Group #22557216 — subgroups

Subgroups — C₆₇×#2785 #22557216

Order 8576.

428 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 176 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	#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#394×C₆₇	C₂	1	3	yes	maximal
3	#393×C₆₇	C₂	1	2	yes	maximal, complemented
4	#388×C₆₇	C₂	1	6	yes	maximal, complemented
5	#382×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
9	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
12	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
13	#95×C₆₇	C₂²	1	3	yes	—
14	#95×C₆₇	C₂²	1	6	yes	—
15	#95×C₆₇	C₂²	1	3	yes	—
16	C₂⁴×C₆₇	C₂³	1	2	yes	—
17	C₂×Q₈×C₆₇	—	4	3	no	—
18	C₂×D₄×C₆₇	—	4	3	no	—
19	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
20	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
21	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
23	C₄⋊C₄×C₆₇	—	4	3	no	—
24	C₄⋊C₄×C₆₇	—	4	3	no	—
25	C₂²⋊C₄×C₆₇	—	2	12	no	—
26	C₂²⋊C₄×C₆₇	—	4	6	no	—
27	C₄²×C₆₇	—	2	6	no	—
28	C₂³×C₆₇	—	2	6	no	—
29	C₂³×C₆₇	—	2	6	no	—
30	C₂³×C₆₇	—	2	2	no	—
31	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
32	Q₈×C₆₇	—	4	4	no	—
33	D₄×C₆₇	—	4	4	no	—
34	C₂×C₄×C₆₇	—	2	3	no	—
35	C₂×C₄×C₆₇	—	2	3	no	—
36	C₂×C₄×C₆₇	—	2	6	no	—
37	C₂×C₄×C₆₇	—	2	6	no	—
38	C₂×C₄×C₆₇	—	2	6	no	—
39	C₂×C₄×C₆₇	—	2	6	no	—
40	C₂×C₄×C₆₇	—	2	3	no	—
41	C₂×C₄×C₆₇	—	2	3	no	—
42	C₂×C₄×C₆₇	#41	1	6	yes	—
43	C₂²×C₆₇	—	4	6	no	—
44	C₂²×C₆₇	—	4	6	no	—
45	C₂²×C₆₇	—	4	2	no	—
46	C₂²×C₆₇	C₂×#41	1	3	yes	—
47	C₂²×C₆₇	H(5,𝔽₂)	1	3	yes	—
48	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
49	C₄×C₆₇	—	2	6	no	—
50	C₄×C₆₇	—	4	6	no	—
51	C₄×C₆₇	—	4	1	no	—
52	C₄×C₆₇	—	4	3	no	—
53	C₂×C₆₇	—	8	2	no	—
54	C₂×C₆₇	#539	1	3	yes	—
55	C₂×C₆₇	#555	1	3	yes	—
56	C₂×C₆₇	H(3,𝔽₄)	1	1	yes	characteristic
57	#2785	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₆₇	#2785	1	1	yes	characteristic, direct product factor, Sylow
59	#395	C₂×C₆₇	1	1	yes	characteristic, complemented
60	#394	C₂×C₆₇	1	3	yes	—
61	#393	C₂×C₆₇	1	2	yes	complemented
62	#388	C₂×C₆₇	1	6	yes	complemented
63	#382	C₂×C₆₇	1	3	yes	complemented
64	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
65	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
67	C₂×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
68	C₂×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
69	C₂×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
70	C₂×C₄²	C₂²×C₆₇	1	3	yes	—
71	#95	C₂²×C₆₇	1	3	yes	—
72	#95	C₂²×C₆₇	1	6	yes	—
73	#95	C₂²×C₆₇	1	3	yes	—
74	C₂⁴	C₂³×C₆₇	1	2	yes	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	4	3	no	—
77	C₂²×C₄	C₂³×C₆₇	1	6	yes	—
78	C₂²×C₄	C₂³×C₆₇	1	3	yes	—
79	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
80	C₂²×C₄	C₂³×C₆₇	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	4	3	no	—
83	C₂²⋊C₄	—	2	12	no	—
84	C₂²⋊C₄	—	4	6	no	—
85	C₄²	—	2	6	no	—
86	C₂³	—	2	6	no	—
87	C₂³	—	2	6	no	—
88	C₂³	—	2	2	no	—
89	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	4	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	6	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#41×C₆₇	1	6	yes	—
101	C₂²	—	4	6	no	—
102	C₂²	—	4	6	no	—
103	C₂²	—	4	2	no	—
104	C₂²	C₂×#41×C₆₇	1	3	yes	—
105	C₂²	H(5,𝔽₂)×C₆₇	1	3	yes	—
106	C₂²	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	6	no	—
109	C₄	—	4	1	no	—
110	C₄	—	4	3	no	—
111	C₂	—	8	2	no	—
112	C₂	#539×C₆₇	1	3	yes	—
113	C₂	#555×C₆₇	1	3	yes	—
114	C₂	H(3,𝔽₄)×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#395×C₆₇

