Finite Group Databasebeta

← #36334702

Subgroups — #36334702

Order 21376.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923803C211yesmaximal, characteristic, complemented
2#24923668C211yesmaximal, characteristic, complemented
3#24923651C211yesmaximal, characteristic, complemented
4#24923626C211yesmaximal, characteristic, complemented
5#24923623C211yesmaximal, characteristic, complemented
6#24923508C211yesmaximal, characteristic, complemented
7#24923503C211yesmaximal, characteristic
8#24923500C211yesmaximal, characteristic, complemented
9#24923496C211yesmaximal, characteristic, complemented
10C4×#14581940C211yesmaximal, characteristic, complemented
11C4×#14581937C211yesmaximal, characteristic, complemented
12C4×#14581938C211yesmaximal, characteristic, complemented
13C4×#14581935C211yesmaximal, characteristic, complemented
14C2×C4×C167⋊C8C211yesmaximal, characteristic, complemented
15#529×C167C211yesmaximal, characteristic, complemented
16C2×#14581940C2211yescharacteristic
17C2×#14581938C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20C22×C167⋊C8C2211yescharacteristic
21C22×C167⋊C8C2211yescharacteristic
22#19128826—21no—
23#19128824C2211yescharacteristic
24#19128820C2211yescharacteristic
25#19128815—21no—
26#19128811C2211yescharacteristic
27#19128806C2211yescharacteristic
28C4×Dic334C2211yescharacteristic
29C4×D668C2211yescharacteristic
30C4×C22⋊D167—21no—
31#19128789—21no—
32#19128765C2211yescharacteristic
33#19128765C2211yescharacteristic
34#19128763C2211yescharacteristic
35#19128763C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128755C2211yescharacteristic
39#19128755C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44C4×C167⋊C8C2211yescharacteristic
45C4×C167⋊C8C2211yescharacteristic
46C4×C167⋊C8C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48#126×C167—21no—
49#125×C167C2211yescharacteristic
50#124×C167C2211yescharacteristic
51#123×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53#121×C167C2211yescharacteristic
54C4×Q8×C167C2211yescharacteristic
55C4×D4×C167—21no—
56C4×D4×C167C2211yescharacteristic
57#117×C167—21no—
58C2×C42×C167C2211yescharacteristic
59C2×Dic334C2311yescharacteristic
60C2×D668C2311yescharacteristic
61C2×C22⋊D167—21no—
62C2×C4×D167—21no—
63#14581940—22no—
64#14581938—22no—
65#14581937—22no—
66#14581935—22no—
67C2×C167⋊C8—22no—
68C2×C167⋊C8—22no—
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C2×C167⋊C8C2311yescharacteristic
73C4⋊Dic167C2311yescharacteristic
74C4⋊Dic167C2311yescharacteristic
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77Dic167⋊C4—21no—
78Dic167⋊C4—21no—
79C22⋊Dic167—21no—
80(C2×D167)⋊C4—21no—
81(C2×D167)⋊C4—21no—
82(C2×D167)⋊C4—21no—
83(C2×D167)⋊C4—21no—
84C4×Dic167—21no—
85C2×Q8×C167C2311yescharacteristic
86C2×D4×C167—21no—
87C2×D4×C167C2311yescharacteristic
88C22×C4×C167—21no—
89C22×C4×C167D411yescharacteristic
90C22×C4×C167D411yescharacteristic
91C22×C4×C167C2311yescharacteristic
92C4⋊C4×C167—21no—
93C4⋊C4×C167—21no—
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167C2311yescharacteristic
97C4⋊C4×C167C2311yescharacteristic
98C22⋊C4×C167—21no—
99C22⋊C4×C167—21no—
100C22⋊C4×C167—21no—
101C22⋊C4×C167—21no—
102C22⋊C4×C167—21no—
103C42×C167—21no—
104C42×C167D411yescharacteristic
105C42×C167D411yescharacteristic
106C42×C167C2311yescharacteristic
107C42×C167C2311yescharacteristic
108C22×D167—21no—
109Dic334—21no—
110Dic334—21no—
111D668—21no—
112D668—21no—
113C22⋊D167—42no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C4×D167—42no—
118C167⋊C8—21no—
119C167⋊C8—21no—
120C167⋊C8#4112yes—
121C167⋊C8#4112yes—
122C23×C167—21no—
123C23×C167C2×D411yescharacteristic
124Q8×C167—21no—
125Q8×C167—21no—
126D4×C167—21no—
127D4×C167—21no—
128D4×C167—42no—
129C2×C4×C167—22no—
130C2×C4×C167—22no—
131C2×C4×C167—21no—
132C2×C4×C167—21no—
133C2×C4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—42no—
137C2×C4×C167C2×D411yescharacteristic
138C2×C4×C167C2×D411yescharacteristic
139C2×C4×C167C2×D411yescharacteristic
140C2×C4×C167C2×D411yescharacteristic
141C2×C4×C167C2×D411yescharacteristic
142C2×C4×C167C2411yescharacteristic
143C22×C167—21no—
144C22×C167—21no—
145C22×C167—41no—
146C22×C167—41no—
147C22×C167—41no—
148C22×C167#13512yes—
149C22×C167C22×D411yescharacteristic
150C2×D167—41no—
151C2×D167—41no—
152C2×D167—41no—
153C4×C167—21no—
154C4×C167—21no—
155C4×C167—21no—
156C4×C167—21no—
157C4×C167—41no—
158C4×C167—41no—
159C4×C167—41no—
