Finite Group Databasebeta

← #22570142

Subgroups — #22570142

Order 8576.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#16863907C211yesmaximal, characteristic
2C2×#14524002C211yesmaximal, characteristic, complemented
3#16863738C214yesmaximal, complemented
4#16863721C214yesmaximal, complemented
5#16863438C212yesmaximal, complemented
6#16863398C212yesmaximal, complemented
7#407×C67C211yesmaximal, characteristic, complemented
8C2×#11761628C2211yescharacteristic
9#14524026—21no—
10#14524022C2212yes—
11#14524017C2212yes—
12#14524012—21no—
13C4×C22⋊D67—22no—
14#14524002—22no—
15#14524002C2214yes—
16#14524002C2211yescharacteristic
17C2×C4⋊Dic67C2212yes—
18C2×C22⋊Dic67—21no—
19C2×C4×Dic67—21no—
20#14523977C2214yescomplemented
21#14523975C2214yescomplemented
22#14523968C2214yescomplemented
23#14523965C2214yescomplemented
24C23×C4×C67C2211yescharacteristic
25C2×M16×C67C2212yes—
26C22⋊C8×C67C2214yes—
27#11761628—24no—
28C2×Dic134C2311yescharacteristic
29C2×D268C2311yescharacteristic
30C2×C22⋊D67—22no—
31C2×C22⋊D67—21no—
32C22×Dic67—22no—
33C2×C4×D67—21no—
34C4⋊Dic67—22no—
35C4⋊Dic67C2314yes—
36C4⋊Dic67C2312yes—
37Dic67⋊C4—22no—
38C22⋊Dic67—24no—
39C22⋊Dic67—21no—
40C22⋊Dic67—22no—
41(C2×D67)⋊C4—22no—
42C4×Dic67—24no—
43C4×Dic67—21no—
44C24×C67D411yescharacteristic
45C22×C4×C67—24no—
46C22×C4×C67D411yescharacteristic
47C22×C4×C67D412yes—
48C22×C4×C67C2312yes—
49C22×C4×C67C2311yescharacteristic
50M16×C67—24no—
51C2×C8×C67C2314yes—
52C23×C67—22no—
53C23×C67—42no—
54C23×C67C2×D412yes—
55C23×C67C2×D411yescharacteristic
56C22×D67—21no—
57C2×C4×C67—22no—
58C2×C4×C67—24no—
59C2×C4×C67—24no—
60C2×C4×C67C2×D412yes—
61C2×C4×C67C2×D411yescharacteristic
62C2×C4×C67#4114yes—
63C2×C4×C67C2411yescharacteristic
64C22⋊D67—44no—
65C22⋊D67—42no—
66C2×Dic67—24no—
67C2×Dic67—22no—
68C2×Dic67—21no—
69C2×Dic67—44no—
70D268—22no—
71C4×D67—42no—
72Dic134—22no—
73C8×C67—24no—
74C22×C67—22no—
75C22×C67—22no—
76C22×C67—22no—
77C22×C67—22no—
78C22×C67—44no—
79C22×C67#12312yes—
80C22×C67C22×D411yescharacteristic
81C2×D67—41no—
82C2×D67—42no—
83C4×C67—22no—
84C4×C67#12312yes—
85C4×C67C2×#4112yes—
86Dic67—41no—
87Dic67—44no—
88Dic67—42no—
89C2×C67—22no—
90C2×C67—42no—
91C2×C67C2×#12311yescharacteristic
92C2×C67#57512yes—
93D67—81no—
94#3093—671nomaximal, Sylow
95C67#309311yescharacteristic, complemented, Sylow
96#525—671no—
97C2×#117—671no—
98#485—674no—
99#482—674no—
100#428—672no—
101#417—672no—
102#407D6711yescharacteristic, complemented
103C2×#41—671no—
104C23×C4C2×D6711yescharacteristic
105C2×M16C2×D6712yes—
106#123—1341no—
107C22⋊Q8—672no—
108#121—672no—
109#120—1341no—
110C4×D4—1342no—
111#117—674no—
112#117—671no—
113#117—1342no—
114C2×C4⋊C4—672no—
115C2×C22⋊C4—1341no—
116C2×C42—1341no—
117#107—674no—
118#106—674no—
119#103—674no—
120#102—674no—
121C22⋊C8C2×D6714yes—
122C24D26811yescharacteristic
123#41—1344no—
124C2×Q8—671no—
125C2×D4—671no—
126C2×D4—1342no—
127C2×D4—1341no—
128C22×C4—24no—
129C22×C4—1342no—
130C22×C4—1341no—
131C22×C4D26811yescharacteristic
132C22×C4D26812yes—
133C22×C4C22×D6712yes—
134C22×C4C22×D6711yescharacteristic
135M16—24no—
136C2×C8C22×D6714yes—
137C4⋊C4—674no—
138C4⋊C4—672no—
139C4⋊C4—1342no—
140C4⋊C4—1342no—
141C22⋊C4—1344no—
142C22⋊C4—1341no—
143C22⋊C4—1342no—
144C22⋊C4—1342no—
145C42—1344no—
146C42—1341no—
147C23—22no—
148C23—42no—
149C23—1341no—
150C23C2×D26812yes—
151C23C2×D26811yescharacteristic
152Q8—1342no—
153D4—2684no—
154D4—2682no—
155D4—1342no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—2684no—
160C2×C4—2682no—
161C2×C4—1344no—
162C2×C4—1342no—
163C2×C4—1341no—
164C2×C4C2×D26812yes—
165C2×C4C2×D26811yescharacteristic
166C2×C4#1176162714yes—
167C2×C4C23×D6711yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—2681no—
175C22—2682no—
176C22#1452402712yes—
177C22C22×D26811yescharacteristic
178C4—22no—
179C4—2681no—
180C4—2684no—
181C4—2682no—
182C4#1452402712yes—
183C4C2×#1176162712yes—
184C2—22no—
185C2—42no—
186C2—5361no—
187C2C2×#1452402711yescharacteristic
188C2#1686419612yes—
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#407×C67C2
socleC22×C67C22×D4
hypercenter#407D67
hypocenterC67#3093
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#14524002C2semidirect
3. #16863738C2semidirect
4. #16863721C2semidirect
5. #16863438C2semidirect
6. #16863398C2semidirect
7. #407×C67C2semidirect
20. #14523977C22semidirect
21. #14523975C22semidirect
22. #14523968C22semidirect
23. #14523965C22semidirect
95. C67#3093semidirect
102. #407D67semidirect