Group #22559813 — subgroups

Subgroups — #22559813

Order 8576.

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

normal subgroups only

This group has 212 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	#16863775	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863772	C₂	1	1	yes	maximal, characteristic
3	#16863744	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863743	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863742	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761618	—	2	2	no	—
9	C₂×#11761613	—	2	2	no	—
10	#14524043	C₂²	1	1	yes	characteristic
11	#14524040	C₂²	1	1	yes	characteristic
12	#14523965	—	2	2	no	—
13	#14523964	—	2	2	no	—
14	#14523963	—	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	#11761618	—	4	4	no	—
26	#11761613	—	4	4	no	—
27	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
28	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×Q₈×C₆₇	—	2	1	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	1	no	—
34	C₂×D₄×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	D₄	1	2	yes	—
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₂³×C₆₇	—	4	1	no	—
40	C₂²×D₆₇	—	4	1	no	—
41	Q₈×C₆₇	—	4	2	no	—
42	D₄×C₆₇	—	4	2	no	—
43	D₄×C₆₇	—	4	2	no	—
44	D₄×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	—	4	1	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₇	—	4	1	no	—
50	D₂₆₈	—	4	2	no	—
51	D₂₆₈	—	4	2	no	—
52	D₂₆₈	—	4	2	no	—
53	Dic₁₃₄	—	4	2	no	—
54	C₈×C₆₇	—	2	2	no	—
55	C₆₇⋊C₈	—	2	2	no	—
56	C₆₇⋊C₈	D₈	1	4	yes	—
57	C₂²×C₆₇	—	8	1	no	—
58	C₂²×C₆₇	—	8	1	no	—
59	C₂²×C₆₇	—	8	1	no	—
60	C₂²×C₆₇	#127	1	1	yes	characteristic
61	C₂×D₆₇	—	8	1	no	—
62	C₂×D₆₇	—	8	1	no	—
63	C₂×D₆₇	—	8	1	no	—
64	C₄×C₆₇	—	8	1	no	—
65	C₄×C₆₇	C₂×D₈	1	2	yes	—
66	C₄×C₆₇	#135	1	2	yes	—
67	C₄×C₆₇	#135	1	2	yes	—
68	Dic₆₇	—	8	1	no	—
69	C₂×C₆₇	—	16	1	no	—
70	C₂×C₆₇	#493	1	1	yes	characteristic
71	C₂×C₆₇	#495	1	1	yes	characteristic
72	C₂×C₆₇	#495	1	1	yes	characteristic
73	D₆₇	—	16	1	no	—
74	#1709	—	67	1	no	maximal, Sylow
75	C₆₇	#1709	1	1	yes	characteristic, complemented, Sylow
76	#494	—	67	1	no	—
77	#493	—	67	1	no	—
78	#486	—	67	1	no	—
79	#486	—	67	1	no	—
80	#486	—	67	1	no	—
81	#486	D₆₇	1	1	yes	characteristic, complemented
82	C₈²	—	67	1	no	—
83	C₂×Q₁₆	—	134	2	no	—
84	C₂×D₈	—	134	2	no	—
85	C₄⋊Q₈	—	67	1	no	—
86	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
87	#127	—	67	1	no	—
88	#127	C₂×D₆₇	1	1	yes	characteristic
89	#102	—	2	2	no	—
90	#102	—	134	2	no	—
91	#102	—	134	2	no	—
92	#102	—	134	2	no	—
93	C₄×C₈	—	67	1	no	—
94	C₄×C₈	—	67	1	no	—
95	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	134	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	134	1	no	—
102	C₂×D₄	—	134	1	no	—
103	C₂×D₄	—	134	1	no	—
104	Q₁₆	—	268	4	no	—
105	D₈	—	268	4	no	—
106	C₂×C₈	—	67	2	no	—
107	C₂×C₈	—	67	2	no	—
108	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	134	1	no	—
112	C₄⋊C₄	—	134	1	no	—
113	C₄²	C₂²×D₆₇	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	268	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	268	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	268	2	no	—
122	D₄	—	268	2	no	—
123	D₄	—	268	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	268	1	no	—
126	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	67	4	no	—
131	C₈	—	134	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	536	1	no	—
136	C₂²	—	536	1	no	—
137	C₂²	—	536	1	no	—
138	C₂²	#14524039	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	536	1	no	—
141	C₄	D₈×D₆₇	1	2	yes	—
142	C₄	#14524063	1	2	yes	—
143	C₄	#14524064	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	1072	1	no	—
146	C₂	#16863771	1	1	yes	characteristic
147	C₂	#16863777	1	1	yes	characteristic
148	C₂	#16863778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863775

C₂

yes

maximal, characteristic, complemented

#16863772

C₂

yes

maximal, characteristic

#16863744

C₂

yes

maximal, characteristic, complemented

#16863743

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, characteristic, complemented

C₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#486×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761618

—

C₂×#11761613

—

#14524043

C₂²

yes

characteristic

#14524040

C₂²

yes

—

—

—

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

—

#11761618

—

#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

D₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#127

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

#135

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1709

—

maximal, Sylow

C₆₇

#1709

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₈²

—

C₂×Q₁₆

—

134

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#127

—

#127

C₂×D₆₇

yes

characteristic

#102

—

#102

—

134

—

#102

—

134

—

#102

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

134

—

102

C₂×D₄

—

134

—

103

C₂×D₄

—

134

—

104

Q₁₆

—

268

—

105

D₈

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₆₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

—

134

—

113

C₄²

C₂²×D₆₇

yes

characteristic

114

C₂³

—

115

C₂³

—

268

—

116

Q₈

—

117

Q₈

—

268

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

268

—

122

D₄

—

268

—

123

D₄

—

268

—

124

C₂×C₄

—

125

C₂×C₄

—

268

—

126

C₂×C₄

D₄×D₆₇

yes

characteristic

127

C₂×C₄

D₄×D₆₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

134

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

536

—

136

C₂²

—

536

—

137

C₂²

—

536

—

138

C₂²

#14524039

yes

characteristic

139

C₄

—

140

C₄

—

536

—

141

C₄

D₈×D₆₇

yes

—

142

C₄

#14524063

yes

—

143

C₄

#14524064

yes

—

144

C₂

—

145

C₂

—

1072

—

146

C₂

#16863771

yes

characteristic

147

C₂

#16863777

yes

characteristic

148

C₂

#16863778

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	#1709
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. #16863775	C₂	semidirect
3. #16863744	C₂	semidirect
4. #16863743	C₂	semidirect
5. #16863742	C₂	semidirect
6. C₈×C₆₇⋊C₈	C₂	semidirect
7. #486×C₆₇	C₂	semidirect
75. C₆₇	#1709	semidirect
81. #486	D₆₇	semidirect

H ≅ G/N

type

1. #16863775

C₂

semidirect

3. #16863744

C₂

semidirect

4. #16863743

C₂

semidirect

5. #16863742

C₂

semidirect

6. C₈×C₆₇⋊C₈

C₂

semidirect

7. #486×C₆₇

C₂

semidirect

75. C₆₇

#1709

semidirect

81. #486

D₆₇

semidirect

Subgroups — #22559813

Conjugacy classes of subgroups

Complements