Finite Group Databasebeta

← #30766020

Subgroups — #30766020

Order 17152.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#22559942C211yesmaximal, characteristic
2#22558523C211yesmaximal, characteristic
3#22558209C214yesmaximal
4C67×#2621C211yesmaximal, characteristic
5C2×#14524002—21no—
6#16863449—21no—
7#16863163—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×C22⋊Q8×C67—21no—
13C2×#117×C67C2211yescharacteristic
14#398×C67—21no—
15#398×C67C412yescomplemented
16#397×C67—21no—
17#397×C67C2212yes—
18#391×C67—21no—
19#382×C67C411yescharacteristic, complemented
20#14524002—24no—
21#14524002—42no—
22C2×C4⋊Dic67—41no—
23C2×C22⋊Dic67—22no—
24C2×C4×Dic67—41no—
25C4⋊(C67⋊C8)—41no—
26C22⋊(C67⋊C8)—22no—
27C4×C67⋊C8—41no—
28C22×Q8×C67—21no—
29C23×C4×C67D411yescharacteristic
30C4⋊Q8×C67—41no—
31C4⋊Q8×C67—41no—
32C22⋊Q8×C67—42no—
33C22⋊Q8×C67—42no—
34#117×C67—41no—
35#117×C67—41no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—22no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67C2×C412yes—
40C2×C4⋊C4×C67C2×C411yescharacteristic
41C2×C4⋊C4×C67D411yescharacteristic
42C2×C22⋊C4×C67—22no—
43C2×C22⋊C4×C67D411yescharacteristic
44C2×C22⋊C4×C67C2311yescharacteristic
45C2×C42×C67C2×C411yescharacteristic
46C2×C42×C67D411yescharacteristic
47#95×C67—21no—
48#95×C67—22no—
49#95×C67—22no—
50#95×C67C2×C412yes—
51C22×Dic67—42no—
52C2×C67⋊C8—42no—
53C4⋊Dic67—44no—
54C22⋊Dic67—24no—
55C22⋊Dic67—42no—
56C4×Dic67—44no—
57C24×C67C2×D411yescharacteristic
58C2×Q8×C67—41no—
59C2×Q8×C67—41no—
60C2×Q8×C67—42no—
61C2×Q8×C67—42no—
62C22×C4×C67—21no—
63C22×C4×C67—21no—
64C22×C4×C67—22no—
65C22×C4×C67—22no—
66C22×C4×C67—22no—
67C22×C4×C67C22⋊C411yescharacteristic
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67D812yes—
72C22×C4×C67SD1612yes—
73C22×C4×C67C22×C411yescharacteristic
74C22×C4×C67C2×D411yescharacteristic
75C4⋊C4×C67—41no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—42no—
78C4⋊C4×C67—42no—
79C4⋊C4×C67—42no—
80C4⋊C4×C67—44no—
81C22⋊C4×C67—41no—
82C22⋊C4×C67—41no—
83C22⋊C4×C67—42no—
84C22⋊C4×C67—42no—
85C42×C67—21no—
86C42×C67—21no—
87C42×C67—21no—
88C42×C67—21no—
89C23×C67—21no—
90C23×C67—21no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67C2×C22⋊C411yescharacteristic
94C23×C67C2×D811yescharacteristic
95C23×C67C2×SD1611yescharacteristic
96Q8×C67—82no—
97Q8×C67—82no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—22no—
101C2×C4×C67—22no—
102C2×C4×C67—22no—
103C2×C4×C67—22no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—41no—
111C2×C4×C67—41no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—42no—
117C2×C4×C67—42no—
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#9911yescharacteristic
120C2×C4×C67#10212yes—
121C2×C4×C67#10212yes—
122C2×C4×C67#10412yes—
123C2×Dic67—44no—
124C2×Dic67—82no—
125C2×Dic67—82no—
126C67⋊C8—82no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—21no—
131C22×C67—41no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—81no—
136C22×C67C2×#9911yescharacteristic
137C22×C67C2×#10211yescharacteristic
138C22×C67C2×#10411yescharacteristic
139C4×C67—22no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—41no—
143C4×C67—41no—
144C4×C67—82no—
145C4×C67—82no—
146C4×C67#32712yes—
147C4×C67#32712yes—
148Dic67—84no—
