×#208	C₂²	1	2	yes	—
10	C₂²×C₃⋊C₈	—	2	4	no	—
11	C₂²×C₃⋊C₈	C₄	1	4	yes	—
12	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₂²⋊C₈	—	2	2	no	—
15	#830	—	2	2	no	—
16	#822	—	2	4	no	—
17	C₂³×C₈	—	3	1	no	—
18	C₂×#108	—	3	2	no	—
19	#407	—	3	1	no	—
20	#407	D₃	1	1	yes	characteristic
21	#341	—	3	2	no	—
22	C₂⁴×C₃	Q₈	1	1	yes	characteristic
23	C₂²×C₃×C₄	—	2	2	no	—
24	C₂²×C₃×C₄	—	2	2	no	—
25	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄	D₄	1	2	yes	—
27	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₃×M₁₆	—	2	2	no	—
30	C₃×M₁₆	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	#208	—	2	2	no	—
33	#208	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	—	2	8	no	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃⋊C₈	D₄	1	4	yes	—
40	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂×M₁₆	—	3	2	no	—
42	C₂×M₁₆	C₂×D₃	1	2	yes	—
43	C₂²×C₈	—	3	1	no	—
44	C₂²×C₈	—	3	4	no	—
45	C₂²×C₈	—	3	1	no	—
46	C₂²×C₈	—	6	4	no	—
47	#108	—	6	4	no	—
48	C₂²⋊C₈	—	2	2	no	—
49	C₂²⋊C₈	—	6	2	no	—
50	C₂³×C₃	—	2	4	no	—
51	C₂³×C₃	—	2	2	no	—
52	C₂³×C₃	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	2	4	no	—
58	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
59	C₂×C₃×C₄	C₂²×C₄	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	C₃×C₈	—	4	2	no	—
64	C₃⋊C₈	—	2	2	no	—
65	C₃⋊C₈	—	4	2	no	—
66	C₃⋊C₈	C₂²⋊C₄	1	4	yes	—
67	C₂⁴	Dic₆	1	1	yes	characteristic
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	2	2	no	—
70	C₂²×C₄	Dic₆	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₃	1	2	yes	—
72	C₂²×C₄	#67	1	2	yes	—
73	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	—	6	2	no	—
77	M₁₆	—	6	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	3	2	no	—
80	C₂×C₈	—	3	4	no	—
81	C₂×C₈	—	3	2	no	—
82	C₂×C₈	—	6	8	no	—
83	C₂×C₈	—	6	2	no	—
84	C₂×C₈	—	6	2	no	—
85	C₂²×C₃	—	2	8	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	—	2	2	no	—
88	C₂²×C₃	—	2	2	no	—
89	C₂²×C₃	#108	1	4	yes	complemented
90	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₃×C₄	#117	1	1	yes	characteristic
95	C₃×C₄	#121	1	2	yes	—
96	C₂³	—	2	4	no	—
97	C₂³	—	2	2	no	—
98	C₂³	Dic₃⋊C₄	1	2	yes	—
99	C₂³	C₂×Dic₆	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	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	—	3	4	no	—
110	C₈	—	4	2	no	—
111	C₈	—	6	2	no	—
112	C₈	—	12	2	no	—
113	C₂×C₃	—	2	4	no	—
114	C₂×C₃	—	2	2	no	—
115	C₂×C₃	#380	1	1	yes	characteristic
116	C₂×C₃	C₂×#108	1	2	yes	—
117	C₂²	—	2	8	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	#822	1	4	yes	complemented
122	C₂²	#878	1	2	yes	—
123	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	#877	1	1	yes	characteristic
126	C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
127	C₄	#939	1	2	yes	—
128	C₃	#1810	1	1	yes	characteristic, complemented, Sylow
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#4942	1	1	yes	characteristic
132	C₂	C₂×#822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₃⋊C₈

C₂

yes

maximal, characteristic

C₃×#407

C₂

yes

maximal, characteristic

#5207

C₂

yes

maximal, characteristic

C₂×#822

C₂

yes

maximal, complemented

#4558

C₂

yes

maximal, complemented

#1810

—

maximal, Sylow

C₂³×C₃×C₄

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

—

C₂×#208

C₂²

yes

—

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

C₄

yes

—

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₃×C₂²⋊C₈

—

#830

—

#822

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₃

yes

characteristic

#341

—

C₂⁴×C₃

Q₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

Q₈

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

C₂³×C₄

C₂×D₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₄⋊C₄

yes

—

C₂³×C₃

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

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

C₂²⋊C₄

yes

—

C₂⁴

Dic₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₆

yes

characteristic

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#108

yes

complemented

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₃×C₄

—

C₃×C₄

C₂×C₂²⋊C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

C₂³

—

C₂³

—

C₂³

Dic₃⋊C₄

yes

—

C₂³

C₂×Dic₆

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

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

#380

yes

characteristic

116

C₂×C₃

C₂×#108

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#822

yes

complemented

122

C₂²

#878

yes

—

123

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

124

C₄

—

125

C₄

#877

yes

characteristic

126

C₄

D₃×C₂²⋊C₄

yes

characteristic

127

C₄

#939

yes

—

128

C₃

#1810

yes

characteristic, complemented, Sylow

129

C₂

—

130

C₂

—

131

C₂

#4942

yes

characteristic

132

C₂

C₂×#822

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#407	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃	Q₈
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#407	D₃
hypocenter	C₃	#1810
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#407

C₂

socle

C₂²×C₃

C₂×C₄⋊C₄

prime radical

C₂⁴×C₃

Q₈

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
4. C₂×#822	C₂	semidirect
5. #4558	C₂	semidirect
89. C₂²×C₃	#108	semidirect
121. C₂²	#822	semidirect
128. C₃	#1810	semidirect

H ≅ G/N

type

4. C₂×#822

C₂

semidirect

5. #4558

C₂

semidirect

89. C₂²×C₃

#108

semidirect

121. C₂²

#822

semidirect

128. C₃

#1810

semidirect

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

subgroup	index
C₂³×C₃⋊C₈	2
C₃×#407	2
#5207	2
C₂×#822	2
#4558	2
#1810	3

subgroup

index

C₂³×C₃⋊C₈

Subgroups — #69982

Conjugacy classes of subgroups

Complements

Maximal subgroups