Finite Group Databasebeta

← #30765211

Subgroups — #30765211

Order 17152.

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

normal subgroups only

This group has 598 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#22558539C212yesmaximal
2#22558143C214yesmaximal, complemented
3C2×#391×C67C211yesmaximal, characteristic
4#16863360—22no—
5#16863122—22no—
6#16863122C2214yes—
7C23×Q8×C67C411yescharacteristic
8C22×C4⋊C4×C67C2211yescharacteristic
9C22×C42×C67C411yescharacteristic
10#391×C67—22no—
11#391×C67C412yes—
12#391×C67C2212yes—
13C2×#95×C67—22no—
14C2×#11761612—42no—
15C22⋊(C67⋊C8)—24no—
16C22⋊(C67⋊C8)—42no—
17C22×Q8×C67—24no—
18C22×Q8×C67—21no—
19C22×Q8×C67—42no—
20C22×Q8×C67—41no—
21C22×Q8×C67—41no—
22C22×Q8×C67C2×C412yes—
23C23×C4×C67—21no—
24C23×C4×C67—21no—
25C23×C4×C67C2×C411yescharacteristic
26C23×C4×C67D412yes—
27C2×C4⋊C4×C67—41no—
28C2×C4⋊C4×C67—41no—
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67C2311yescharacteristic
32C2×C42×C67—21no—
33C2×C42×C67—42no—
34C2×C42×C67C2×C412yes—
35#95×C67—24no—
36#95×C67—24no—
37#11761612—84no—
38C2×C67⋊C8—44no—
39C24×C67C22⋊C411yescharacteristic
40C2×Q8×C67—24no—
41C2×Q8×C67—22no—
42C2×Q8×C67—48no—
43C2×Q8×C67—48no—
44C2×Q8×C67—42no—
45C2×Q8×C67—42no—
46C2×Q8×C67—42no—
47C2×Q8×C67—42no—
48C2×Q8×C67M1614yescomplemented
49C22×C4×C67—21no—
50C22×C4×C67—21no—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—24no—
54C22×C4×C67—24no—
55C22×C4×C67—22no—
56C22×C4×C67—22no—
57C22×C4×C67—22no—
58C22×C4×C67—22no—
59C22×C4×C67—42no—
60C22×C4×C67—42no—
61C22×C4×C67—42no—
62C22×C4×C67—41no—
63C22×C4×C67—41no—
64C22×C4×C67—44no—
65C22×C4×C67C22⋊C411yescharacteristic
66C22×C4×C67C22⋊C412yes—
67C22×C4×C67C22×C411yescharacteristic
68C22×C4×C67C2×D412yes—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—42no—
71C4⋊C4×C67—41no—
72C42×C67—42no—
73C42×C67—42no—
74C23×C67—22no—
75C23×C67—22no—
76C23×C67—21no—
77C23×C67—21no—
78C23×C67#9911yescharacteristic
79C23×C67#9911yescharacteristic
80C23×C67C2×C22⋊C411yescharacteristic
81Q8×C67—44no—
82Q8×C67—48no—
83Q8×C67—44no—
84C2×C4×C67—24no—
85C2×C4×C67—24no—
86C2×C4×C67—21no—
87C2×C4×C67—21no—
88C2×C4×C67—21no—
89C2×C4×C67—21no—
90C2×C4×C67—24no—
91C2×C4×C67—24no—
92C2×C4×C67—22no—
93C2×C4×C67—22no—
94C2×C4×C67—48no—
95C2×C4×C67—42no—
96C2×C4×C67—44no—
97C2×C4×C67—44no—
98C2×C4×C67—44no—
99C2×C4×C67—42no—
100C2×C4×C67—42no—
101C2×C4×C67—42no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—82no—
105C2×C4×C67#10012yes—
106C2×C4×C67C2×M1612yes—
107C67⋊C8—84no—
108C22×C67—24no—
109C22×C67—24no—
110C22×C67—21no—
111C22×C67—21no—
112C22×C67—21no—
113C22×C67—21no—
114C22×C67—21no—
115C22×C67—21no—
116C22×C67#35412yes—
117C22×C67#35612yes—
118C22×C67#40711yescharacteristic
119C22×C67C2×#9911yescharacteristic
120C22×C67C2×#10011yescharacteristic
121C4×C67—22no—
122C4×C67—22no—
123C4×C67—44no—
124C4×C67—42no—
125C4×C67—42no—
126C4×C67—44no—
127#11506—671nomaximal, Sylow
128C2×C67—21no—
129C2×C67—22no—
130C2×C67—22no—
131C2×C67—21no—
132C2×C67#146211yescharacteristic
133C2×C67C2×#35411yescharacteristic
134C2×C67C2×#35611yescharacteristic
135C2×#391D6711yescharacteristic
136#1463—672no—
137#1319—674no—
138C67#1150611yescharacteristic, complemented, Sylow
139C23×Q8Dic6711yescharacteristic
140C22×C4⋊C4C2×D6711yescharacteristic
141C22×C42Dic6711yescharacteristic
142#407—1342no—
143#391—22no—
144#391Dic6712yes—
145#391C2×D6712yes—
146C2×#95—22no—
147#324—674no—
148#324—1342no—
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—42no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8C2×Dic6712yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C2×Dic6711yescharacteristic
158C23×C4C22⋊D6712yes—
159C2×M16—2682no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C2×Dic6711yescharacteristic
163C2×C4⋊C4C22⋊D6712yes—
164C2×C4⋊C4C22×D6711yescharacteristic
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C2×Dic6712yes—
168C22⋊C8—2682no—
169C22⋊C8—1344no—
170#95—24no—
171#95—24no—
172C24C22⋊Dic6711yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8#1176161214yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—44no—
198C22×C4C22⋊Dic6711yescharacteristic
199C22×C4C22⋊Dic6712yes—
200C22×C4C22×Dic6711yescharacteristic
201C22×C4C2×C22⋊D6712yes—
202M16—5364no—
203C2×C8—2684no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C42—42no—
208C42—42no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23#1452395811yescharacteristic
214C23#1452395811yescharacteristic
215C23C2×C22⋊Dic6711yescharacteristic
216Q8—44no—
217Q8—48no—
218Q8—44no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—48no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4#1452396012yes—
241C2×C4C2×#1176161212yes—
242C8—5364no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1686319112yes—
252C22#1686319512yes—
253C22#1686336011yescharacteristic
254C22C2×#1452395811yescharacteristic
255C22C2×#1452396011yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—44no—
260C4—42no—
261C4—42no—
262C2—21no—
263C2—21no—
264C2—22no—
265C2—22no—
266C2#2255853611yescharacteristic
267C2C2×#1686319111yescharacteristic
268C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC2×#391×C67C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenterC2×#391D67
hypocenterC67#11506
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #22558143C2semidirect
48. C2×Q8×C67M16semidirect
138. C67#11506semidirect
181. C2×Q8#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585392
#225581432
C2×#391×C672
#1150667