Group #69555 — subgroups

Subgroups — #69555

Order 384.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×#803	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	4	yes	maximal
5	#1993	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	—	2	1	no	—
7	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	—	2	1	no	—
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
12	#869	—	2	2	no	—
13	#869	C₄	1	4	yes	complemented
14	C₃×#95	—	2	1	no	—
15	C₃×#95	—	2	1	no	—
16	#803	—	2	2	no	—
17	C₂×C₄⋊Q₈	—	3	1	no	—
18	C₂×#97	—	3	1	no	—
19	#382	D₃	1	1	yes	characteristic
20	#340	—	3	4	no	—
21	C₂²×C₃×C₄	—	2	2	no	—
22	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃	—	2	2	no	—
26	C₂×Dic₆	—	2	4	no	—
27	C₂×Dic₆	—	4	4	no	—
28	C₃×C₄⋊C₄	—	2	4	no	—
29	C₃×C₄²	C₂×C₄	1	2	yes	—
30	C₃×C₄²	D₄	1	2	yes	—
31	C₄⋊Dic₃	—	2	4	no	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	—	2	2	no	—
36	C₂×C₃⋊C₈	Q₈	1	4	yes	complemented
37	C₂²×Q₈	—	6	1	no	—
38	C₂²×C₈	—	3	2	no	—
39	C₄⋊Q₈	—	3	4	no	—
40	C₄⋊Q₈	—	6	2	no	—
41	C₂×C₄⋊C₄	—	3	2	no	—
42	C₂×C₄⋊C₄	—	6	1	no	—
43	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
44	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
45	#97	—	6	2	no	—
46	#95	—	2	1	no	—
47	#95	—	2	1	no	—
48	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄	—	2	2	no	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	—	4	2	no	—
52	C₂×C₃×C₄	—	4	2	no	—
53	C₂×C₃×C₄	—	4	2	no	—
54	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
59	C₂×C₃×C₄	#41	1	2	yes	—
60	C₂×Dic₃	—	2	4	no	—
61	C₂×Dic₃	—	4	4	no	—
62	Dic₆	—	4	8	no	—
63	C₃⋊C₈	—	2	4	no	—
64	C₂×Q₈	—	6	4	no	—
65	C₂×Q₈	—	12	4	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	6	2	no	—
68	C₂²×C₄	#67	1	1	yes	characteristic
69	C₂²×C₄	#67	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₂×C₈	—	3	4	no	—
72	C₂×C₈	—	6	2	no	—
73	C₂×C₈	—	6	2	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	3	4	no	—
76	C₄⋊C₄	—	6	4	no	—
77	C₄⋊C₄	—	6	2	no	—
78	C₄²	C₄×D₃	1	2	yes	—
79	C₄²	D₁₂	1	2	yes	—
80	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₃	#124	1	1	yes	characteristic
82	C₂²×C₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₂²×C₃	#137	1	2	yes	—
85	C₃×C₄	—	2	2	no	—
86	C₃×C₄	—	2	2	no	—
87	C₃×C₄	—	8	2	no	—
88	C₃×C₄	C₄×Q₈	1	2	yes	—
89	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
90	Dic₃	—	4	4	no	—
91	C₂³	C₂×#67	1	1	yes	characteristic
92	Q₈	—	12	8	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	6	4	no	—
99	C₂×C₄	—	12	4	no	—
100	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
101	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
102	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
103	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
104	C₂×C₄	#237	1	2	yes	—
105	C₂×C₄	D₃×Q₈	1	2	yes	—
106	C₈	—	6	4	no	—
107	C₂×C₃	#391	1	1	yes	characteristic
108	C₂×C₃	#418	1	1	yes	characteristic
109	C₂×C₃	#419	1	1	yes	characteristic
110	C₂×C₃	#489	1	2	yes	—
111	C₂×C₃	#501	1	2	yes	—
112	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
113	C₂²	#936	1	1	yes	characteristic
114	C₂²	#944	1	1	yes	characteristic
115	C₂²	#949	1	2	yes	—
116	C₂²	#951	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	8	2	no	—
120	C₄	—	12	4	no	—
121	C₄	#887	1	2	yes	—
122	C₄	#897	1	2	yes	—
123	C₃	#1993	1	1	yes	characteristic, complemented, Sylow
124	C₂	#4973	1	1	yes	characteristic
125	C₂	#5001	1	1	yes	characteristic
126	C₂	#5004	1	1	yes	characteristic
127	C₂	#5055	1	2	yes	—
128	C₂	#5064	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₃×#382

C₂

yes

maximal, characteristic

C₂×#803

C₂

yes

maximal, characteristic

#4529

C₂

yes

maximal

#1993

—

maximal, Sylow

C₂²×Dic₆

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

#869

—

#869

C₄

yes

complemented

C₃×#95

—

C₃×#95

—

#803

—

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

D₃

yes

characteristic

#340

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₄

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

Q₈

yes

complemented

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

—

—

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

#124

yes

characteristic

C₂²×C₃

C₄⋊Q₈

yes

characteristic

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×Q₈

yes

—

C₃×C₄

C₂²⋊Q₈

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

101

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

102

C₂×C₄

C₂×C₄×D₃

yes

characteristic

103

C₂×C₄

C₂×D₁₂

yes

characteristic

104

C₂×C₄

#237

yes

—

105

C₂×C₄

D₃×Q₈

yes

—

106

C₈

—

107

C₂×C₃

#391

yes

characteristic

108

C₂×C₃

#418

yes

characteristic

109

C₂×C₃

#419

yes

characteristic

110

C₂×C₃

#489

yes

—

111

C₂×C₃

#501

yes

—

112

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

113

C₂²

#936

yes

characteristic

114

C₂²

#944

yes

characteristic

115

C₂²

#949

yes

—

116

C₂²

#951

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#887

yes

—

122

C₄

#897

yes

—

123

C₃

#1993

yes

characteristic, complemented, Sylow

124

C₂

#4973

yes

characteristic

125

C₂

#5001

yes

characteristic

126

C₂

#5004

yes

characteristic

127

C₂

#5055

yes

—

128

C₂

#5064

yes

—

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#382	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#382	D₃
hypocenter	C₃	#1993
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#382

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
13. #869	C₄	semidirect
36. C₂×C₃⋊C₈	Q₈	semidirect
123. C₃	#1993	semidirect

H ≅ G/N

type

13. #869

C₄

semidirect

36. C₂×C₃⋊C₈

Q₈

semidirect

123. C₃

#1993

semidirect

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

subgroup	index
C₂×#869	2
C₃×#382	2
C₂×#803	2
#4529	2
#1993	3

subgroup

index

Subgroups — #69555

Conjugacy classes of subgroups

Complements

Maximal subgroups