Group #69575 — subgroups

Subgroups — #69575

Order 384.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₃×#384	C₂	1	1	yes	maximal, characteristic
2	#4936	C₂	1	1	yes	maximal, characteristic
3	C₂×#803	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	4	yes	maximal
5	#1836	—	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	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
11	C₃⋊#95	—	2	2	no	—
12	#803	C₄	1	4	yes	complemented
13	C₂×#97	—	3	1	no	—
14	#384	D₃	1	1	yes	characteristic
15	#382	—	3	1	no	—
16	#340	—	3	4	no	—
17	C₂²×C₃×C₄	—	2	2	no	—
18	C₂²×C₃×C₄	D₄	1	2	yes	—
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃	—	2	2	no	—
21	C₃×C₄⋊C₄	—	2	4	no	—
22	C₃×C₄⋊C₄	—	4	4	no	—
23	C₃×C₄²	C₂×C₄	1	2	yes	—
24	C₃×C₄²	Q₈	1	2	yes	—
25	C₄⋊Dic₃	—	2	4	no	—
26	C₂×C₃⋊C₈	—	2	2	no	—
27	C₂×C₃⋊C₈	—	2	2	no	—
28	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₈	—	3	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	—	3	2	no	—
32	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	#97	—	3	4	no	—
35	#95	—	6	2	no	—
36	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₃×C₄	—	2	2	no	—
38	C₂×C₃×C₄	—	2	2	no	—
39	C₂×C₃×C₄	—	4	2	no	—
40	C₂×C₃×C₄	—	4	4	no	—
41	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
42	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₃×C₄	#41	1	2	yes	—
45	C₂×C₃×C₄	#41	1	2	yes	—
46	C₂×Dic₃	—	4	2	no	—
47	C₂×Dic₃	—	4	2	no	—
48	C₂×Dic₃	—	4	2	no	—
49	C₃⋊C₈	—	2	4	no	—
50	C₂²×C₄	—	2	2	no	—
51	C₂²×C₄	—	6	2	no	—
52	C₂²×C₄	#67	1	2	yes	—
53	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
54	C₂×C₈	—	3	4	no	—
55	C₂×C₈	—	6	2	no	—
56	C₂×C₈	—	6	2	no	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	4	4	no	—
59	C₄⋊C₄	—	6	4	no	—
60	C₄²	Dic₆	1	2	yes	—
61	C₄²	C₄×D₃	1	2	yes	—
62	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃	#124	1	1	yes	characteristic
64	C₂²×C₃	#124	1	1	yes	characteristic
65	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
66	C₂²×C₃	#137	1	2	yes	—
67	C₃×C₄	—	2	4	no	—
68	C₃×C₄	—	8	2	no	—
69	C₃×C₄	#117	1	2	yes	—
70	C₃×C₄	#125	1	2	yes	—
71	Dic₃	—	8	2	no	—
72	C₂³	C₂×#67	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	12	2	no	—
78	C₂×C₄	—	12	2	no	—
79	C₂×C₄	—	12	2	no	—
80	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
81	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
82	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
83	C₂×C₄	#237	1	2	yes	—
84	C₂×C₄	#239	1	2	yes	—
85	C₈	—	6	4	no	—
86	C₂×C₃	#382	1	1	yes	characteristic
87	C₂×C₃	#428	1	2	yes	—
88	C₂×C₃	#489	1	2	yes	—
89	C₂×C₃	#490	1	1	yes	characteristic
90	C₂×C₃	#491	1	1	yes	characteristic
91	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
92	C₂²	#932	1	2	yes	—
93	C₂²	#936	1	1	yes	characteristic
94	C₂²	#942	1	2	yes	—
95	C₂²	#947	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	8	2	no	—
98	C₄	—	24	2	no	—
99	C₄	#890	1	2	yes	—
100	C₄	#892	1	2	yes	—
101	C₃	#1836	1	1	yes	characteristic, complemented, Sylow
102	C₂	#4964	1	2	yes	—
103	C₂	#4977	1	1	yes	characteristic
104	C₂	#5071	1	1	yes	characteristic
105	C₂	#5082	1	1	yes	characteristic
106	C₂	#5089	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#384

C₂

yes

maximal, characteristic

#4936

C₂

yes

maximal, characteristic

C₂×#803

C₂

yes

maximal, characteristic

#4529

C₂

yes

maximal

#1836

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

C₃⋊#95

—

#803

C₄

yes

complemented

C₂×#97

—

#384

D₃

yes

characteristic

#382

—

#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

Q₈

yes

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

#97

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

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

—

C₄²

Dic₆

yes

—

C₄²

C₄×D₃

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#117

yes

—

C₃×C₄

#125

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₃⋊C₄

yes

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₄

#237

yes

—

C₂×C₄

#239

yes

—

C₈

—

C₂×C₃

#382

yes

characteristic

C₂×C₃

#428

yes

—

C₂×C₃

#489

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

C₂²

#932

yes

—

C₂²

#936

yes

characteristic

C₂²

#942

yes

—

C₂²

#947

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#890

yes

—

100

C₄

#892

yes

—

101

C₃

#1836

yes

characteristic, complemented, Sylow

102

C₂

#4964

yes

—

103

C₂

#4977

yes

characteristic

104

C₂

#5071

yes

characteristic

105

C₂

#5082

yes

characteristic

106

C₂

#5089

yes

—

	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	#1836
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
12. #803	C₄	semidirect
101. C₃	#1836	semidirect

H ≅ G/N

type

12. #803

C₄

semidirect

101. C₃

#1836

semidirect

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

subgroup	index
C₃×#384	2
#4936	2
C₂×#803	2
#4529	2
#1836	3

subgroup

index

Subgroups — #69575

Conjugacy classes of subgroups

Complements

Maximal subgroups