Finite Group Databasebeta

← #30766704

Subgroups — #30766704

Order 17152.

390 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 87 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#22560286C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558230C212yesmaximal
4#22558209C212yesmaximal
5C67×#2599C211yesmaximal, characteristic
6#16863449—21no—
7#16863259—21no—
8#16863163—21no—
9#16863163C2212yes—
10#16863122—21no—
11#16863120C2212yes—
12C2×C4×Q8×C67C411yescharacteristic, complemented
13C2×#117×C67C2211yescharacteristic
14#398×C67C411yescharacteristic, complemented
15#397×C67C2211yescharacteristic
16#395×C67C411yescharacteristic, complemented
17#393×C67C2211yescharacteristic
18#388×C67—21no—
19#387×C67—21no—
20#387×C67—21no—
21#385×C67—21no—
22#383×C67C411yescharacteristic, complemented
23C2×C22⋊Dic67—22no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—41no—
27C67⋊#95—41no—
28C67⋊#95—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4×Q8×C67—41no—
32C4×Q8×C67—41no—
33#117×C67—41no—
34#117×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67C2×C411yescharacteristic
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C22⋊C4×C67—21no—
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67C2311yescharacteristic
45C2×C42×C67—21no—
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67D411yescharacteristic
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#95×C67C2×C411yescharacteristic
55#95×C67C2×C411yescharacteristic
56C22×Dic67—42no—
57C2×C67⋊C8—42no—
58C22⋊Dic67—42no—
59C22⋊Dic67—42no—
60C24×C67C2×D411yescharacteristic
61C2×Q8×C67—42no—
62C2×Q8×C67—41no—
63C22×C4×C67—21no—
64C22×C4×C67—22no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—41no—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22×C411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—41no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—42no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67C2×C22⋊C411yescharacteristic
98C23×C67#13512yes—
99Q8×C67—42no—
100Q8×C67—42no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—22no—
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—42no—
112C2×C4×C67—42no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
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#9911yescharacteristic
129C2×C4×C67#9911yescharacteristic
130C2×Dic67—82no—
131C2×Dic67—82no—
132C2×Dic67—82no—
133C67⋊C8—82no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—22no—
137C22×C67—42no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—81no—
141C22×C67C2×#9911yescharacteristic
142C22×C67#41611yescharacteristic
143C22×C67#42111yescharacteristic
144C4×C67—21no—
145C4×C67—21no—
146C4×C67—41no—
147C4×C67—42no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67—81no—
153C4×C67—81no—
154C4×C67—81no—
155Dic67—162no—
156#11842—671nomaximal, Sylow
157C2×C67—21no—
158C2×C67—21no—
159C2×C67—42no—
160C2×C67#151611yescharacteristic
161C2×C67#212011yescharacteristic
162C2×C67#212911yescharacteristic
163#2599D6711yescharacteristic
164#1820—671no—
165#1458—671no—
166#1351—672no—
167#1344—672no—
168C67#1184211yescharacteristic, complemented, Sylow
169C2×C4×Q8Dic6711yescharacteristic, complemented
170C2×#117C2×D6711yescharacteristic
171#431—1341no—
172#398Dic6711yescharacteristic, complemented
173#397C2×D6711yescharacteristic
174#395Dic6711yescharacteristic, complemented
175#393C2×D6711yescharacteristic
176#388—21no—
177#387—21no—
178#387—21no—
179#385—21no—
180#383Dic6711yescharacteristic, complemented
181#381—1341no—
182#342—672no—
183#342—1341no—
184#324—1341no—
185#323—672no—
186C22×Q8C2×Dic6711yescharacteristic
187C23×C4C22⋊D6711yescharacteristic
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic6711yescharacteristic
196C2×C4⋊C4C2×Dic6711yescharacteristic
197C2×C4⋊C4C22⋊D6711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—1342no—
201C2×C22⋊C4C22⋊D6711yescharacteristic
202C2×C22⋊C4C22×D6711yescharacteristic
203C2×C42—21no—
204C2×C42C2×Dic6711yescharacteristic
205C2×C42C22⋊D6711yescharacteristic
206C4⋊C8—2681no—
207C22⋊C8—1342no—
208C4×C8—2681no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—2681no—
216#95—2681no—
217#95C2×Dic6711yescharacteristic
218#95C2×Dic6711yescharacteristic
219C24C2×C22⋊D6711yescharacteristic
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—2682no—
231C22×C4C22⋊Dic6711yescharacteristic
232C22×C4C22⋊Dic6711yescharacteristic
233C22×C4C22⋊Dic6711yescharacteristic
234C22×C4C22⋊Dic6711yescharacteristic
235C22×C4C22×Dic6711yescharacteristic
236C22×C4C2×C22⋊D6711yescharacteristic
237C2×C8—2682no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—2682no—
250C22⋊C4—2682no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×C22⋊Dic6711yescharacteristic
261C23#1452406312yes—
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—5362no—
292C2×C4—5362no—
293C2×C4—5362no—
294C2×C4#1452395811yescharacteristic
295C2×C4#1452395811yescharacteristic
296C8—5362no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1452395811yescharacteristic
305C22#1686339211yescharacteristic
306C22#1686341311yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—10722no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2255876511yescharacteristic
323C2#2256185011yescharacteristic
324C2#2256188411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2599C2
socleC22×C67C2×#99
prime radicalC24×C67C2×D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2599D67
hypocenterC67#11842
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C67C4semidirect
14. #398×C67C4semidirect
16. #395×C67C4semidirect
22. #383×C67C4semidirect
168. C67#11842semidirect
169. C2×C4×Q8Dic67semidirect
172. #398Dic67semidirect
174. #395Dic67semidirect
180. #383Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602862
#225585232
#225582302
#225582092
C67×#25992
#1184267