Group #22559839 — subgroups

Subgroups — #22559839

Order 8576.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16863792	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863772	C₂	1	1	yes	maximal, characteristic
3	#16863770	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863769	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863742	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863118	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761617	—	2	2	no	—
9	C₂×#11761615	—	2	2	no	—
10	C₂×#11761613	—	2	2	no	—
11	#14524043	C₂²	1	1	yes	characteristic
12	#14524040	C₂²	1	1	yes	characteristic
13	#14523974	—	2	2	no	—
14	#14523964	—	2	2	no	—
15	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	#127×C₆₇	C₂²	1	1	yes	characteristic
19	#102×C₆₇	—	2	2	no	—
20	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂×D₂₆₈	—	2	1	no	—
23	C₂×D₂₆₈	—	2	1	no	—
24	C₂×D₂₆₈	—	2	1	no	—
25	#11761617	—	4	4	no	—
26	#11761615	—	4	4	no	—
27	#11761613	—	4	4	no	—
28	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×Q₈×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	1	no	—
34	C₂×D₄×C₆₇	—	2	1	no	—
35	C₂×D₄×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	D₄	1	2	yes	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂³×C₆₇	—	4	1	no	—
41	C₂²×D₆₇	—	4	1	no	—
42	Q₈×C₆₇	—	4	2	no	—
43	D₄×C₆₇	—	4	2	no	—
44	D₄×C₆₇	—	4	2	no	—
45	D₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	—	4	1	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₆₇	—	4	1	no	—
51	D₂₆₈	—	4	2	no	—
52	D₂₆₈	—	4	2	no	—
53	D₂₆₈	—	4	2	no	—
54	Dic₁₃₄	—	4	2	no	—
55	C₈×C₆₇	—	2	2	no	—
56	C₆₇⋊C₈	—	2	2	no	—
57	C₆₇⋊C₈	SD₁₆	1	4	yes	complemented
58	C₂²×C₆₇	—	8	1	no	—
59	C₂²×C₆₇	—	8	1	no	—
60	C₂²×C₆₇	—	8	1	no	—
61	C₂²×C₆₇	#127	1	1	yes	characteristic
62	C₂×D₆₇	—	8	1	no	—
63	C₂×D₆₇	—	8	1	no	—
64	C₂×D₆₇	—	8	1	no	—
65	C₄×C₆₇	—	8	1	no	—
66	C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
67	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₆₇	—	8	1	no	—
70	C₂×C₆₇	—	16	1	no	—
71	C₂×C₆₇	#492	1	1	yes	characteristic
72	C₂×C₆₇	#496	1	1	yes	characteristic
73	C₂×C₆₇	#496	1	1	yes	characteristic
74	D₆₇	—	16	1	no	—
75	#1715	—	67	1	no	maximal, Sylow
76	C₆₇	#1715	1	1	yes	characteristic, complemented, Sylow
77	#498	—	67	1	no	—
78	#493	—	67	1	no	—
79	#492	—	67	1	no	—
80	#492	—	67	1	no	—
81	#486	—	67	1	no	—
82	#486	D₆₇	1	1	yes	characteristic, complemented
83	#322	—	67	1	no	—
84	C₂×SD₁₆	—	134	2	no	—
85	C₂×SD₁₆	—	134	2	no	—
86	C₂×D₈	—	134	2	no	—
87	C₄⋊Q₈	—	67	1	no	—
88	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
89	#127	—	67	1	no	—
90	#127	C₂×D₆₇	1	1	yes	characteristic
91	#106	—	134	2	no	—
92	#102	—	2	2	no	—
93	#102	—	134	2	no	—
94	C₄×C₈	—	67	1	no	—
95	C₄×C₈	—	67	1	no	—
96	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	134	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	134	1	no	—
103	C₂×D₄	—	134	1	no	—
104	C₂×D₄	—	134	1	no	—
105	SD₁₆	—	268	4	no	—
106	SD₁₆	—	268	4	no	—
107	D₈	—	268	4	no	—
108	C₂×C₈	—	67	2	no	—
109	C₂×C₈	—	67	2	no	—
110	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	134	1	no	—
114	C₄⋊C₄	—	134	1	no	—
115	C₄²	C₂²×D₆₇	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	268	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	268	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	268	2	no	—
124	D₄	—	268	2	no	—
125	D₄	—	268	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	268	1	no	—
128	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	67	4	no	—
133	C₈	—	134	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	536	1	no	—
138	C₂²	—	536	1	no	—
139	C₂²	—	536	1	no	—
140	C₂²	#14524039	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	536	1	no	—
143	C₄	SD₁₆×D₆₇	1	2	yes	—
144	C₄	#14524067	1	2	yes	—
145	C₄	#14524068	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	1072	1	no	—
148	C₂	#16863767	1	1	yes	characteristic
149	C₂	#16863784	1	1	yes	characteristic
150	C₂	#16863786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863792

