Finite Group Databasebeta

← C2×#480×C67 #22557516

Subgroups — C2×#480×C67 #22557516

Order 8576.

512 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×C67C211yesmaximal, characteristic
2C22×C4⋊C4×C67C211yesmaximal, characteristic, complemented
3#480×C67C218yesmaximal, direct product factor
4C2×#107×C67C212yesmaximal, complemented
5C2×#102×C67C212yesmaximal, complemented
6C2×C22⋊C8×C67C211yesmaximal, characteristic, complemented
7C22×D4×C67—21no—
8C22×D4×C67C2211yescharacteristic
9C23×C4×C67—21no—
10C23×C4×C67C2211yescharacteristic
11C22×C8×C67C2212yes—
12#121×C67—22no—
13#121×C67C2214yes—
14C2×C4⋊C4×C67—22no—
15C2×C4⋊C4×C67—21no—
16C2×C4⋊C4×C67C2212yes—
17C2×C4⋊C4×C67C2211yescharacteristic
18C2×C4⋊C4×C67C2214yescomplemented
19C2×C22⋊C4×C67—21no—
20#107×C67C2218yescomplemented
21#102×C67C2218yescomplemented
22C22⋊C8×C67C2214yescomplemented
23C24×C67—21no—
24C24×C67D411yescharacteristic
25C2×D4×C67—24no—
26C2×D4×C67—21no—
27C2×D4×C67—24no—
28C2×D4×C67—44no—
29C2×D4×C67C2312yes—
30C22×C4×C67—24no—
31C22×C4×C67—24no—
32C22×C4×C67—21no—
33C22×C4×C67—22no—
34C22×C4×C67—44no—
35C22×C4×C67D412yes—
36C22×C4×C67D411yescharacteristic
37C22×C4×C67C2312yes—
38C22×C4×C67C2311yescharacteristic
39C2×C8×C67—24no—
40C2×C8×C67C2314yes—
41C4⋊C4×C67—22no—
42C4⋊C4×C67—24no—
43C4⋊C4×C67—21no—
44C4⋊C4×C67C2314yes—
45C4⋊C4×C67C2312yes—
46C22⋊C4×C67—24no—
47C23×C67—22no—
48C23×C67—22no—
49C23×C67—22no—
50C23×C67—42no—
51C23×C67—42no—
52C23×C67—42no—
53C23×C67D814yes—
54C23×C67C2×D412yes—
55C23×C67C2×D411yescharacteristic
56D4×C67—24no—
57D4×C67—48no—
58D4×C67—42no—
59C2×C4×C67—22no—
60C2×C4×C67—24no—
61C2×C4×C67—24no—
62C2×C4×C67—22no—
63C2×C4×C67—24no—
64C2×C4×C67—48no—
65C2×C4×C67—44no—
66C2×C4×C67—42no—
67C2×C4×C67C2×D412yes—
68C2×C4×C67C2×D411yescharacteristic
69C2×C4×C67#4114yes—
70C2×C4×C67C2411yescharacteristic
71C8×C67—24no—
72C22×C67—22no—
73C22×C67—22no—
74C22×C67—28no—
75C22×C67—42no—
76C22×C67—42no—
77C22×C67—42no—
78C22×C67—84no—
79C22×C67#12312yes—
80C22×C67C2×D814yes—
81C22×C67C2×D812yes—
82C22×C67#13712yes—
83C22×C67C22×D411yescharacteristic
84C4×C67—22no—
85C4×C67—44no—
86C4×C67—42no—
87C4×C67#12312yes—
88C4×C67C2×#4112yes—
89C2×C67—24no—
90C2×C67—82no—
91C2×C67#48014yesdirect product factor
92C2×C67C2×#12311yescharacteristic
93C2×C67C22×D811yescharacteristic
94C2×C67C2×#13711yescharacteristic
95C2×#480C6711yesmaximal, characteristic, direct product factor, Sylow
96C67C2×#48011yescharacteristic, direct product factor, Sylow
97C2×#121C2×C6711yescharacteristic
98C22×C4⋊C4C2×C6711yescharacteristic, complemented
99#480C2×C6718yesdirect product factor
100C2×#107C2×C6712yescomplemented
101C2×#102C2×C6712yescomplemented
