Group #67710 — subgroups

Subgroups — #67710

Order 384.

150 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 65 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×#423	C₂	1	1	yes	maximal, characteristic
2	#4925	C₂	1	1	yes	maximal, characteristic
3	C₄×#208	C₂	1	1	yes	maximal, characteristic
4	#1299	—	3	1	no	maximal, Sylow
5	C₂×C₃×M₁₆	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂×#208	—	2	1	no	—
8	C₃×C₄⋊C₈	—	2	1	no	—
9	C₃×C₄⋊C₈	—	2	1	no	—
10	#830	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
12	#803	C₄	1	2	yes	—
13	C₄×C₃⋊C₈	—	2	1	no	—
14	C₄×C₃⋊C₈	—	2	1	no	—
15	#431	—	3	1	no	—
16	#423	D₃	1	1	yes	characteristic
17	C₄×M₁₆	—	3	1	no	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
21	C₃×M₁₆	—	4	2	no	—
22	C₂×C₃×C₈	—	2	2	no	—
23	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₃×C₄²	D₄	1	1	yes	characteristic
26	C₃×C₄²	D₄	1	1	yes	characteristic
27	#208	—	2	4	no	—
28	C₂×C₃⋊C₈	—	2	2	no	—
29	C₂×C₃⋊C₈	—	2	2	no	—
30	C₂×M₁₆	—	6	1	no	—
31	C₂×M₁₆	Dic₃	1	2	yes	—
32	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	3	2	no	—
36	C₂²⋊C₈	—	6	1	no	—
37	#97	—	3	2	no	—
38	C₄×C₈	—	6	1	no	—
39	C₄×C₈	—	6	1	no	—
40	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₃×C₄	—	2	1	no	—
42	C₂×C₃×C₄	—	2	1	no	—
43	C₂×C₃×C₄	—	2	2	no	—
44	C₂×C₃×C₄	—	2	2	no	—
45	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₃×C₈	—	4	2	no	—
52	C₃⋊C₈	—	2	4	no	—
53	C₃⋊C₈	—	4	2	no	—
54	C₂²×C₄	Dic₆	1	1	yes	characteristic
55	C₂²×C₄	D₁₂	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
57	M₁₆	—	4	2	no	—
58	M₁₆	—	6	4	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	6	2	no	—
61	C₂×C₈	—	6	2	no	—
62	C₄²	C₄×D₃	1	1	yes	characteristic
63	C₄²	C₄×D₃	1	1	yes	characteristic
64	C₄²	#67	1	1	yes	characteristic
65	C₄²	#67	1	1	yes	characteristic
66	C₂²×C₃	—	2	1	no	—
67	C₂²×C₃	—	2	1	no	—
68	C₂²×C₃	—	2	1	no	—
69	C₂²×C₃	#95	1	1	yes	characteristic
70	C₃×C₄	—	2	1	no	—
71	C₃×C₄	—	2	1	no	—
72	C₃×C₄	—	2	1	no	—
73	C₃×C₄	#100	1	1	yes	characteristic
74	C₃×C₄	#101	1	1	yes	characteristic
75	C₃×C₄	#104	1	2	yes	—
76	C₃×C₄	#108	1	2	yes	—
77	C₂³	C₄⋊Dic₃	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
83	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
84	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
85	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	#217	1	1	yes	characteristic
88	C₈	—	4	2	no	—
89	C₈	—	6	4	no	—
90	C₈	—	12	2	no	—
91	C₂×C₃	—	4	1	no	—
92	C₂×C₃	#339	1	1	yes	characteristic
93	C₂×C₃	#341	1	1	yes	characteristic
94	C₂×C₃	#343	1	1	yes	characteristic
95	C₂²	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₃⋊#95	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	#822	1	2	yes	—
103	C₄	#825	1	2	yes	—
104	C₄	#833	1	1	yes	characteristic
105	C₄	#836	1	1	yes	characteristic
106	C₃	#1299	1	1	yes	characteristic, complemented, Sylow
107	C₂	—	4	1	no	—
108	C₂	#4530	1	1	yes	characteristic
109	C₂	#4555	1	1	yes	characteristic
110	C₂	#4558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#423

C₂

yes

maximal, characteristic

#4925

C₂

yes

maximal, characteristic

C₄×#208

C₂

yes

maximal, characteristic

#1299

—

maximal, Sylow

C₂×C₃×M₁₆

C₄

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

#830

—

C₄⋊(C₃⋊C₈)

C₄

yes

—

#803

C₄

yes

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

#431

—

#423

D₃

yes

characteristic

C₄×M₁₆

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

Q₈

yes

characteristic

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

Dic₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₄⋊C₄

yes

characteristic

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

Dic₆

yes

characteristic

C₂²×C₄

D₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₃

yes

characteristic

C₄²

C₄×D₃

yes

C₄²

yes

C₄²

yes

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#95

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

C₃×C₄

#108

yes

—

C₂³

C₄⋊Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₃

yes

characteristic

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

Dic₃⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₃)⋊C₄

yes

characteristic

C₂×C₄

#217

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

C₃⋊#95

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#822

yes

—

103

C₄

#825

yes

—

104

C₄

#833

yes

characteristic

105

C₄

#836

yes

characteristic

106

C₃

#1299

yes

characteristic, complemented, Sylow

107

C₂

—

108

C₂

#4530

yes

characteristic

109

C₂

#4555

yes

characteristic

110

C₂

#4558

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂×C₃×C₄	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#423	C₂
socle	C₂²×C₃	#95
prime radical	C₂³×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#423	D₃
hypocenter	C₃	#1299
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

#217

commutator

C₂×C₃×C₄

C₄²

Frattini

C₂×C₄²

C₂×D₃

C₂

C₂²×C₃

C₂³×C₃

C₄⋊C₄

cosocle

C₂×C₃×C₄²

C₂²

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
106. C₃	#1299	semidirect

H ≅ G/N

type

106. C₃

#1299

semidirect

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

subgroup	index
C₃×#423	2
#4925	2
C₄×#208	2
#1299	3

subgroup

index

Subgroups — #67710

Conjugacy classes of subgroups

Complements

Maximal subgroups