C₂

yes

maximal, characteristic, complemented

#394×C₆₇

C₂

yes

maximal

#393×C₆₇

C₂

yes

maximal, complemented

#388×C₆₇

C₂

yes

maximal, complemented

#382×C₆₇

C₂

yes

maximal, complemented

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

C₂⁴×C₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#539

yes

—

C₂×C₆₇

#555

yes

—

C₂×C₆₇

H(3,𝔽₄)

yes

characteristic

#2785

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#2785

yes

characteristic, direct product factor, Sylow

#395

C₂×C₆₇

yes

characteristic, complemented

#394

C₂×C₆₇

yes

—

#393

C₂×C₆₇

yes

complemented

#388

C₂×C₆₇

yes

complemented

#382

C₂×C₆₇

yes

complemented

C₂²×Q₈

C₂²×C₆₇

yes

characteristic

C₂²×D₄

C₂²×C₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₆₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₆₇

yes

—

C₂×C₄²

C₂²×C₆₇

yes

—

#95

C₂²×C₆₇

yes

—

#95

C₂²×C₆₇

yes

—

#95

C₂²×C₆₇

yes

—

C₂⁴

C₂³×C₆₇

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

C₂³×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

characteristic

C₂²×C₄

C₂³×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₆₇

yes

characteristic

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#41×C₆₇

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×#41×C₆₇

yes

—

105

C₂²

H(5,𝔽₂)×C₆₇

yes

—

106

C₂²

2₋¹⁺⁴×C₆₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

#539×C₆₇

yes

—

113

C₂

#555×C₆₇

yes

—

114

C₂

H(3,𝔽₄)×C₆₇

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₆₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₆₇

prime radical

C₂²×D₄×C₆₇

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ G

N	H ≅ G/N	type
1. #395×C₆₇	C₂	semidirect
3. #393×C₆₇	C₂	semidirect
4. #388×C₆₇	C₂	semidirect
5. #382×C₆₇	C₂	semidirect
57. #2785	C₆₇	direct (N × H)
58. C₆₇	#2785	direct (N × H)
59. #395	C₂×C₆₇	semidirect
61. #393	C₂×C₆₇	semidirect
62. #388	C₂×C₆₇	semidirect
63. #382	C₂×C₆₇	semidirect

H ≅ G/N

type

1. #395×C₆₇

C₂

semidirect

3. #393×C₆₇

C₂

semidirect

4. #388×C₆₇

C₂

semidirect

5. #382×C₆₇

C₂

semidirect

57. #2785

C₆₇

direct (N × H)

58. C₆₇

#2785

direct (N × H)

59. #395

C₂×C₆₇

semidirect

61. #393

C₂×C₆₇

semidirect

62. #388

C₂×C₆₇

semidirect

63. #382

C₂×C₆₇

semidirect

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

subgroup	index
#395×C₆₇	2
#394×C₆₇	2
#393×C₆₇	2
#388×C₆₇	2
#382×C₆₇	2
#2785	67

subgroup

index

#395×C₆₇

#394×C₆₇

#393×C₆₇

#388×C₆₇

#382×C₆₇

#2785

Subgroups — C₆₇×#2785 #22557216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#2785 #22557216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2785 #22557216