Group #71420 — subgroups

Subgroups — #71420

Order 384.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#812	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	#2150	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
8	C₂²×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₃×M₃₂	—	2	2	no	—
10	C₂×C₃×C₁₆	C₂²	1	2	yes	—
11	C₈×Dic₃	C₄	1	2	yes	—
12	C₈×Dic₃	C₂²	1	2	yes	—
13	#812	—	2	2	no	—
14	C₂×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×M₃₂	—	3	1	no	—
16	C₂×M₃₂	D₃	1	1	yes	characteristic
17	C₂×C₄×C₈	—	3	1	no	—
18	C₄⋊C₁₆	—	3	4	no	—
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₃	C₈	1	2	yes	—
21	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₈	D₄	1	2	yes	—
23	C₂×C₃×C₈	Q₈	1	2	yes	—
24	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₄×Dic₃	C₈	1	1	yes	characteristic
26	C₄×Dic₃	C₈	1	1	yes	characteristic
27	C₄×Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₃⋊C₈	—	2	2	no	—
29	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₃×C₁₆	—	2	2	no	—
31	C₃⋊C₁₆	—	2	2	no	—
32	C₂²×C₈	—	3	1	no	—
33	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
34	C₂×C₄²	—	3	1	no	—
35	M₃₂	—	2	2	no	—
36	M₃₂	—	6	2	no	—
37	C₂×C₁₆	—	3	2	no	—
38	C₂×C₁₆	C₂×D₃	1	2	yes	—
39	C₄×C₈	—	3	2	no	—
40	C₄×C₈	—	3	2	no	—
41	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
44	C₂×C₃×C₄	M₁₆	1	2	yes	—
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₃	—	2	4	no	—
47	C₂×Dic₃	C₂×C₈	1	2	yes	—
48	C₂×Dic₃	C₂×C₈	1	2	yes	—
49	C₃×C₈	C₄⋊C₄	1	2	yes	—
50	C₃×C₈	C₂×D₄	1	1	yes	characteristic
51	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₃⋊C₈	—	2	2	no	—
53	C₂²×C₄	—	3	2	no	—
54	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
55	C₂×C₈	—	3	2	no	—
56	C₂×C₈	—	6	2	no	—
57	C₂×C₈	Dic₆	1	2	yes	—
58	C₂×C₈	C₄×D₃	1	1	yes	characteristic
59	C₂×C₈	#67	1	2	yes	—
60	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
61	C₄²	—	3	1	no	—
62	C₄²	—	3	1	no	—
63	C₄²	—	3	2	no	—
64	C₁₆	—	2	2	no	—
65	C₁₆	—	6	2	no	—
66	C₂²×C₃	—	2	2	no	—
67	C₂²×C₃	C₄⋊C₈	1	2	yes	—
68	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
69	C₃×C₄	C₄⋊C₈	1	2	yes	—
70	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
72	Dic₃	—	2	2	no	—
73	Dic₃	M₃₂	1	4	yes	complemented
74	C₂³	D₃×C₈	1	1	yes	characteristic
75	C₂×C₄	—	3	2	no	—
76	C₂×C₄	—	3	2	no	—
77	C₂×C₄	—	6	4	no	—
78	C₂×C₄	D₃×C₈	1	1	yes	characteristic
79	C₂×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
80	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
81	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
82	C₈	—	6	2	no	—
83	C₈	Dic₃⋊C₄	1	2	yes	—
84	C₈	C₂×Dic₆	1	1	yes	characteristic
85	C₈	C₂×#67	1	1	yes	characteristic
86	C₂×C₃	—	2	2	no	—
87	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂×C₃	C₂×M₃₂	1	2	yes	—
89	C₂²	—	2	2	no	—
90	C₂²	Dic₃⋊C₈	1	2	yes	—
91	C₂²	C₂×D₃×C₈	1	1	yes	characteristic
92	C₄	—	3	4	no	—
93	C₄	—	6	2	no	—
94	C₄	Dic₃⋊C₈	1	2	yes	—
95	C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
96	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
97	C₃	#2150	1	1	yes	characteristic, complemented, Sylow
98	C₂	—	2	2	no	—
99	C₂	D₃×M₃₂	1	2	yes	—
100	C₂	C₂×Dic₃⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×M₃₂

