Group #22569615 — subgroups

Subgroups — #22569615

Order 8576.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#16863967	C₂	1	3	yes	maximal, complemented
2	#16863586	C₂	1	8	yes	maximal, complemented
3	#16863370	C₂	1	3	yes	maximal, complemented
4	#585×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic, complemented
6	#14524041	—	2	3	no	—
7	#14524030	—	2	3	no	—
8	#14524024	C₂²	1	12	yes	complemented
9	#14524022	—	2	6	no	—
10	#14524002	C₂²	1	3	yes	complemented
11	#14523958	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₆₇	—	2	3	no	—
13	H(5,𝔽₂)×C₆₇	—	2	3	no	—
14	H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	—
15	C₂×#41×C₆₇	—	2	6	no	—
16	C₂×#41×C₆₇	C₂²	1	3	yes	—
17	C₂×Dic₁₃₄	—	2	3	no	—
18	C₂×Dic₁₃₄	—	2	3	no	—
19	C₂²×Dic₆₇	C₂³	1	2	yes	—
20	C₄⋊Dic₆₇	—	2	3	no	—
21	Dic₆₇⋊C₄	—	2	12	no	—
22	C₂²⋊Dic₆₇	—	2	12	no	—
23	C₂²⋊Dic₆₇	C₂³	1	6	yes	—
24	C₄×Dic₆₇	—	2	3	no	—
25	#41×C₆₇	—	2	4	no	—
26	#41×C₆₇	—	2	12	no	—
27	#41×C₆₇	—	2	6	no	—
28	#41×C₆₇	—	2	12	no	—
29	#41×C₆₇	—	2	6	no	—
30	C₂×Q₈×C₆₇	—	2	6	no	—
31	C₂×Q₈×C₆₇	D₄	1	3	yes	—
32	C₂×D₄×C₆₇	—	2	6	no	—
33	C₂×D₄×C₆₇	—	2	12	no	—
34	C₂×D₄×C₆₇	D₄	1	3	yes	—
35	C₂×D₄×C₆₇	C₂³	1	6	yes	—
36	C₂²×C₄×C₆₇	—	2	4	no	—
37	C₂²×C₄×C₆₇	D₄	1	6	yes	—
38	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂³×C₆₇	—	2	4	no	—
40	C₂³×C₆₇	C₂×D₄	1	6	yes	—
41	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₆₇	—	2	4	no	—
43	Q₈×C₆₇	—	2	6	no	—
44	D₄×C₆₇	—	2	12	no	—
45	D₄×C₆₇	—	2	12	no	—
46	D₄×C₆₇	—	2	6	no	—
47	C₂×C₄×C₆₇	—	2	12	no	—
48	C₂×C₄×C₆₇	—	2	6	no	—
49	C₂×C₄×C₆₇	—	2	6	no	—
50	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
53	C₂×Dic₆₇	—	2	6	no	—
54	C₂×Dic₆₇	—	2	6	no	—
55	Dic₁₃₄	—	4	1	no	—
56	Dic₁₃₄	—	4	3	no	—
57	C₂²×C₆₇	—	2	6	no	—
58	C₂²×C₆₇	—	4	1	no	—
59	C₂²×C₆₇	—	4	6	no	—
60	C₂²×C₆₇	—	8	4	no	—
61	C₂²×C₆₇	C₂²×D₄	1	3	yes	—
62	C₄×C₆₇	—	2	6	no	—
63	C₄×C₆₇	#120	1	1	yes	characteristic
64	C₄×C₆₇	#120	1	3	yes	—
65	Dic₆₇	—	4	6	no	—
66	Dic₆₇	—	4	2	no	—
67	C₂×C₆₇	—	2	3	no	—
68	C₂×C₆₇	—	4	6	no	—
69	C₂×C₆₇	C₂×#120	1	1	yes	characteristic
70	#3028	—	67	1	no	maximal, Sylow
71	C₆₇	#3028	1	1	yes	characteristic, complemented, Sylow
72	#585	D₆₇	1	1	yes	characteristic
73	#536	—	67	3	no	—
74	#458	—	67	8	no	—
75	#410	—	67	3	no	—
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₆₇	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₆₇	1	3	yes	—
81	C₂²×Q₈	—	67	1	no	—
82	C₄⋊Q₈	—	134	3	no	—
83	#124	—	134	3	no	—
84	#123	—	67	12	no	—
85	C₂²⋊Q₈	—	134	6	no	—
86	#117	—	67	3	no	—
87	#99	—	67	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	134	3	no	—
95	C₂×Q₈	—	134	3	no	—
96	C₂×Q₈	C₂²⋊D₆₇	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₆₇	1	3	yes	—
100	C₂×D₄	C₂²×D₆₇	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	67	2	no	—
103	C₂²×C₄	C₂²⋊D₆₇	1	6	yes	—
104	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
105	C₄⋊C₄	—	134	3	no	—
106	C₄⋊C₄	—	134	12	no	—
107	C₂²⋊C₄	—	67	6	no	—
108	C₂²⋊C₄	—	134	12	no	—
109	C₄²	—	134	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₆₇	1	6	yes	—
112	C₂³	C₂³×D₆₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	268	1	no	—
116	Q₈	—	268	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	134	6	no	—
124	C₂×C₄	—	134	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₆₇	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	268	6	no	—
135	C₄	—	268	2	no	—
136	C₄	#14524012	1	1	yes	characteristic
137	C₄	#14524012	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#14524012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863967

