Group #30856081 — subgroups

Subgroups — #30856081

Order 17152.

1062 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 411 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	#22560352	C₂	1	8	yes	maximal, complemented
2	#22559903	C₂	1	6	yes	maximal
3	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	C₂²	1	4	yes	—
5	#16863151	C₂²	1	24	yes	—
6	C₂²×C₄²×C₆₇	C₄	1	4	yes	—
7	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
8	C₄³×C₆₇	C₄	1	4	yes	—
9	C₄³×C₆₇	C₂²	1	4	yes	—
10	C₂×#11761612	—	2	24	no	—
11	C₂²×C₆₇⋊C₈	C₂³	1	8	yes	—
12	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	—	2	12	no	—
15	C₂×C₄²×C₆₇	—	2	16	no	—
16	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
20	#11761612	—	4	16	no	—
21	C₂×C₆₇⋊C₈	—	2	24	no	—
22	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	—	2	12	no	—
24	C₂²×C₄×C₆₇	—	2	24	no	—
25	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
26	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	#41	1	12	yes	—
28	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
29	C₄²×C₆₇	—	2	12	no	—
30	C₄²×C₆₇	—	2	24	no	—
31	C₄²×C₆₇	—	2	12	no	—
32	C₄²×C₆₇	M₁₆	1	16	yes	complemented
33	C₂³×C₆₇	—	2	4	no	—
34	C₂³×C₆₇	#117	1	6	yes	—
35	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	—	2	48	no	—
37	C₂×C₄×C₆₇	—	2	12	no	—
38	C₂×C₄×C₆₇	—	2	12	no	—
39	C₂×C₄×C₆₇	#117	1	6	yes	—
40	C₂×C₄×C₆₇	#126	1	16	yes	—
41	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	—
42	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
43	C₆₇⋊C₈	—	4	8	no	—
44	C₂²×C₆₇	—	2	12	no	—
45	C₂²×C₆₇	#383	1	4	yes	—
46	C₂²×C₆₇	C₂×#117	1	3	yes	—
47	C₂²×C₆₇	C₂²×M₁₆	1	4	yes	—
48	C₄×C₆₇	—	2	24	no	—
49	C₄×C₆₇	#383	1	4	yes	—
50	C₄×C₆₇	C₂×#126	1	4	yes	—
51	#20772	—	67	1	no	maximal, Sylow
52	C₂×C₆₇	—	2	4	no	—
53	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
54	C₂×C₆₇	#2920	1	6	yes	—
55	C₂×C₄³	D₆₇	1	1	yes	characteristic
56	#1839	—	67	8	no	—
57	#1728	—	67	6	no	—
58	C₆₇	#20772	1	1	yes	characteristic, complemented, Sylow
59	C₂²×M₁₆	—	67	4	no	—
60	C₂²×C₄²	Dic₆₇	1	4	yes	—
61	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
62	C₄³	Dic₆₇	1	4	yes	—
63	C₄³	C₂×D₆₇	1	4	yes	—
64	#336	—	67	24	no	—
65	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
66	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
67	C₂×M₁₆	—	134	24	no	—
68	C₂²×C₈	—	67	8	no	—
69	C₂×C₄²	—	2	12	no	—
70	C₂×C₄²	—	2	16	no	—
71	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
72	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
73	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
74	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
75	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	2	24	no	—
78	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
79	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
80	C₂²×C₄	#11761628	1	12	yes	—
81	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
82	M₁₆	—	268	16	no	—
83	C₂×C₈	—	134	24	no	—
84	C₄²	—	2	12	no	—
85	C₄²	—	2	24	no	—
86	C₄²	—	2	12	no	—
87	C₄²	#11761612	1	16	yes	complemented
88	C₂³	—	2	4	no	—
89	C₂³	#14524002	1	6	yes	—
90	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
91	C₂×C₄	—	2	48	no	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	12	no	—
94	C₂×C₄	#14524002	1	6	yes	—
95	C₂×C₄	#14524037	1	16	yes	—
96	C₂×C₄	C₂×#11761612	1	24	yes	—
97	C₂×C₄	C₂×#11761628	1	6	yes	—
98	C₈	—	268	8	no	—
99	C₂²	—	2	12	no	—
100	C₂²	#16863264	1	4	yes	—
101	C₂²	C₂×#14524002	1	3	yes	—
102	C₂²	C₂²×#11761612	1	4	yes	—
103	C₄	—	2	24	no	—
104	C₄	#16863264	1	4	yes	—
105	C₄	C₂×#14524037	1	4	yes	—
106	C₂	—	2	4	no	—
107	C₂	C₂×#16863264	1	1	yes	characteristic
108	C₂	#22568842	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560352