C₂

yes

maximal, characteristic, complemented

#16863772

C₂

yes

maximal, characteristic

#16863770

C₂

yes

maximal, characteristic, complemented

#16863769

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, characteristic, complemented

#16863118

C₂

yes

maximal, characteristic, complemented

#486×C₆₇

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#14524040

C₂²

yes

characteristic

#14523974

—

#14523964

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#127×C₆₇

C₂²

yes

characteristic

#102×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

#11761617

—

#11761615

—

#11761613

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#127

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1715

—

maximal, Sylow

C₆₇

#1715

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic, complemented

#322

—

C₂×SD₁₆

—

134

—

C₂×SD₁₆

—

134

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#127

—

#127

C₂×D₆₇

yes

characteristic

#106

—

134

—

#102

—

#102

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

134

—

103

C₂×D₄

—

134

—

104

C₂×D₄

—

134

—

105

SD₁₆

—

268

—

106

SD₁₆

—

268

—

107

D₈

—

268

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₆₇

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

134

—

114

C₄⋊C₄

—

134

—

115

C₄²

C₂²×D₆₇

yes

characteristic

116

C₂³

—

117

C₂³

—

268

—

118

Q₈

—

119

Q₈

—

268

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

268

—

124

D₄

—

268

—

125

D₄

—

268

—

126

C₂×C₄

—

127

C₂×C₄

—

268

—

128

C₂×C₄

D₄×D₆₇

yes

characteristic

129

C₂×C₄

D₄×D₆₇

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

536

—

138

C₂²

—

536

—

139

C₂²

—

536

—

140

C₂²

#14524039

yes

characteristic

141

C₄

—

142

C₄

—

536

—

143

C₄

SD₁₆×D₆₇

yes

—

144

C₄

#14524067

yes

—

145

C₄

#14524068

yes

—

146

C₂

—

147

C₂

—

1072

—

148

C₂

#16863767

yes

characteristic

149

C₂

#16863784

yes

characteristic

150

C₂

#16863786

yes

characteristic

	subgroup	G/H
center	C₂²	#14524039
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#486×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#486	D₆₇
hypocenter	C₆₇	#1715
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14524039

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#486×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16863792	C₂	semidirect
3. #16863770	C₂	semidirect
4. #16863769	C₂	semidirect
5. #16863742	C₂	semidirect
6. #16863118	C₂	semidirect
7. #486×C₆₇	C₂	semidirect
57. C₆₇⋊C₈	SD₁₆	semidirect
76. C₆₇	#1715	semidirect
82. #486	D₆₇	semidirect

H ≅ G/N

type

1. #16863792

C₂

semidirect

3. #16863770

C₂

semidirect

4. #16863769

C₂

semidirect

5. #16863742

C₂

semidirect

6. #16863118

C₂

semidirect

7. #486×C₆₇

C₂

semidirect

57. C₆₇⋊C₈

SD₁₆

semidirect

76. C₆₇

#1715

semidirect

82. #486

D₆₇

semidirect

Subgroups — #22559839

Conjugacy classes of subgroups

Complements