Group #68692 — subgroups

Subgroups — #68692

Order 384.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#817	C₂	1	2	yes	maximal
2	C₂×#816	C₂	1	2	yes	maximal
3	#4935	C₂	1	1	yes	maximal, characteristic
4	#4929	C₂	1	1	yes	maximal, characteristic
5	C₃×#336	C₂	1	1	yes	maximal, characteristic
6	#2096	—	3	1	no	maximal, Sylow
7	C₂²×Dic₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₃⋊#95	—	2	2	no	—
14	#817	—	2	4	no	—
15	#816	—	2	4	no	—
16	C₂×#106	—	3	2	no	—
17	C₂×#103	—	3	2	no	—
18	#391	—	3	1	no	—
19	#384	—	3	1	no	—
20	#336	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄	D₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃	—	2	1	no	—
24	C₂²×Dic₃	—	2	1	no	—
25	C₂²×Dic₃	—	2	2	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	—	2	2	no	—
28	C₂×Dic₆	—	2	1	no	—
29	C₂×Dic₆	Q₈	1	2	yes	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₂×C₃×C₈	D₄	1	4	yes	—
33	C₃×C₄²	—	2	2	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	—	2	2	no	—
36	C₄⋊Dic₃	—	2	2	no	—
37	C₄⋊Dic₃	—	4	4	no	—
38	C₂²×Q₈	—	3	1	no	—
39	C₂²×C₈	C₂×D₃	1	2	yes	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	3	1	no	—
42	C₂×C₄⋊C₄	—	6	2	no	—
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	#106	—	6	4	no	—
45	#103	—	6	4	no	—
46	#95	—	6	2	no	—
47	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	2	no	—
49	C₂×C₃×C₄	—	2	2	no	—
50	C₂×C₃×C₄	SD₁₆	1	4	yes	—
51	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
52	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₃×C₄	#41	1	2	yes	—
54	C₂×C₃×C₄	#41	1	1	yes	characteristic
55	C₂×Dic₃	—	2	2	no	—
56	C₂×Dic₃	—	4	1	no	—
57	C₂×Dic₃	—	4	1	no	—
58	C₂×Dic₃	—	4	2	no	—
59	C₂×Dic₃	—	4	1	no	—
60	C₂×Dic₃	—	4	4	no	—
61	C₂×Dic₃	—	4	2	no	—
62	Dic₆	—	2	4	no	—
63	Dic₆	—	4	2	no	—
64	C₃×C₈	—	2	4	no	—
65	C₂×Q₈	—	3	2	no	—
66	C₂×Q₈	—	6	1	no	—
67	C₂×Q₈	—	6	2	no	—
68	C₂×Q₈	—	6	2	no	—
69	C₂²×C₄	—	6	1	no	—
70	C₂²×C₄	—	6	1	no	—
71	C₂²×C₄	—	6	2	no	—
72	C₂²×C₄	D₁₂	1	2	yes	—
73	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	D₁₂	1	4	yes	—
77	C₄⋊C₄	—	6	2	no	—
78	C₄⋊C₄	—	6	2	no	—
79	C₄⋊C₄	—	6	2	no	—
80	C₄⋊C₄	—	12	4	no	—
81	C₄²	—	2	2	no	—
82	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂²×C₃	#123	1	1	yes	characteristic
84	C₂²×C₃	#127	1	1	yes	characteristic
85	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
86	C₂²×C₃	#137	1	2	yes	—
87	C₃×C₄	—	2	4	no	—
88	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
89	C₃×C₄	#123	1	2	yes	—
90	Dic₃	—	4	2	no	—
91	Dic₃	—	8	2	no	—
92	Dic₃	—	8	1	no	—
93	C₂³	C₂×D₁₂	1	1	yes	characteristic
94	Q₈	—	6	4	no	—
95	Q₈	—	12	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	6	2	no	—
99	C₂×C₄	—	12	1	no	—
100	C₂×C₄	—	12	1	no	—
101	C₂×C₄	—	12	2	no	—
102	C₂×C₄	—	12	1	no	—
103	C₂×C₄	—	12	4	no	—
104	C₂×C₄	—	12	2	no	—
105	C₂×C₄	#204	1	4	yes	—
106	C₂×C₄	C₂×D₁₂	1	2	yes	—
107	C₂×C₄	#237	1	1	yes	characteristic
108	C₂×C₄	#237	1	2	yes	—
109	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₂×C₃	#399	1	1	yes	characteristic
112	C₂×C₃	#475	1	2	yes	—
113	C₂×C₃	#483	1	2	yes	—
114	C₂×C₃	#492	1	1	yes	characteristic
115	C₂×C₃	#497	1	1	yes	characteristic
116	C₂²	#874	1	1	yes	characteristic
117	C₂²	#886	1	1	yes	characteristic
118	C₂²	#897	1	1	yes	characteristic
119	C₂²	C₂×#204	1	2	yes	—
120	C₂²	#909	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	12	2	no	—
123	C₄	—	24	2	no	—
124	C₄	—	24	1	no	—
125	C₄	#886	1	2	yes	—
126	C₄	#897	1	2	yes	—
127	C₃	#2096	1	1	yes	characteristic, complemented, Sylow
128	C₂	#4677	1	1	yes	characteristic
129	C₂	#4693	1	1	yes	characteristic
130	C₂	#4715	1	1	yes	characteristic
131	C₂	#4721	1	2	yes	—
132	C₂	#4851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#817

C₂

yes

maximal

C₂×#816

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

#4929

C₂

yes

maximal, characteristic

C₃×#336

C₂

yes

maximal, characteristic

#2096

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

—

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

Q₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

—

—

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

C₂²⋊Q₈

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

C₂×SD₁₆

yes

—

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#123

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

C₂×D₁₂

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#204

yes

—

106

C₂×C₄

C₂×D₁₂

yes

—

107

C₂×C₄

#237

yes

characteristic

108

C₂×C₄

#237

yes

—

109

C₂×C₄

D₃×Q₈

yes

characteristic

110

C₈

—

111

C₂×C₃

#399

yes

characteristic

112

C₂×C₃

#475

yes

—

113

C₂×C₃

#483

yes

—

114

C₂×C₃

#492

yes

characteristic

115

C₂×C₃

#497

yes

characteristic

116

C₂²

#874

yes

characteristic

117

C₂²

#886

yes

characteristic

118

C₂²

#897

yes

characteristic

119

C₂²

C₂×#204

yes

—

120

C₂²

#909

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#886

yes

—

126

C₄

#897

yes

—

127

C₃

#2096

yes

characteristic, complemented, Sylow

128

C₂

#4677

yes

characteristic

129

C₂

#4693

yes

characteristic

130

C₂

#4715

yes

characteristic

131

C₂

#4721

yes

—

132

C₂

#4851

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#336	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	#336	D₃
hypocenter	C₃	#2096
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₂

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#336

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
127. C₃	#2096	semidirect

H ≅ G/N

type

127. C₃

#2096

semidirect

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

subgroup	index
C₂×#817	2
C₂×#816	2
#4935	2
#4929	2
C₃×#336	2
#2096	3

subgroup

index

Subgroups — #68692

Conjugacy classes of subgroups

Complements

Maximal subgroups