Finite Group Databasebeta

← #30765323

Subgroups — #30765323

Order 17152.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#22558547C212yesmaximal
2#22558155C212yesmaximal, complemented
3#22558143C212yesmaximal, complemented
4C67×#2326C211yesmaximal, characteristic
5#16863360—22no—
6#16863122C2214yes—
7#16863120—22no—
8C2×#124×C67C411yescharacteristic
9C2×C4×Q8×C67C411yescharacteristic
10C2×C4×D4×C67C2211yescharacteristic
11#391×C67C2211yescharacteristic
12#390×C67C411yescharacteristic
13#388×C67—22no—
14#385×C67—22no—
15#384×C67C2211yescharacteristic
16#376×C67C411yescharacteristic
17C2×#11761612—42no—
18C22⋊(C67⋊C8)—24no—
19C22⋊(C67⋊C8)—42no—
20C22×Q8×C67C2×C411yescharacteristic
21C22×D4×C67C2×C411yescharacteristic
22C23×C4×C67D412yes—
23#124×C67—22no—
24#124×C67—22no—
25C4×Q8×C67—42no—
26C4×D4×C67—42no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67—22no—
29C2×C4⋊C4×C67C2×C411yescharacteristic
30C2×C4⋊C4×C67C2311yescharacteristic
31C2×C22⋊C4×C67—22no—
32C2×C22⋊C4×C67—22no—
33C2×C22⋊C4×C67D412yes—
34C2×C42×C67—21no—
35C2×C42×C67C2×C411yescharacteristic
36C2×C42×C67C2×C411yescharacteristic
37C2×C42×C67C2×C411yescharacteristic
38#95×C67—21no—
39#95×C67—21no—
40#95×C67—22no—
41#11761612—84no—
42C2×C67⋊C8—44no—
43C24×C67C22⋊C412yes—
44C2×Q8×C67—21no—
45C2×Q8×C67—41no—
46C2×Q8×C67—41no—
47C2×Q8×C67M1612yescomplemented
48C2×D4×C67—21no—
49C2×D4×C67—21no—
50C2×D4×C67—41no—
51C2×D4×C67—41no—
52C22×C4×C67—21no—
53C22×C4×C67—21no—
54C22×C4×C67—22no—
55C22×C4×C67—22no—
56C22×C4×C67—21no—
57C22×C4×C67—21no—
58C22×C4×C67—82no—
59C22×C4×C67C22⋊C411yescharacteristic
60C22×C4×C67C22⋊C411yescharacteristic
61C22×C4×C67C22×C411yescharacteristic
62C22×C4×C67C2×D412yes—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—41no—
65C4⋊C4×C67—42no—
66C4⋊C4×C67—82no—
67C22⋊C4×C67—28no—
68C22⋊C4×C67—42no—
69C22⋊C4×C67—82no—
70C42×C67—22no—
71C42×C67—21no—
72C42×C67—41no—
73C42×C67—42no—
74C42×C67M1612yescomplemented
75C23×C67—22no—
76C23×C67—22no—
77C23×C67—22no—
78C23×C67—42no—
79C23×C67—42no—
80C23×C67C2×C22⋊C411yescharacteristic
81Q8×C67—42no—
82Q8×C67—42no—
83D4×C67—42no—
84D4×C67—42no—
85C2×C4×C67—21no—
86C2×C4×C67—21no—
87C2×C4×C67—22no—
88C2×C4×C67—22no—
89C2×C4×C67—21no—
90C2×C4×C67—22no—
91C2×C4×C67—41no—
92C2×C4×C67—44no—
93C2×C4×C67—41no—
94C2×C4×C67—41no—
95C2×C4×C67—41no—
96C2×C4×C67—41no—
97C2×C4×C67—42no—
98C2×C4×C67—41no—
99C2×C4×C67—41no—
100C2×C4×C67—42no—
101C2×C4×C67—41no—
102C2×C4×C67—41no—
103C2×C4×C67—41no—
104C2×C4×C67—82no—
105C2×C4×C67#9912yes—
106C2×C4×C67C2×M1611yescharacteristic
107C2×C4×C67C2×M1611yescharacteristic
108C67⋊C8—84no—
109C22×C67—21no—
110C22×C67—21no—
111C22×C67—44no—
112C22×C67—42no—
113C22×C67—42no—
114C22×C67—42no—
115C22×C67—82no—
116C22×C67#40711yescharacteristic
117C22×C67C2×#9911yescharacteristic
118C22×C67#41311yescharacteristic
119C4×C67—22no—
120C4×C67—21no—
121C4×C67—21no—
122C4×C67—42no—
123C4×C67—42no—
124C4×C67—41no—
125C4×C67—42no—
126C4×C67—81no—
127C4×C67—81no—
128#11535—671nomaximal, Sylow
129C2×C67—21no—
130C2×C67—21no—
131C2×C67—82no—
132C2×C67#146511yescharacteristic
133C2×C67#212711yescharacteristic
134C2×C67#213511yescharacteristic
135#2326D6711yescharacteristic
136#1465—672no—
137#1325—672no—
138#1319—672no—
139C67#1153511yescharacteristic, complemented, Sylow
140C2×#124Dic6711yescharacteristic
141C2×C4×Q8Dic6711yescharacteristic
142C2×C4×D4C2×D6711yescharacteristic
143#407—1342no—
144#391C2×D6711yescharacteristic
145#390Dic6711yescharacteristic
146#388—22no—
147#385—22no—
148#384C2×D6711yescharacteristic
149#376Dic6711yescharacteristic
150#324—674no—
151#323—1342no—
152C22×Q8C2×Dic6711yescharacteristic
153C22×D4C2×Dic6711yescharacteristic
154C23×C4C22⋊D6712yes—
155C2×M16—2682no—
156#124—22no—
157#124—22no—
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×Dic6711yescharacteristic
163C2×C4⋊C4C22×D6711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D6712yes—
167C2×C42—21no—
168C2×C42C2×Dic6711yescharacteristic
169C2×C42C2×Dic6711yescharacteristic
170C2×C42C2×Dic6711yescharacteristic
171C22⋊C8—2682no—
172C22⋊C8—1344no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊Dic6712yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1176161212yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊Dic6711yescharacteristic
193C22×C4C22⋊Dic6711yescharacteristic
194C22×C4C22×Dic6711yescharacteristic
195C22×C4C2×C22⋊D6712yes—
196M16—5364no—
197C2×C8—2684no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42#1176161212yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic6711yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#1452395812yes—
241C2×C4C2×#1176161211yescharacteristic
242C2×C4C2×#1176161211yescharacteristic
243C8—5364no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1686336011yescharacteristic
252C22C2×#1452395811yescharacteristic
253C22#1686338011yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2255854711yescharacteristic
267C2#2256187811yescharacteristic
268C2#2256190411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2326C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2326D67
hypocenterC67#11535
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #22558155C2semidirect
3. #22558143C2semidirect
47. C2×Q8×C67M16semidirect
74. C42×C67M16semidirect
139. C67#11535semidirect
180. C2×Q8#11761612semidirect
209. C42#11761612semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225581552
#225581432
C67×#23262
#1153567