Finite Group Databasebeta

← #30765448

Subgroups — #30765448

Order 17152.

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

normal subgroups only

This group has 444 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#22558547C211yesmaximal, characteristic
2#22558530C211yesmaximal, characteristic
3#22558139C214yesmaximal, complemented
4C67×#2486C211yesmaximal, characteristic
5#16863449—21no—
6#16863360—21no—
7#16863122—21no—
8#16863120—21no—
9#16863120C2212yes—
10#16863120C2212yes—
11C2×#123×C67C412yes—
12C2×C4×D4×C67C2211yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic
14#398×C67—21no—
15#397×C67—21no—
16#389×C67—21no—
17#381×C67C2212yes—
18#380×C67—21no—
19#380×C67C411yescharacteristic
20C2×#11761612—41no—
21C4⋊(C67⋊C8)—41no—
22C22⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24C22⋊(C67⋊C8)—41no—
25C4×C67⋊C8—41no—
26C22×D4×C67C2×C411yescharacteristic
27C23×C4×C67C2×C412yes—
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67D411yescharacteristic
30#123×C67—44no—
31C4×D4×C67—41no—
32C4×D4×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67—41no—
39C2×C4⋊C4×C67D411yescharacteristic
40C2×C22⋊C4×C67—22no—
41C2×C22⋊C4×C67C2×C412yes—
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C22⋊C4×C67C2311yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—21no—
46#95×C67—22no—
47#95×C67—22no—
48#11761612—82no—
49C2×C67⋊C8—42no—
50C2×C67⋊C8—42no—
51C24×C67C22⋊C411yescharacteristic
52C24×C67C22×C411yescharacteristic
53C2×D4×C67—22no—
54C2×D4×C67—22no—
55C2×D4×C67—41no—
56C22×C4×C67—21no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—24no—
63C22×C4×C67—42no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C2×D411yescharacteristic
71C22×C4×C67C2×D411yescharacteristic
72C4⋊C4×C67—41no—
73C4⋊C4×C67—42no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—44no—
78C22⋊C4×C67—41no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—44no—
81C22⋊C4×C67—42no—
82C42×C67—21no—
83C42×C67—21no—
84C23×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67C2×C22⋊C411yescharacteristic
94C23×C67#13112yes—
95D4×C67—44no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—22no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—42no—
103C2×C4×C67—42no—
104C2×C4×C67—42no—
105C2×C4×C67—41no—
106C2×C4×C67—41no—
107C2×C4×C67—41no—
108C2×C4×C67—42no—
109C2×C4×C67—42no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—81no—
115C2×C4×C67—84no—
116C2×C4×C67#9911yescharacteristic
117C2×C4×C67#9911yescharacteristic
118C2×C4×C67#10012yes—
119C67⋊C8—82no—
120C67⋊C8—82no—
121C22×C67—21no—
122C22×C67—21no—
123C22×C67—22no—
124C22×C67—42no—
125C22×C67—42no—
126C22×C67—42no—
127C22×C67—41no—
128C22×C67—41no—
129C22×C67—41no—
130C22×C67—41no—
131C22×C67—41no—
132C22×C67—81no—
133C22×C67#40811yescharacteristic
134C22×C67C2×#9911yescharacteristic
135C22×C67C2×#10011yescharacteristic
136C4×C67—21no—
137C4×C67—22no—
138C4×C67—41no—
139C4×C67—41no—
140C4×C67—41no—
141C4×C67—82no—
142C4×C67—82no—
143C4×C67#35212yes—
144#11562—671nomaximal, Sylow
145C2×C67—21no—
146C2×C67—21no—
147C2×C67—42no—
148C2×C67—81no—
149C2×C67#147111yescharacteristic
150C2×C67C2×#35211yescharacteristic
151C2×C67#212311yescharacteristic
152#2486D6711yescharacteristic
153#1465—671no—
154#1460—671no—
155#1317—674no—
156C67#1156211yescharacteristic, complemented, Sylow
157C2×#123Dic6712yes—
158C2×C4×D4C2×D6711yescharacteristic
159C22×C4⋊C4Dic6711yescharacteristic
160#431—1341no—
161#407—1341no—
162#398—21no—
163#397—21no—
164#389—21no—
165#381C2×D6712yes—
166#380—21no—
167#380Dic6711yescharacteristic
168#324—1341no—
169#323—672no—
170#323—672no—
171#323—1341no—
172C22×D4C2×Dic6711yescharacteristic
173C23×C4C2×Dic6712yes—
174C23×C4C2×Dic6711yescharacteristic
175C23×C4C22⋊D6711yescharacteristic
176C2×M16—2681no—
177#123—44no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C22⋊D6711yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4C2×Dic6712yes—
189C2×C22⋊C4C22⋊D6711yescharacteristic
190C2×C22⋊C4C22×D6711yescharacteristic
191C2×C42C22⋊D6711yescharacteristic
192C4⋊C8—2681no—
193C22⋊C8—2681no—
194C22⋊C8—1342no—
195C22⋊C8—1342no—
196C4×C8—2681no—
197#95—21no—
198#95—22no—
199#95—22no—
200C24C22⋊Dic6711yescharacteristic
201C24C22×Dic6711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4C22⋊Dic6711yescharacteristic
217C22×C4C22⋊Dic6711yescharacteristic
218C22×C4C22⋊Dic6711yescharacteristic
219C22×C4C2×C22⋊D6711yescharacteristic
220C22×C4C2×C22⋊D6711yescharacteristic
221M16—5362no—
222C2×C8—2682no—
223C2×C8—2682no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—44no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23C2×C22⋊Dic6711yescharacteristic
246C23#1452405012yes—
247D4—44no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—84no—
268C2×C4#1452395811yescharacteristic
269C2×C4#1452395811yescharacteristic
270C2×C4#1452396012yes—
271C8—5362no—
272C8—5362no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1686336311yescharacteristic
286C22C2×#1452395811yescharacteristic
287C22C2×#1452396011yescharacteristic
288C4—21no—
289C4—22no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#1686318612yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—81no—
300C2#2255856911yescharacteristic
301C2C2×#1686318611yescharacteristic
302C2#2256186311yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2486C2
socleC22×C67C2×#99
prime radicalC22×D4×C67C2×C4
cosocleC2×C22⋊C4×C67C23
hypercenter#2486D67
hypocenterC67#11562
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #22558139C2semidirect
156. C67#11562semidirect

Maximal subgroups

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

subgroupindex
#225585472
#225585302
#225581392
C67×#24862
#1156267