Finite Group Databasebeta

← #30744240

Subgroups — #30744240

Order 17152.

468 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560277C211yesmaximal, characteristic
2C4×C22⋊(C67⋊C8)C211yesmaximal, characteristic
3#22558032C212yesmaximal, complemented
4#22558032C212yesmaximal, complemented
5C67×#2851C211yesmaximal, characteristic
6C2×C22⋊(C67⋊C8)—21no—
7C2×C22⋊(C67⋊C8)—21no—
8C2×C4×C67⋊C8—21no—
9#16863151—21no—
10#16863151C2212yes—
11#16863151C2212yes—
12C2×#125×C67C411yescharacteristic
13C22×C42×C67C2211yescharacteristic
14#389×C67C2211yescharacteristic
15#389×C67C2211yescharacteristic
16#387×C67—21no—
17#387×C67—21no—
18#385×C67—21no—
19#385×C67—21no—
20#382×C67C411yescharacteristic
21#380×C67C411yescharacteristic
22#380×C67C411yescharacteristic
23C22×C67⋊C8—22no—
24C22×C67⋊C8—22no—
25C22⋊(C67⋊C8)—24no—
26C22⋊(C67⋊C8)—42no—
27C4×C67⋊C8—42no—
28C23×C4×C67C2×C411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31#125×C67—42no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67C2×C411yescharacteristic
37C2×C4⋊C4×C67C2×C411yescharacteristic
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—21no—
42C2×C42×C67—42no—
43C2×C42×C67C2×C411yescharacteristic
44C2×C42×C67D411yescharacteristic
45C2×C42×C67D411yescharacteristic
46C2×C42×C67C2311yescharacteristic
47#95×C67—21no—
48#95×C67—21no—
49#95×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53C2×C67⋊C8—24no—
54C2×C67⋊C8—42no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C24×C67C22⋊C411yescharacteristic
60C22×C4×C67—22no—
61C22×C4×C67—22no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—41no—
71C22×C4×C67—41no—
72C22×C4×C67—41no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
78C22×C4×C67C2×D411yescharacteristic
79C4⋊C4×C67—22no—
80C4⋊C4×C67—22no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—42no—
84C4⋊C4×C67—42no—
85C4⋊C4×C67—42no—
86C4⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C22⋊C4×C67—42no—
89C42×C67—22no—
90C42×C67—22no—
91C42×C67—41no—
92C42×C67—41no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67—21no—
98C23×C67—21no—
99C23×C67#10412yes—
100C23×C67C2×C22⋊C411yescharacteristic
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—44no—
115C2×C4×C67—44no—
116C2×C4×C67—44no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67#9912yes—
130C2×C4×C67#10112yes—
131C2×C4×C67#13112yes—
132C2×C4×C67#13512yes—
133C2×C4×C67#13512yes—
134C67⋊C8—44no—
135C67⋊C8—82no—
136C22×C67—22no—
137C22×C67—22no—
138C22×C67—22no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67—41no—
143C22×C67#40811yescharacteristic
144C22×C67C2×#9911yescharacteristic
145C22×C67C2×#10111yescharacteristic
146C22×C67#41611yescharacteristic
147C22×C67#41611yescharacteristic
148C22×C67C2×#10411yescharacteristic
149C22×C67#42111yescharacteristic
150C4×C67—22no—
151C4×C67—22no—
152C4×C67—22no—
153C4×C67—22no—
154C4×C67—22no—
155C4×C67—41no—
156C4×C67—81no—
157C4×C67—81no—
158C4×C67—81no—
159C4×C67—81no—
160#7671—671nomaximal, Sylow
161C2×C67—42no—
162C2×C67#147211yescharacteristic
163C2×C67#148211yescharacteristic
164C2×C67#149411yescharacteristic
165C2×C67#150111yescharacteristic
166C2×C67#151611yescharacteristic
167C2×C67#152811yescharacteristic
168C2×C67#154411yescharacteristic
169#2851D6711yescharacteristic
170#1817—671no—
171C4×C22⋊C8—671no—
172#1271—672no—
173#1271—672no—
174C67#767111yescharacteristic, complemented, Sylow
175C2×#125Dic6711yescharacteristic
176C22×C42C2×D6711yescharacteristic
177C2×C22⋊C8—1341no—
178C2×C22⋊C8—1341no—
179C2×C4×C8—1341no—
180#389C2×D6711yescharacteristic
181#389C2×D6711yescharacteristic
182#387—21no—
183#387—21no—
184#385—21no—
185#385—21no—
186#382Dic6711yescharacteristic
187#380Dic6711yescharacteristic
188#380Dic6711yescharacteristic
189#336—672no—
190#336—672no—
191#336—1341no—
192C23×C4C2×Dic6711yescharacteristic
193C23×C4C22⋊D6711yescharacteristic
194C23×C4C22⋊D6711yescharacteristic
195C22×C8—1342no—
196C22×C8—1342no—
197#125—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×Dic6711yescharacteristic
203C2×C4⋊C4C2×Dic6711yescharacteristic
204C2×C4⋊C4C2×Dic6711yescharacteristic
205C2×C4⋊C4C2×Dic6711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—42no—
209C2×C42C2×Dic6711yescharacteristic
210C2×C42C22⋊D6711yescharacteristic
211C2×C42C22⋊D6711yescharacteristic
212C2×C42C22×D6711yescharacteristic
213C22⋊C8—2682no—
214C22⋊C8—1344no—
215C4×C8—2682no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222C24C22⋊Dic6711yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22⋊Dic6711yescharacteristic
238C22×C4C22⋊Dic6711yescharacteristic
239C22×C4C22×Dic6711yescharacteristic
240C22×C4C2×C22⋊D6711yescharacteristic
241C22×C4C2×C22⋊D6711yescharacteristic
242C2×C8—2682no—
243C2×C8—2682no—
244C2×C8—2682no—
245C2×C8—2682no—
246C2×C8—2682no—
247C2×C8—1344no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—22no—
259C42—22no—
260C42—41no—
261C42—41no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1452397012yes—
269C23C2×C22⋊Dic6711yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4#1452395812yes—
299C2×C4#1452396212yes—
300C2×C4#1452405012yes—
301C2×C4#1452406312yes—
302C2×C4#1452406312yes—
303C8—2684no—
304C8—5362no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22#1686336311yescharacteristic
313C22C2×#1452395811yescharacteristic
314C22C2×#1452396211yescharacteristic
315C22#1686339211yescharacteristic
316C22#1686339211yescharacteristic
317C22C2×#1452397011yescharacteristic
318C22#1686341311yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—42no—
330C2#2255857311yescharacteristic
331C2#2255861711yescharacteristic
332C2#2255866911yescharacteristic
333C2#2255870011yescharacteristic
334C2#2255876511yescharacteristic
335C2#2255881111yescharacteristic
336C2#2255887411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2851C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2851D67
hypocenterC67#7671
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. #22558032C2semidirect
4. #22558032C2semidirect
174. C67#7671semidirect

Maximal subgroups

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

subgroupindex
#225602772
C4×C22⋊(C67⋊C8)2
#225580322
C67×#28512
#767167