Group #22555880 — subgroups

Subgroups — C₈×M₁₆×C₆₇ #22555880

Order 8576.

224 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 184 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	#431×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#322×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₈²×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×C₆₇	C₄	1	2	yes	—
7	C₂²×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
9	C₄⋊C₈×C₆₇	C₄	1	2	yes	—
10	C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
11	#97×C₆₇	C₄	1	2	yes	—
12	C₄×C₈×C₆₇	C₄	1	2	yes	—
13	C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
20	M₁₆×C₆₇	C₈	1	8	yes	direct product factor
21	C₂×C₈×C₆₇	—	2	4	no	—
22	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₂³×C₆₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₆₇	—	2	2	no	—
32	C₂×C₄×C₆₇	—	2	2	no	—
33	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₆₇	—	2	2	no	—
41	C₈×C₆₇	—	2	4	no	—
42	C₈×C₆₇	C₂×C₈	1	8	yes	complemented
43	C₈×C₆₇	M₁₆	1	4	yes	direct product factor
44	C₂²×C₆₇	—	2	1	no	—
45	C₂²×C₆₇	—	2	1	no	—
46	C₂²×C₆₇	C₄×C₈	1	2	yes	—
47	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₆₇	—	2	1	no	—
49	C₄×C₆₇	—	2	1	no	—
50	C₄×C₆₇	C₄×C₈	1	2	yes	—
51	C₄×C₆₇	C₂²×C₈	1	2	yes	—
52	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
53	C₄×C₆₇	#131	1	2	yes	—
54	C₂×C₆₇	—	2	2	no	—
55	C₂×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
56	C₂×C₆₇	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₆₇	#405	1	1	yes	characteristic
58	C₈×M₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₆₇	C₈×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
60	#431	C₂×C₆₇	1	1	yes	characteristic
61	C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
63	#322	C₂×C₆₇	1	2	yes	complemented
64	C₈²	C₂×C₆₇	1	2	yes	complemented
65	C₂×M₁₆	C₄×C₆₇	1	2	yes	—
66	C₂²×C₈	C₄×C₆₇	1	2	yes	—
67	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
68	C₄⋊C₈	C₄×C₆₇	1	2	yes	—
69	C₂²⋊C₈	C₄×C₆₇	1	2	yes	—
70	#97	C₄×C₆₇	1	2	yes	—
71	C₄×C₈	C₄×C₆₇	1	2	yes	—
72	C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
73	C₄×C₈	C₂²×C₆₇	1	2	yes	—
74	C₄×C₈	C₂²×C₆₇	1	2	yes	—
75	C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
78	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
79	M₁₆	C₈×C₆₇	1	8	yes	direct product factor
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
82	C₂×C₈	C₂×C₄×C₆₇	1	4	yes	—
83	C₂×C₈	C₂×C₄×C₆₇	1	4	yes	—
84	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
85	C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
86	C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
87	C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
88	C₄²	C₂³×C₆₇	1	1	yes	characteristic
89	C₂³	C₄²×C₆₇	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄²×C₆₇	1	1	yes	characteristic
93	C₂×C₄	C₄²×C₆₇	1	1	yes	characteristic
94	C₂×C₄	C₄²×C₆₇	1	1	yes	characteristic
95	C₂×C₄	C₂×C₈×C₆₇	1	4	yes	—
96	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	2	4	no	—
101	C₈	C₂×C₈×C₆₇	1	8	yes	complemented
102	C₈	M₁₆×C₆₇	1	4	yes	direct product factor
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	C₄×C₈×C₆₇	1	2	yes	—
106	C₂²	C₂×C₄²×C₆₇	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	C₄×C₈×C₆₇	1	2	yes	—
110	C₄	C₂²×C₈×C₆₇	1	2	yes	—
111	C₄	C₂×M₁₆×C₆₇	1	2	yes	—
112	C₄	#131×C₆₇	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×C₄×C₈×C₆₇	1	1	yes	characteristic
115	C₂	C₄×M₁₆×C₆₇	1	1	yes	characteristic
116	C₂	#405×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#431×C₆₇