160C4×C167#13512yes—
161C4×C167C2×#4111yescharacteristic
162C4×C167C2×#4111yescharacteristic
163Dic167—41no—
164Dic167—41no—
165Dic167—41no—
166C2×C167—22no—
167C2×C167—81no—
168C2×C167#53211yescharacteristic
169C2×C167C2×#13511yescharacteristic
170C2×C167C2×#13511yescharacteristic
171D167—81no—
172C167#316811yescharacteristic, complemented, Sylow
173#3168—1671nomaximal, Sylow
174#529—1671no—
175#529D16711yescharacteristic, complemented
176#499—1671no—
177#495—1671no—
178#488—1671no—
179#488—1671no—
180#467—1671no—
181#466—1671no—
182#465—1671no—
183#465—1671no—
184C4×Q16—1671no—
185C4×SD16—1671no—
186C4×SD16—1671no—
187C4×D8—1671no—
188C2×C4×C8—1671no—
189C2×Q16—1671no—
190C2×SD16—1671no—
191C2×SD16—1671no—
192C2×D8—1671no—
193C22×C8—1671no—
194C22×C8—1671no—
195#126—21no—
196#126—3341no—
197#125—1671no—
198#125C2×D16711yescharacteristic
199#124—1671no—
200#124C2×D16711yescharacteristic
201#123—21no—
202#123—3341no—
203C22⋊Q8—1671no—
204C22⋊Q8C2×D16711yescharacteristic
205#121—1671no—
206#121C2×D16711yescharacteristic
207C4×Q8—1671no—
208C4×Q8C2×D16711yescharacteristic
209C4×D4—21no—
210C4×D4—3341no—
211C4×D4—1671no—
212C4×D4C2×D16711yescharacteristic
213#117—21no—
214#117—3341no—
215C2×C42C2×D16711yescharacteristic
216#107—1671no—
217#107—1671no—
218#106—1671no—
219#106—1671no—
220#103—1671no—
221#103—1671no—
222#103—1671no—
223#103—1671no—
224#102—1671no—
225#102—1671no—
226#102—1671no—
227#102—1671no—
228C4×C8—1671no—
229C4×C8—1671no—
230C4×C8—1671no—
231C4×C8—1671no—
232C2×Q8—1671no—
233C2×Q8C22×D16711yescharacteristic
234C2×D4—21no—
235C2×D4—3341no—
236C2×D4—1671no—
237C2×D4C22×D16711yescharacteristic
238C22×C4—21no—
239C22×C4—3341no—
240C22×C4C22⋊D16711yescharacteristic
241C22×C4C22⋊D16711yescharacteristic
242C22×C4C22×D16711yescharacteristic
243Q16—3342no—
244SD16—3342no—
245SD16—3342no—
246D8—3342no—
247C2×C8—3342no—
248C2×C8—3342no—
249C2×C8—1671no—
250C2×C8—1671no—
251C2×C8—1671no—
252C2×C8—1671no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—3341no—
258C4⋊C4—3341no—
259C4⋊C4—3341no—
260C4⋊C4—3341no—
261C4⋊C4—1671no—
262C4⋊C4—1671no—
263C4⋊C4C22×D16711yescharacteristic
264C4⋊C4C22×D16711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3341no—
271C22⋊C4—3341no—
272C22⋊C4—3341no—
273C22⋊C4—3341no—
274C22⋊C4—3341no—
275C42—21no—
276C42—3341no—
277C42C22⋊D16711yescharacteristic
278C42C22⋊D16711yescharacteristic
279C42C22×D16711yescharacteristic
280C42C22×D16711yescharacteristic
281C23—21no—
282C23—3341no—
283C23C2×C22⋊D16711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—3341no—
287Q8—3341no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—3341no—
292D4—3341no—
293D4—6682no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3341no—
303C2×C4—3341no—
304C2×C4—3341no—
305C2×C4—6682no—
306C2×C4C2×C22⋊D16711yescharacteristic
307C2×C4C2×C22⋊D16711yescharacteristic
308C2×C4C2×C22⋊D16711yescharacteristic
309C2×C4C2×C22⋊D16711yescharacteristic
310C2×C4C2×C22⋊D16711yescharacteristic
311C2×C4C23×D16711yescharacteristic
312C8—3341no—
313C8—3341no—
314C8—1672no—
315C8—1672no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—6681no—
322C22—6681no—
323C22—6681no—
324C22#1912885212yes—
325C22C22×C22⋊D16711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—6681no—
334C4—6681no—
335C4—6681no—
336C4#1912885212yes—
337C4#41×D16711yescharacteristic
338C4#41×D16711yescharacteristic
339C2—22no—
340C2—81no—
341C2—13361no—
342C2#2492381511yescharacteristic
343C2C2×#1912885211yescharacteristic
344C2C2×#1912885211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#529×C167C2
socleC22×C167C22×D4
prime radical#24923503C2
hypercenter#529D167
hypocenterC167#3168
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #24923803C2semidirect
2. #24923668C2semidirect
3. #24923651C2semidirect
4. #24923626C2semidirect
5. #24923623C2semidirect
6. #24923508C2semidirect
8. #24923500C2semidirect
9. #24923496C2semidirect
10. C4×#14581940C2semidirect
11. C4×#14581937C2semidirect
12. C4×#14581938C2semidirect
13. C4×#14581935C2semidirect
14. C2×C4×C167⋊C8C2semidirect
15. #529×C167C2semidirect
172. C167#3168semidirect
175. #529D167semidirect

Maximal subgroups

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

subgroupindex
#249238032
#249236682
#249236512
#249236262
#249236232
#249235082
#249235032
#249235002
#249234962
C4×#145819402
C4×#145819372
C4×#145819382
C4×#145819352
C2×C4×C167⋊C82
#529×C1672
#3168167