Finite Group Databasebeta

← #30766452

Subgroups — #30766452

Order 17152.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14523958C211yesmaximal, characteristic
2#22558547C211yesmaximal, characteristic
3#22558233C214yesmaximal, complemented
4C67×#2542C211yesmaximal, characteristic
5C2×#14523958—21no—
6#16863360—21no—
7C4×C22⋊Dic67—21no—
8#16863163C2212yes—
9#16863122—21no—
10#16863120C2212yes—
11C2×C4⋊Q8×C67C411yescharacteristic, complemented
12C2×C22⋊Q8×C67—21no—
13C2×C22⋊Q8×C67—21no—
14C2×C4×D4×C67C2211yescharacteristic
15#399×C67C412yescomplemented
16#399×C67C2212yes—
17#390×C67—21no—
18#384×C67—21no—
19C4×C4⋊C4×C67C411yescharacteristic, complemented
20C2×#11761612—41no—
21C2×C22⋊Dic67—22no—
22C2×C4×Dic67—41no—
23#14523958—24no—
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—
33C22⋊Q8×C67—44no—
34C22⋊Q8×C67—44no—
35C4×D4×C67—41no—
36C4×D4×C67—41no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—22no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—22no—
41C2×C4⋊C4×C67C2×C412yes—
42C2×C4⋊C4×C67D411yescharacteristic
43C2×C22⋊C4×C67—22no—
44C2×C22⋊C4×C67—22no—
45C2×C22⋊C4×C67C2×C411yescharacteristic
46C2×C22⋊C4×C67C2311yescharacteristic
47C2×C42×C67—21no—
48C2×C42×C67C2×C411yescharacteristic
49#95×C67C2×C412yes—
50C22×Dic67—42no—
51#11761612—82no—
52C2×C67⋊C8—42no—
53C22⋊Dic67—24no—
54C22⋊Dic67—42no—
55C4×Dic67—82no—
56C24×C67C22⋊C411yescharacteristic
57C24×C67C2×D411yescharacteristic
58C2×Q8×C67—42no—
59C2×Q8×C67—42no—
60C2×Q8×C67—41no—
61C2×Q8×C67—41no—
62C2×D4×C67—22no—
63C2×D4×C67—22no—
64C2×D4×C67—41no—
65C22×C4×C67—22no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
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—24no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—44no—
82C4⋊C4×C67—44no—
83C4⋊C4×C67—42no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—44no—
87C22⋊C4×C67—44no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—42no—
91C23×C67—21no—
92C23×C67—21no—
93C23×C67—21no—
94C23×C67—21no—
95C23×C67—21no—
96C23×C67—41no—
97C23×C67—41no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67C2×C22⋊C411yescharacteristic
101C23×C67#13712yes—
102Q8×C67—82no—
103Q8×C67—82no—
104D4×C67—44no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—24no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—42no—
120C2×C4×C67—42no—
121C2×C4×C67—42no—
122C2×C4×C67—81no—
123C2×C4×C67—81no—
124C2×C4×C67#9911yescharacteristic
125C2×C4×C67#9911yescharacteristic
126C2×C4×C67#10412yes—
127C2×Dic67—44no—
128C2×Dic67—82no—
129C2×Dic67—82no—
130C67⋊C8—82no—
131C22×C67—21no—
132C22×C67—21no—
133C22×C67—22no—
134C22×C67—42no—
135C22×C67—42no—
136C22×C67—41no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—41no—
141C22×C67—81no—
142C22×C67—81no—
143C22×C67C2×#9911yescharacteristic
144C22×C67#41911yescharacteristic
145C22×C67C2×#10411yescharacteristic
146C4×C67—21no—
147C4×C67—22no—
148C4×C67—44no—
149C4×C67—41no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67—82no—
153C4×C67#35312yes—
154Dic67—84no—
155#11788—671nomaximal, Sylow
156C2×C67—21no—
157C2×C67—21no—
158C2×C67—42no—
159C2×C67—81no—
160C2×C67#150811yescharacteristic
161C2×C67#212311yescharacteristic
162C2×C67C2×#35311yescharacteristic
163#2542D6711yescharacteristic
164C4×#99—671no—
165#1465—671no—
166#1352—674no—
167C67#1178811yescharacteristic, complemented, Sylow
168C2×C4⋊Q8Dic6711yescharacteristic, complemented
169C2×C22⋊Q8—21no—
170C2×C22⋊Q8—21no—
171C2×C4×D4C2×D6711yescharacteristic
172C2×#99—1341no—
173#407—1341no—
174#399Dic6712yescomplemented
175#399C2×D6712yes—
176#390—21no—
177#384—21no—
178C4×C4⋊C4Dic6711yescharacteristic, complemented
179C4×C22⋊C4—1341no—
180#342—672no—
181#324—1341no—
182#323—672no—
183C22×Q8—21no—
184C22×D4C22⋊D6711yescharacteristic
185C23×C4C22⋊D6711yescharacteristic
186C23×C4C22⋊D6711yescharacteristic
187C2×M16—2681no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C2×Dic6712yes—
199C2×C4⋊C4C22⋊D6711yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—1342no—
203C2×C22⋊C4C2×Dic6711yescharacteristic
204C2×C22⋊C4C22×D6711yescharacteristic
205C2×C42—21no—
206C2×C42—2681no—
207C2×C42C2×Dic6711yescharacteristic
208#99—1344no—
209C22⋊C8—2681no—
210C22⋊C8—1342no—
211#95—2681no—
212#95C2×Dic6712yes—
213C24C22⋊Dic6711yescharacteristic
214C24C2×C22⋊D6711yescharacteristic
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—2682no—
231C22×C4C22⋊Dic6711yescharacteristic
232C22×C4C22⋊Dic6711yescharacteristic
233C22×C4C22⋊Dic6711yescharacteristic
234C22×C4C22×Dic6711yescharacteristic
235C22×C4C2×C22⋊D6711yescharacteristic
236M16—5362no—
237C2×C8—2682no—
238C4⋊C4—24no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—2682no—
249C22⋊C4—1344no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—5362no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊Dic6711yescharacteristic
264C23#1452407412yes—
265Q8—82no—
266Q8—82no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—2684no—
288C2×C4—5362no—
289C2×C4—5362no—
290C2×C4#1452395811yescharacteristic
291C2×C4#1452395811yescharacteristic
292C2×C4#1452397012yes—
293C8—5362no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×#1452395811yescharacteristic
307C22#1686340511yescharacteristic
308C22C2×#1452397011yescharacteristic
309C4—22no—
310C4—21no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—82no—
316C4—5364no—
317C4#1686318912yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#2255872911yescharacteristic
323C2#2256186311yescharacteristic
324C2C2×#1686318911yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2542C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2542D67
hypocenterC67#11788
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #22558233C2semidirect
11. C2×C4⋊Q8×C67C4semidirect
15. #399×C67C4semidirect
19. C4×C4⋊C4×C67C4semidirect
167. C67#11788semidirect
168. C2×C4⋊Q8Dic67semidirect
174. #399Dic67semidirect
178. C4×C4⋊C4Dic67semidirect

Maximal subgroups

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

subgroupindex
C4×#145239582
#225585472
#225582332
C67×#25422
#1178867