Finite Group Databasebeta

← #30766868

Subgroups — #30766868

Order 17152.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#22560197C211yesmaximal, characteristic
2#22558527C211yesmaximal, characteristic
3#22558209C214yesmaximal, complemented
4C67×#2547C211yesmaximal, characteristic
5C2×#14523958—21no—
6C2×C22⋊(C67⋊C8)—21no—
7#16863256—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×#124×C67—21no—
13C2×C4×D4×C67C2211yescharacteristic
14#397×C67C412yescomplemented
15#397×C67C2212yes—
16#391×C67—21no—
17#380×C67—21no—
18#380×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic, complemented
20C22×C67⋊C8—41no—
21C2×C4⋊Dic67—41no—
22C2×C22⋊Dic67—22no—
23#14523958—42no—
24C22⋊(C67⋊C8)—22no—
25C22⋊(C67⋊C8)—41no—
26C67⋊#95—41no—
27C22×Q8×C67—21no—
28C22×D4×C67D411yescharacteristic
29C23×C4×C67D411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4⋊Q8×C67—41no—
32C4⋊Q8×C67—41no—
33#124×C67—42no—
34#124×C67—42no—
35C4×D4×C67—41no—
36C4×D4×C67—41no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67C2×C411yescharacteristic
44C2×C22⋊C4×C67C2311yescharacteristic
45C2×C42×C67—21no—
46C2×C42×C67C2×C411yescharacteristic
47#95×C67—22no—
48#95×C67—22no—
49#95×C67C2×C412yes—
50C22×Dic67—42no—
51C2×C67⋊C8—42no—
52C2×C67⋊C8—82no—
53C4⋊Dic67—82no—
54C22⋊Dic67—42no—
55C22⋊Dic67—42no—
56C24×C67C22⋊C411yescharacteristic
57C24×C67C2×D411yescharacteristic
58C2×Q8×C67—42no—
59C2×Q8×C67—42no—
60C2×Q8×C67—42no—
61C2×D4×C67—41no—
62C2×D4×C67—41no—
63C2×D4×C67D812yes—
64C2×D4×C67SD1612yes—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—22no—
70C22×C4×C67—22no—
71C22×C4×C67—42no—
72C22×C4×C67—42no—
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22⋊C411yescharacteristic
75C22×C4×C67C22⋊C411yescharacteristic
76C22×C4×C67C22×C411yescharacteristic
77C22×C4×C67C2×D411yescharacteristic
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—44no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—42no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—41no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67#10212yes—
100C23×C67C2×C22⋊C411yescharacteristic
101C23×C67C2×D811yescharacteristic
102C23×C67C2×SD1611yescharacteristic
103Q8×C67—84no—
104D4×C67—42no—
105D4×C67—42no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—22no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—42no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67—42no—
123C2×C4×C67—81no—
124C2×C4×C67—81no—
125C2×C4×C67#9911yescharacteristic
126C2×C4×C67#9911yescharacteristic
127C2×C4×C67#10212yes—
128C2×Dic67—82no—
129C2×Dic67—82no—
130C2×Dic67—82no—
131C67⋊C8—82no—
132C22×C67—21no—
133C22×C67—21no—
134C22×C67—21no—
135C22×C67—22no—
136C22×C67—21no—
137C22×C67—42no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67—41no—
142C22×C67—41no—
143C22×C67—41no—
144C22×C67—81no—
145C22×C67—81no—
146C22×C67C2×#9911yescharacteristic
147C22×C67C2×#10211yescharacteristic
148C22×C67#42111yescharacteristic
149C4×C67—21no—
150C4×C67—22no—
151C4×C67—41no—
152C4×C67—41no—
153C4×C67—41no—
154C4×C67—82no—
155C4×C67—82no—
156C4×C67#35412yes—
157Dic67—162no—
158#11878—671nomaximal, Sylow
159C2×C67—21no—
160C2×C67—21no—
161C2×C67—41no—
162C2×C67—42no—
163C2×C67—41no—
164C2×C67#151911yescharacteristic
165C2×C67#212311yescharacteristic
166C2×C67C2×#35411yescharacteristic
167#2547D6711yescharacteristic
168#1799—671no—
169#1459—671no—
170#1344—674no—
171C67#1187811yescharacteristic, complemented, Sylow
172C2×C4⋊Q8Dic6711yescharacteristic, complemented
173C2×#124—21no—
174C2×C4×D4C2×D6711yescharacteristic
175C2×#99—1341no—
176C2×C22⋊C8—1341no—
177#397Dic6712yescomplemented
178#397C2×D6712yes—
179#391—21no—
180#380—21no—
181#380—21no—
182#380—1341no—
183C4×C4⋊C4Dic6711yescharacteristic, complemented
184#342—672no—
185#324—1341no—
186#323—672no—
187C22×Q8—21no—
188C22×D4C22⋊D6711yescharacteristic
189C23×C4C22⋊D6711yescharacteristic
190C23×C4C22⋊D6711yescharacteristic
191C22×C8—2681no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194#124—42no—
195#124—42no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—2681no—
201C2×C4⋊C4C2×Dic6712yes—
202C2×C4⋊C4C22⋊D6711yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—1342no—
206C2×C22⋊C4C2×Dic6711yescharacteristic
207C2×C22⋊C4C22×D6711yescharacteristic
208C2×C42—21no—
209C2×C42C2×Dic6711yescharacteristic
210#99—2682no—
211C22⋊C8—2681no—
212C22⋊C8—1342no—
213#95—22no—
214#95—22no—
215#95—2681no—
216#95C2×Dic6712yes—
217C24C22⋊Dic6711yescharacteristic
218C24C2×C22⋊D6711yescharacteristic
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#1176161312yes—
225C2×D4#1176161612yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—2682no—
235C22×C4C22⋊Dic6711yescharacteristic
236C22×C4C22⋊Dic6711yescharacteristic
237C22×C4C22⋊Dic6711yescharacteristic
238C22×C4C22×Dic6711yescharacteristic
239C22×C4C2×C22⋊D6711yescharacteristic
240C2×C8—2682no—
241C2×C8—5362no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—5362no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—2682no—
254C22⋊C4—2682no—
255C42—21no—
256C42—21no—
257C42—42no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1452396412yes—
267C23C2×C22⋊Dic6711yescharacteristic
268C23C2×#1176161311yescharacteristic
269C23C2×#1176161611yescharacteristic
270Q8—84no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—5362no—
293C2×C4—5362no—
294C2×C4—5362no—
295C2×C4#1452395811yescharacteristic
296C2×C4#1452395811yescharacteristic
297C2×C4#1452396412yes—
298C8—5362no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#1452395811yescharacteristic
314C22C2×#1452396411yescharacteristic
315C22#1686341311yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—10722no—
324C4#1686319112yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2#2255877711yescharacteristic
331C2#2256186311yescharacteristic
332C2C2×#1686319111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2547C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2547D67
hypocenterC67#11878
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #22558209C2semidirect
11. C2×C4⋊Q8×C67C4semidirect
14. #397×C67C4semidirect
19. C4×C4⋊C4×C67C4semidirect
171. C67#11878semidirect
172. C2×C4⋊Q8Dic67semidirect
177. #397Dic67semidirect
183. C4×C4⋊C4Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585272
#225582092
C67×#25472
#1187867