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃

C₂

yes

maximal, characteristic

C₂×#812

C₂

yes

maximal, characteristic

Dic₃⋊C₁₆

C₂

yes

maximal, complemented

#2150

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₄

yes

characteristic

C₂²×C₃⋊C₈

C₄

yes

characteristic

C₃×M₃₂

—

C₂×C₃×C₁₆

C₂²

yes

—

C₈×Dic₃

C₄

yes

—

C₈×Dic₃

C₂²

yes

—

#812

—

C₂×C₃⋊C₁₆

C₂²

yes

—

C₂×M₃₂

—

C₂×M₃₂

D₃

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×Dic₃

C₈

yes

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

Q₈

yes

—

C₂×C₃×C₈

C₂³

yes

characteristic

C₄×Dic₃

C₈

yes

characteristic

C₄×Dic₃

C₈

yes

characteristic

C₄×Dic₃

C₂×C₄

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₂×C₈

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

characteristic

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

C₂×C₈

yes

—

C₂×Dic₃

C₂×C₈

yes

—

C₃×C₈

C₄⋊C₄

yes

—

C₃×C₈

C₂×D₄

yes

characteristic

C₃×C₈

C₂×Q₈

yes

characteristic

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆

yes

—

C₂×C₈

C₄×D₃

yes

characteristic

C₂×C₈

#67

yes

—

C₂×C₈

C₂²×D₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

C₄⋊C₈

yes

—

C₂²×C₃

C₂²×C₈

yes

characteristic

C₃×C₄

C₄⋊C₈

yes

—

C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

C₃×C₄

C₂×M₁₆

yes

characteristic

Dic₃

—

Dic₃

M₃₂

yes

complemented

C₂³

D₃×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×C₈

yes

characteristic

C₂×C₄

ΓL(1,𝔽₂₅)

yes

—

C₂×C₄

Dic₃⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₈

—

C₈

Dic₃⋊C₄

yes

—

C₈

C₂×Dic₆

yes

characteristic

C₈

C₂×#67

yes

characteristic

C₂×C₃

—

C₂×C₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₃

C₂×M₃₂

yes

—

C₂²

—

C₂²

Dic₃⋊C₈

yes

—

C₂²

C₂×D₃×C₈

yes

characteristic

C₄

—

C₄

—

C₄

Dic₃⋊C₈

yes

—

C₄

C₂×ΓL(1,𝔽₂₅)

yes

characteristic

C₄

C₂×Dic₃⋊C₄

yes

characteristic

C₃

#2150

yes

characteristic, complemented, Sylow

C₂

—

C₂

D₃×M₃₂

yes

—

100

C₂

C₂×Dic₃⋊C₈

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₂×C₃×M₃₂	C₂
prime radical	C₂³×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	#2150
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃

commutator, socle

C₂²×C₃

C₂²×C₈

Fitting

C₂×C₃×M₃₂

C₂

prime radical

C₂³×C₃

C₂×C₈

cosocle

C₂×C₃×C₈

C₂³

hypercenter

C₂×M₃₂

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
4. Dic₃⋊C₁₆	C₂	semidirect
73. Dic₃	M₃₂	semidirect
97. C₃	#2150	semidirect

H ≅ G/N

type

4. Dic₃⋊C₁₆

C₂

semidirect

73. Dic₃

M₃₂

semidirect

97. C₃

#2150

semidirect

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

subgroup	index
C₂×C₃×M₃₂	2
C₂×C₈×Dic₃	2
C₂×#812	2
Dic₃⋊C₁₆	2
#2150	3

subgroup

index

C₂×C₃×M₃₂

C₂×C₈×Dic₃

C₂×#812

Dic₃⋊C₁₆

#2150

Subgroups — #71420

Conjugacy classes of subgroups

Complements

Maximal subgroups