C₂

yes

maximal, characteristic

C₄×M₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

#322×C₆₇

C₂

yes

maximal, complemented

C₈²×C₆₇

C₂

yes

maximal, complemented

C₂×M₁₆×C₆₇

C₄

yes

—

C₂²×C₈×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₄

yes

—

C₂²⋊C₈×C₆₇

C₄

yes

—

#97×C₆₇

C₄

yes

—

C₄×C₈×C₆₇

C₄

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

M₁₆×C₆₇

C₈

yes

direct product factor

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×C₈

yes

complemented

C₈×C₆₇

M₁₆

yes

direct product factor

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₈

yes

—

C₂²×C₆₇

C₂×C₄²

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×C₈

yes

—

C₄×C₆₇

C₂²×C₈

yes

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

#131

yes

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×C₆₇

C₄×M₁₆

yes

characteristic

C₂×C₆₇

#405

yes

characteristic

C₈×M₁₆

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

C₈×M₁₆

yes

characteristic, direct product factor, Sylow

#431

C₂×C₆₇

yes

characteristic

C₄×M₁₆

C₂×C₆₇

yes

characteristic

C₂×C₄×C₈

C₂×C₆₇

yes

characteristic

#322

C₂×C₆₇

yes

complemented

C₈²

C₂×C₆₇

yes

complemented

C₂×M₁₆

C₄×C₆₇

yes

—

C₂²×C₈

C₄×C₆₇

yes

—

C₂×C₄²

C₂²×C₆₇

yes

characteristic

C₄⋊C₈

C₄×C₆₇

yes

—

C₂²⋊C₈

C₄×C₆₇

yes

—

#97

C₄×C₆₇

yes

—

C₄×C₈

C₄×C₆₇

yes

—

C₄×C₈

C₂²×C₆₇

yes

characteristic

C₄×C₈

C₂²×C₆₇

yes

—

C₄×C₈

C₂²×C₆₇

yes

—

C₄×C₈

C₂²×C₆₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₇

yes

characteristic

M₁₆

C₈×C₆₇

yes

direct product factor

C₂×C₈

—

C₂×C₈

C₂×C₄×C₆₇

yes

—

C₂×C₈

C₂×C₄×C₆₇

yes

—

C₂×C₈

C₂×C₄×C₆₇

yes

—

C₂×C₈

C₂×C₄×C₆₇

yes

—

C₄²

C₂×C₄×C₆₇

yes

characteristic

C₄²

C₂×C₄×C₆₇

yes

characteristic

C₄²

C₂×C₄×C₆₇

yes

characteristic

C₄²

C₂³×C₆₇

yes

characteristic

C₂³

C₄²×C₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄²×C₆₇

yes

characteristic

C₂×C₄

C₄²×C₆₇

yes

characteristic

C₂×C₄

C₄²×C₆₇

yes

characteristic

C₂×C₄

C₂×C₈×C₆₇

yes

—

C₂×C₄

C₂²×C₄×C₆₇

yes

characteristic

C₂×C₄

C₂²×C₄×C₆₇

yes

characteristic

C₂×C₄

C₂²×C₄×C₆₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

C₂×C₈×C₆₇

yes

complemented

102

C₈

M₁₆×C₆₇

yes

direct product factor

103

C₂²

—

104

C₂²

—

105

C₂²

C₄×C₈×C₆₇

yes

—

106

C₂²

C₂×C₄²×C₆₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

C₄×C₈×C₆₇

yes

—

110

C₄

C₂²×C₈×C₆₇

yes

—

111

C₄

C₂×M₁₆×C₆₇

yes

—

112

C₄

#131×C₆₇

yes

—

113

C₂

—

114

C₂

C₂×C₄×C₈×C₆₇

yes

characteristic

115

C₂

C₄×M₁₆×C₆₇

yes

characteristic

116

C₂

#405×C₆₇

yes

characteristic

	subgroup	G/H
center	C₄×C₈×C₆₇	C₂²
commutator	C₂	C₂×C₄×C₈×C₆₇
Frattini, cosocle	C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×C₄²
prime radical	C₂³×C₆₇	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₆₇ ◁ G

subgroup

G/H

center

C₄×C₈×C₆₇

C₂²

commutator

C₂

C₂×C₄×C₈×C₆₇

Frattini, cosocle

C₄²

C₂³×C₆₇

socle

C₂²×C₆₇

C₂×C₄²

prime radical

C₂³×C₆₇

C₄²

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
4. #322×C₆₇	C₂	semidirect
5. C₈²×C₆₇	C₂	semidirect
20. M₁₆×C₆₇	C₈	direct (N × H)
42. C₈×C₆₇	C₂×C₈	semidirect
43. C₈×C₆₇	M₁₆	direct (N × H)
58. C₈×M₁₆	C₆₇	direct (N × H)
59. C₆₇	C₈×M₁₆	direct (N × H)
63. #322	C₂×C₆₇	semidirect
64. C₈²	C₂×C₆₇	semidirect
79. M₁₆	C₈×C₆₇	direct (N × H)
101. C₈	C₂×C₈×C₆₇	semidirect
102. C₈	M₁₆×C₆₇	direct (N × H)

H ≅ G/N

type

4. #322×C₆₇

C₂

semidirect

5. C₈²×C₆₇

C₂

semidirect

20. M₁₆×C₆₇

C₈

direct (N × H)

42. C₈×C₆₇

C₂×C₈

semidirect

43. C₈×C₆₇

M₁₆

direct (N × H)

58. C₈×M₁₆

C₆₇

direct (N × H)

59. C₆₇

C₈×M₁₆

direct (N × H)

63. #322

C₂×C₆₇

semidirect

64. C₈²

C₂×C₆₇

semidirect

79. M₁₆

C₈×C₆₇

direct (N × H)

101. C₈

C₂×C₈×C₆₇

semidirect

102. C₈

M₁₆×C₆₇

direct (N × H)

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

subgroup	index
#431×C₆₇	2
C₄×M₁₆×C₆₇	2
C₂×C₄×C₈×C₆₇	2
#322×C₆₇	2
C₈²×C₆₇	2
C₈×M₁₆	67

subgroup

index

#431×C₆₇

C₄×M₁₆×C₆₇

C₂×C₄×C₈×C₆₇

#322×C₆₇

C₈²×C₆₇

C₈×M₁₆

Subgroups — C₈×M₁₆×C₆₇ #22555880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×M16×C67 #22555880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×M₁₆×C₆₇ #22555880