Group #30743312 — subgroups

Subgroups — #30743312

Order 17152.

110 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22562117	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₆₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1397	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
5	C₂×C₆₇⋊C₃₂	—	2	1	no	—
6	C₄⋊(C₆₇⋊C₁₆)	C₄	1	1	yes	characteristic
7	Dic₆₇⋊C₁₆	—	2	1	no	—
8	#16863171	—	2	1	no	—
9	C₂×C₃₂×C₆₇	—	2	1	no	—
10	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₃₄	C₈	1	1	yes	characteristic
12	C₂×C₆₇⋊C₁₆	—	2	1	no	—
13	C₄⋊(C₆₇⋊C₈)	C₈	1	1	yes	characteristic
14	Dic₆₇⋊C₈	—	2	1	no	—
15	C₈×Dic₆₇	—	2	1	no	—
16	C₆₇⋊C₃₂	—	4	1	no	—
17	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
19	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₆₇	—	2	2	no	—
21	C₂×Dic₁₃₄	C₁₆	1	1	yes	characteristic
22	C₂×C₆₇⋊C₈	—	2	1	no	—
23	C₄⋊Dic₆₇	C₁₆	1	1	yes	characteristic
24	Dic₆₇⋊C₄	—	2	1	no	—
25	C₄×Dic₆₇	—	2	1	no	—
26	C₆₇⋊C₁₆	—	4	1	no	—
27	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
29	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
30	C₁₆×C₆₇	—	2	1	no	—
31	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
32	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
35	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
36	C₂×Dic₆₇	—	2	1	no	—
37	C₂×Dic₆₇	—	2	1	no	—
38	Dic₁₃₄	—	4	1	no	—
39	C₈×C₆₇	—	2	1	no	—
40	C₈×C₆₇	#103	1	1	yes	characteristic
41	C₈×C₆₇	#104	1	1	yes	characteristic
42	C₆₇⋊C₈	—	4	1	no	—
43	C₂²×C₆₇	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄×C₆₇	—	2	1	no	—
45	C₄×C₆₇	—	2	1	no	—
46	C₄×C₆₇	Q₈⋊C₈	1	1	yes	characteristic
47	C₄×C₆₇	#350	1	1	yes	characteristic
48	Dic₆₇	—	4	1	no	—
49	Dic₆₇	—	4	1	no	—
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	—	67	1	no	—
55	C₄⋊C₃₂	—	67	1	no	—
56	#1397	D₆₇	1	1	yes	characteristic
57	C₆₇	#7393	1	1	yes	characteristic, complemented, Sylow
58	C₈×Q₈	—	67	1	no	—
59	C₂×C₃₂	—	2	1	no	—
60	C₂×C₃₂	—	134	1	no	—
61	C₄⋊C₁₆	—	67	1	no	—
62	C₄⋊C₁₆	—	134	1	no	—
63	#346	—	134	1	no	—
64	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
65	C₄×Q₈	—	67	1	no	—
66	C₂×C₁₆	—	134	1	no	—
67	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
68	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
69	C₄⋊C₈	—	67	1	no	—
70	C₄⋊C₈	—	134	1	no	—
71	C₄×C₈	—	134	1	no	—
72	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
73	C₃₂	—	2	2	no	—
74	C₃₂	—	268	1	no	—
75	C₂×Q₈	—	67	1	no	—
76	C₂×C₈	—	134	1	no	—
77	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
78	C₂×C₈	#11761611	1	1	yes	characteristic
79	C₄⋊C₄	—	67	1	no	—
80	C₄⋊C₄	—	134	1	no	—
81	C₄²	—	134	1	no	—
82	C₄²	C₈×D₆₇	1	1	yes	characteristic
83	C₁₆	—	2	1	no	—
84	C₁₆	—	268	1	no	—
85	C₁₆	#11761617	1	1	yes	characteristic
86	C₁₆	Dic₂₆₈	1	1	yes	characteristic
87	Q₈	—	268	1	no	—
88	C₂×C₄	—	134	1	no	—
89	C₂×C₄	—	134	1	no	—
90	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
91	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
92	C₂×C₄	#14523982	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	268	1	no	—
95	C₈	#14523968	1	1	yes	characteristic
96	C₈	#14523971	1	1	yes	characteristic
97	C₂²	(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	268	1	no	—
101	C₄	—	268	1	no	—
102	C₄	#16863128	1	1	yes	characteristic
103	C₄	#16863179	1	1	yes	characteristic
104	C₂	#22558189	1	1	yes	characteristic
105	C₂	#22558348	1	1	yes	characteristic
106	C₂	#22558352	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562117

