Finite Group Databasebeta

← #30766759

Subgroups — #30766759

Order 17152.

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

normal subgroups only

This group has 368 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#22560292C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558236C212yesmaximal, complemented
4#22558215C212yesmaximal, complemented
5C67×#2611C211yesmaximal, characteristic
6#16863449—21no—
7#16863360—21no—
8#16863165—21no—
9#16863165C2212yes—
10#16863122C2212yes—
11#16863120—21no—
12C2×#125×C67C411yescharacteristic
13C2×#117×C67C2211yescharacteristic
14#400×C67C2211yescharacteristic
15#399×C67C411yescharacteristic
16#398×C67C2211yescharacteristic
17#397×C67C411yescharacteristic
18#389×C67—21no—
19#388×C67—21no—
20#387×C67—21no—
21#385×C67—21no—
22#376×C67C411yescharacteristic
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22⋊(C67⋊C8)—41no—
28C4×C67⋊C8—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×C67D411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
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
56#11761612—42no—
57#11761612—42no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—42no—
60C24×C67C22⋊C411yescharacteristic
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—22no—
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×C67C2×D411yescharacteristic
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—41no—
82C4⋊C4×C67—41no—
83C4⋊C4×C67—42no—
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—21no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67C2×C22⋊C411yescharacteristic
99C2×C4×C67—21no—
100C2×C4×C67—21no—
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—42no—
109C2×C4×C67—42no—
110C2×C4×C67—41no—
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#9911yescharacteristic
126C2×C4×C67#9911yescharacteristic
127C2×C4×C67#13512yes—
128C67⋊C8—82no—
129C67⋊C8—82no—
130C22×C67—21no—
131C22×C67—21no—
132C22×C67—42no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—41no—
136C22×C67—81no—
137C22×C67C2×#9911yescharacteristic
138C22×C67#41611yescharacteristic
139C22×C67#42111yescharacteristic
140C4×C67—22no—
141C4×C67—21no—
142C4×C67—21no—
143C4×C67—42no—
144C4×C67—41no—
145C4×C67—41no—
146C4×C67—41no—
147C4×C67—41no—
148C4×C67—81no—
149C4×C67—81no—
150C4×C67—81no—
151#11853—671nomaximal, Sylow
152C2×C67—21no—
153C2×C67—21no—
154C2×C67—81no—
155C2×C67#151611yescharacteristic
156C2×C67#212311yescharacteristic
157C2×C67#213211yescharacteristic
158#2611D6711yescharacteristic
159#1822—671no—
160#1458—671no—
161#1353—672no—
162#1346—672no—
163C67#1185311yescharacteristic, complemented, Sylow
164C2×#125Dic6711yescharacteristic
165C2×#117C2×D6711yescharacteristic
166#431—1341no—
167#407—1341no—
168#400C2×D6711yescharacteristic
169#399Dic6711yescharacteristic
170#398C2×D6711yescharacteristic
171#397Dic6711yescharacteristic
172#389—21no—
173#388—21no—
174#387—21no—
175#385—21no—
176#376Dic6711yescharacteristic
177#343—672no—
178#343—1341no—
179#324—672no—
180#323—1341no—
181C23×C4C22⋊D6711yescharacteristic
182C2×M16—1342no—
183#125—41no—
184#125—41no—
185#117—41no—
186#117—41no—
187C2×C4⋊C4—21no—
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×C4⋊C4C22×D6711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4C2×Dic6711yescharacteristic
198C2×C22⋊C4C22⋊D6711yescharacteristic
199C2×C42—21no—
200C2×C42C2×Dic6711yescharacteristic
201C2×C42C22⋊D6711yescharacteristic
202C4⋊C8—2681no—
203C22⋊C8—2681no—
204C22⋊C8—2681no—
205C22⋊C8—1342no—
206C4×C8—2681no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95C2×Dic6711yescharacteristic
213#95C2×Dic6711yescharacteristic
214C24C22⋊Dic6711yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4C22⋊Dic6711yescharacteristic
224C22×C4C22⋊Dic6711yescharacteristic
225C22×C4C22⋊Dic6711yescharacteristic
226C22×C4C22×Dic6711yescharacteristic
227C22×C4C2×C22⋊D6711yescharacteristic
228C22×C4C2×C22⋊D6711yescharacteristic
229M16—2682no—
230M16—2682no—
231C2×C8—2682no—
232C2×C8—2682no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊Dic6711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
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#1452395811yescharacteristic
284C2×C4#1452395811yescharacteristic
285C2×C4#1452406312yes—
286C8—5362no—
287C8—5362no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1452395811yescharacteristic
296C22#1686339211yescharacteristic
297C22#1686341311yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#2255876511yescharacteristic
313C2#2256186311yescharacteristic
314C2#2256189411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2611C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2611D67
hypocenterC67#11853
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
3. #22558236C2semidirect
4. #22558215C2semidirect
163. C67#11853semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225585232
#225582362
#225582152
C67×#26112
#1185367