Group #22556544 — subgroups

Subgroups — C₆₇×#2113 #22556544

Order 8576.

228 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 120 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×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#41×C₆₇	C₄	1	1	yes	characteristic
6	#131×C₆₇	—	2	2	no	—
7	#131×C₆₇	—	2	2	no	—
8	C₂×M₁₆×C₆₇	C₄	1	1	yes	characteristic
9	C₂×M₁₆×C₆₇	C₄	1	2	yes	—
10	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₆₇	—	2	2	no	—
13	C₂×C₁₆×C₆₇	—	2	2	no	—
14	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×C₁₆×C₆₇	C₂²	1	2	yes	—
16	#41×C₆₇	—	2	2	no	—
17	#41×C₆₇	—	2	2	no	—
18	C₂×Q₈×C₆₇	C₈	1	1	yes	characteristic
19	C₂×D₄×C₆₇	C₈	1	2	yes	—
20	C₂×D₄×C₆₇	C₈	1	1	yes	characteristic
21	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	M₁₆×C₆₇	—	2	2	no	—
24	M₁₆×C₆₇	—	2	4	no	—
25	C₂×C₈×C₆₇	—	2	2	no	—
26	C₂×C₈×C₆₇	—	2	2	no	—
27	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₆₇	D₄	1	2	yes	—
30	C₂×C₈×C₆₇	D₄	1	2	yes	—
31	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₁₆×C₆₇	—	2	2	no	—
33	C₁₆×C₆₇	—	2	2	no	—
34	C₂³×C₆₇	C₂×C₈	1	2	yes	—
35	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
36	Q₈×C₆₇	—	2	2	no	—
37	D₄×C₆₇	—	2	4	no	—
38	D₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	—	2	2	no	—
41	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₆₇	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₆₇	M₁₆	1	2	yes	—
45	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₆₇	—	2	2	no	—
47	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
48	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₆₇	—	2	2	no	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	—	2	4	no	—
53	C₂²×C₆₇	C₂²⋊C₈	1	2	yes	—
54	C₂²×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
57	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₆₇	—	2	2	no	—
60	C₂×C₆₇	—	4	2	no	—
61	C₂×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₆₇	#504	1	2	yes	—
63	#2113	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₆₇	#2113	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×#131	C₂×C₆₇	1	1	yes	characteristic
66	C₂×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
67	C₂²×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
68	C₂²⋊C₁₆	C₂×C₆₇	1	4	yes	complemented
69	C₂×#41	C₄×C₆₇	1	1	yes	characteristic
70	#131	—	2	2	no	—
71	#131	—	2	2	no	—
72	C₂×M₁₆	C₄×C₆₇	1	1	yes	characteristic
73	C₂×M₁₆	C₄×C₆₇	1	2	yes	—
74	C₂²×C₈	C₂²×C₆₇	1	2	yes	—
75	C₂²×C₈	C₂²×C₆₇	1	1	yes	characteristic
76	M₃₂	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	C₂²×C₆₇	1	2	yes	—
79	C₂×C₁₆	C₂²×C₆₇	1	2	yes	—
80	#41	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	C₈×C₆₇	1	1	yes	characteristic
83	C₂×D₄	C₈×C₆₇	1	2	yes	—
84	C₂×D₄	C₈×C₆₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₄×C₆₇	1	2	yes	—
86	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
92	C₂×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
93	C₂×C₈	D₄×C₆₇	1	2	yes	—
94	C₂×C₈	D₄×C₆₇	1	2	yes	—
95	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	2	2	no	—
98	C₂³	C₂×C₈×C₆₇	1	2	yes	—
99	C₂³	C₂×C₈×C₆₇	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
106	C₂×C₄	C₂×C₈×C₆₇	1	2	yes	—
107	C₂×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
108	C₂×C₄	M₁₆×C₆₇	1	2	yes	—
109	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
112	C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
113	C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂²⋊C₈×C₆₇	1	2	yes	—
118	C₂²	C₂²×C₈×C₆₇	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	C₂²⋊C₈×C₆₇	1	2	yes	—
121	C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
122	C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
126	C₂	#504×C₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#131×C₆₇

