Finite Group Databasebeta

← #30773224

Subgroups — #30773224

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14523983C211yesmaximal, characteristic, complemented
2C2×#16863206C212yesmaximal, complemented
3#22560234C211yesmaximal, characteristic, complemented
4#22558314C212yesmaximal, complemented
5C67×#3050C211yesmaximal, characteristic
6C2×#14523983—24no—
7C2×#14523983—21no—
8C2×#14523983C2212yes—
9C22×C67⋊C16—21no—
10C2×#14523975C2212yes—
11#16863256—21no—
12#16863206—24no—
13#16863159—21no—
14C2×#135×C67C2211yescharacteristic
15C23×C8×C67C2211yescharacteristic
16#525×C67—21no—
17#467×C67C411yescharacteristic, complemented
18#467×C67C411yescharacteristic, complemented
19#466×C67C411yescharacteristic, complemented
20#466×C67C411yescharacteristic, complemented
21#465×C67—21no—
22#465×C67—21no—
23#427×C67C2211yescharacteristic
24#416×C67—21no—
25C2×C4⋊Dic67—22no—
26C2×C22⋊Dic67—41no—
27#14523983—24no—
28#14523983—22no—
29#14523983D414yescomplemented
30C2×C67⋊C16—24no—
31C2×C67⋊C16—22no—
32#14523975—22no—
33#14523975—22no—
34C67⋊#95—41no—
35C2×#41×C67—21no—
36C23×C4×C67D411yescharacteristic
37#135×C67—22no—
38#135×C67—22no—
39C2×Q16×C67C2×C411yescharacteristic
40C2×SD16×C67—21no—
41C2×D8×C67C2×C411yescharacteristic
42C22×C8×C67—22no—
43C22×C8×C67—22no—
44C22×C8×C67C2×C411yescharacteristic
45C22×C8×C67C2×C411yescharacteristic
46C22×C8×C67D411yescharacteristic
47C22×C8×C67D411yescharacteristic
48C22×C8×C67D411yescharacteristic
49C22×C8×C67C2311yescharacteristic
50#123×C67—41no—
51C22⋊Q8×C67—21no—
52C22⋊Q8×C67—21no—
53#121×C67—21no—
54#121×C67—21no—
55#120×C67—41no—
56C4×D4×C67—41no—
57C4×D4×C67—41no—
58#117×C67—21no—
59#107×C67C2×C411yescharacteristic
60#107×C67C2×C411yescharacteristic
61#106×C67—21no—
62#103×C67—21no—
63#103×C67—21no—
64#102×C67—21no—
65#102×C67—21no—
66C22×Dic67—42no—
67C4⋊Dic67—42no—
68C4⋊Dic67—42no—
69C22⋊Dic67—82no—
70C67⋊C16—24no—
71C24×C67D811yescharacteristic
72#41×C67—42no—
73#41×C67—42no—
74C2×Q8×C67—21no—
75C2×D4×C67—21no—
76C2×D4×C67—41no—
77C2×D4×C67—41no—
78C2×D4×C67—41no—
79C22×C4×C67—22no—
80C22×C4×C67—22no—
81C22×C4×C67—41no—
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67D811yescharacteristic
85C22×C4×C67SD1611yescharacteristic
86C22×C4×C67SD1611yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88Q16×C67—22no—
89SD16×C67—41no—
90SD16×C67—41no—
91D8×C67—22no—
92C2×C8×C67—22no—
93C2×C8×C67—22no—
94C2×C8×C67—22no—
95C2×C8×C67—21no—
96C2×C8×C67—21no—
97C2×C8×C67—21no—
98C2×C8×C67—21no—
99C2×C8×C67C22⋊C411yescharacteristic
100C2×C8×C67C22⋊C411yescharacteristic
101C2×C8×C67C22×C411yescharacteristic
102C2×C8×C67C2×D412yes—
103C2×C8×C67C2×D411yescharacteristic
104C2×C8×C67#4112yes—
105C4⋊C4×C67—21no—
106C4⋊C4×C67—21no—
107C4⋊C4×C67—41no—
108C4⋊C4×C67—41no—
109C22⋊C4×C67—41no—
110C22⋊C4×C67—41no—
111C22⋊C4×C67—41no—
112C22⋊C4×C67—41no—
113C22⋊C4×C67—41no—
114C42×C67—41no—
115C23×C67—22no—
116C23×C67—22no—
117C23×C67—41no—
118C23×C67—41no—
119C23×C67#10211yescharacteristic
120C23×C67#10211yescharacteristic
121C23×C67C2×D811yescharacteristic
122Q8×C67—41no—
123Q8×C67—41no—
124D4×C67—41no—
125D4×C67—41no—
126D4×C67—82no—
127D4×C67—82no—
128D4×C67—82no—
129C2×C4×C67—22no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—22no—
135C2×C4×C67—22no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—82no—
140C2×C4×C67#10211yescharacteristic
141C2×C4×C67#10211yescharacteristic
142C2×C4×C67C2×C22⋊C411yescharacteristic
143C2×C4×C67#12112yes—
144C2×C4×C67C2×SD1611yescharacteristic
145C2×C4×C67#13512yes—
146C2×Dic67—82no—
147C2×Dic67—82no—
148C2×Dic67—82no—
149C8×C67—21no—
150C8×C67—21no—
151C8×C67C4×D412yes—
152C8×C67#12011yescharacteristic
153C8×C67#12311yescharacteristic
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—44no—
162C22×C67—81no—
163C22×C67—81no—
164C22×C67—81no—
165C22×C67#36012yescomplemented
166C22×C67C2×#10211yescharacteristic
167C22×C67#46612yes—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—81no—
171C4×C67—81no—
172C4×C67—81no—
173C4×C67#38611yescharacteristic
174C4×C67#41611yescharacteristic
175C4×C67#46512yes—
176Dic67—162no—
177#13065—671nomaximal, Sylow
178C2×C67—22no—
179C2×C67—22no—
180C2×C67—41no—
181C2×C67—161no—
182C2×C67#189311yescharacteristic
183C2×C67C2×#36011yescharacteristic
