Finite Group Databasebeta

← #22565951

Subgroups — #22565951

Order 8576.

632 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14524026C212yesmaximal, complemented
2C2×#14524017C211yesmaximal, characteristic
3#16863326C212yesmaximal, complemented
4#16863318C214yesmaximal, complemented
5#16863313C212yesmaximal, complemented
6#16863275C212yesmaximal, complemented
7#16863256C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C67C211yesmaximal, characteristic, complemented
9C22×D268C2211yescharacteristic
10C22×C22⋊D67C2212yes—
11#14524026—28no—
12#14524017—24no—
13C2×C4⋊Dic67C2211yescharacteristic
14C2×C4⋊Dic67C2212yes—
15C2×Dic67⋊C4C2214yes—
16C2×C22⋊Dic67C2212yes—
17C2×(C2×D67)⋊C4C2214yes—
18C2×(C2×D67)⋊C4C2212yes—
19C2×(C2×D67)⋊C4C2214yes—
20C67⋊#95C2212yes—
21C67⋊#95C2214yes—
22C23×C4×C67C2211yescharacteristic
23C23×C4×C67C2211yescharacteristic
24C23×C4×C67C2211yescharacteristic
25C2×C4⋊C4×C67—24no—
26C2×C4⋊C4×C67C2212yes—
27C2×C4⋊C4×C67C2212yes—
28C23×D67C2312yes—
29C2×D268—22no—
30C2×D268—42no—
31C2×C22⋊D67—24no—
32C2×C22⋊D67—24no—
33C2×C22⋊D67—24no—
34C22×Dic67C2314yes—
35C22×Dic67C2312yes—
36C4⋊Dic67—22no—
37C4⋊Dic67—42no—
38Dic67⋊C4—28no—
39C22⋊Dic67—24no—
40(C2×D67)⋊C4—24no—
41(C2×D67)⋊C4—28no—
42(C2×D67)⋊C4—44no—
43C24×C67C2311yescharacteristic
44C22×C4×C67—22no—
45C22×C4×C67—22no—
46C22×C4×C67—22no—
47C22×C4×C67—42no—
48C22×C4×C67—42no—
49C22×C4×C67D414yes—
50C22×C4×C67D414yescomplemented
51C22×C4×C67C2312yes—
52C22×C4×C67C2311yescharacteristic
53C22×C4×C67C2312yes—
54C22×C4×C67C2311yescharacteristic
55C4⋊C4×C67—24no—
56C4⋊C4×C67—24no—
57C23×C67—22no—
58C23×C67—22no—
59C23×C67C2×D412yes—
60C23×C67C2×D412yes—
61C23×C67#4112yes—
62C23×C67C2411yescharacteristic
63C22×D67—22no—
64C22×D67—22no—
65C22×D67—22no—
66C22×D67—24no—
67C22×D67—24no—
68C2×C4×C67—28no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—24no—
75C2×C4×C67—44no—
76C2×C4×C67—44no—
77C2×C4×C67C2×D412yes—
78C2×C4×C67C2×D412yes—
79C2×C4×C67C2×D414yes—
80C22⋊D67—48no—
81C2×Dic67—24no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—24no—
86C2×Dic67—24no—
87D268—44no—
88C22×C67—22no—
89C22×C67—22no—
90C22×C67—24no—
91C22×C67—24no—
92C22×C67#12114yes—
93C22×C67C22×D411yescharacteristic
94C22×C67C22×D411yescharacteristic
95C22×C67C2×#4111yescharacteristic
96C22×C67H(5,𝔽2)12yes—
97C22×C672−1+412yes—
98C2×D67—42no—
99C2×D67—42no—
100C2×D67—42no—
101C2×D67—44no—
102C2×D67—44no—
103C4×C67—24no—
104C4×C67—22no—
105C4×C67—22no—
106C4×C67—42no—
107Dic67—42no—
108Dic67—44no—
109C2×C67—24no—
110C2×C67C2×#12111yescharacteristic
111C2×C67#53411yescharacteristic
112C2×C67#53611yescharacteristic
113C2×C67#53712yes—
114C2×C67#53912yes—
115D67—82no—
116#2618—671nomaximal, Sylow
117C67#261811yescharacteristic, complemented, Sylow
118C2×#123—672no—
119C2×#121—671no—
120C22×C4⋊C4D6711yescharacteristic, complemented
121#398—672no—
122#396—674no—
123#394—672no—
124#386—672no—
125#380—671no—
126C22×D4—672no—
127C22×D4—671no—
128C23×C4C2×D6711yescharacteristic
129C23×C4C2×D6711yescharacteristic
130C23×C4C2×D6711yescharacteristic
131#123—1348no—
132#121—1344no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—671no—
136C2×C4⋊C4—674no—
137C2×C4⋊C4C2×D6712yes—
138C2×C4⋊C4C2×D6712yes—
139C2×C22⋊C4—674no—
140C2×C22⋊C4—672no—
141C2×C22⋊C4—672no—
142C2×C22⋊C4—674no—
143#95—672no—
144#95—674no—
145C24—672no—
146C24C22×D6711yescharacteristic
147C2×D4—2682no—
148C2×D4—1344no—
149C2×D4—1344no—
150C2×D4—1344no—
151C2×D4—1342no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—674no—
158C22×C4—672no—
159C22×C4D26814yescomplemented
160C22×C4C22⋊D6714yes—
161C22×C4C22×D6712yes—
162C22×C4C22×D6711yescharacteristic
163C22×C4C22×D6711yescharacteristic
164C22×C4C22×D6712yes—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—2682no—
168C4⋊C4—1342no—
169C4⋊C4—1348no—
170C22⋊C4—2684no—
171C22⋊C4—1344no—
172C22⋊C4—1344no—
173C22⋊C4—1348no—
174C23—22no—
175C23—22no—
176C23—1342no—
177C23—1342no—
178C23—1342no—
179C23—1344no—
180C23—1344no—
181C23C2×C22⋊D6712yes—
182C23C2×D26812yes—
183C23#1176162812yes—
184C23C23×D6711yescharacteristic
185D4—2684no—
186D4—2688no—
187C2×C4—28no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—1342no—
197C2×C4—1344no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1344no—
201C2×C4—1344no—
202C2×C4C2×C22⋊D6712yes—
203C2×C4C2×C22⋊D6712yes—
204C2×C4C2×D26814yes—
205C22—22no—
206C22—22no—
207C22—24no—
208C22—24no—
209C22—2682no—
210C22—2682no—
211C22—2682no—
212C22—2684no—
213C22—2684no—
214C22#1452401714yes—
215C22C22×C22⋊D6711yescharacteristic
216C22C22×D26811yescharacteristic
217C22C2×#1176162811yescharacteristic
218C22#1452408812yes—
219C22#1452409112yes—
220C4—24no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—2682no—
225C4—2684no—
226C2—24no—
227C2—5362no—
228C2C2×#1452401711yescharacteristic
229C2#1686395511yescharacteristic
230C2#1686397011yescharacteristic
231C2#1686397712yes—
232C2#1686399312yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC22×C4⋊C4×C67C2
hypercenterC22×C4⋊C4D67
hypocenterC67#2618
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#14524026C2semidirect
3. #16863326C2semidirect
4. #16863318C2semidirect
5. #16863313C2semidirect
6. #16863275C2semidirect
7. #16863256C2semidirect
8. C22×C4⋊C4×C67C2semidirect
50. C22×C4×C67D4semidirect
117. C67#2618semidirect
120. C22×C4⋊C4D67semidirect
159. C22×C4D268semidirect