Finite Group Databasebeta

← #22565376

Subgroups — #22565376

Order 8576.

532 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 153 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×C4×D268C211yesmaximal, characteristic
2#16863335C212yesmaximal, complemented
3#16863313C212yesmaximal
4#16863286C214yesmaximal, complemented
5#16863253C212yesmaximal, complemented
6#16863253C212yesmaximal, complemented
7C4×C4⋊Dic67C211yesmaximal, characteristic, complemented
8C2×#125×C67C211yesmaximal, characteristic, complemented
9C22×D268C2211yescharacteristic
10C22×C4×D67C2212yes—
11C4×D268—24no—
12C2×C4⋊Dic67C2212yes—
13C2×C4⋊Dic67C2211yescharacteristic
14C2×(C2×D67)⋊C4C2214yes—
15C2×(C2×D67)⋊C4C2212yes—
16C2×(C2×D67)⋊C4C2214yes—
17C2×C4×Dic67C2212yes—
18C67⋊#95C2212yes—
19C67⋊#95C2214yes—
20C67⋊#95C2214yes—
21#125×C67—24no—
22C2×C4⋊C4×C67C2211yescharacteristic
23C2×C4⋊C4×C67C2211yescharacteristic
24C2×C4⋊C4×C67C2212yes—
25C2×C4⋊C4×C67C2212yes—
26C2×C42×C67C2211yescharacteristic
27C23×D67C2312yes—
28C2×D268—24no—
29C2×D268—22no—
30C22×Dic67C2312yes—
31C22×Dic67C2314yes—
32C2×C4×D67—24no—
33C2×C4×D67—24no—
34C2×C4×D67—24no—
35C4⋊Dic67—22no—
36C4⋊Dic67—24no—
37(C2×D67)⋊C4—28no—
38(C2×D67)⋊C4—24no—
39(C2×D67)⋊C4—44no—
40C4×Dic67—42no—
41C22×C4×C67C2312yes—
42C22×C4×C67C2312yes—
43C22×C4×C67C2311yescharacteristic
44C22×C4×C67C2311yescharacteristic
45C22×C4×C67C2311yescharacteristic
46C4⋊C4×C67—24no—
47C4⋊C4×C67—24no—
48C4⋊C4×C67—22no—
49C4⋊C4×C67—22no—
50C42×C67D414yes—
51C23×C67C2411yescharacteristic
52C22×D67—22no—
53C22×D67—22no—
54C22×D67—22no—
55C22×D67—22no—
56C22×D67—22no—
57C22×D67—22no—
58C22×D67#4114yes—
59C2×C4×C67—22no—
60C2×C4×C67—21no—
61C2×C4×C67—21no—
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—24no—
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67C2×D412yes—
68C2×C4×C67C2×D412yes—
69C2×C4×C67C2×D412yes—
70C2×C4×C67#4114yes—
71C2×Dic67—22no—
72C2×Dic67—24no—
73C2×Dic67—24no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67—24no—
77D268—28no—
78C4×D67—48no—
79C22×C67C22×D411yescharacteristic
80C22×C67C2×#4112yes—
81C22×C67C2×#4112yes—
82C22×C67H(5,𝔽2)11yescharacteristic
83C22×C672−1+411yescharacteristic
84C2×D67—24no—
85C2×D67—24no—
86C2×D67—24no—
87C2×D67—42no—
88C2×D67—42no—
89C2×D67—42no—
90C2×D67—42no—
91C4×C67—22no—
92C4×C67—22no—
93C4×C67—42no—
94C4×C67—42no—
95C4×C67#12414yes—
96Dic67—42no—
97Dic67—44no—
98C2×C67#52511yescharacteristic
99C2×C67C2×#12411yescharacteristic
100C2×C67#53711yescharacteristic
101C2×C67#55212yes—
102C2×C67#55312yes—
103D67—44no—
104#2562—671nomaximal, Sylow
105C67#256211yescharacteristic, complemented, Sylow
106C2×#125D6711yescharacteristic, complemented
107C2×C4×D4—671no—
108#400—672no—
109#394—672no—
110#388—674no—
111#380—672no—
112#380—672no—
113C4×C4⋊C4—671no—
114C22×D4—671no—
115C23×C4—672no—
116#125—24no—
117C4×D4—1344no—
118C2×C4⋊C4—671no—
119C2×C4⋊C4—672no—
120C2×C4⋊C4C2×D6711yescharacteristic
121C2×C4⋊C4C2×D6711yescharacteristic
122C2×C4⋊C4C2×D6712yes—
123C2×C4⋊C4C2×D6712yes—
124C2×C22⋊C4—674no—
125C2×C22⋊C4—672no—
126C2×C22⋊C4—674no—
127C2×C42—672no—
128C2×C42C2×D6711yescharacteristic
129#95—672no—
130#95—674no—
131#95—674no—
132C24—672no—
133C2×D4—1344no—
134C2×D4—1342no—
135C22×C4—672no—
136C22×C4—674no—
137C22×C4—1344no—
138C22×C4—1344no—
139C22×C4—1344no—
140C22×C4C22×D6712yes—
141C22×C4C22×D6712yes—
142C22×C4C22×D6711yescharacteristic
143C22×C4C22×D6711yescharacteristic
144C22×C4C22×D6711yescharacteristic
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—1342no—
150C4⋊C4—1344no—
151C22⋊C4—2684no—
152C22⋊C4—1348no—
153C22⋊C4—1344no—
154C42—2682no—
155C42C22⋊D6714yes—
156C23—674no—
157C23—1342no—
158C23—1342no—
159C23—1342no—
160C23—1342no—
161C23—1342no—
162C23—1342no—
163C23C23×D6711yescharacteristic
164D4—1348no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—2688no—
174C2×C4—1342no—
175C2×C4—1344no—
176C2×C4—1344no—
177C2×C4—1342no—
178C2×C4—1342no—
179C2×C4—1344no—
180C2×C4C2×C22⋊D6712yes—
181C2×C4C2×C22⋊D6712yes—
182C2×C4C2×C22⋊D6712yes—
183C2×C4#1176162914yes—
184C22—2682no—
185C22—2682no—
186C22—2682no—
187C22—2682no—
188C22—1344no—
189C22—1344no—
190C22—1344no—
191C22C22×C22⋊D6711yescharacteristic
192C22#41×D6712yes—
193C22C2×#1176162912yes—
194C22#1452408911yescharacteristic
195C22#1452409011yescharacteristic
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—2684no—
201C4—2682no—
202C4#1452403214yes—
203C2—2684no—
204C2#1686390311yescharacteristic
205C2C2×#1452403211yescharacteristic
206C2#1686397411yescharacteristic
207C2#1686408412yes—
208C2#1686409912yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#125×C67C2
hypercenterC2×#125D67
hypocenterC67#2562
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #16863335C2semidirect
4. #16863286C2semidirect
5. #16863253C2semidirect
6. #16863253C2semidirect
7. C4×C4⋊Dic67C2semidirect
8. C2×#125×C67C2semidirect
105. C67#2562semidirect
106. C2×#125D67semidirect