Finite Group Databasebeta

← #30766444

Subgroups — #30766444

Order 17152.

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

normal subgroups only

This group has 462 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#22558550C211yesmaximal, characteristic
3#22558233C214yesmaximal, complemented
4C67×#2487C211yesmaximal, characteristic
5#16863453—21no—
6C2×#14523958—21no—
7C4×C22⋊Dic67—21no—
8#16863163C2212yes—
9#16863120—21no—
10#16863120C2212yes—
11C2×C4×D4×C67C2211yescharacteristic
12C22×C4⋊C4×C67C411yescharacteristic, complemented
13#400×C67C412yescomplemented
14#399×C67C2212yes—
15#389×C67—21no—
16#386×C67—21no—
17#385×C67—21no—
18#380×C67—21no—
19#380×C67C411yescharacteristic, complemented
20C2×C22⋊Dic67—22no—
21C2×C4×Dic67—41no—
22C4⋊(C67⋊C8)—41no—
23#14523958—24no—
24C22⋊(C67⋊C8)—22no—
25#14523954—41no—
26C67⋊#95—41no—
27C22×D4×C67D411yescharacteristic
28C23×C4×C67—21no—
29C23×C4×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4×D4×C67—41no—
32C4×D4×C67—41no—
33C2×C4⋊C4×C67—21no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67—41no—
37C2×C4⋊C4×C67—41no—
38C2×C4⋊C4×C67C2×C412yes—
39C2×C4⋊C4×C67C2×C411yescharacteristic
40C2×C22⋊C4×C67—21no—
41C2×C22⋊C4×C67—22no—
42C2×C22⋊C4×C67D411yescharacteristic
43C2×C22⋊C4×C67C2311yescharacteristic
44C2×C42×C67D411yescharacteristic
45#95×C67—22no—
46#95×C67—22no—
47#95×C67C2×C412yes—
48C22×Dic67—42no—
49C2×C67⋊C8—42no—
50C22⋊Dic67—24no—
51C22⋊Dic67—42no—
52C4×Dic67—82no—
53C24×C67C22⋊C411yescharacteristic
54C24×C67C2×D411yescharacteristic
55C2×D4×C67—22no—
56C2×D4×C67—22no—
57C2×D4×C67—41no—
58C22×C4×C67—21no—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—22no—
63C22×C4×C67—22no—
64C22×C4×C67—42no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—82no—
68C22×C4×C67C22⋊C411yescharacteristic
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22×C411yescharacteristic
72C22×C4×C67C2×D411yescharacteristic
73C4⋊C4×C67—41no—
74C4⋊C4×C67—42no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—41no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—82no—
79C22⋊C4×C67—41no—
80C22⋊C4×C67—41no—
81C22⋊C4×C67—42no—
82C22⋊C4×C67—82no—
83C42×C67—21no—
84C42×C67—21no—
85C23×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—41no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67#10412yes—
94C23×C67C2×C22⋊C411yescharacteristic
95C23×C67#13712yes—
96D4×C67—44no—
97C2×C4×C67—22no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—42no—
105C2×C4×C67—42no—
106C2×C4×C67—42no—
107C2×C4×C67—41no—
108C2×C4×C67—41no—
109C2×C4×C67—41no—
110C2×C4×C67—42no—
111C2×C4×C67—42no—
112C2×C4×C67—42no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—42no—
116C2×C4×C67—81no—
117C2×C4×C67—84no—
118C2×C4×C67#9911yescharacteristic
119C2×C4×C67#9911yescharacteristic
120C2×Dic67—44no—
121C2×Dic67—82no—
122C2×Dic67—82no—
123C67⋊C8—82no—
124C22×C67—21no—
125C22×C67—21no—
126C22×C67—22no—
127C22×C67—22no—
128C22×C67—42no—
129C22×C67—42no—
130C22×C67—42no—
131C22×C67—41no—
132C22×C67—41no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—81no—
136C22×C67C2×#9911yescharacteristic
137C22×C67#41911yescharacteristic
138C22×C67C2×#10411yescharacteristic
139C4×C67—21no—
140C4×C67—41no—
141C4×C67—41no—
142C4×C67—41no—
143C4×C67—41no—
144C4×C67—82no—
145C4×C67—82no—
146C4×C67#35212yes—
147Dic67—84no—
148#11786—671nomaximal, Sylow
149C2×C67—21no—
150C2×C67—21no—
151C2×C67—42no—
152C2×C67—42no—
153C2×C67#150811yescharacteristic
154C2×C67C2×#35211yescharacteristic
155C2×C67#212811yescharacteristic
156#2487D6711yescharacteristic
157C4×#99—671no—
158#1466—671no—
159#1352—674no—
160C67#1178611yescharacteristic, complemented, Sylow
161C2×C4×D4C2×D6711yescharacteristic
162C22×C4⋊C4Dic6711yescharacteristic, complemented
163#432—1341no—
164C2×#99—1341no—
165#400Dic6712yescomplemented
166#399C2×D6712yes—
167#389—21no—
168#386—21no—
169#385—21no—
170#380—21no—
171#380Dic6711yescharacteristic, complemented
172C4×C22⋊C4—1341no—
173#342—672no—
174#323—672no—
175#323—1341no—
176C22×D4C22⋊D6711yescharacteristic
177C23×C4—21no—
178C23×C4C2×Dic6711yescharacteristic
179C23×C4C22⋊D6711yescharacteristic
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×Dic6712yes—
188C2×C4⋊C4C2×Dic6711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—1342no—
192C2×C22⋊C4C22⋊D6711yescharacteristic
193C2×C22⋊C4C22×D6711yescharacteristic
194C2×C42—2681no—
195C2×C42C22⋊D6711yescharacteristic
196C4⋊C8—2681no—
197#99—1344no—
198C22⋊C8—1342no—
199#97—2681no—
200#95—22no—
201#95—22no—
202#95—2681no—
203#95C2×Dic6712yes—
204C24C22⋊Dic6711yescharacteristic
205C24C2×C22⋊D6711yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—2682no—
220C22×C4C22⋊Dic6711yescharacteristic
221C22×C4C22⋊Dic6711yescharacteristic
222C22×C4C22⋊Dic6711yescharacteristic
223C22×C4C22×Dic6711yescharacteristic
224C22×C4C2×C22⋊D6711yescharacteristic
225C2×C8—2682no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—2682no—
237C22⋊C4—1344no—
238C42—21no—
239C42—21no—
240C42—5362no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#1452397012yes—
250C23C2×C22⋊Dic6711yescharacteristic
251C23#1452407412yes—
252D4—44no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4—2684no—
275C2×C4—5362no—
276C2×C4—5362no—
277C2×C4#1452395811yescharacteristic
278C2×C4#1452395811yescharacteristic
279C8—5362no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#1452395811yescharacteristic
293C22#1686340511yescharacteristic
294C22C2×#1452397011yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4—5364no—
303C4#1686318612yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2255872911yescharacteristic
309C2C2×#1686318611yescharacteristic
310C2#2256188111yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2487C2
socleC22×C67C2×#99
prime radicalC22×D4×C67D4
cosocleC2×C22⋊C4×C67C23
hypercenter#2487D67
hypocenterC67#11786
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #22558233C2semidirect
12. C22×C4⋊C4×C67C4semidirect
13. #400×C67C4semidirect
19. #380×C67C4semidirect
160. C67#11786semidirect
162. C22×C4⋊C4Dic67semidirect
165. #400Dic67semidirect
171. #380Dic67semidirect

Maximal subgroups

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

subgroupindex
C4×#145239582
#225585502
#225582332
C67×#24872
#1178667