Finite Group Databasebeta

← #30744352

Subgroups — #30744352

Order 17152.

494 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560277C211yesmaximal, characteristic
2#22559903C211yesmaximal, characteristic
3#22558035C212yesmaximal, complemented
4#22558032C212yesmaximal, complemented
5C67×#2855C211yesmaximal, characteristic
6C22×#11761612—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863151—21no—
9#16863151—21no—
10#16863151C2212yes—
11#16863151C2212yes—
12C2×C22⋊Q8×C67C411yescharacteristic
13C22×C42×C67C2211yescharacteristic
14#391×C67C411yescharacteristic
15#391×C67C2211yescharacteristic
16#389×C67C411yescharacteristic
17#389×C67C2211yescharacteristic
18#387×C67—21no—
19#387×C67—21no—
20#385×C67—21no—
21#381×C67—21no—
22#380×C67C411yescharacteristic
23C2×#11761612—24no—
24C2×#11761612—42no—
25C2×#11761612—42no—
26C22×C67⋊C8—22no—
27C22×C67⋊C8—22no—
28C22⋊(C67⋊C8)—42no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C22⋊Q8×C67—42no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
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—
53#95×C67—21no—
54#11761612—48no—
55C2×C67⋊C8—24no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—42no—
61C24×C67C22⋊C411yescharacteristic
62C2×Q8×C67—22no—
63C2×Q8×C67—41no—
64C2×Q8×C67—41no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—22no—
70C22×C4×C67—22no—
71C22×C4×C67—22no—
72C22×C4×C67—22no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—41no—
78C22×C4×C67—41no—
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22×C411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C22×C4×C67C2×D411yescharacteristic
85C4⋊C4×C67—22no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—42no—
89C4⋊C4×C67—42no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C22⋊C4×C67—42no—
93C22⋊C4×C67—42no—
94C42×C67—41no—
95C42×C67—41no—
96C42×C67—41no—
97C42×C67—41no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67#9912yes—
105C23×C67C2×C22⋊C411yescharacteristic
106Q8×C67—82no—
107C2×C4×C67—24no—
108C2×C4×C67—24no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—44no—
124C2×C4×C67—44no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67#10412yes—
138C2×C4×C67#13112yes—
139C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C67#13711yescharacteristic
141C2×C4×C67#13712yes—
142C67⋊C8—44no—
143C67⋊C8—82no—
144C22×C67—22no—
145C22×C67—22no—
146C22×C67—22no—
147C22×C67—41no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67—41no—
151C22×C67#40811yescharacteristic
152C22×C67C2×#9911yescharacteristic
153C22×C67#41311yescharacteristic
154C22×C67#41711yescharacteristic
155C22×C67#41911yescharacteristic
156C22×C67C2×#10411yescharacteristic
157C22×C67#42111yescharacteristic
158C4×C67—22no—
159C4×C67—21no—
160C4×C67—22no—
161C4×C67—21no—
162C4×C67—22no—
163C4×C67—41no—
164C4×C67—41no—
165C4×C67—81no—
166C4×C67—81no—
167C4×C67—81no—
168C4×C67—81no—
169#7695—671nomaximal, Sylow
170C2×C67—42no—
171C2×C67#146811yescharacteristic
172C2×C67#148511yescharacteristic
173C2×C67#149511yescharacteristic
174C2×C67#150411yescharacteristic
175C2×C67#151111yescharacteristic
176C2×C67#152911yescharacteristic
177C2×C67#155911yescharacteristic
178#2855D6711yescharacteristic
179#1817—671no—
180#1728—671no—
181#1272—672no—
182#1271—672no—
183C67#769511yescharacteristic, complemented, Sylow
184C22×M16—1341no—
185C2×C22⋊Q8Dic6711yescharacteristic
186C22×C42C2×D6711yescharacteristic
187C2×C22⋊C8—1341no—
188#391Dic6711yescharacteristic
189#391C2×D6711yescharacteristic
190#389Dic6711yescharacteristic
191#389C2×D6711yescharacteristic
192#387—21no—
193#387—21no—
194#385—21no—
195#381—21no—
196#380Dic6711yescharacteristic
197#336—672no—
198#336—672no—
199#336—1341no—
200#336—1341no—
201C22×Q8C2×Dic6711yescharacteristic
202C23×C4C2×Dic6711yescharacteristic
203C23×C4C22⋊D6711yescharacteristic
204C23×C4C22⋊D6711yescharacteristic
205C2×M16—2682no—
206C2×M16—2682no—
207C2×M16—1344no—
208C22×C8—1342no—
209C22×C8—1342no—
210C22⋊Q8—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic6711yescharacteristic
215C2×C4⋊C4C2×Dic6711yescharacteristic
216C2×C4⋊C4C2×Dic6711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—42no—
220C2×C42C2×Dic6711yescharacteristic
221C2×C42C22⋊D6711yescharacteristic
222C2×C42C22⋊D6711yescharacteristic
223C2×C42C22×D6711yescharacteristic
224C22⋊C8—2682no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24C22⋊Dic6711yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C22⋊Dic6711yescharacteristic
251C22×C4C22⋊Dic6711yescharacteristic
252C22×C4C22⋊Dic6711yescharacteristic
253C22×C4C22×Dic6711yescharacteristic
254C22×C4C2×C22⋊D6711yescharacteristic
255C22×C4C2×C22⋊D6711yescharacteristic
256M16—2688no—
257C2×C8—2682no—
258C2×C8—2682no—
259C2×C8—2682no—
260C2×C8—2682no—
261C2×C8—2682no—
262C2×C8—1344no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#1452395812yes—
283C23C2×C22⋊Dic6711yescharacteristic
284Q8—82no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#1452397012yes—
316C2×C4#1452405012yes—
317C2×C4#1452406811yescharacteristic
318C2×C4#1452407311yescharacteristic
319C2×C4#1452407412yes—
320C8—2684no—
321C8—5362no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#1686336311yescharacteristic
330C22C2×#1452395811yescharacteristic
331C22#1686338011yescharacteristic
332C22#1686339711yescharacteristic
333C22#1686340511yescharacteristic
334C22C2×#1452397011yescharacteristic
335C22#1686341311yescharacteristic
336C4—22no—
337C4—22no—
338C4—21no—
339C4—22no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C2—42no—
348C2#2255855711yescharacteristic
349C2#2255863011yescharacteristic
350C2#2255867411yescharacteristic
351C2#2255871311yescharacteristic
352C2#2255874211yescharacteristic
353C2#2255881611yescharacteristic
354C2#2255893911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#2855C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#2855D67
hypocenterC67#7695
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #22558035C2semidirect
4. #22558032C2semidirect
183. C67#7695semidirect

Maximal subgroups

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

subgroupindex
#225602772
#225599032
#225580352
#225580322
C67×#28552
#769567