Group #22556520 — subgroups

Subgroups — C₆₇×#2089 #22556520

Order 8576.

176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 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	#428×C₆₇	C₂	1	2	yes	maximal
2	C₂×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#424×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#343×C₆₇	C₂	1	2	yes	maximal
5	#341×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	#117×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
11	#107×C₆₇	—	2	2	no	—
12	#107×C₆₇	—	2	2	no	—
13	#106×C₆₇	—	2	2	no	—
14	C₄⋊C₈×C₆₇	—	2	2	no	—
15	C₂²×C₄×C₆₇	—	2	2	no	—
16	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
17	M₁₆×C₆₇	—	2	2	no	—
18	M₁₆×C₆₇	—	2	2	no	—
19	M₁₆×C₆₇	Q₈	1	4	yes	—
20	C₂×C₈×C₆₇	—	2	2	no	—
21	C₂×C₈×C₆₇	—	2	2	no	—
22	C₂×C₈×C₆₇	—	2	1	no	—
23	C₂×C₈×C₆₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₆₇	—	2	1	no	—
29	C₄²×C₆₇	—	2	1	no	—
30	C₂³×C₆₇	#41	1	1	yes	characteristic
31	C₂×C₄×C₆₇	—	2	1	no	—
32	C₂×C₄×C₆₇	—	2	1	no	—
33	C₂×C₄×C₆₇	—	4	2	no	—
34	C₂×C₄×C₆₇	—	8	2	no	—
35	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₇	#41	1	2	yes	—
39	C₈×C₆₇	—	2	4	no	—
40	C₈×C₆₇	—	2	2	no	—
41	C₈×C₆₇	—	4	1	no	—
42	C₂²×C₆₇	—	2	1	no	—
43	C₂²×C₆₇	—	2	1	no	—
44	C₂²×C₆₇	#121	1	1	yes	characteristic
45	C₂²×C₆₇	#125	1	2	yes	—
46	C₄×C₆₇	—	4	1	no	—
47	C₄×C₆₇	—	4	1	no	—
48	C₄×C₆₇	—	8	2	no	—
49	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₆₇	#123	1	1	yes	characteristic
51	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₆₇	—	2	2	no	—
53	C₂×C₆₇	#398	1	1	yes	characteristic
54	C₂×C₆₇	#472	1	1	yes	characteristic
55	C₂×C₆₇	#473	1	1	yes	characteristic
56	#2089	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₆₇	#2089	1	1	yes	characteristic, direct product factor, Sylow
58	#428	C₂×C₆₇	1	2	yes	—
59	C₂×#107	C₂×C₆₇	1	1	yes	characteristic
60	#424	C₂×C₆₇	1	1	yes	characteristic
61	#343	C₂×C₆₇	1	2	yes	—
62	#341	C₂×C₆₇	1	1	yes	characteristic
63	C₂×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
64	C₂×M₁₆	C₂²×C₆₇	1	2	yes	—
65	C₂²×C₈	C₂²×C₆₇	1	1	yes	characteristic
66	#117	C₂²×C₆₇	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
68	#107	—	2	2	no	—
69	#107	—	2	2	no	—
70	#106	—	2	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	Q₈×C₆₇	1	4	yes	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	D₄×C₆₇	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
84	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#41×C₆₇	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
93	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
94	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
95	C₂×C₄	#41×C₆₇	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	4	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#121×C₆₇	1	1	yes	characteristic
102	C₂²	#125×C₆₇	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	8	2	no	—
106	C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
107	C₄	#123×C₆₇	1	1	yes	characteristic
108	C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#398×C₆₇	1	1	yes	characteristic
111	C₂	#472×C₆₇	1	1	yes	characteristic
112	C₂	#473×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#428×C₆₇

C₂

yes

maximal

C₂×#107×C₆₇

C₂

yes

maximal, characteristic

#424×C₆₇

C₂

yes

maximal, characteristic

#343×C₆₇

C₂

yes

maximal

#341×C₆₇

C₂

yes

maximal, characteristic

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#107×C₆₇

—

#107×C₆₇

—

#106×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

Q₈

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

#125

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#123

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#2089

yes

characteristic, direct product factor, Sylow

#428

C₂×C₆₇

yes

—

C₂×#107

C₂×C₆₇

yes

characteristic

#424

C₂×C₆₇

yes

characteristic

#343

C₂×C₆₇

yes

—

#341

C₂×C₆₇

yes

characteristic

C₂×M₁₆

C₂²×C₆₇

yes

characteristic

C₂×M₁₆

C₂²×C₆₇

yes

—

C₂²×C₈

C₂²×C₆₇

yes

characteristic

#117

C₂²×C₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Q₈×C₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₆₇

yes

characteristic

C₄⋊C₄

D₄×C₆₇

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#41×C₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₆₇

yes

—

C₂×C₄

#41×C₆₇

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

—

101

C₂²

#121×C₆₇

yes

characteristic

102

C₂²

#125×C₆₇

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂²⋊Q₈×C₆₇

yes

—

107

C₄

#123×C₆₇

yes

characteristic

108

C₄

C₄⋊Q₈×C₆₇

yes

characteristic

109

C₂

—

110

C₂

#398×C₆₇

yes

characteristic

111

C₂

#472×C₆₇

yes

characteristic

112

C₂

#473×C₆₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₆₇
prime radical	C₂³×C₆₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₆₇

C₂³×C₆₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

N	H ≅ G/N	type
56. #2089	C₆₇	direct (N × H)
57. C₆₇	#2089	direct (N × H)

H ≅ G/N

type

56. #2089

C₆₇

direct (N × H)

57. C₆₇

#2089

direct (N × H)

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

subgroup	index
#428×C₆₇	2
C₂×#107×C₆₇	2
#424×C₆₇	2
#343×C₆₇	2
#341×C₆₇	2
#2089	67

subgroup

index

#428×C₆₇

C₂×#107×C₆₇

#424×C₆₇

#343×C₆₇

#341×C₆₇

#2089

Subgroups — C₆₇×#2089 #22556520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#2089 #22556520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2089 #22556520