Group #30699591 — subgroups

Subgroups — C₆₇×#20340 #30699591

Order 17152.

1956 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 1114 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×#568×C₆₇	C₂	1	3	yes	maximal
2	C₂×#338×C₆₇	C₂	1	12	yes	maximal, complemented
3	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
4	#568×C₆₇	—	2	24	no	—
5	C₂×#131×C₆₇	C₄	1	24	yes	—
6	C₂²×M₁₆×C₆₇	C₂²	1	18	yes	—
7	#338×C₆₇	C₂²	1	16	yes	—
8	C₂×#41×C₆₇	C₂×C₄	1	12	yes	—
9	C₂×#41×C₆₇	D₄	1	12	yes	—
10	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
11	C₂²×D₄×C₆₇	Q₈	1	3	yes	—
12	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
13	#131×C₆₇	—	2	96	no	—
14	C₂×M₁₆×C₆₇	—	2	72	no	—
15	C₂×M₁₆×C₆₇	C₂×C₄	1	36	yes	—
16	C₂×M₁₆×C₆₇	C₂³	1	12	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	36	yes	—
18	C₂⁴×C₆₇	C₂×Q₈	1	3	yes	—
19	#41×C₆₇	C₄²	1	16	yes	—
20	#41×C₆₇	C₂²⋊C₄	1	48	yes	—
21	C₂×Q₈×C₆₇	C₄⋊C₄	1	12	yes	—
22	C₂×D₄×C₆₇	C₄⋊C₄	1	36	yes	—
23	C₂²×C₄×C₆₇	C₂²×C₄	1	18	yes	—
24	C₂²×C₄×C₆₇	C₂×D₄	1	18	yes	—
25	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
26	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
27	M₁₆×C₆₇	—	2	72	no	—
28	M₁₆×C₆₇	—	2	24	no	—
29	C₂×C₈×C₆₇	—	2	72	no	—
30	C₂×C₈×C₆₇	C₂²×C₄	1	24	yes	—
31	C₂³×C₆₇	—	8	3	no	—
32	C₂³×C₆₇	C₂×C₄⋊C₄	1	18	yes	—
33	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
34	Q₈×C₆₇	#95	1	16	yes	—
35	D₄×C₆₇	#95	1	48	yes	—
36	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
37	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	36	yes	—
38	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	18	yes	—
39	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
40	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
41	C₈×C₆₇	—	2	24	no	—
42	C₂²×C₆₇	—	4	1	no	—
43	C₂²×C₆₇	—	4	18	no	—
44	C₂²×C₆₇	C₂×#95	1	12	yes	—
45	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
46	C₄×C₆₇	C₂×#95	1	12	yes	—
47	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
48	C₄×C₆₇	C₂²×C₂²⋊C₄	1	3	yes	—
49	#20340	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₆₇	—	2	3	no	—
51	C₂×C₆₇	—	2	12	no	—
52	C₂×C₆₇	C₂²×#95	1	1	yes	characteristic
53	C₂×#568	C₂×C₆₇	1	3	yes	—
54	C₂×#338	C₂×C₆₇	1	12	yes	complemented
55	C₆₇	#20340	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
57	#568	—	2	24	no	—
58	C₂×#131	C₄×C₆₇	1	24	yes	—
59	C₂²×M₁₆	C₂²×C₆₇	1	18	yes	—
60	#338	C₂²×C₆₇	1	16	yes	—
61	C₂×#41	C₂×C₄×C₆₇	1	12	yes	—
62	C₂×#41	D₄×C₆₇	1	12	yes	—
63	C₂²×Q₈	Q₈×C₆₇	1	1	yes	characteristic
64	C₂²×D₄	Q₈×C₆₇	1	3	yes	—
65	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
66	#131	—	2	96	no	—
67	C₂×M₁₆	—	2	72	no	—
68	C₂×M₁₆	C₂×C₄×C₆₇	1	36	yes	—
69	C₂×M₁₆	C₂³×C₆₇	1	12	yes	—
70	C₂²×C₈	C₂×C₄×C₆₇	1	36	yes	—
71	C₂⁴	C₂×Q₈×C₆₇	1	3	yes	—
72	#41	C₄²×C₆₇	1	16	yes	—
73	#41	C₂²⋊C₄×C₆₇	1	48	yes	—
74	C₂×Q₈	C₄⋊C₄×C₆₇	1	12	yes	—
75	C₂×D₄	C₄⋊C₄×C₆₇	1	36	yes	—
76	C₂²×C₄	C₂²×C₄×C₆₇	1	18	yes	—
77	C₂²×C₄	C₂×D₄×C₆₇	1	18	yes	—
78	C₂²×C₄	C₂×Q₈×C₆₇	1	3	yes	—
79	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
80	M₁₆	—	2	72	no	—
81	M₁₆	—	2	24	no	—
82	C₂×C₈	—	2	72	no	—
83	C₂×C₈	C₂²×C₄×C₆₇	1	24	yes	—
84	C₂³	—	8	3	no	—
85	C₂³	C₂×C₄⋊C₄×C₆₇	1	18	yes	—
86	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
87	Q₈	#95×C₆₇	1	16	yes	—
88	D₄	#95×C₆₇	1	48	yes	—
89	C₂×C₄	C₂×C₄²×C₆₇	1	12	yes	—
90	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	36	yes	—
91	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	18	yes	—
92	C₂×C₄	C₂³×C₄×C₆₇	1	3	yes	—
93	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
94	C₈	—	2	24	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	18	no	—
97	C₂²	C₂×#95×C₆₇	1	12	yes	—
98	C₂²	C₂²×C₄⋊C₄×C₆₇	1	3	yes	—
99	C₄	C₂×#95×C₆₇	1	12	yes	—
100	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
101	C₄	C₂²×C₂²⋊C₄×C₆₇	1	3	yes	—
102	C₂	—	2	3	no	—
103	C₂	—	2	12	no	—
104	C₂	C₂²×#95×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#568×C₆₇

