Group #22562288 — subgroups

Subgroups — #22562288

Order 8576.

150 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16863836	C₂	1	1	yes	maximal, characteristic
2	#16863774	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863513	C₂	1	1	yes	maximal, characteristic
4	#16863216	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863213	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863212	C₂	1	1	yes	maximal, characteristic, complemented
7	#510×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11761618	—	2	1	no	—
10	#14524050	—	2	1	no	—
11	#14524041	—	2	1	no	—
12	#14523989	—	2	1	no	—
13	#14523987	—	2	1	no	—
14	#14523983	C₂²	1	1	yes	characteristic
15	#14523979	C₂²	1	1	yes	characteristic
16	#14523970	—	2	1	no	—
17	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
18	#135×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	Q₃₂×C₆₇	—	2	1	no	—
21	SD₃₂×C₆₇	—	2	1	no	—
22	M₃₂×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	Dic₂₆₈	—	2	1	no	—
25	Dic₂₆₈	—	2	1	no	—
26	#11761618	—	4	1	no	—
27	#11761618	—	4	1	no	—
28	#11761612	—	2	1	no	—
29	#11761612	—	4	1	no	—
30	C₂×C₆₇⋊C₈	—	4	1	no	—
31	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
32	Dic₆₇⋊C₄	—	4	2	no	—
33	C₄×Dic₆₇	D₄	1	1	yes	characteristic
34	C₆₇⋊C₁₆	—	2	1	no	—
35	#41×C₆₇	—	2	1	no	—
36	C₂×Q₈×C₆₇	—	2	1	no	—
37	Q₁₆×C₆₇	—	2	1	no	—
38	Q₁₆×C₆₇	—	2	1	no	—
39	Q₁₆×C₆₇	D₄	1	1	yes	characteristic
40	SD₁₆×C₆₇	—	2	1	no	—
41	D₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₁₆×C₆₇	—	2	1	no	—
44	Q₈×C₆₇	—	2	1	no	—
45	Q₈×C₆₇	—	4	1	no	—
46	Q₈×C₆₇	—	4	1	no	—
47	D₄×C₆₇	—	2	1	no	—
48	D₄×C₆₇	—	4	1	no	—
49	C₂×C₄×C₆₇	—	4	1	no	—
50	C₂×C₄×C₆₇	—	4	1	no	—
51	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₆₇	—	2	1	no	—
53	C₂×Dic₆₇	—	4	1	no	—
54	Dic₁₃₄	—	4	1	no	—
55	Dic₁₃₄	—	4	1	no	—
56	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₆₇	#41	1	1	yes	characteristic
58	C₆₇⋊C₈	—	4	1	no	—
59	C₆₇⋊C₈	D₈	1	1	yes	characteristic
60	C₆₇⋊C₈	D₈	1	1	yes	characteristic
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₄×C₆₇	—	4	1	no	—
64	C₄×C₆₇	—	8	1	no	—
65	C₄×C₆₇	#121	1	1	yes	characteristic
66	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
67	Dic₆₇	—	2	2	no	—
68	Dic₆₇	—	8	1	no	—
69	C₂×C₆₇	—	2	1	no	—
70	C₂×C₆₇	—	8	1	no	—
71	C₂×C₆₇	#459	1	1	yes	characteristic
72	#2214	—	67	1	no	maximal, Sylow
73	C₆₇	#2214	1	1	yes	characteristic, complemented, Sylow
74	#510	—	67	1	no	—
75	#510	D₆₇	1	1	yes	characteristic
76	#494	—	67	1	no	—
77	#443	—	67	1	no	—
78	#364	—	67	1	no	—
79	#362	—	67	1	no	—
80	#362	—	67	1	no	—
81	#135	C₂×D₆₇	1	1	yes	characteristic
82	C₂×Q₁₆	—	67	1	no	—
83	C₂×Q₁₆	—	134	1	no	—
84	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
85	#131	—	134	1	no	—
86	C₄⋊Q₈	—	134	1	no	—
87	Q₃₂	—	2	1	no	—
88	Q₃₂	—	134	1	no	—
89	SD₃₂	—	2	1	no	—
90	SD₃₂	—	134	1	no	—
91	M₃₂	—	67	1	no	—
92	M₃₂	C₂×D₆₇	1	1	yes	characteristic
93	#108	—	67	1	no	—
94	#104	—	134	1	no	—
95	C₄×C₈	—	67	1	no	—
96	#41	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	134	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	268	1	no	—
102	Q₁₆	—	268	1	no	—
103	Q₁₆	—	134	1	no	—
104	Q₁₆	—	134	1	no	—
105	Q₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
106	SD₁₆	—	2	1	no	—
107	D₈	C₂²⋊D₆₇	1	1	yes	characteristic
108	M₁₆	—	268	1	no	—
109	M₁₆	—	134	1	no	—
110	C₂×C₈	—	268	1	no	—
111	C₂×C₈	—	67	1	no	—
112	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
113	C₄⋊C₄	—	268	2	no	—
114	C₄²	—	67	1	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	—	134	1	no	—
117	Q₈	—	2	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	268	1	no	—
121	Q₈	—	268	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	268	1	no	—
127	C₂×C₄	—	134	1	no	—
128	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
129	C₈	—	268	1	no	—
130	C₈	—	67	1	no	—
131	C₈	—	67	1	no	—
132	C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
133	C₈	#11761627	1	1	yes	characteristic
134	C₂²	—	4	1	no	—
135	C₂²	#14524067	1	1	yes	characteristic
136	C₄	—	4	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	536	1	no	—
139	C₄	—	134	2	no	—
140	C₄	#14524015	1	1	yes	characteristic
141	C₄	D₈×D₆₇	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	8	1	no	—
144	C₂	#16863589	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863836

