Finite Group Databasebeta

← #30765422

Subgroups — #30765422

Order 17152.

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

normal subgroups only

This group has 366 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#22558543C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558153C212yesmaximal, complemented
4#22558143C212yesmaximal, complemented
5C67×#2499C211yesmaximal, characteristic
6#16863453—21no—
7#16863449—21no—
8#16863122C2212yes—
9#16863122C2212yes—
10#16863120—21no—
11#16863120—21no—
12C2×C22⋊Q8×C67C411yescharacteristic
13C2×#117×C67C2211yescharacteristic
14#397×C67—21no—
15#397×C67—21no—
16#391×C67C411yescharacteristic
17#391×C67C2211yescharacteristic
18#387×C67—21no—
19#387×C67—21no—
20#384×C67C411yescharacteristic
21#382×C67C2211yescharacteristic
22C4×C22⋊C4×C67C411yescharacteristic
23C4⋊(C67⋊C8)—41no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—22no—
27#14523954—41no—
28C4×C67⋊C8—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C22⋊Q8×C67—41no—
32C22⋊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×C67C2311yescharacteristic
41C2×C22⋊C4×C67—21no—
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67D411yescharacteristic
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67C2×C411yescharacteristic
47C2×C42×C67D411yescharacteristic
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×C67—21no—
55#95×C67—21no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—42no—
58C24×C67C22⋊C411yescharacteristic
59C2×Q8×C67—22no—
60C2×Q8×C67—41no—
61C2×Q8×C67—41no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—41no—
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—41no—
82C4⋊C4×C67—41no—
83C22⋊C4×C67—41no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—42no—
86C22⋊C4×C67—42no—
87C42×C67—21no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—22no—
92C42×C67—41no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67C2×C22⋊C411yescharacteristic
99Q8×C67—82no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—22no—
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×C4×C67#13112yes—
129C67⋊C8—82no—
130C67⋊C8—82no—
131C22×C67—21no—
132C22×C67—21no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—41no—
136C22×C67—81no—
137C22×C67—81no—
138C22×C67#40811yescharacteristic
139C22×C67C2×#9911yescharacteristic
140C22×C67#41311yescharacteristic
141C4×C67—21no—
142C4×C67—21no—
143C4×C67—22no—
144C4×C67—42no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—81no—
150C4×C67—81no—
151C4×C67—81no—
152#11556—671nomaximal, Sylow
153C2×C67—21no—
154C2×C67—21no—
155C2×C67—81no—
156C2×C67#146811yescharacteristic
157C2×C67#212911yescharacteristic
158C2×C67#213211yescharacteristic
159#2499D6711yescharacteristic
160#1464—671no—
161#1458—671no—
162#1324—672no—
163#1319—672no—
164C67#1155611yescharacteristic, complemented, Sylow
165C2×C22⋊Q8Dic6711yescharacteristic
166C2×#117C2×D6711yescharacteristic
167#432—1341no—
168#431—1341no—
169#397—21no—
170#397—21no—
171#391Dic6711yescharacteristic
172#391C2×D6711yescharacteristic
173#387—21no—
174#387—21no—
175#384Dic6711yescharacteristic
176#382C2×D6711yescharacteristic
177C4×C22⋊C4Dic6711yescharacteristic
178#324—672no—
179#324—672no—
180#323—1341no—
181#323—1341no—
182C22×Q8C2×Dic6711yescharacteristic
183C23×C4C2×Dic6711yescharacteristic
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic6711yescharacteristic
192C2×C4⋊C4C2×Dic6711yescharacteristic
193C2×C4⋊C4C22×D6711yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4C22⋊D6711yescharacteristic
198C2×C42C2×Dic6711yescharacteristic
199C2×C42C2×Dic6711yescharacteristic
200C2×C42C22⋊D6711yescharacteristic
201C2×C42C22⋊D6711yescharacteristic
202C4⋊C8—2681no—
203C4⋊C8—2681no—
204C22⋊C8—1342no—
205C22⋊C8—1342no—
206#97—2681no—
207C4×C8—2681no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215C24C22⋊Dic6711yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22×Dic6711yescharacteristic
231C22×C4C2×C22⋊D6711yescharacteristic
232C22×C4C2×C22⋊D6711yescharacteristic
233C2×C8—2682no—
234C2×C8—2682no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—41no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊Dic6711yescharacteristic
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
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#1452395811yescharacteristic
286C2×C4#1452395811yescharacteristic
287C2×C4#1452405012yes—
288C8—5362no—
289C8—5362no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1686336311yescharacteristic
298C22C2×#1452395811yescharacteristic
299C22#1686338011yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—81no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2255855711yescharacteristic
315C2#2256188411yescharacteristic
316C2#2256189411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2499C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2499D67
hypocenterC67#11556
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2499

Complements

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

NH ≅ G/Ntype
3. #22558153C2semidirect
4. #22558143C2semidirect
164. C67#11556semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225585232
#225581532
#225581432
C67×#24992
#1155667