Finite Group Databasebeta

← #36334749

Subgroups — #36334749

Order 21376.

362 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#24923890C211yesmaximal, characteristic, complemented
2#24923665C211yesmaximal, characteristic, complemented
3#24923651C211yesmaximal, characteristic, complemented
4#24923631C211yesmaximal, characteristic, complemented
5#24923628C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923520C211yesmaximal, characteristic
8#24923500C211yesmaximal, characteristic, complemented
9#24923496C211yesmaximal, characteristic, complemented
10#24923379C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12C4×#14581937C211yesmaximal, characteristic, complemented
13C4×#14581938C211yesmaximal, characteristic, complemented
14#24923225C211yesmaximal, characteristic, complemented
15#542×C167C211yesmaximal, characteristic, complemented
16C2×#14581940C2211yescharacteristic
17C2×#14581938C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20C2×#14581934C2211yescharacteristic
21C22×C167⋊C8C2211yescharacteristic
22#19128832C2211yescharacteristic
23#19128827—21no—
24#19128820C2211yescharacteristic
25#19128818—21no—
26#19128814—21no—
27#19128811C2211yescharacteristic
28#19128810—21no—
29#19128806C2211yescharacteristic
30C4×Dic334C2211yescharacteristic
31C4×D668C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34#19128763C2211yescharacteristic
35#19128763C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128755C2211yescharacteristic
39#19128755C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44#19128743C2211yescharacteristic
45#19128743C2211yescharacteristic
46C4×C167⋊C8C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48C4⋊Q8×C167C2211yescharacteristic
49#126×C167—21no—
50#124×C167—21no—
51#124×C167C2211yescharacteristic
52#123×C167—21no—
53C22⋊Q8×C167—21no—
54C22⋊Q8×C167C2211yescharacteristic
55#121×C167C2211yescharacteristic
56C4×Q8×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58#117×C167C2211yescharacteristic
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C4×D167—21no—
64#14581940—22no—
65#14581938—22no—
66#14581937—22no—
67#14581935—22no—
68#14581934—22no—
69C2×C167⋊C8—22no—
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C4⋊Dic167—21no—
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
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×C167—21no—
86C2×Q8×C167C2311yescharacteristic
87C2×D4×C167—21no—
88C2×D4×C167C2311yescharacteristic
89C22×C4×C167—21no—
90C22×C4×C167C2311yescharacteristic
91C4⋊C4×C167—21no—
92C4⋊C4×C167—21no—
93C4⋊C4×C167—21no—
94C4⋊C4×C167D411yescharacteristic
95C4⋊C4×C167D411yescharacteristic
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—
103C22⋊C4×C167D411yescharacteristic
104C22⋊C4×C167D411yescharacteristic
105C42×C167—21no—
106C42×C167C2311yescharacteristic
107C42×C167C2311yescharacteristic
108C22×D167—21no—
109Dic334—21no—
110Dic334—21no—
111Dic334—81no—
112D668—21no—
113D668—21no—
114C22⋊D167—81no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C4×D167—81no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8#4112yes—
122C167⋊C8#4112yes—
123C23×C167—21no—
124C23×C167C2×D411yescharacteristic
125Q8×C167—21no—
126Q8×C167—21no—
127Q8×C167—81no—
128D4×C167—21no—
129D4×C167—21no—
130D4×C167—81no—
131C2×C4×C167—21no—
132C2×C4×C167—21no—
133C2×C4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—81no—
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—21no—
146C22×C167—41no—
147C22×C167—41no—
148C22×C167—41no—
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—21no—
158C4×C167—41no—
159C4×C167—41no—
160C4×C167—41no—
161C4×C167C2×#4111yescharacteristic
162C4×C167C2×#4111yescharacteristic
163Dic167—41no—
