Group #5684 — subgroups

Subgroups — #5684

Order 192.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	#946	C₂	1	2	yes	maximal, complemented
3	#939	C₂	1	2	yes	maximal, complemented
4	#886	C₂	1	2	yes	maximal, complemented
5	#884	C₂	1	2	yes	maximal, complemented
6	#883	C₂	1	2	yes	maximal, complemented
7	#878	C₂	1	2	yes	maximal, complemented
8	#875	C₂	1	1	yes	maximal, characteristic, complemented
9	#870	C₂	1	1	yes	maximal, characteristic, complemented
10	#562	—	3	1	no	maximal, Sylow
11	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂×#67	C₂²	1	2	yes	—
14	C₂×#67	C₂²	1	2	yes	—
15	C₂²×Dic₃	C₂²	1	2	yes	—
16	C₂×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃	C₂²	1	2	yes	—
18	C₂×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²	C₂²	1	1	yes	characteristic
22	#217	C₂²	1	2	yes	—
23	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₃	C₂²	1	2	yes	—
29	Dic₃⋊C₄	C₂²	1	2	yes	—
30	Dic₃⋊C₄	C₂²	1	2	yes	—
31	Dic₃⋊C₄	C₂²	1	2	yes	—
32	#125	—	3	1	no	—
33	#124	—	3	1	no	—
34	#124	D₃	1	1	yes	characteristic, complemented
35	#123	—	3	2	no	—
36	#123	—	3	2	no	—
37	C₂²⋊Q₈	—	3	2	no	—
38	C₂²⋊Q₈	—	3	2	no	—
39	#121	—	3	2	no	—
40	#121	—	3	2	no	—
41	C₂³×C₃	C₂³	1	2	yes	—
42	C₂²×D₃	C₂³	1	2	yes	—
43	C₃×Q₈	—	4	1	no	—
44	C₃×D₄	—	4	1	no	—
45	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂³	1	2	yes	—
48	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
49	#67	—	4	2	no	—
50	#67	—	4	2	no	—
51	C₂×Dic₃	—	4	2	no	—
52	C₂×Dic₃	C₂³	1	2	yes	—
53	C₂×Dic₃	C₂³	1	2	yes	—
54	C₂×Dic₃	C₂³	1	2	yes	—
55	D₁₂	—	4	1	no	—
56	C₄×D₃	—	4	2	no	—
57	Dic₆	—	4	1	no	—
58	C₂×Q₈	—	3	1	no	—
59	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
60	C₂×D₄	—	3	2	no	—
61	C₂×D₄	—	3	2	no	—
62	C₂×D₄	—	3	1	no	—
63	C₂×D₄	C₂×D₃	1	1	yes	characteristic
64	C₂²×C₄	—	3	2	no	—
65	C₂²×C₄	—	3	2	no	—
66	C₄⋊C₄	—	3	2	no	—
67	C₄⋊C₄	—	3	2	no	—
68	C₄⋊C₄	—	3	2	no	—
69	C₄⋊C₄	—	3	2	no	—
70	C₄⋊C₄	—	3	1	no	—
71	C₄⋊C₄	—	3	1	no	—
72	C₂²⋊C₄	—	3	2	no	—
73	C₂²⋊C₄	—	3	2	no	—
74	C₂²⋊C₄	—	3	2	no	—
75	C₂²⋊C₄	—	3	2	no	—
76	C₂²⋊C₄	C₂×D₃	1	2	yes	—
77	C₂²⋊C₄	C₂×D₃	1	2	yes	—
78	C₄²	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₃	—	2	2	no	—
80	C₂²×C₃	—	2	2	no	—
81	C₂²×C₃	—	2	2	no	—
82	C₂²×C₃	C₂⁴	1	1	yes	characteristic
83	C₂×D₃	—	2	2	no	—
84	C₂×D₃	—	2	2	no	—
85	C₂×D₃	—	2	2	no	—
86	C₃×C₄	—	2	1	no	—
87	C₃×C₄	—	2	2	no	—
88	C₃×C₄	—	2	1	no	—
89	C₃×C₄	—	2	1	no	—
90	Dic₃	—	2	2	no	—
91	Dic₃	—	2	2	no	—
92	Dic₃	—	2	2	no	—
93	C₂³	—	3	2	no	—
94	C₂³	C₂²×D₃	1	2	yes	—
95	Q₈	—	4	1	no	—
96	Q₈	—	12	1	no	—
97	D₄	—	4	1	no	—
98	D₄	—	12	1	no	—
99	D₄	—	12	2	no	—
100	D₄	—	12	2	no	—
101	C₂×C₄	—	3	2	no	—
102	C₂×C₄	—	3	2	no	—
103	C₂×C₄	—	3	2	no	—
104	C₂×C₄	—	12	2	no	—
105	C₂×C₄	—	12	2	no	—
106	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₄	C₂²×D₃	1	2	yes	—
110	C₂×C₃	—	4	2	no	—
111	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
114	D₃	—	4	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	6	2	no	—
119	C₂²	—	6	2	no	—
120	C₂²	—	6	2	no	—
121	C₂²	C₂³×D₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	6	2	no	—
127	C₄	—	6	2	no	—
128	C₄	—	6	2	no	—
129	C₃	#562	1	1	yes	characteristic, complemented, Sylow
130	C₂	—	4	2	no	—
131	C₂	—	12	2	no	—
132	C₂	#1004	1	1	yes	characteristic
133	C₂	#1009	1	1	yes	characteristic
134	C₂	#1010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#124