149#11694—671nomaximal, Sylow
150C2×C67—21no—
151C2×C67—21no—
152C2×C67—41no—
153C2×C67—41no—
154C2×C67C2×#32711yescharacteristic
155C2×C67#212311yescharacteristic
156C2×C67#212911yescharacteristic
157#2621D6711yescharacteristic
158#1740—671no—
159#1458—671no—
160#1344—674no—
161C67#1169411yescharacteristic, complemented, Sylow
162C2×C4⋊Q8Dic6711yescharacteristic, complemented
163C2×C22⋊Q8—21no—
164C2×#117—1341no—
165C2×#117C2×D6711yescharacteristic
166#431—1341no—
167#398—21no—
168#398Dic6712yescomplemented
169#397—21no—
170#397C2×D6712yes—
171#391—21no—
172#382Dic6711yescharacteristic, complemented
173#342—672no—
174#342—1341no—
175#324—1341no—
176#323—672no—
177C22×Q8—21no—
178C23×C4C22⋊D6711yescharacteristic
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—41no—
184#117—41no—
185#117—2682no—
186#117—1344no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—2681no—
191C2×C4⋊C4C2×Dic6712yes—
192C2×C4⋊C4C2×Dic6711yescharacteristic
193C2×C4⋊C4C22⋊D6711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1342no—
196C2×C22⋊C4C22⋊D6711yescharacteristic
197C2×C22⋊C4C22×D6711yescharacteristic
198C2×C42—2681no—
199C2×C42C2×Dic6711yescharacteristic
200C2×C42C22⋊D6711yescharacteristic
201C4⋊C8—2681no—
202C22⋊C8—1342no—
203C4×C8—2681no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95C2×Dic6712yes—
208C24C2×C22⋊D6711yescharacteristic
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—2682no—
219C22×C4C22⋊Dic6711yescharacteristic
220C22×C4C22⋊Dic6711yescharacteristic
221C22×C4C22⋊Dic6711yescharacteristic
222C22×C4C22⋊Dic6711yescharacteristic
223C22×C4#1176161312yes—
224C22×C4#1176161612yes—
225C22×C4C22×Dic6711yescharacteristic
226C22×C4C2×C22⋊D6711yescharacteristic
227C2×C8—2682no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—2684no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—2682no—
240C22⋊C4—1344no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—2684no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×C22⋊Dic6711yescharacteristic
251C23C2×#1176161311yescharacteristic
252C23C2×#1176161611yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—2684no—
276C2×C4—5362no—
277C2×C4—5362no—
278C2×C4#1452395811yescharacteristic
279C2×C4#1452395811yescharacteristic
280C2×C4#1452396412yes—
281C2×C4#1452396412yes—
282C2×C4#1452397012yes—
283C8—5362no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#1452395811yescharacteristic
294C22C2×#1452396411yescharacteristic
295C22C2×#1452397011yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—82no—
303C4—5364no—
304C4#1686313012yes—
305C4#1686313012yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#1686313011yescharacteristic
311C2#2256186311yescharacteristic
312C2#2256188411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2621C2
socleC22×C67C2×#99
prime radicalC24×C67C2×D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2621D67
hypocenterC67#11694
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C67C4semidirect
15. #398×C67C4semidirect
19. #382×C67C4semidirect
161. C67#11694semidirect
162. C2×C4⋊Q8Dic67semidirect
168. #398Dic67semidirect
172. #382Dic67semidirect

Maximal subgroups

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

subgroupindex
#225599422
#225585232
#225582092
C67×#26212
#1169467