Group #22556099 — subgroups

Subgroups — C₆₇×#1668 #22556099

Order 8576.

188 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 72 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	#493×C₆₇	C₂	1	1	yes	maximal, characteristic
2	#460×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#331×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₆₇	—	2	2	no	—
7	C₂×D₈×C₆₇	—	2	1	no	—
8	C₂×D₈×C₆₇	—	2	1	no	—
9	#127×C₆₇	C₂²	1	2	yes	—
10	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₆₇	—	2	1	no	—
12	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
13	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#102×C₆₇	—	2	2	no	—
15	C₄×C₈×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₆₇	—	2	2	no	—
19	C₂×D₄×C₆₇	—	2	2	no	—
20	C₂×D₄×C₆₇	—	2	2	no	—
21	SD₁₆×C₆₇	—	4	4	no	—
22	D₈×C₆₇	—	4	2	no	—
23	D₈×C₆₇	—	4	2	no	—
24	C₂×C₈×C₆₇	—	2	2	no	—
25	C₂×C₈×C₆₇	—	2	1	no	—
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₄²×C₆₇	—	2	1	no	—
31	C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₆₇	—	4	2	no	—
33	Q₈×C₆₇	—	2	1	no	—
34	Q₈×C₆₇	D₈	1	2	yes	complemented
35	D₄×C₆₇	—	4	2	no	—
36	D₄×C₆₇	—	4	2	no	—
37	D₄×C₆₇	—	4	2	no	—
38	D₄×C₆₇	—	4	2	no	—
39	D₄×C₆₇	—	8	2	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	2	1	no	—
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
45	C₈×C₆₇	—	2	1	no	—
46	C₈×C₆₇	—	4	2	no	—
47	C₈×C₆₇	—	4	1	no	—
48	C₈×C₆₇	SD₁₆	1	2	yes	complemented
49	C₂²×C₆₇	—	8	2	no	—
50	C₂²×C₆₇	—	8	2	no	—
51	C₂²×C₆₇	—	8	2	no	—
52	C₂²×C₆₇	#121	1	1	yes	characteristic
53	C₄×C₆₇	—	2	2	no	—
54	C₄×C₆₇	—	2	1	no	—
55	C₄×C₆₇	—	4	1	no	—
56	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
57	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×C₆₇	#135	1	1	yes	characteristic
59	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₆₇	—	16	2	no	—
61	C₂×C₆₇	#460	1	1	yes	characteristic
62	C₂×C₆₇	#466	1	1	yes	characteristic
63	C₂×C₆₇	#472	1	1	yes	characteristic
64	#1668	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₆₇	#1668	1	1	yes	characteristic, direct product factor, Sylow
66	#493	C₂×C₆₇	1	1	yes	characteristic
67	#460	C₂×C₆₇	1	2	yes	complemented
68	C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
69	#335	C₂×C₆₇	1	1	yes	characteristic, complemented
70	#331	C₂×C₆₇	1	2	yes	complemented
71	C₂×SD₁₆	—	2	2	no	—
72	C₂×D₈	—	2	1	no	—
73	C₂×D₈	—	2	1	no	—
74	#127	C₂²×C₆₇	1	2	yes	—
75	C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
78	C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
79	#102	—	2	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
82	C₂×Q₈	D₄×C₆₇	1	1	yes	characteristic
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	SD₁₆	—	4	4	no	—
87	D₈	—	4	2	no	—
88	D₈	—	4	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
92	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂³×C₆₇	1	1	yes	characteristic
97	C₂³	—	4	2	no	—
98	Q₈	—	2	1	no	—
99	Q₈	D₈×C₆₇	1	2	yes	complemented
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	8	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
108	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
109	C₂×C₄	#41×C₆₇	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	4	1	no	—
113	C₈	SD₁₆×C₆₇	1	2	yes	complemented
114	C₂²	—	8	2	no	—
115	C₂²	—	8	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	#121×C₆₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
122	C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
123	C₄	#135×C₆₇	1	1	yes	characteristic
124	C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
125	C₂	—	16	2	no	—
126	C₂	#460×C₆₇	1	1	yes	characteristic
127	C₂	#466×C₆₇	1	1	yes	characteristic
128	C₂	#472×C₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#493×C₆₇

