Finite Group Databasebeta

← #30765191

Subgroups — #30765191

Order 17152.

706 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 107 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#22558536C212yesmaximal
2#22558139C214yesmaximal, complemented
3C2×#381×C67C211yesmaximal, characteristic
4#16863360—22no—
5#16863120—22no—
6#16863120C2214yes—
7C24×C4×C67C411yescharacteristic
8C22×C22⋊C4×C67C411yescharacteristic
9C22×C22⋊C4×C67C2211yescharacteristic
10#381×C67—22no—
11#381×C67C412yes—
12#381×C67C2212yes—
13C2×#95×C67—22no—
14C2×#11761612—42no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—42no—
17C25×C67C2×C411yescharacteristic
18C23×C4×C67—21no—
19C23×C4×C67—21no—
20C23×C4×C67—24no—
21C23×C4×C67—21no—
22C23×C4×C67—42no—
23C23×C4×C67—42no—
24C23×C4×C67C2×C412yes—
25C23×C4×C67D412yes—
26C2×C22⋊C4×C67—21no—
27C2×C22⋊C4×C67—42no—
28C2×C22⋊C4×C67—42no—
29C2×C22⋊C4×C67C2×C412yes—
30C2×C22⋊C4×C67C2×C411yescharacteristic
31C2×C22⋊C4×C67D412yes—
32C2×C22⋊C4×C67C2311yescharacteristic
33#95×C67—24no—
34#95×C67—24no—
35#11761612—84no—
36C2×C67⋊C8—44no—
37C24×C67—22no—
38C24×C67—22no—
39C24×C67—22no—
40C24×C67—44no—
41C24×C67C22⋊C411yescharacteristic
42C24×C67C22⋊C411yescharacteristic
43C24×C67C22×C411yescharacteristic
44C22×C4×C67—24no—
45C22×C4×C67—24no—
46C22×C4×C67—22no—
47C22×C4×C67—22no—
48C22×C4×C67—22no—
49C22×C4×C67—22no—
50C22×C4×C67—22no—
51C22×C4×C67—416no—
52C22×C4×C67—42no—
53C22×C4×C67—42no—
54C22×C4×C67—42no—
55C22×C4×C67—42no—
56C22×C4×C67—42no—
57C22×C4×C67—42no—
58C22×C4×C67—44no—
59C22×C4×C67—42no—
60C22×C4×C67—44no—
61C22×C4×C67C22⋊C412yes—
62C22×C4×C67M1614yescomplemented
63C22×C4×C67C2×D412yes—
64C22⋊C4×C67—41no—
65C22⋊C4×C67—42no—
66C22⋊C4×C67—41no—
67C22⋊C4×C67—42no—
68C22⋊C4×C67—42no—
69C23×C67—24no—
70C23×C67—21no—
71C23×C67—21no—
72C23×C67—28no—
73C23×C67—21no—
74C23×C67—21no—
75C23×C67—21no—
76C23×C67—21no—
77C23×C67—42no—
78C23×C67—42no—
79C23×C67—48no—
80C23×C67—48no—
81C23×C67—44no—
82C23×C67—44no—
83C23×C67#9912yes—
84C23×C67#10012yes—
85C23×C67C2×C22⋊C411yescharacteristic
86C23×C67C2×M1612yes—
87C2×C4×C67—24no—
88C2×C4×C67—22no—
89C2×C4×C67—22no—
90C2×C4×C67—48no—
91C2×C4×C67—48no—
92C2×C4×C67—44no—
93C2×C4×C67—44no—
94C2×C4×C67—42no—
95C2×C4×C67—42no—
96C2×C4×C67—44no—
97C2×C4×C67—42no—
98C2×C4×C67—42no—
99C2×C4×C67—44no—
100C2×C4×C67—42no—
101C2×C4×C67—82no—
102C2×C4×C67—84no—
103C67⋊C8—84no—
104C22×C67—24no—
105C22×C67—24no—
106C22×C67—22no—
107C22×C67—22no—
108C22×C67—22no—
109C22×C67—22no—
110C22×C67—22no—
111C22×C67—42no—
112C22×C67—42no—
113C22×C67—42no—
114C22×C67—416no—
115C22×C67—42no—
116C22×C67—42no—
117C22×C67—42no—
118C22×C67#35214yes—
119C22×C67#40711yescharacteristic
120C22×C67C2×#9911yescharacteristic
121C22×C67C2×#10011yescharacteristic
122C4×C67—42no—
123C4×C67—42no—
124C4×C67—44no—
125C4×C67—82no—
126#11500—671nomaximal, Sylow
127C2×C67—21no—
128C2×C67—24no—
129C2×C67—21no—
130C2×C67—42no—
131C2×C67—42no—
132C2×C67#146211yescharacteristic
133C2×C67C2×#35212yes—
134C2×#381D6711yescharacteristic
135#1462—672no—
136#1317—674no—
137C67#1150011yescharacteristic, complemented, Sylow
138C24×C4Dic6711yescharacteristic
139C22×C22⋊C4Dic6711yescharacteristic
140C22×C22⋊C4C2×D6711yescharacteristic
141#407—1342no—
142#381—22no—
143#381Dic6712yes—
144#381C2×D6712yes—
145C2×#95—22no—
146#323—674no—
147#323—1342no—
148C25C2×Dic6711yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4C2×Dic6712yes—
156C23×C4C22⋊D6712yes—
157C2×M16—2682no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4C2×Dic6712yes—
162C2×C22⋊C4C2×Dic6711yescharacteristic
163C2×C22⋊C4C22⋊D6712yes—
164C2×C22⋊C4C22×D6711yescharacteristic
165C22⋊C8—2682no—
166C22⋊C8—1344no—
167#95—24no—
168#95—24no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—44no—
173C24C22⋊Dic6711yescharacteristic
174C24C22⋊Dic6711yescharacteristic
175C24C22×Dic6711yescharacteristic
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—416no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4C22⋊Dic6712yes—
194C22×C4#1176161214yescomplemented
195C22×C4C2×C22⋊D6712yes—
196M16—5364no—
197C2×C8—2684no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C23—21no—
204C23—21no—
205C23—24no—
206C23—28no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—42no—
212C23—42no—
213C23—48no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23#1452395812yes—
218C23#1452396012yes—
219C23C2×C22⋊Dic6711yescharacteristic
220C23C2×#1176161212yes—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—84no—
237C8—5364no—
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—416no—
250C22—42no—
251C22—42no—
252C22#1686318614yes—
253C22#1686336011yescharacteristic
254C22C2×#1452395811yescharacteristic
255C22C2×#1452396011yescharacteristic
256C4—42no—
257C4—42no—
258C4—44no—
259C4—82no—
260C2—21no—
261C2—21no—
262C2—24no—
263C2—42no—
264C2—42no—
265C2#2255853611yescharacteristic
266C2C2×#1686318612yes—
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC2×#381×C67C2
socleC22×C67C2×#99
prime radicalC25×C67C2×C4
cosocleC2×C22⋊C4×C67C23
hypercenterC2×#381D67
hypocenterC67#11500
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #22558139C2semidirect
62. C22×C4×C67M16semidirect
137. C67#11500semidirect
194. C22×C4#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585362
#225581392
C2×#381×C672
#1150067