Finite Group Databasebeta

← #36334322

Subgroups — #36334322

Order 21376.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923890C211yesmaximal, characteristic, complemented
2#24923610C211yesmaximal, characteristic, complemented
3#24923593C211yesmaximal, characteristic, complemented
4#24923559C211yesmaximal, characteristic, complemented
5#24923549C211yesmaximal, characteristic, complemented
6#24923484C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic, complemented
8#24923420C211yesmaximal, characteristic, complemented
9#24923403C211yesmaximal, characteristic
10#24923379C211yesmaximal, characteristic, complemented
11#24923369C211yesmaximal, characteristic, complemented
12C4×#14581938C211yesmaximal, characteristic, complemented
13C4×#14581935C211yesmaximal, characteristic, complemented
14#24923330C211yesmaximal, characteristic, complemented
15#518×C167C211yesmaximal, characteristic, complemented
16C2×#14581938C2211yescharacteristic
17C2×#14581938C2211yescharacteristic
18C2×#14581935C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20#19128832C2211yescharacteristic, complemented
21#19128827—21no—
22#19128820C2211yescharacteristic, complemented
23#19128818—21no—
24#19128814—21no—
25#19128811C2211yescharacteristic
26#19128810—21no—
27#19128806C2211yescharacteristic
28C4×Dic334C2211yescharacteristic, complemented
29C4×D668C2211yescharacteristic, complemented
30#19128765C2211yescharacteristic, complemented
31#19128765C2211yescharacteristic, complemented
32#19128763C2211yescharacteristic, complemented
33#19128763C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35C4⋊(C167⋊C8)C2211yescharacteristic, complemented
36#19128755C2211yescharacteristic, complemented
37#19128755C2211yescharacteristic, complemented
38#19128753C2211yescharacteristic
39#19128753C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128752C2211yescharacteristic, complemented
43#19128752C2211yescharacteristic, complemented
44C22⋊(C167⋊C8)C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46#19128743C2211yescharacteristic, complemented
47C4×C167⋊C8C2211yescharacteristic, complemented
48C22×D4×C167C2211yescharacteristic
49C4×D4×C167—21no—
50C4×D4×C167—21no—
51C4×D4×C167C2211yescharacteristic
52C4×D4×C167C2211yescharacteristic
53C4×D4×C167C2211yescharacteristic
54C4×D4×C167C2211yescharacteristic
55#117×C167C2211yescharacteristic
56#117×C167C2211yescharacteristic
57C2×C22⋊C4×C167—21no—
58C2×C22⋊C4×C167—21no—
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C4×D167—21no—
64#14581938—22no—
65#14581938—22no—
66#14581935—22no—
67#14581935—22no—
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C4⋊Dic167—21no—
73C4⋊Dic167C2311yescharacteristic
74C4⋊Dic167C2311yescharacteristic
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77C22⋊Dic167—21no—
78(C2×D167)⋊C4—21no—
79(C2×D167)⋊C4—21no—
80(C2×D167)⋊C4—21no—
81(C2×D167)⋊C4—21no—
82C4×Dic167—21no—
83C24×C167—21no—
84C2×D4×C167—22no—
85C2×D4×C167—22no—
86C2×D4×C167—21no—
87C2×D4×C167C2311yescharacteristic
88C2×D4×C167C2311yescharacteristic
89C22×C4×C167—21no—
90C22×C4×C167—21no—
91C22×C4×C167—21no—
92C22×C4×C167—21no—
93C22×C4×C167C2311yescharacteristic
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—21no—
122C167⋊C8—21no—
123C23×C167—24no—
124C23×C167—21no—
125C23×C167—21no—
126C23×C167—81no—
127C23×C167—81no—
128C23×C167C2×D411yescharacteristic
129D4×C167—24no—
130D4×C167—21no—
131D4×C167—21no—
132D4×C167#4112yes—
133D4×C167#4112yes—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—81no—
139C2×C4×C167—81no—
140C2×C4×C167—81no—
141C2×C4×C167—81no—
142C2×C4×C167C2×D411yescharacteristic
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2411yescharacteristic
148C22×C167—21no—
149C22×C167—22no—
150C22×C167—22no—
151C22×C167—21no—
152C22×C167—21no—
153C22×C167—41no—
154C22×C167—41no—
155C22×C167—41no—
156C22×C167—41no—
157C22×C167—84no—
158C22×C167C22×D411yescharacteristic
159C2×D167—41no—
160C2×D167—41no—
161C2×D167—41no—
162C4×C167—21no—
163C4×C167—21no—
164C4×C167—21no—
165C4×C167—21no—
166C4×C167—21no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167C2×#4111yescharacteristic
170C4×C167C2×#4111yescharacteristic
171Dic167—41no—
172Dic167—41no—
173Dic167—41no—
174C2×C167—41no—
175C2×C167—42no—
176C2×C167—42no—
177C2×C167#52511yescharacteristic
178C2×C167#57611yescharacteristic
179C2×C167#57711yescharacteristic
180D167—81no—