C₂

yes

maximal, characteristic

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₁₆×C₆₇

C₂

yes

maximal, complemented

C₂×#41×C₆₇

C₄

yes

characteristic

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₄

yes

characteristic

C₂×M₁₆×C₆₇

C₄

yes

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂²

yes

—

C₂×C₁₆×C₆₇

C₂²

yes

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₈

yes

characteristic

C₂×D₄×C₆₇

C₈

yes

—

C₂×D₄×C₆₇

C₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×C₈

yes

—

C₂³×C₆₇

C₂×C₈

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²⋊C₈

yes

—

C₂²×C₆₇

C₂²×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊C₈

yes

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₂×M₁₆

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₆₇

#504

yes

—

#2113

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#2113

yes

characteristic, direct product factor, Sylow

C₂×#131

C₂×C₆₇

yes

characteristic

C₂×M₃₂

C₂×C₆₇

yes

characteristic, complemented

C₂²×C₁₆

C₂×C₆₇

yes

characteristic, complemented

C₂²⋊C₁₆

C₂×C₆₇

yes

complemented

C₂×#41

C₄×C₆₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

C₄×C₆₇

yes

characteristic

C₂×M₁₆

C₄×C₆₇

yes

—

C₂²×C₈

C₂²×C₆₇

yes

—

C₂²×C₈

C₂²×C₆₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₆₇

yes

—

C₂×C₁₆

C₂²×C₆₇

yes

—

#41

—

#41

—

C₂×Q₈

C₈×C₆₇

yes

characteristic

C₂×D₄

C₈×C₆₇

yes

—

C₂×D₄

C₈×C₆₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₆₇

yes

—

C₂²×C₄

C₂×C₄×C₆₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₆₇

yes

—

C₂×C₈

C₂×C₄×C₆₇

yes

characteristic

C₂×C₈

D₄×C₆₇

yes

—

C₂×C₈

D₄×C₆₇

yes

—

C₂×C₈

C₂³×C₆₇

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×C₈×C₆₇

yes

—

C₂³

C₂×C₈×C₆₇

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊C₄×C₆₇

yes

—

106

C₂×C₄

C₂×C₈×C₆₇

yes

—

107

C₂×C₄

C₂×C₈×C₆₇

yes

characteristic

108

C₂×C₄

M₁₆×C₆₇

yes

—

109

C₂×C₄

C₂²×C₄×C₆₇

yes

characteristic

110

C₈

—

111

C₈

C₂²⋊C₄×C₆₇

yes

—

112

C₈

C₂×D₄×C₆₇

yes

characteristic

113

C₈

C₂×D₄×C₆₇

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂²⋊C₈×C₆₇

yes

—

118

C₂²

C₂²×C₈×C₆₇

yes

characteristic

119

C₄

—

120

C₄

C₂²⋊C₈×C₆₇

yes

—

121

C₄

C₂×C₂²⋊C₄×C₆₇

yes

characteristic

122

C₄

C₂×M₁₆×C₆₇

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×C₂²⋊C₈×C₆₇

yes

characteristic

126

C₂

#504×C₆₇

yes

—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂³
commutator	C₂²	C₂²×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂²×C₈
prime radical	C₂×#41×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₈×C₆₇

C₂³

commutator

C₂²

C₂²×C₈×C₆₇

Frattini, cosocle

C₂×C₈

C₂³×C₆₇

socle

C₂²×C₆₇

C₂²×C₈

prime radical

C₂×#41×C₆₇

C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
2. C₂×M₃₂×C₆₇	C₂	semidirect
3. C₂²×C₁₆×C₆₇	C₂	semidirect
4. C₂²⋊C₁₆×C₆₇	C₂	semidirect
63. #2113	C₆₇	direct (N × H)
64. C₆₇	#2113	direct (N × H)
66. C₂×M₃₂	C₂×C₆₇	semidirect
67. C₂²×C₁₆	C₂×C₆₇	semidirect
68. C₂²⋊C₁₆	C₂×C₆₇	semidirect

H ≅ G/N

type

2. C₂×M₃₂×C₆₇

C₂

semidirect

3. C₂²×C₁₆×C₆₇

C₂

semidirect

4. C₂²⋊C₁₆×C₆₇

C₂

semidirect

63. #2113

C₆₇

direct (N × H)

64. C₆₇

#2113

direct (N × H)

66. C₂×M₃₂

C₂×C₆₇

semidirect

67. C₂²×C₁₆

C₂×C₆₇

semidirect

68. C₂²⋊C₁₆

C₂×C₆₇

semidirect

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

subgroup	index
C₂×#131×C₆₇	2
C₂×M₃₂×C₆₇	2
C₂²×C₁₆×C₆₇	2
C₂²⋊C₁₆×C₆₇	2
#2113	67

subgroup

index

C₂×#131×C₆₇

C₂×M₃₂×C₆₇

C₂²×C₁₆×C₆₇

C₂²⋊C₁₆×C₆₇

#2113

Subgroups — C₆₇×#2113 #22556544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#2113 #22556544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2113 #22556544