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₃₂)

C₂

yes

maximal, characteristic

C₆₇×#1397

C₂

yes

maximal, characteristic

C₈×Dic₁₃₄

C₄

yes

characteristic

C₂×C₆₇⋊C₃₂

—

C₄⋊(C₆₇⋊C₁₆)

C₄

yes

characteristic

Dic₆₇⋊C₁₆

—

#16863171

—

C₂×C₃₂×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₈

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₄⋊(C₆₇⋊C₈)

C₈

yes

characteristic

Dic₆₇⋊C₈

—

C₈×Dic₆₇

—

C₆₇⋊C₃₂

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₃₂×C₆₇

—

C₂×Dic₁₃₄

C₁₆

yes

characteristic

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

C₁₆

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

M₁₆

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

SD₁₆

yes

characteristic

C₁₆×C₆₇

Q₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

M₃₂

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²⋊C₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

Q₈⋊C₈

yes

characteristic

C₄×C₆₇

#350

yes

characteristic

Dic₆₇

—

Dic₆₇

—

#7393

—

maximal, Sylow

C₂×C₆₇

Q₈⋊C₁₆

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₄⋊C₃₂

—

#1397

D₆₇

yes

characteristic

C₆₇

#7393

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

134

—

#346

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

134

—

C₂×C₁₆

D₂₆₈

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄×C₈

—

134

—

C₄×C₈

C₄×D₆₇

yes

characteristic

C₃₂

—

C₃₂

—

268

—

C₂×Q₈

—

C₂×C₈

—

134

—

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₈

#11761611

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₈×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

268

—

C₁₆

#11761617

yes

characteristic

C₁₆

Dic₂₆₈

yes

characteristic

Q₈

—

268

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

(C₂×D₆₇)⋊C₈

yes

characteristic

C₂×C₄

C₁₆×D₆₇

yes

characteristic

C₂×C₄

#14523982

yes

characteristic

C₈

—

C₈

—

268

—

C₈

yes

C₈

yes

C₂²

(C₂×D₆₇)⋊C₁₆

yes

characteristic

C₄

—

C₄

—

100

C₄

—

268

—

101

C₄

—

268

—

102

C₄

#16863128

yes

characteristic

103

C₄

#16863179

yes

characteristic

104

C₂

#22558189

yes

characteristic

105

C₂

#22558348

yes

characteristic

106

C₂

#22558352

yes

characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×#1397	C₂
socle, prime radical	C₂²×C₆₇	C₂²⋊C₁₆
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	#1397	D₆₇
hypocenter	C₆₇	#7393
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂×C₈

(C₂×D₆₇)⋊C₄

commutator

C₂×C₄×C₆₇

C₂×C₁₆

Frattini

C₄×C₁₆

C₂×D₆₇

Fitting

C₆₇×#1397

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 ◁ #1397

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
#22562117	2
C₄⋊(C₆₇⋊C₃₂)	2
C₆₇×#1397	2
#7393	67

subgroup

index

#22562117

C₄⋊(C₆₇⋊C₃₂)

C₆₇×#1397

#7393

Subgroups — #30743312

Conjugacy classes of subgroups

Complements

Maximal subgroups