181C167#311811yescharacteristic, complemented, Sylow
182#3118—1671nomaximal, Sylow
183#542—1671no—
184#518D16711yescharacteristic, complemented
185#485—1671no—
186#482—1671no—
187#476—1671no—
188#474—1671no—
189#463—1671no—
190#463—1671no—
191#450—1671no—
192#447—1671no—
193#442—1671no—
194#440—1671no—
195C4×SD16—1671no—
196C4×D8—1671no—
197#433—1671no—
198C22×D4C2×D16711yescharacteristic
199C2×SD16—1671no—
200C2×SD16—1671no—
201C2×D8—1671no—
202C2×D8—1671no—
203C4⋊Q8—1671no—
204#126—3341no—
205#124—3341no—
206#124—1671no—
207#123—3341no—
208C22⋊Q8—3341no—
209C22⋊Q8—1671no—
210#121—1671no—
211C4×Q8—1671no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1671no—
215C4×D4C2×D16711yescharacteristic
216C4×D4C2×D16711yescharacteristic
217C4×D4C2×D16711yescharacteristic
218C4×D4C2×D16711yescharacteristic
219#117C2×D16711yescharacteristic
220#117C2×D16711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1671no—
224#107—1671no—
225#106—1671no—
226#106—1671no—
227C4⋊C8—1671no—
228C4⋊C8—1671no—
229#103—1671no—
230#103—1671no—
231#102—1671no—
232#102—1671no—
233#102—1671no—
234#102—1671no—
235#102—1671no—
236#102—1671no—
237C22⋊C8—1671no—
238C22⋊C8—1671no—
239#97—1671no—
240C4×C8—1671no—
241C24—21no—
242C2×Q8—3341no—
243C2×Q8—1671no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3341no—
248C2×D4—1671no—
249C2×D4C22×D16711yescharacteristic
250C2×D4C22×D16711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3341no—
256C22×C4C22×D16711yescharacteristic
257SD16—3342no—
258SD16—3342no—
259D8—3342no—
260D8—3342no—
261C2×C8—1671no—
262C2×C8—1671no—
263C2×C8—1671no—
264C2×C8—1671no—
265C4⋊C4—3341no—
266C4⋊C4—3341no—
267C4⋊C4—3341no—
268C4⋊C4—1671no—
269C4⋊C4—1671no—
270C4⋊C4C22⋊D16711yescharacteristic
271C4⋊C4C22⋊D16711yescharacteristic
272C4⋊C4C22×D16711yescharacteristic
273C4⋊C4C22×D16711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C22⋊C4C22⋊D16711yescharacteristic
285C22⋊C4C22⋊D16711yescharacteristic
286C42—21no—
287C42—3341no—
288C42C22×D16711yescharacteristic
289C42C22×D16711yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3341no—
296C23C2×C22⋊D16711yescharacteristic
297Q8—13361no—
298Q8—3341no—
299Q8—3341no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—13361no—
304D4—3341no—
305D4—3341no—
306D4#1458195012yes—
307D4#1458195012yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—13361no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4C2×C22⋊D16711yescharacteristic
321C2×C4C2×C22⋊D16711yescharacteristic
322C2×C4C2×C22⋊D16711yescharacteristic
323C2×C4C2×C22⋊D16711yescharacteristic
324C2×C4C2×C22⋊D16711yescharacteristic
325C2×C4C23×D16711yescharacteristic
326C8—3341no—
327C8—3341no—
328C8—3341no—
329C8—3341no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—6681no—
341C22—6681no—
342C22—6681no—
343C22C22×C22⋊D16711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—6681no—
352C4—6681no—
353C4—6681no—
354C4C2×#1458195011yescharacteristic
355C4C2×#1458195011yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—13361no—
360C2#2492378011yescharacteristic
361C2#2492407211yescharacteristic
362C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#518×C167C2
socleC22×C167C22×D4
prime radical#24923403C2
hypercenter#518D167
hypocenterC167#3118
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #24923890C2semidirect
2. #24923610C2semidirect
3. #24923593C2semidirect
4. #24923559C2semidirect
5. #24923549C2semidirect
6. #24923484C2semidirect
7. #24923484C2semidirect
8. #24923420C2semidirect
10. #24923379C2semidirect
11. #24923369C2semidirect
12. C4×#14581938C2semidirect
13. C4×#14581935C2semidirect
14. #24923330C2semidirect
15. #518×C167C2semidirect
20. #19128832C22semidirect
22. #19128820C22semidirect
28. C4×Dic334C22semidirect
29. C4×D668C22semidirect
30. #19128765C22semidirect
31. #19128765C22semidirect
32. #19128763C22semidirect
33. #19128763C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. C4⋊(C167⋊C8)C22semidirect
36. #19128755C22semidirect
37. #19128755C22semidirect
42. #19128752C22semidirect
43. #19128752C22semidirect
46. #19128743C22semidirect
47. C4×C167⋊C8C22semidirect
181. C167#3118semidirect
184. #518D167semidirect

Maximal subgroups

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

subgroupindex
#249238902
#249236102
#249235932
#249235592
#249235492
#249234842
#249234202
#249234032
#249233792
#249233692
C4×#145819382
C4×#145819352
#249233302
#518×C1672
#3118167