C₂

yes

maximal, complemented

#16863586

C₂

yes

maximal, complemented

#16863370

C₂

yes

maximal, complemented

#585×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic, complemented

#14524041

—

#14524030

—

#14524024

C₂²

yes

complemented

#14524022

—

#14524002

C₂²

yes

complemented

#14523958

C₂²

yes

complemented

2₋¹⁺⁴×C₆₇

—

H(5,𝔽₂)×C₆₇

—

H(5,𝔽₂)×C₆₇

C₂²

yes

—

C₂×#41×C₆₇

—

C₂×#41×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

characteristic

C₄×C₆₇

#120

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#585

D₆₇

yes

—

—

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₆₇

yes

—

C₂×#41

—

C₂×#41

C₂×D₆₇

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

134

—

#124

—

134

—

#123

—

C₂²⋊Q₈

—

134

—

—

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

C₂²⋊D₆₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₆₇

yes

—

100

C₂×D₄

C₂²×D₆₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₆₇

yes

—

104

C₂²×C₄

C₂²×D₆₇

yes

characteristic

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

134

—

109

C₄²

—

134

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₆₇

yes

—

112

C₂³

C₂³×D₆₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

268

—

116

Q₈

—

268

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

126

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

127

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₆₇

yes

—

133

C₄

—

134

C₄

—

268

—

135

C₄

—

268

—

136

C₄

#14524012

yes

characteristic

137

C₄

#14524012

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#14524012

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14524012
commutator	C₂³×C₆₇	C₂⁴
Frattini	C₂³	C₂³×D₆₇
Fitting	#585×C₆₇	C₂
socle	C₂×C₆₇	C₂×#120
hypercenter	#585	D₆₇
hypocenter	C₆₇	#3028
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#14524012

commutator

C₂³×C₆₇

C₂⁴

Frattini

C₂³

C₂³×D₆₇

Fitting

#585×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #16863967	C₂	semidirect
2. #16863586	C₂	semidirect
3. #16863370	C₂	semidirect
5. C₂²×Dic₁₃₄	C₂²	semidirect
8. #14524024	C₂²	semidirect
10. #14524002	C₂²	semidirect
11. #14523958	C₂²	semidirect
71. C₆₇	#3028	semidirect

H ≅ G/N

type

1. #16863967

C₂

semidirect

2. #16863586

C₂

semidirect

3. #16863370

C₂

semidirect

5. C₂²×Dic₁₃₄

C₂²

semidirect

8. #14524024

C₂²

semidirect

10. #14524002

C₂²

semidirect

11. #14523958

C₂²

semidirect

71. C₆₇

#3028

semidirect

Subgroups — #22569615

Conjugacy classes of subgroups

Complements