164Dic167—41no—
165Dic167—41no—
166C2×C167—41no—
167C2×C167—81no—
168C2×C167#53211yescharacteristic
169C2×C167#57711yescharacteristic
170C2×C167#57711yescharacteristic
171D167—81no—
172C167#317311yescharacteristic, complemented, Sylow
173#3173—1671nomaximal, Sylow
174#542—1671no—
175#542D16711yescharacteristic, complemented
176#498—1671no—
177#495—1671no—
178#489—1671no—
179#489—1671no—
180#470—1671no—
181#469—1671no—
182#465—1671no—
183#465—1671no—
184#442—1671no—
185#441—1671no—
186C4×SD16—1671no—
187C4×SD16—1671no—
188#405—1671no—
189C2×Q16—1671no—
190C2×SD16—1671no—
191C2×SD16—1671no—
192C2×D8—1671no—
193C2×M16—1671no—
194C22×C8—1671no—
195C4⋊Q8—1671no—
196C4⋊Q8C2×D16711yescharacteristic
197#126—21no—
198#126—3341no—
199#124—21no—
200#124—3341no—
201#124—1671no—
202#124C2×D16711yescharacteristic
203#123—21no—
204#123—3341no—
205C22⋊Q8—21no—
206C22⋊Q8—3341no—
207C22⋊Q8—1671no—
208C22⋊Q8C2×D16711yescharacteristic
209#121—1671no—
210#121C2×D16711yescharacteristic
211C4×Q8—1671no—
212C4×Q8C2×D16711yescharacteristic
213C4×D4—1671no—
214C4×D4C2×D16711yescharacteristic
215#117C2×D16711yescharacteristic
216#106—1671no—
217#106—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—
228#97—1671no—
229#97—1671no—
230C4×C8—1671no—
231C4×C8—1671no—
232C2×Q8—21no—
233C2×Q8—3341no—
234C2×Q8—1671no—
235C2×Q8C22×D16711yescharacteristic
236C2×D4—21no—
237C2×D4—3341no—
238C2×D4—1671no—
239C2×D4C22×D16711yescharacteristic
240C22×C4—21no—
241C22×C4—3341no—
242C22×C4C22×D16711yescharacteristic
243Q16—3342no—
244SD16—3342no—
245SD16—3342no—
246D8—3342no—
247M16—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—3341no—
257C4⋊C4—3341no—
258C4⋊C4—3341no—
259C4⋊C4—1671no—
260C4⋊C4—1671no—
261C4⋊C4C22⋊D16711yescharacteristic
262C4⋊C4C22⋊D16711yescharacteristic
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—
275C22⋊C4C22⋊D16711yescharacteristic
276C22⋊C4C22⋊D16711yescharacteristic
277C42—21no—
278C42—3341no—
279C42C22×D16711yescharacteristic
280C42C22×D16711yescharacteristic
281C23—21no—
282C23—3341no—
283C23C2×C22⋊D16711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—13361no—
288Q8—3341no—
289Q8—3341no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—13361no—
294D4—3341no—
295D4—3341no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—13361no—
303C2×C4—3341no—
304C2×C4—3341no—
305C2×C4—3341no—
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—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—6681no—
323C22—6681no—
324C22—6681no—
325C22C22×C22⋊D16711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—6681no—
335C4—6681no—
336C4—6681no—
337C4#41×D16711yescharacteristic
338C4#41×D16711yescharacteristic
339C2—41no—
340C2—81no—
341C2—13361no—
342C2#2492381511yescharacteristic
343C2#2492407711yescharacteristic
344C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#542×C167C2
socleC22×C167C22×D4
prime radical#24923520C2
hypercenter#542D167
hypocenterC167#3173
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #24923890C2semidirect
2. #24923665C2semidirect
3. #24923651C2semidirect
4. #24923631C2semidirect
5. #24923628C2semidirect
6. #24923525C2semidirect
8. #24923500C2semidirect
9. #24923496C2semidirect
10. #24923379C2semidirect
11. #24923374C2semidirect
12. C4×#14581937C2semidirect
13. C4×#14581938C2semidirect
14. #24923225C2semidirect
15. #542×C167C2semidirect
172. C167#3173semidirect
175. #542D167semidirect

Maximal subgroups

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

subgroupindex
#249238902
#249236652
#249236512
#249236312
#249236282
#249235252
#249235202
#249235002
#249234962
#249233792
#249233742
C4×#145819372
C4×#145819382
#249232252
#542×C1672
#3173167