C₂

yes

maximal, characteristic, complemented

#946

C₂

yes

maximal, complemented

#939

C₂

yes

maximal, complemented

#886

C₂

yes

maximal, complemented

#884

C₂

yes

maximal, complemented

#883

C₂

yes

maximal, complemented

#878

C₂

yes

maximal, complemented

#875

C₂

yes

maximal, characteristic, complemented

#870

C₂

yes

maximal, characteristic, complemented

#562

—

maximal, Sylow

C₂×C₃×Q₈

C₂²

yes

characteristic

C₂×C₃×D₄

C₂²

yes

characteristic

C₂×#67

C₂²

yes

—

C₂×#67

C₂²

yes

—

C₂²×Dic₃

C₂²

yes

—

C₂×D₁₂

C₂²

yes

characteristic

C₂×C₄×D₃

C₂²

yes

—

C₂×Dic₆

C₂²

yes

characteristic

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₂²⋊C₄

C₂²

yes

—

C₃×C₄²

C₂²

yes

characteristic

#217

C₂²

yes

—

(C₂×D₃)⋊C₄

C₂²

yes

—

(C₂×D₃)⋊C₄

C₂²

yes

—

(C₂×D₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₃

C₂²

yes

characteristic

C₄⋊Dic₃

C₂²

yes

characteristic

C₄⋊Dic₃

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

—

Dic₃⋊C₄

C₂²

yes

—

#125

—

#124

—

#124

D₃

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂³×C₃

C₂³

yes

—

C₂²×D₃

C₂³

yes

—

C₃×Q₈

—

C₃×D₄

—

C₂×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₄

C₂³

yes

—

C₂×C₃×C₄

C₂³

yes

characteristic

#67

—

#67

—

C₂×Dic₃

—

C₂×Dic₃

C₂³

yes

—

C₂×Dic₃

C₂³

yes

—

C₂×Dic₃

C₂³

yes

—

D₁₂

—

C₄×D₃

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃

yes

—

C₂²⋊C₄

C₂×D₃

yes

—

C₄²

C₂×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂⁴

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

—

C₂³

C₂²×D₃

yes

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²×D₃

yes

characteristic

107

C₂×C₄

C₂²×D₃

yes

characteristic

108

C₂×C₄

C₂²×D₃

yes

characteristic

109

C₂×C₄

C₂²×D₃

yes

—

110

C₂×C₃

—

111

C₂×C₃

H(5,𝔽₂)

yes

characteristic

112

C₂×C₃

H(5,𝔽₂)

yes

characteristic

113

C₂×C₃

2₋¹⁺⁴

yes

characteristic

114

D₃

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×D₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₃

#562

yes

characteristic, complemented, Sylow

130

C₂

—

131

C₂

—

132

C₂

#1004

yes

characteristic

133

C₂

#1009

yes

characteristic

134

C₂

#1010

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#124	C₂
hypercenter	#124	D₃
hypocenter	C₃	#562
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃

commutator, socle, cosocle

C₂²×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₃×#124	C₂	semidirect
2. #946	C₂	semidirect
3. #939	C₂	semidirect
4. #886	C₂	semidirect
5. #884	C₂	semidirect
6. #883	C₂	semidirect
7. #878	C₂	semidirect
8. #875	C₂	semidirect
9. #870	C₂	semidirect
34. #124	D₃	semidirect
129. C₃	#562	semidirect

H ≅ G/N

type

1. C₃×#124

C₂

semidirect

2. #946

C₂

semidirect

3. #939

C₂

semidirect

4. #886

C₂

semidirect

5. #884

C₂

semidirect

6. #883

C₂

semidirect

7. #878

C₂

semidirect

8. #875

C₂

semidirect

9. #870

C₂

semidirect

34. #124

D₃

semidirect

129. C₃

#562

semidirect

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

subgroup	index
C₃×#124	2
#946	2
#939	2
#886	2
#884	2
#883	2
#878	2
#875	2
#870	2
#562	3

subgroup

index

Subgroups — #5684

Conjugacy classes of subgroups

Complements

Maximal subgroups