184C2×C67#214511yescharacteristic
185C22×M32—671no—
186#3050D6711yescharacteristic
187C2×#360—672no—
188#1807—671no—
189#1386—672no—
190C67#1306511yescharacteristic, complemented, Sylow
191C2×#135C2×D6711yescharacteristic
192C23×C8C2×D6711yescharacteristic
193#525—21no—
194C2×M32—672no—
195C2×M32—1344no—
196C2×M32—1341no—
197C22×C16—1341no—
198#467Dic6711yescharacteristic, complemented
199#467Dic6711yescharacteristic, complemented
200#466Dic6711yescharacteristic, complemented
201#466Dic6711yescharacteristic, complemented
202#465—21no—
203#465—21no—
204#427C2×D6711yescharacteristic
205C2×#106—672no—
206#416—21no—
207#380—1341no—
208#360—1344no—
209#340—1341no—
210C2×#41—21no—
211C23×C4C22⋊D6711yescharacteristic
212#135—22no—
213#135—22no—
214C2×Q16C2×Dic6711yescharacteristic
215C2×SD16—21no—
216C2×D8C2×Dic6711yescharacteristic
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C2×Dic6711yescharacteristic
220C22×C8C2×Dic6711yescharacteristic
221C22×C8C22⋊D6711yescharacteristic
222C22×C8C22⋊D6711yescharacteristic
223C22×C8C22⋊D6711yescharacteristic
224C22×C8C22×D6711yescharacteristic
225#123—41no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#121—21no—
229#121—21no—
230#120—41no—
231C4×D4—41no—
232C4×D4—41no—
233#117—21no—
234C2×C4⋊C4—1342no—
235C2×C22⋊C4—2681no—
236M32—674no—
237M32—1344no—
238M32—1342no—
239C2×C16—1344no—
240C2×C16—1342no—
241#107C2×Dic6711yescharacteristic
242#107C2×Dic6711yescharacteristic
243#106—21no—
244#106—1342no—
245#106—1342no—
246#103—21no—
247#103—21no—
248#102—21no—
249#102—21no—
250#95—2681no—
251C24#1176161311yescharacteristic
252#41—42no—
253#41—42no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—2682no—
263C22×C4C22⋊Dic6711yescharacteristic
264C22×C4C22⋊Dic6711yescharacteristic
265C22×C4#1176161311yescharacteristic
266C22×C4#1176161611yescharacteristic
267C22×C4#1176161611yescharacteristic
268C22×C4C2×C22⋊D6711yescharacteristic
269Q16—22no—
270SD16—41no—
271SD16—41no—
272D8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C22⋊Dic6711yescharacteristic
281C2×C8C22⋊Dic6711yescharacteristic
282C2×C8C22×Dic6711yescharacteristic
283C2×C8D4×D6712yes—
284C2×C8C2×C22⋊D6711yescharacteristic
285C2×C8#1176162712yes—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—2682no—
291C4⋊C4—2682no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—5362no—
298C42—41no—
299C16—1344no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23#1452396411yescharacteristic
305C23#1452396411yescharacteristic
306C23C2×#1176161311yescharacteristic
307Q8—41no—
308Q8—41no—
309D4—41no—
310D4—41no—
311D4—82no—
312D4—82no—
313D4—82no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—5362no—
326C2×C4—5362no—
327C2×C4—5362no—
328C2×C4#1452396411yescharacteristic
329C2×C4#1452396411yescharacteristic
330C2×C4C2×C22⋊Dic6711yescharacteristic
331C2×C4#1452401512yes—
332C2×C4C2×#1176161611yescharacteristic
333C2×C4#1452406312yes—
334C8—21no—
335C8—21no—
336C8D4×Dic6712yes—
337C8#1452401111yescharacteristic
338C8#1452402411yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#1686320612yescomplemented
351C22C2×#1452396411yescharacteristic
352C22#1686363012yes—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—10722no—
359C4#1686327611yescharacteristic
360C4#1686339211yescharacteristic
361C4#1686362312yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—161no—
366C2#2256063811yescharacteristic
367C2C2×#1686320611yescharacteristic
368C2#2256194511yescharacteristic
subgroupG/H
centerC22C2×#14523964
commutatorC2×C8×C67C22×C4
FrattiniC22×C8C22×D67
Fitting, prime radicalC67×#3050C2
socleC22×C67C2×#102
cosocleC22×C8×C67C23
hypercenter#3050D67
hypocenterC67#13065
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C22×#14523983C2semidirect
2. C2×#16863206C2semidirect
3. #22560234C2semidirect
4. #22558314C2semidirect
17. #467×C67C4semidirect
18. #467×C67C4semidirect
19. #466×C67C4semidirect
20. #466×C67C4semidirect
29. #14523983D4semidirect
165. C22×C67#360semidirect
190. C67#13065semidirect
198. #467Dic67semidirect
199. #467Dic67semidirect
200. #466Dic67semidirect
201. #466Dic67semidirect
350. C22#16863206semidirect

Maximal subgroups

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

subgroupindex
C22×#145239832
C2×#168632062
#225602342
#225583142
C67×#30502
#1306567