Finite Group Databasebeta

← #30766724

Subgroups — #30766724

Order 17152.

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

normal subgroups only

This group has 382 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#22558233C212yesmaximal
4#22558209C212yesmaximal
5C67×#2611C211yesmaximal, characteristic
6#16863449—21no—
7#16863259—21no—
8#16863163—21no—
9#16863163C2212yes—
10#16863122—21no—
11#16863120C2212yes—
12C2×#125×C67C411yescharacteristic, complemented
13C2×#117×C67C2211yescharacteristic
14#400×C67C411yescharacteristic, complemented
15#399×C67C2211yescharacteristic
16#398×C67C411yescharacteristic, complemented
17#397×C67C2211yescharacteristic
18#389×C67—21no—
19#388×C67—21no—
20#387×C67—21no—
21#385×C67—21no—
22#376×C67C411yescharacteristic, complemented
23C2×C22⋊Dic67—22no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C4×C67⋊C8—41no—
27C67⋊#95—41no—
28C67⋊#95—41no—
29C23×C4×C67D411yescharacteristic
30#125×C67—41no—
31#125×C67—41no—
32#117×C67—41no—
33#117×C67—41no—
34C2×C4⋊C4×C67—21no—
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×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C22⋊C4×C67C2311yescharacteristic
46C2×C42×C67—21no—
47C2×C42×C67C2×C411yescharacteristic
48C2×C42×C67D411yescharacteristic
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
61C22×C4×C67—21no—
62C22×C4×C67—22no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—41no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—42no—
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—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—22no—
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—42no—
110C2×C4×C67—42no—
111C2×C4×C67—41no—
112C2×C4×C67—41no—
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#9911yescharacteristic
127C2×C4×C67#9911yescharacteristic
128C2×Dic67—82no—
129C2×Dic67—82no—
130C2×Dic67—82no—
131C67⋊C8—82no—
132C22×C67—21no—
133C22×C67—21no—
134C22×C67—22no—
135C22×C67—42no—
136C22×C67—41no—
137C22×C67—41no—
138C22×C67—81no—
139C22×C67C2×#9911yescharacteristic
140C22×C67#41611yescharacteristic
141C22×C67#42111yescharacteristic
142C4×C67—21no—
143C4×C67—21no—
144C4×C67—42no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—81no—
151C4×C67—81no—
152C4×C67—81no—
153Dic67—162no—
154#11846—671nomaximal, Sylow
155C2×C67—21no—
156C2×C67—21no—
157C2×C67—42no—
158C2×C67#151611yescharacteristic
159C2×C67#212311yescharacteristic
160C2×C67#212611yescharacteristic
161#2611D6711yescharacteristic
162#1820—671no—
163#1458—671no—
164#1352—672no—
165#1344—672no—
166C67#1184611yescharacteristic, complemented, Sylow
167C2×#125Dic6711yescharacteristic, complemented
168C2×#117C2×D6711yescharacteristic
169#431—1341no—
170#400Dic6711yescharacteristic, complemented
171#399C2×D6711yescharacteristic
172#398Dic6711yescharacteristic, complemented
173#397C2×D6711yescharacteristic
174#389—21no—
175#388—21no—
176#387—21no—
177#385—21no—
178#381—1341no—
179#376Dic6711yescharacteristic, complemented
180#342—672no—
181#342—1341no—
182#324—1341no—
183#323—672no—
184C23×C4C22⋊D6711yescharacteristic
185#125—41no—
186#125—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic6711yescharacteristic
194C2×C4⋊C4C2×Dic6711yescharacteristic
195C2×C4⋊C4C2×Dic6711yescharacteristic
196C2×C4⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—1342no—
200C2×C22⋊C4C22⋊D6711yescharacteristic
201C2×C22⋊C4C22×D6711yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic6711yescharacteristic
204C2×C42C22⋊D6711yescharacteristic
205C4⋊C8—2681no—
206C22⋊C8—1342no—
207C4×C8—2681no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—2681no—
214#95—2681no—
215#95C2×Dic6711yescharacteristic
216#95C2×Dic6711yescharacteristic
217C24C2×C22⋊D6711yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—2682no—
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22⋊Dic6711yescharacteristic
231C22×C4C22×Dic6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233C2×C8—2682no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—2682no—
248C22⋊C4—2682no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic6711yescharacteristic
259C23#1452406312yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
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—5362no—
288C2×C4—5362no—
289C2×C4—5362no—
290C2×C4#1452395811yescharacteristic
291C2×C4#1452395811yescharacteristic
292C8—5362no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1452395811yescharacteristic
301C22#1686339211yescharacteristic
302C22#1686341311yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—10722no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2255876511yescharacteristic
319C2#2256186311yescharacteristic
320C2#2256187411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2611C2
socleC22×C67C2×#99
prime radicalC24×C67C2×D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2611D67
hypocenterC67#11846
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C67C4semidirect
14. #400×C67C4semidirect
16. #398×C67C4semidirect
22. #376×C67C4semidirect
166. C67#11846semidirect
167. C2×#125Dic67semidirect
170. #400Dic67semidirect
172. #398Dic67semidirect
179. #376Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602862
#225585232
#225582332
#225582092
C67×#26112
#1184667