C₂

yes

maximal, characteristic

#16863774

C₂

yes

maximal, characteristic, complemented

#16863513

C₂

yes

maximal, characteristic

#16863216

C₂

yes

maximal, characteristic, complemented

#16863213

C₂

yes

maximal, characteristic, complemented

#16863212

C₂

yes

maximal, characteristic, complemented

#510×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#14523979

C₂²

yes

characteristic

#14523970

—

C₈×Dic₆₇

C₂²

yes

characteristic

#135×C₆₇

C₂²

yes

characteristic

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

Q₃₂×C₆₇

—

SD₃₂×C₆₇

—

M₃₂×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

—

—

—

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

D₄

yes

characteristic

SD₁₆×C₆₇

—

D₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

D₈

yes

characteristic

C₆₇⋊C₈

D₈

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#510

—

#510

D₆₇

yes

—

—

—

—

—

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

#131

—

134

—

C₄⋊Q₈

—

134

—

Q₃₂

—

Q₃₂

—

134

—

SD₃₂

—

SD₃₂

—

134

—

M₃₂

—

M₃₂

C₂×D₆₇

yes

characteristic

#108

—

#104

—

134

—

C₄×C₈

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

134

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

268

—

102

Q₁₆

—

268

—

103

Q₁₆

—

134

—

104

Q₁₆

—

134

—

105

Q₁₆

C₂²⋊D₆₇

yes

characteristic

106

SD₁₆

—

107

D₈

C₂²⋊D₆₇

yes

characteristic

108

M₁₆

—

268

—

109

M₁₆

—

134

—

110

C₂×C₈

—

268

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₆₇

yes

characteristic

113

C₄⋊C₄

—

268

—

114

C₄²

—

115

C₁₆

—

116

C₁₆

—

134

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

268

—

121

Q₈

—

268

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

268

—

127

C₂×C₄

—

134

—

128

C₂×C₄

D₄×D₆₇

yes

characteristic

129

C₈

—

268

—

130

C₈

—

131

C₈

—

132

C₈

C₂×C₂²⋊D₆₇

yes

characteristic

133

C₈

#11761627

yes

characteristic

134

C₂²

—

135

C₂²

#14524067

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

536

—

139

C₄

—

134

—

140

C₄

#14524015

yes

characteristic

141

C₄

D₈×D₆₇

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

#16863589

yes

characteristic

	subgroup	G/H
center	C₂	#16863589
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#510×C₆₇	C₂
socle	C₂×C₆₇	#459
hypercenter	#510	D₆₇
hypocenter	C₆₇	#2214
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

subgroup

G/H

center

C₂

#16863589

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#510×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

N	H ≅ G/N	type
2. #16863774	C₂	semidirect
4. #16863216	C₂	semidirect
5. #16863213	C₂	semidirect
6. #16863212	C₂	semidirect
73. C₆₇	#2214	semidirect

H ≅ G/N

type

2. #16863774

C₂

semidirect

4. #16863216

C₂

semidirect

5. #16863213

C₂

semidirect

6. #16863212

C₂

semidirect

73. C₆₇

#2214

semidirect

Subgroups — #22562288

Conjugacy classes of subgroups

Complements