Group #68638 — subgroups

Subgroups — #68638

Order 384.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#816	C₂	1	2	yes	maximal
2	#4936	C₂	1	1	yes	maximal, characteristic
3	#4929	C₂	1	1	yes	maximal, characteristic
4	C₃×#336	C₂	1	1	yes	maximal, characteristic
5	#4551	C₂	1	2	yes	maximal
6	#2074	—	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	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃⋊#95	—	2	2	no	—
13	C₃⋊#95	—	2	2	no	—
14	#816	—	2	4	no	—
15	C₂×#103	—	3	2	no	—
16	#391	—	3	1	no	—
17	#382	—	3	1	no	—
18	#340	—	3	2	no	—
19	#336	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄	D₄	1	2	yes	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	—	2	1	no	—
23	C₂²×Dic₃	—	2	2	no	—
24	C₂²×Dic₃	—	2	1	no	—
25	C₂×Dic₆	—	2	2	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	—	2	1	no	—
28	C₂×Dic₆	D₄	1	2	yes	—
29	C₂×C₃×C₈	—	2	2	no	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₂×C₃×C₈	—	2	2	no	—
32	C₃×C₄²	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	—	2	2	no	—
36	C₂²×Q₈	—	3	1	no	—
37	C₂²×C₈	C₂×D₃	1	2	yes	—
38	C₂×C₄⋊C₄	—	3	1	no	—
39	C₂×C₄⋊C₄	—	3	2	no	—
40	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
41	#103	—	6	4	no	—
42	#95	—	6	2	no	—
43	#95	—	6	2	no	—
44	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃×C₄	—	2	2	no	—
46	C₂×C₃×C₄	—	2	2	no	—
47	C₂×C₃×C₄	—	2	2	no	—
48	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄	#41	1	2	yes	—
50	C₂×C₃×C₄	#41	1	2	yes	—
51	C₂×C₃×C₄	#41	1	1	yes	characteristic
52	C₂×Dic₃	—	2	2	no	—
53	C₂×Dic₃	—	4	1	no	—
54	C₂×Dic₃	—	4	1	no	—
55	C₂×Dic₃	—	4	2	no	—
56	C₂×Dic₃	—	4	1	no	—
57	C₂×Dic₃	—	4	2	no	—
58	C₂×Dic₃	—	4	2	no	—
59	C₂×Dic₃	—	4	2	no	—
60	Dic₆	—	2	4	no	—
61	Dic₆	—	4	2	no	—
62	C₃×C₈	—	4	2	no	—
63	C₂×Q₈	—	3	2	no	—
64	C₂×Q₈	—	6	1	no	—
65	C₂×Q₈	—	6	2	no	—
66	C₂×Q₈	—	6	2	no	—
67	C₂²×C₄	—	6	1	no	—
68	C₂²×C₄	—	6	1	no	—
69	C₂²×C₄	—	6	2	no	—
70	C₂²×C₄	D₁₂	1	2	yes	—
71	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₄⋊C₄	—	6	2	no	—
76	C₄⋊C₄	—	6	2	no	—
77	C₄⋊C₄	—	6	2	no	—
78	C₄²	—	2	2	no	—
79	C₂²×C₃	#121	1	1	yes	characteristic
80	C₂²×C₃	#123	1	1	yes	characteristic
81	C₂²×C₃	#124	1	1	yes	characteristic
82	C₂²×C₃	#135	1	2	yes	—
83	C₂²×C₃	#137	1	2	yes	—
84	C₃×C₄	—	4	2	no	—
85	C₃×C₄	#123	1	2	yes	—
86	C₃×C₄	#126	1	2	yes	—
87	Dic₃	—	4	2	no	—
88	Dic₃	—	8	2	no	—
89	Dic₃	—	8	1	no	—
90	C₂³	C₂×D₁₂	1	1	yes	characteristic
91	Q₈	—	6	4	no	—
92	Q₈	—	12	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	6	2	no	—
97	C₂×C₄	—	12	1	no	—
98	C₂×C₄	—	12	1	no	—
99	C₂×C₄	—	12	1	no	—
100	C₂×C₄	—	12	2	no	—
101	C₂×C₄	—	12	2	no	—
102	C₂×C₄	—	12	2	no	—
103	C₂×C₄	—	12	2	no	—
104	C₂×C₄	#235	1	2	yes	—
105	C₂×C₄	D₃×D₄	1	1	yes	characteristic
106	C₂×C₄	#237	1	2	yes	—
107	C₂×C₄	#239	1	1	yes	characteristic
108	C₈	—	4	2	no	—
109	C₂×C₃	#397	1	1	yes	characteristic
110	C₂×C₃	#464	1	2	yes	—
111	C₂×C₃	#485	1	2	yes	—
112	C₂×C₃	#488	1	1	yes	characteristic
113	C₂×C₃	#491	1	1	yes	characteristic
114	C₂²	#875	1	1	yes	characteristic
115	C₂²	#886	1	1	yes	characteristic
116	C₂²	#895	1	1	yes	characteristic
117	C₂²	#904	1	2	yes	—
118	C₂²	#909	1	2	yes	—
119	C₄	—	4	2	no	—
120	C₄	—	12	2	no	—
121	C₄	—	24	1	no	—
122	C₄	—	24	2	no	—
123	C₄	#883	1	2	yes	—
124	C₄	#898	1	2	yes	—
125	C₃	#2074	1	1	yes	characteristic, complemented, Sylow
126	C₂	#4674	1	1	yes	characteristic
127	C₂	#4697	1	1	yes	characteristic
128	C₂	#4703	1	1	yes	characteristic
129	C₂	#4723	1	2	yes	—
130	C₂	#4845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#816

C₂

yes

maximal

#4936

C₂

yes

maximal, characteristic

#4929

C₂

yes

maximal, characteristic

C₃×#336

C₂

yes

maximal, characteristic

#4551

C₂

yes

maximal

#2074

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

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

D₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₃

yes

—

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

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

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

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

#123

yes

—

C₃×C₄

#126

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

C₂×D₁₂

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#235

yes

—

105

C₂×C₄

D₃×D₄

yes

characteristic

106

C₂×C₄

#237

yes

—

107

C₂×C₄

#239

yes

characteristic

108

C₈

—

109

C₂×C₃

#397

yes

characteristic

110

C₂×C₃

#464

yes

—

111

C₂×C₃

#485

yes

—

112

C₂×C₃

#488

yes

characteristic

113

C₂×C₃

#491

yes

characteristic

114

C₂²

#875

yes

characteristic

115

C₂²

#886

yes

characteristic

116

C₂²

#895

yes

characteristic

117

C₂²

#904

yes

—

118

C₂²

#909

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#883

yes

—

124

C₄

#898

yes

—

125

C₃

#2074

yes

characteristic, complemented, Sylow

126

C₂

#4674

yes

characteristic

127

C₂

#4697

yes

characteristic

128

C₂

#4703

yes

characteristic

129

C₂

#4723

yes

—

130

C₂

#4845

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	#2074
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
125. C₃	#2074	semidirect

H ≅ G/N

type

125. C₃

#2074

semidirect

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

subgroup	index
C₂×#816	2
#4936	2
#4929	2
C₃×#336	2
#4551	2
#2074	3

subgroup

index

Subgroups — #68638

Conjugacy classes of subgroups

Complements

Maximal subgroups