C₂

yes

maximal, complemented

#22559903

C₂

yes

maximal

C₂×C₄³×C₆₇

C₂

yes

maximal, characteristic

C₂²×#11761612

C₂²

yes

—

#16863151

C₂²

yes

—

C₂²×C₄²×C₆₇

C₄

yes

—

C₂²×C₄²×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₄

yes

—

C₄³×C₆₇

C₂²

yes

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

C₂³

yes

—

C₂³×C₄×C₆₇

C₂×C₄

yes

—

C₂³×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂⁴×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

M₁₆

yes

complemented

C₂³×C₆₇

—

C₂³×C₆₇

#117

yes

—

C₂³×C₆₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#126

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#383

yes

—

C₂²×C₆₇

C₂×#117

yes

—

C₂²×C₆₇

C₂²×M₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

#383

yes

—

C₄×C₆₇

C₂×#126

yes

—

#20772

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₂×#383

yes

characteristic

C₂×C₆₇

#2920

yes

—

C₂×C₄³

D₆₇

yes

characteristic

#1839

—

#1728

—

C₆₇

#20772

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂²×C₄²

Dic₆₇

yes

—

C₂²×C₄²

C₂×D₆₇

yes

—

C₄³

Dic₆₇

yes

—

C₄³

C₂×D₆₇

yes

—

#336

—

C₂³×C₄

C₂×Dic₆₇

yes

—

C₂³×C₄

C₂²×D₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

—

C₂⁴

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₆₇

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

#11761628

yes

—

C₂²×C₄

C₂³×D₆₇

yes

characteristic

M₁₆

—

268

—

C₂×C₈

—

134

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11761612

yes

complemented

C₂³

—

C₂³

#14524002

yes

—

C₂³

C₂³×Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14524002

yes

—

C₂×C₄

#14524037

yes

—

C₂×C₄

C₂×#11761612

yes

—

C₂×C₄

C₂×#11761628

yes

—

C₈

—

268

—

C₂²

—

100

C₂²

#16863264

yes

—

101

C₂²

C₂×#14524002

yes

—

102

C₂²

C₂²×#11761612

yes

—

103

C₄

—

104

C₄

#16863264

yes

—

105

C₄

C₂×#14524037

yes

—

106

C₂

—

107

C₂

C₂×#16863264

yes

characteristic

108

C₂

#22568842

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₂×C₄³×C₆₇	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	#20772
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂³×C₄

Fitting

C₂×C₄³×C₆₇

C₂

prime radical

C₂⁴×C₆₇

C₂²×C₄

cosocle

C₂²×C₄×C₆₇

C₂⁴

hypercenter

C₂×C₄³

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

N	H ≅ G/N	type
1. #22560352	C₂	semidirect
32. C₄²×C₆₇	M₁₆	semidirect
58. C₆₇	#20772	semidirect
87. C₄²	#11761612	semidirect

H ≅ G/N

type

1. #22560352

C₂

semidirect

32. C₄²×C₆₇

M₁₆

semidirect

58. C₆₇

#20772

semidirect

87. C₄²

#11761612

semidirect

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

subgroup	index
#22560352	2
#22559903	2
C₂×C₄³×C₆₇	2
#20772	67

subgroup

index

#22560352

#22559903

C₂×C₄³×C₆₇

#20772

Subgroups — #30856081

Conjugacy classes of subgroups

Complements

Maximal subgroups