C₂

yes

maximal, characteristic

#460×C₆₇

C₂

yes

maximal, complemented

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic, complemented

#335×C₆₇

C₂

yes

maximal, characteristic, complemented

#331×C₆₇

C₂

yes

maximal, complemented

C₂×SD₁₆×C₆₇

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

#127×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#102×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

D₈

yes

complemented

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#1668

yes

characteristic, direct product factor, Sylow

#493

C₂×C₆₇

yes

characteristic

#460

C₂×C₆₇

yes

complemented

C₈×Q₈

C₂×C₆₇

yes

characteristic, complemented

#335

C₂×C₆₇

yes

characteristic, complemented

#331

C₂×C₆₇

yes

complemented

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

#127

C₂²×C₆₇

yes

—

C₄×Q₈

C₂²×C₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₆₇

yes

characteristic

C₄⋊C₈

C₂²×C₆₇

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂²×C₆₇

yes

characteristic

C₂×Q₈

D₄×C₆₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₆₇

yes

characteristic

C₂×C₈

D₄×C₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₄×C₆₇

yes

characteristic

C₄²

—

C₄²

C₂³×C₆₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

D₈×C₆₇

yes

complemented

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

108

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

109

C₂×C₄

#41×C₆₇

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

SD₁₆×C₆₇

yes

complemented

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#121×C₆₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×D₈×C₆₇

yes

characteristic

122

C₄

C₂×SD₁₆×C₆₇

yes

characteristic

123

C₄

#135×C₆₇

yes

characteristic

124

C₄

AGL(1,ℤ/8)×C₆₇

yes

characteristic

125

C₂

—

126

C₂

#460×C₆₇

yes

characteristic

127

C₂

#466×C₆₇

yes

characteristic

128

C₂

#472×C₆₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₆₇
prime radical	#493×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₆₇

prime radical

#493×C₆₇

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
2. #460×C₆₇	C₂	semidirect
3. C₈×Q₈×C₆₇	C₂	semidirect
4. #335×C₆₇	C₂	semidirect
5. #331×C₆₇	C₂	semidirect
34. Q₈×C₆₇	D₈	semidirect
48. C₈×C₆₇	SD₁₆	semidirect
64. #1668	C₆₇	direct (N × H)
65. C₆₇	#1668	direct (N × H)
67. #460	C₂×C₆₇	semidirect
68. C₈×Q₈	C₂×C₆₇	semidirect
69. #335	C₂×C₆₇	semidirect
70. #331	C₂×C₆₇	semidirect
99. Q₈	D₈×C₆₇	semidirect
113. C₈	SD₁₆×C₆₇	semidirect

H ≅ G/N

type

2. #460×C₆₇

C₂

semidirect

3. C₈×Q₈×C₆₇

C₂

semidirect

4. #335×C₆₇

C₂

semidirect

5. #331×C₆₇

C₂

semidirect

34. Q₈×C₆₇

D₈

semidirect

48. C₈×C₆₇

SD₁₆

semidirect

64. #1668

C₆₇

direct (N × H)

65. C₆₇

#1668

direct (N × H)

67. #460

C₂×C₆₇

semidirect

68. C₈×Q₈

C₂×C₆₇

semidirect

69. #335

C₂×C₆₇

semidirect

70. #331

C₂×C₆₇

semidirect

99. Q₈

D₈×C₆₇

semidirect

113. C₈

SD₁₆×C₆₇

semidirect

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

subgroup	index
#493×C₆₇	2
#460×C₆₇	2
C₈×Q₈×C₆₇	2
#335×C₆₇	2
#331×C₆₇	2
#1668	67

subgroup

index

#493×C₆₇

#460×C₆₇

C₈×Q₈×C₆₇

#335×C₆₇

#331×C₆₇

#1668

Subgroups — C₆₇×#1668 #22556099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#1668 #22556099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1668 #22556099