C₂

yes

maximal

C₂×#338×C₆₇

C₂

yes

maximal, complemented

C₂²×#41×C₆₇

C₂²

yes

characteristic

#568×C₆₇

—

C₂×#131×C₆₇

C₄

yes

—

C₂²×M₁₆×C₆₇

C₂²

yes

—

#338×C₆₇

C₂²

yes

—

C₂×#41×C₆₇

C₂×C₄

yes

—

C₂×#41×C₆₇

D₄

yes

—

C₂²×Q₈×C₆₇

Q₈

yes

characteristic

C₂²×D₄×C₆₇

Q₈

yes

—

C₂³×C₄×C₆₇

C₂³

yes

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

C₂³

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂⁴×C₆₇

C₂×Q₈

yes

—

#41×C₆₇

C₄²

yes

—

#41×C₆₇

C₂²⋊C₄

yes

—

C₂×Q₈×C₆₇

C₄⋊C₄

yes

—

C₂×D₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×Q₈

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×C₄⋊C₄

yes

—

C₂³×C₆₇

C₂²×Q₈

yes

characteristic

Q₈×C₆₇

#95

yes

—

D₄×C₆₇

#95

yes

—

C₂×C₄×C₆₇

C₂×C₄²

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂³×C₄

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#95

yes

—

C₂²×C₆₇

C₂²×C₄⋊C₄

yes

—

C₄×C₆₇

C₂×#95

yes

—

C₄×C₆₇

C₂²×C₄²

yes

characteristic

C₄×C₆₇

C₂²×C₂²⋊C₄

yes

—

#20340

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×#95

yes

characteristic

C₂×#568

C₂×C₆₇

yes

—

C₂×#338

C₂×C₆₇

yes

complemented

C₆₇

#20340

yes

characteristic, direct product factor, Sylow

C₂²×#41

C₂²×C₆₇

yes

characteristic

#568

—

C₂×#131

C₄×C₆₇

yes

—

C₂²×M₁₆

C₂²×C₆₇

yes

—

#338

C₂²×C₆₇

yes

—

C₂×#41

C₂×C₄×C₆₇

yes

—

C₂×#41

D₄×C₆₇

yes

—

C₂²×Q₈

Q₈×C₆₇

yes

characteristic

C₂²×D₄

Q₈×C₆₇

yes

—

C₂³×C₄

C₂³×C₆₇

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₆₇

yes

—

C₂×M₁₆

C₂³×C₆₇

yes

—

C₂²×C₈

C₂×C₄×C₆₇

yes

—

C₂⁴

C₂×Q₈×C₆₇

yes

—

#41

C₄²×C₆₇

yes

—

#41

C₂²⋊C₄×C₆₇

yes

—

C₂×Q₈

C₄⋊C₄×C₆₇

yes

—

C₂×D₄

C₄⋊C₄×C₆₇

yes

—

C₂²×C₄

C₂²×C₄×C₆₇

yes

—

C₂²×C₄

C₂×D₄×C₆₇

yes

—

C₂²×C₄

C₂×Q₈×C₆₇

yes

—

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×C₄×C₆₇

yes

—

C₂³

—

C₂³

C₂×C₄⋊C₄×C₆₇

yes

—

C₂³

C₂²×Q₈×C₆₇

yes

characteristic

Q₈

#95×C₆₇

yes

—

D₄

#95×C₆₇

yes

—

C₂×C₄

C₂×C₄²×C₆₇

yes

—

C₂×C₄

C₂×C₂²⋊C₄×C₆₇

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₆₇

yes

—

C₂×C₄

C₂³×C₄×C₆₇

yes

—

C₂×C₄

C₂²×D₄×C₆₇

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×#95×C₆₇

yes

—

C₂²

C₂²×C₄⋊C₄×C₆₇

yes

—

C₄

C₂×#95×C₆₇

yes

—

100

C₄

C₂²×C₄²×C₆₇

yes

characteristic

101

C₄

C₂²×C₂²⋊C₄×C₆₇

yes

—

102

C₂

—

103

C₂

—

104

C₂

C₂²×#95×C₆₇

yes

characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂²×#95
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ G

subgroup

G/H

center

C₄×C₆₇

C₂²×C₄²

commutator

C₄

C₂²×C₄²×C₆₇

Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

socle

C₂×C₆₇

C₂²×#95

prime radical

C₂²×#41×C₆₇

C₂²

LCS

G ▷ C₄ ▷ C₁

UCS

C₁ ◁ C₄×C₆₇ ◁ G

N	H ≅ G/N	type
2. C₂×#338×C₆₇	C₂	semidirect
49. #20340	C₆₇	direct (N × H)
54. C₂×#338	C₂×C₆₇	semidirect
55. C₆₇	#20340	direct (N × H)

H ≅ G/N

type

2. C₂×#338×C₆₇

C₂

semidirect

49. #20340

C₆₇

direct (N × H)

54. C₂×#338

C₂×C₆₇

semidirect

55. C₆₇

#20340

direct (N × H)

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

subgroup	index
C₂×#568×C₆₇	2
C₂×#338×C₆₇	2
#20340	67

subgroup

index

C₂×#568×C₆₇

C₂×#338×C₆₇

#20340

Subgroups — C₆₇×#20340 #30699591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#20340 #30699591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20340 #30699591