Group #30743311 — subgroups

Subgroups — #30743311

Order 17152.

110 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₄⋊(C₆₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	#22558343	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊C₃₂	—	2	1	no	—
5	C₂×C₆₇⋊C₃₂	—	2	1	no	—
6	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₆₇	—	2	1	no	—
8	C₄⋊C₁₆×C₆₇	C₄	1	1	yes	characteristic
9	#346×C₆₇	—	2	1	no	—
10	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₆₇⋊C₃₂	—	2	2	no	—
12	C₆₇⋊C₃₂	—	4	1	no	—
13	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₆₇	—	2	1	no	—
15	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
17	C₄⋊C₈×C₆₇	—	2	1	no	—
18	C₄⋊C₈×C₆₇	C₈	1	1	yes	characteristic
19	C₄×C₈×C₆₇	—	2	1	no	—
20	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Q₈×C₆₇	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₆₇	—	2	1	no	—
23	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
25	C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₄⋊C₄×C₆₇	C₁₆	1	1	yes	characteristic
27	C₄²×C₆₇	—	2	1	no	—
28	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
29	C₁₆×C₆₇	—	2	1	no	—
30	C₁₆×C₆₇	—	4	1	no	—
31	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
32	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
33	Q₈×C₆₇	—	4	1	no	—
34	C₂×C₄×C₆₇	—	2	1	no	—
35	C₂×C₄×C₆₇	—	2	1	no	—
36	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
39	C₈×C₆₇	—	2	1	no	—
40	C₈×C₆₇	—	4	1	no	—
41	C₈×C₆₇	#103	1	1	yes	characteristic
42	C₈×C₆₇	#104	1	1	yes	characteristic
43	C₂²×C₆₇	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄×C₆₇	—	2	1	no	—
45	C₄×C₆₇	—	2	1	no	—
46	C₄×C₆₇	—	4	1	no	—
47	C₄×C₆₇	—	4	1	no	—
48	C₄×C₆₇	Q₈⋊C₈	1	1	yes	characteristic
49	C₄×C₆₇	#350	1	1	yes	characteristic
50	#7393	—	67	1	no	maximal, Sylow
51	C₂×C₆₇	Q₈⋊C₁₆	1	1	yes	characteristic
52	C₂×C₆₇	#1400	1	1	yes	characteristic
53	C₂×C₆₇	#1401	1	1	yes	characteristic
54	#2183	D₆₇	1	1	yes	characteristic
55	C₄⋊C₃₂	—	67	1	no	—
56	#1397	—	67	1	no	—
57	C₆₇	#7393	1	1	yes	characteristic, complemented, Sylow
58	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
59	C₂×C₃₂	—	134	1	no	—
60	C₂×C₃₂	—	134	1	no	—
61	C₄⋊C₁₆	—	2	1	no	—
62	C₄⋊C₁₆	Dic₆₇	1	1	yes	characteristic
63	#346	—	2	1	no	—
64	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
65	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
68	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₆₇⋊C₈	1	1	yes	characteristic
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
73	C₃₂	—	268	1	no	—
74	C₃₂	—	134	2	no	—
75	C₂×Q₈	C₆₇⋊C₁₆	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
78	C₂×C₈	#11761612	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	C₆₇⋊C₁₆	1	1	yes	characteristic
81	C₄²	—	2	1	no	—
82	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	—	4	1	no	—
85	C₁₆	#11761615	1	1	yes	characteristic
86	C₁₆	#11761618	1	1	yes	characteristic
87	Q₈	—	4	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
91	C₂×C₄	C₂×C₆₇⋊C₁₆	1	1	yes	characteristic
92	C₂×C₄	#14523983	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	1	no	—
95	C₈	#14523967	1	1	yes	characteristic
96	C₈	#14523970	1	1	yes	characteristic
97	C₂²	C₂²⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	Q₈⋊(C₆₇⋊C₈)	1	1	yes	characteristic
103	C₄	#16863180	1	1	yes	characteristic
104	C₂	Q₈⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
105	C₂	#22558349	1	1	yes	characteristic
106	C₂	#22558351	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₆₇⋊C₃₂)

C₂

yes

maximal, characteristic

#22558343

C₂

yes

maximal, characteristic

C₆₇×#2183

C₂

yes

maximal, characteristic

C₂×C₆₇⋊C₃₂

—

C₂×C₆₇⋊C₃₂

—

C₈×Q₈×C₆₇

C₄

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₄⋊C₁₆×C₆₇

C₄

yes

characteristic

#346×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₆₇⋊C₃₂

—

C₆₇⋊C₃₂

—

C₄×Q₈×C₆₇

C₈

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₈

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₇

C₁₆

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

M₁₆

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₁₆

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

SD₁₆

yes

characteristic

C₁₆×C₆₇

Q₁₆

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

M₃₂

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₂²×C₆₇

C₂²⋊C₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

Q₈⋊C₈

yes

C₄×C₆₇

yes

—

C₂×C₆₇

Q₈⋊C₁₆

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

C₄⋊C₃₂

—

#1397

—

C₆₇

#7393

yes

characteristic, complemented, Sylow

C₈×Q₈

Dic₆₇

yes

characteristic

C₂×C₃₂

—

134

—

C₂×C₃₂

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Dic₆₇

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₄×Q₈

C₆₇⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₆₇⋊C₈

yes

characteristic

C₄×C₈

—

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₃₂

—

268

—

C₃₂

—

134

—

C₂×Q₈

C₆₇⋊C₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₈

#11761612

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₆₇⋊C₁₆

yes

characteristic

C₄²

—

C₄²

C₂×C₆₇⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₆₇⋊C₈)

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

characteristic

C₂×C₄

#14523983

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

C₂²⋊(C₆₇⋊C₁₆)

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

Q₈⋊(C₆₇⋊C₈)

yes

characteristic

103

C₄

#16863180

yes

characteristic

104

C₂

Q₈⋊(C₆₇⋊C₁₆)

yes

characteristic

105

C₂

#22558349

yes

characteristic

106

C₂

#22558351

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×#2183	C₂
socle, prime radical	C₂²×C₆₇	C₂²⋊C₁₆
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	#2183	D₆₇
hypocenter	C₆₇	#7393
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #2183

subgroup

G/H

center

C₂×C₈

C₂²⋊Dic₆₇

commutator

C₂×C₄×C₆₇

C₂×C₁₆

Frattini

C₄×C₁₆

C₂×D₆₇

Fitting

C₆₇×#2183

C₂

socle, prime radical

C₂²×C₆₇

C₂²⋊C₁₆

cosocle

C₄×C₁₆×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #2183

N	H ≅ G/N	type
57. C₆₇	#7393	semidirect

H ≅ G/N

type

57. C₆₇

#7393

semidirect

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

subgroup	index
C₄⋊(C₆₇⋊C₃₂)	2
#22558343	2
C₆₇×#2183	2
#7393	67

subgroup

index

C₄⋊(C₆₇⋊C₃₂)

#22558343

C₆₇×#2183

#7393

Subgroups — #30743311

Conjugacy classes of subgroups

Complements

Maximal subgroups