Group #69583 — subgroups

Subgroups — #69583

Order 384.

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

normal subgroups only

This group has 260 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×#870	C₂	1	1	yes	maximal, characteristic
2	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×#803	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	4	yes	maximal
5	#1992	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂×Dic₃⋊C₄	—	2	2	no	—
11	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
12	#870	—	2	2	no	—
13	#870	C₄	1	4	yes	complemented
14	#803	—	2	2	no	—
15	C₂×#125	—	3	1	no	—
16	C₂×#97	—	3	1	no	—
17	#384	D₃	1	1	yes	characteristic
18	#340	—	3	4	no	—
19	C₂²×C₃×C₄	—	2	2	no	—
20	C₂²×C₃×C₄	D₄	1	2	yes	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	—	2	2	no	—
23	C₃×C₄⋊C₄	—	2	4	no	—
24	C₃×C₄⋊C₄	—	4	4	no	—
25	C₃×C₄²	C₂×C₄	1	2	yes	—
26	C₃×C₄²	D₄	1	2	yes	—
27	C₄⋊Dic₃	—	2	2	no	—
28	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
29	Dic₃⋊C₄	—	2	8	no	—
30	C₂×C₃⋊C₈	—	2	2	no	—
31	C₂×C₃⋊C₈	—	2	2	no	—
32	C₂×C₃⋊C₈	Q₈	1	4	yes	—
33	C₂²×C₈	—	3	2	no	—
34	#125	—	3	4	no	—
35	#125	—	6	2	no	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	—	3	2	no	—
38	C₂×C₄⋊C₄	—	6	2	no	—
39	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
40	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
41	#97	—	6	2	no	—
42	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	2	2	no	—
45	C₂×C₃×C₄	—	4	2	no	—
46	C₂×C₃×C₄	—	4	4	no	—
47	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
48	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
51	C₂×C₃×C₄	#41	1	2	yes	—
52	C₂×Dic₃	—	2	4	no	—
53	C₂×Dic₃	—	4	2	no	—
54	C₂×Dic₃	—	4	2	no	—
55	C₃⋊C₈	—	2	4	no	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	—	6	2	no	—
58	C₂²×C₄	#67	1	2	yes	—
59	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₈	—	3	4	no	—
61	C₂×C₈	—	6	2	no	—
62	C₂×C₈	—	6	2	no	—
63	C₄⋊C₄	—	2	4	no	—
64	C₄⋊C₄	—	3	4	no	—
65	C₄⋊C₄	—	4	4	no	—
66	C₄⋊C₄	—	6	8	no	—
67	C₄⋊C₄	—	6	2	no	—
68	C₄²	C₄×D₃	1	2	yes	—
69	C₄²	D₁₂	1	2	yes	—
70	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₃	#124	1	1	yes	characteristic
72	C₂²×C₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₂²×C₃	#137	1	2	yes	—
75	C₃×C₄	—	2	4	no	—
76	C₃×C₄	—	8	2	no	—
77	C₃×C₄	C₄×Q₈	1	2	yes	—
78	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
79	Dic₃	—	4	4	no	—
80	C₂³	C₂×#67	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	6	4	no	—
86	C₂×C₄	—	12	2	no	—
87	C₂×C₄	—	12	2	no	—
88	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
90	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
91	C₂×C₄	#237	1	2	yes	—
92	C₂×C₄	D₃×Q₈	1	2	yes	—
93	C₈	—	6	4	no	—
94	C₂×C₃	#391	1	1	yes	characteristic
95	C₂×C₃	#417	1	2	yes	—
96	C₂×C₃	#490	1	1	yes	characteristic
97	C₂×C₃	#491	1	1	yes	characteristic
98	C₂×C₃	#501	1	2	yes	—
99	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
100	C₂²	#932	1	2	yes	—
101	C₂²	#936	1	1	yes	characteristic
102	C₂²	#942	1	2	yes	—
103	C₂²	#944	1	1	yes	characteristic
104	C₄	—	2	4	no	—
105	C₄	—	8	2	no	—
106	C₄	—	12	4	no	—
107	C₄	#887	1	2	yes	—
108	C₄	#897	1	2	yes	—
109	C₃	#1992	1	1	yes	characteristic, complemented, Sylow
110	C₂	#4966	1	2	yes	—
111	C₂	#4973	1	1	yes	characteristic
112	C₂	#5071	1	1	yes	characteristic
113	C₂	#5081	1	2	yes	—
114	C₂	#5082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#870

C₂

yes

maximal, characteristic

C₃×#384

C₂

yes

maximal, characteristic

C₂×#803

C₂

yes

maximal, characteristic

#4529

C₂

yes

maximal

#1992

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

—

C₂²×C₃⋊C₈

C₂²

yes

—

#870

—

#870

C₄

yes

complemented

#803

—

C₂×#125

—

C₂×#97

—

#384

D₃

yes

characteristic

#340

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

Q₈

yes

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

#97

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

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×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

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

C₄×Q₈

yes

—

C₃×C₄

C₂²⋊Q₈

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

C₂×C₄

—

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

#237

yes

—

C₂×C₄

D₃×Q₈

yes

—

C₈

—

C₂×C₃

#391

yes

characteristic

C₂×C₃

#417

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

—

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

100

C₂²

#932

yes

—

101

C₂²

#936

yes

characteristic

102

C₂²

#942

yes

—

103

C₂²

#944

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#887

yes

—

108

C₄

#897

yes

—

109

C₃

#1992

yes

characteristic, complemented, Sylow

110

C₂

#4966

yes

—

111

C₂

#4973

yes

characteristic

112

C₂

#5071

yes

characteristic

113

C₂

#5081

yes

—

114

C₂

#5082

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#384	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#384	D₃
hypocenter	C₃	#1992
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#384

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 ◁ #384

N	H ≅ G/N	type
13. #870	C₄	semidirect
109. C₃	#1992	semidirect

H ≅ G/N

type

13. #870

C₄

semidirect

109. C₃

#1992

semidirect

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

subgroup	index
C₂×#870	2
C₃×#384	2
C₂×#803	2
#4529	2
#1992	3

subgroup

index

Subgroups — #69583

Conjugacy classes of subgroups

Complements

Maximal subgroups