Group #69552 — subgroups

Subgroups — #69552

Order 384.

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

normal subgroups only

This group has 308 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×#810	C₂	1	4	yes	maximal
4	C₂×#803	C₂	1	1	yes	maximal, characteristic
5	#1960	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	—	2	2	no	—
10	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
11	#869	—	2	2	no	—
12	#869	C₄	1	4	yes	complemented
13	C₃×#95	—	2	2	no	—
14	#810	—	2	8	no	—
15	#803	—	2	2	no	—
16	C₂×C₄⋊Q₈	—	3	1	no	—
17	C₂×#103	—	3	4	no	—
18	C₂×#97	—	3	1	no	—
19	#382	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄	—	2	2	no	—
21	C₂²×C₃×C₄	D₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃	—	2	2	no	—
24	C₂×Dic₆	—	2	4	no	—
25	C₂×Dic₆	—	2	4	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	C₂×C₄	1	4	yes	—
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	8	no	—
32	C₂×C₃⋊C₈	—	2	2	no	—
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃⋊C₈	D₄	1	4	yes	—
35	C₂²×Q₈	—	3	2	no	—
36	C₂²×C₈	—	3	2	no	—
37	C₄⋊Q₈	—	3	4	no	—
38	C₄⋊Q₈	—	6	2	no	—
39	C₂×C₄⋊C₄	—	6	2	no	—
40	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
41	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
42	#103	—	6	8	no	—
43	#97	—	6	2	no	—
44	#95	—	2	2	no	—
45	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄	—	2	4	no	—
47	C₂×C₃×C₄	—	4	2	no	—
48	C₂×C₃×C₄	—	4	4	no	—
49	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
50	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
53	C₂×C₃×C₄	#41	1	2	yes	—
54	C₂×Dic₃	—	2	4	no	—
55	C₂×Dic₃	—	4	2	no	—
56	C₂×Dic₃	—	4	2	no	—
57	Dic₆	—	2	8	no	—
58	Dic₆	—	4	4	no	—
59	C₃⋊C₈	—	2	4	no	—
60	C₂×Q₈	—	3	4	no	—
61	C₂×Q₈	—	6	2	no	—
62	C₂×Q₈	—	6	4	no	—
63	C₂×Q₈	—	6	4	no	—
64	C₂²×C₄	—	2	2	no	—
65	C₂²×C₄	—	6	2	no	—
66	C₂²×C₄	#67	1	2	yes	—
67	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂×C₈	—	3	4	no	—
69	C₂×C₈	—	6	2	no	—
70	C₂×C₈	—	6	2	no	—
71	C₄⋊C₄	—	2	4	no	—
72	C₄⋊C₄	—	6	8	no	—
73	C₄²	C₄×D₃	1	2	yes	—
74	C₄²	D₁₂	1	2	yes	—
75	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₃	#124	1	1	yes	characteristic
77	C₂²×C₃	#127	1	1	yes	characteristic
78	C₂²×C₃	#137	1	4	yes	—
79	C₃×C₄	—	2	4	no	—
80	C₃×C₄	—	8	2	no	—
81	C₃×C₄	C₄×D₄	1	2	yes	—
82	C₃×C₄	#121	1	2	yes	—
83	Dic₃	—	4	4	no	—
84	C₂³	C₂×#67	1	1	yes	characteristic
85	Q₈	—	6	8	no	—
86	Q₈	—	12	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	—	6	4	no	—
91	C₂×C₄	—	12	2	no	—
92	C₂×C₄	—	12	2	no	—
93	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
94	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
95	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
96	C₂×C₄	D₃×D₄	1	2	yes	—
97	C₂×C₄	#239	1	2	yes	—
98	C₈	—	6	4	no	—
99	C₂×C₃	#390	1	1	yes	characteristic
100	C₂×C₃	#419	1	2	yes	—
101	C₂×C₃	#491	1	2	yes	—
102	C₂×C₃	#497	1	2	yes	—
103	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
104	C₂²	#940	1	1	yes	characteristic
105	C₂²	#947	1	1	yes	characteristic
106	C₂²	#951	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	8	2	no	—
109	C₄	—	12	4	no	—
110	C₄	#893	1	2	yes	—
111	C₄	#895	1	2	yes	—
112	C₃	#1960	1	1	yes	characteristic, complemented, Sylow
113	C₂	#4988	1	1	yes	characteristic
114	C₂	#5004	1	2	yes	—
115	C₂	#5060	1	2	yes	—
116	C₂	#5069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₃×#382

C₂

yes

maximal, characteristic

C₂×#810

C₂

yes

maximal

C₂×#803

C₂

yes

maximal, characteristic

#1960

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

C₂²

yes

—

#869

—

#869

C₄

yes

complemented

C₃×#95

—

#810

—

#803

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

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

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×D₄

yes

—

C₃×C₄

#121

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

D₃×D₄

yes

—

C₂×C₄

#239

yes

—

C₈

—

C₂×C₃

#390

yes

characteristic

100

C₂×C₃

#419

yes

—

101

C₂×C₃

#491

yes

—

102

C₂×C₃

#497

yes

—

103

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

104

C₂²

#940

yes

characteristic

105

C₂²

#947

yes

characteristic

106

C₂²

#951

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#893

yes

—

111

C₄

#895

yes

—

112

C₃

#1960

yes

characteristic, complemented, Sylow

113

C₂

#4988

yes

characteristic

114

C₂

#5004

yes

—

115

C₂

#5060

yes

—

116

C₂

#5069

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	#1960
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
12. #869	C₄	semidirect
112. C₃	#1960	semidirect

H ≅ G/N

type

12. #869

C₄

semidirect

112. C₃

#1960

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×#810	2
C₂×#803	2
#1960	3

subgroup

index

Subgroups — #69552

Conjugacy classes of subgroups

Complements

Maximal subgroups