Finite Group Databasebeta

← #30766959

Subgroups — #30766959

Order 17152.

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

normal subgroups only

This group has 452 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#22558530C211yesmaximal, characteristic
3#22558233C214yesmaximal, complemented
4C67×#2487C211yesmaximal, characteristic
5#16863449—21no—
6C2×#14523958—21no—
7#16863256—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×C4⋊Dic67—41no—
21C2×C22⋊Dic67—22no—
22C4⋊(C67⋊C8)—41no—
23#14523958—42no—
24C22⋊(C67⋊C8)—22no—
25C4×C67⋊C8—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—
50C4⋊Dic67—82no—
51C22⋊Dic67—42no—
52C22⋊Dic67—42no—
53C24×C67C22⋊C411yescharacteristic
54C24×C67C2×D411yescharacteristic
55C2×D4×C67—41no—
56C2×D4×C67—41no—
57C2×D4×C67SD1612yes—
58C2×D4×C67Q1612yes—
59C22×C4×C67—21no—
60C22×C4×C67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67—22no—
64C22×C4×C67—22no—
65C22×C4×C67—42no—
66C22×C4×C67—42no—
67C22×C4×C67—42no—
68C22×C4×C67—82no—
69C22×C4×C67C22⋊C411yescharacteristic
70C22×C4×C67C22⋊C411yescharacteristic
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22×C411yescharacteristic
73C22×C4×C67C2×D411yescharacteristic
74C4⋊C4×C67—41no—
75C4⋊C4×C67—42no—
76C4⋊C4×C67—42no—
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—82no—
80C22⋊C4×C67—41no—
81C22⋊C4×C67—41no—
82C22⋊C4×C67—42no—
83C22⋊C4×C67—82no—
84C42×C67—21no—
85C42×C67—21no—
86C23×C67—21no—
87C23×C67—21no—
88C23×C67—21no—
89C23×C67—21no—
90C23×C67—41no—
91C23×C67—41no—
92C23×C67—41no—
93C23×C67—41no—
94C23×C67#10312yes—
95C23×C67C2×C22⋊C411yescharacteristic
96C23×C67C2×SD1611yescharacteristic
97C23×C67C2×Q1611yescharacteristic
98D4×C67—42no—
99D4×C67—42no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—22no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—42no—
107C2×C4×C67—42no—
108C2×C4×C67—42no—
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—81no—
119C2×C4×C67—84no—
120C2×C4×C67#9911yescharacteristic
121C2×C4×C67#9911yescharacteristic
122C2×C4×C67#10312yes—
123C2×Dic67—82no—
124C2×Dic67—82no—
125C2×Dic67—82no—
126C67⋊C8—82no—
127C22×C67—21no—
128C22×C67—21no—
129C22×C67—21no—
130C22×C67—22no—
131C22×C67—21no—
132C22×C67—42no—
133C22×C67—41no—
134C22×C67—41no—
135C22×C67—42no—
136C22×C67—41no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—81no—
141C22×C67C2×#9911yescharacteristic
142C22×C67C2×#10311yescharacteristic
143C22×C67#42111yescharacteristic
144C4×C67—21no—
145C4×C67—22no—
146C4×C67—41no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—82no—
150C4×C67—82no—
151C4×C67#35212yes—
152Dic67—162no—
153#11899—671nomaximal, Sylow
154C2×C67—21no—
155C2×C67—21no—
156C2×C67—41no—
157C2×C67—42no—
158C2×C67—41no—
159C2×C67#152011yescharacteristic
160C2×C67C2×#35211yescharacteristic
161C2×C67#212811yescharacteristic
162#2487D6711yescharacteristic
163#1799—671no—
164#1460—671no—
165#1352—674no—
166C67#1189911yescharacteristic, complemented, Sylow
167C2×C4×D4C2×D6711yescharacteristic
168C22×C4⋊C4Dic6711yescharacteristic, complemented
169#431—1341no—
170C2×#99—1341no—
171#400Dic6712yescomplemented
172#399C2×D6712yes—
173#389—21no—
174#386—21no—
175#385—21no—
176#380—21no—
177#380—1341no—
178#380Dic6711yescharacteristic, complemented
179#342—672no—
180#323—672no—
181#323—1341no—
182C22×D4C22⋊D6711yescharacteristic
183C23×C4—21no—
184C23×C4C2×Dic6711yescharacteristic
185C23×C4C22⋊D6711yescharacteristic
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—2681no—
194C2×C4⋊C4C2×Dic6712yes—
195C2×C4⋊C4C2×Dic6711yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1342no—
199C2×C22⋊C4C22⋊D6711yescharacteristic
200C2×C22⋊C4C22×D6711yescharacteristic
201C2×C42C22⋊D6711yescharacteristic
202C4⋊C8—2681no—
203#99—2682no—
204C22⋊C8—1342no—
205C4×C8—2681no—
206#95—22no—
207#95—22no—
208#95—2681no—
209#95C2×Dic6712yes—
210C24C22⋊Dic6711yescharacteristic
211C24C2×C22⋊D6711yescharacteristic
212C2×D4—41no—
213C2×D4—41no—
214C2×D4#1176161512yes—
215C2×D4#1176161812yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—2682no—
227C22×C4C22⋊Dic6711yescharacteristic
228C22×C4C22⋊Dic6711yescharacteristic
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22×Dic6711yescharacteristic
231C22×C4C2×C22⋊D6711yescharacteristic
232C2×C8—2682no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—82no—
239C4⋊C4—5362no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—2682no—
245C22⋊C4—2682no—
246C42—21no—
247C42—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#1452396712yes—
257C23C2×C22⋊Dic6711yescharacteristic
258C23C2×#1176161511yescharacteristic
259C23C2×#1176161811yescharacteristic
260D4—42no—
261D4—42no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4—5362no—
283C2×C4—5362no—
284C2×C4—5362no—
285C2×C4#1452395811yescharacteristic
286C2×C4#1452395811yescharacteristic
287C2×C4#1452396712yes—
288C8—5362no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1452395811yescharacteristic
304C22C2×#1452396711yescharacteristic
305C22#1686341311yescharacteristic
306C4—21no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—10722no—
314C4#1686318612yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#2255878111yescharacteristic
321C2C2×#1686318611yescharacteristic
322C2#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#11899
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
166. C67#11899semidirect
168. C22×C4⋊C4Dic67semidirect
171. #400Dic67semidirect
178. #380Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601972
#225585302
#225582332
C67×#24872
#1189967