102C2×C22⋊C8C2×C6711yescharacteristic, complemented
103C22×D4—21no—
104C22×D4C22×C6711yescharacteristic
105C23×C4—21no—
106C23×C4C22×C6711yescharacteristic
107C22×C8C22×C6712yes—
108#121—22no—
109#121C22×C6714yes—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C22×C6712yes—
113C2×C4⋊C4C22×C6711yescharacteristic
114C2×C4⋊C4C22×C6714yescomplemented
115C2×C22⋊C4—21no—
116#107C22×C6718yescomplemented
117#102C22×C6718yescomplemented
118C22⋊C8C22×C6714yescomplemented
119C24—21no—
120C24D4×C6711yescharacteristic
121C2×D4—24no—
122C2×D4—21no—
123C2×D4—24no—
124C2×D4—44no—
125C2×D4C23×C6712yes—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—44no—
131C22×C4D4×C6711yescharacteristic
132C22×C4D4×C6712yes—
133C22×C4C23×C6712yes—
134C22×C4C23×C6711yescharacteristic
135C2×C8—24no—
136C2×C8C23×C6714yes—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4C23×C6714yes—
141C4⋊C4C23×C6712yes—
142C22⋊C4—24no—
143C23—22no—
144C23—22no—
145C23—22no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23D8×C6714yes—
150C23C2×D4×C6712yes—
151C23C2×D4×C6711yescharacteristic
152D4—24no—
153D4—48no—
154D4—42no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—48no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4C2×D4×C6712yes—
164C2×C4C2×D4×C6711yescharacteristic
165C2×C4#41×C6714yes—
166C2×C4C24×C6711yescharacteristic
167C8—24no—
168C22—28no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—84no—
175C22#123×C6712yes—
176C22C2×D8×C6714yes—
177C22C2×D8×C6712yes—
178C22#137×C6712yes—
179C22C22×D4×C6711yescharacteristic
180C4—22no—
181C4—44no—
182C4—42no—
183C4#123×C6712yes—
184C4C2×#41×C6712yes—
185C2—24no—
186C2—82no—
187C2#480×C6714yesdirect product factor
188C2C2×#123×C6711yescharacteristic
189C2C22×D8×C6711yescharacteristic
190C2C2×#137×C6711yescharacteristic
subgroupG/H
center, socleC23×C67C2×D4
commutator, Frattini, cosocleC2×C4C24×C67
prime radicalC2×#121×C67C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×C4⋊C4×C67C2semidirect
3. #480×C67C2direct (N × H)
4. C2×#107×C67C2semidirect
5. C2×#102×C67C2semidirect
6. C2×C22⋊C8×C67C2semidirect
18. C2×C4⋊C4×C67C22semidirect
20. #107×C67C22semidirect
21. #102×C67C22semidirect
22. C22⋊C8×C67C22semidirect
91. C2×C67#480direct (N × H)
95. C2×#480C67direct (N × H)
96. C67C2×#480direct (N × H)
98. C22×C4⋊C4C2×C67semidirect
99. #480C2×C67direct (N × H)
100. C2×#107C2×C67semidirect
101. C2×#102C2×C67semidirect
102. C2×C22⋊C8C2×C67semidirect
114. C2×C4⋊C4C22×C67semidirect
116. #107C22×C67semidirect
117. #102C22×C67semidirect
118. C22⋊C8C22×C67semidirect
187. C2#480×C67direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#121×C672
C22×C4⋊C4×C672
#480×C672
C2×#107×C672
C2×#102×C672
C2×C22⋊C8×C672
C2×#48067