Finite Group Databasebeta

← #36336219

Subgroups — #36336219

Order 21376.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923976C211yesmaximal, characteristic, complemented
2#24923628C211yesmaximal, characteristic, complemented
3#24923602C211yesmaximal, characteristic, complemented
4#24923580C211yesmaximal, characteristic, complemented
5#24923549C211yesmaximal, characteristic, complemented
6#24923513C211yesmaximal, characteristic, complemented
7#24923508C211yesmaximal, characteristic, complemented
8#24923488C211yesmaximal, characteristic, complemented
9#24923431C211yesmaximal, characteristic, complemented
10#24923420C211yesmaximal, characteristic, complemented
11#24923369C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923270C211yesmaximal, characteristic, complemented
14C2×#19128753C211yesmaximal, characteristic, complemented
15#546×C167C211yesmaximal, characteristic
16C2×#14581940C2211yescharacteristic, complemented
17C2×#14581938C2211yescharacteristic
18C2×#14581938C2211yescharacteristic
19C2×#14581934C2211yescharacteristic, complemented
20C22×C167⋊C8C2211yescharacteristic
21#19128832C2211yescharacteristic, complemented
22#19128823—21no—
23#19128811C2211yescharacteristic
24#19128811C2211yescharacteristic, complemented
25#19128809—21no—
26#19128809—21no—
27C4×Dic334C2211yescharacteristic, complemented
28D4×Dic167—21no—
29C2×C4⋊Dic167C2211yescharacteristic
30C2×C4⋊Dic167C2211yescharacteristic, complemented
31#19128765C2211yescharacteristic, complemented
32#19128765C2211yescharacteristic, complemented
33#19128765C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35#19128756C2211yescharacteristic, complemented
36#19128756C2211yescharacteristic, complemented
37#19128756C2211yescharacteristic, complemented
38#19128755C2211yescharacteristic, complemented
39#19128753C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128753C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic, complemented
47#19128743C2211yescharacteristic, complemented
48C2×#41×C167C2211yescharacteristic
49C22×D4×C167C2211yescharacteristic
50#124×C167C2211yescharacteristic
51#123×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53#121×C167—21no—
54#121×C167C2211yescharacteristic
55#120×C167—21no—
56C4×D4×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58C2×C22⋊C4×C167—21no—
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61C22×Dic167—21no—
62C22×Dic167—21no—
63#14581940—41no—
64#14581938—22no—
65#14581938—41no—
66#14581934—41no—
67C2×C167⋊C8—41no—
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C4⋊Dic167—21no—
73C4⋊Dic167—21no—
74C4⋊Dic167—21no—
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77C4⋊Dic167C2311yescharacteristic
78Dic167⋊C4—21no—
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81C22⋊Dic167—21no—
82C22⋊Dic167—21no—
83C4×Dic167—21no—
84C24×C167—21no—
85#41×C167—22no—
86#41×C167—41no—
87C2×Q8×C167C2311yescharacteristic
88C2×D4×C167—22no—
89C2×D4×C167—21no—
90C2×D4×C167—21no—
91C2×D4×C167—21no—
92C2×D4×C167—41no—
93C2×D4×C167D411yescharacteristic
94C2×D4×C167C2311yescharacteristic
95C2×D4×C167C2311yescharacteristic
96C22×C4×C167—21no—
97C22×C4×C167—21no—
98C22×C4×C167C2311yescharacteristic
99C22×C4×C167C2311yescharacteristic
100C4⋊C4×C167—21no—
101C4⋊C4×C167D411yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C22⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C42×C167C2311yescharacteristic
111Dic334—21no—
112Dic334—21no—
113Dic334—81no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—81no—
119C2×Dic167—81no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C167⋊C8—21no—
123C167⋊C8—21no—
124C23×C167—21no—
125C23×C167—21no—
126C23×C167—42no—
127C23×C167—42no—
128C23×C167—81no—
129C23×C167C2×D411yescharacteristic
130C23×C167C2×D411yescharacteristic
131Q8×C167—21no—
132Q8×C167—21no—
133D4×C167—22no—
134D4×C167—21no—
135D4×C167—21no—
136D4×C167—21no—
137D4×C167—42no—
138D4×C167—42no—
139D4×C167—81no—
140D4×C167#4112yes—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—42no—
148C2×C4×C167—81no—
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2×D411yescharacteristic
152C2×C4×C167C2×D411yescharacteristic
153C2×C4×C167C2411yescharacteristic
154C22×C167—21no—
155C22×C167—21no—
156C22×C167—21no—
157C22×C167—22no—
158C22×C167—21no—
159C22×C167—21no—
160C22×C167—41no—
161C22×C167—41no—
162C22×C167—41no—
163C22×C167—41no—
164C22×C167—41no—
165C22×C167—84no—
166C22×C167#13712yes—
167C22×C167C22×D411yescharacteristic
168C4×C167—21no—
169C4×C167—21no—
170C4×C167—21no—
171C4×C167—21no—
172C4×C167—41no—
173C4×C167—41no—
174C4×C167C2×#4111yescharacteristic
175C4×C167H(5,𝔽2)11yescharacteristic
176Dic167—41no—
177Dic167—41no—
178Dic167—41no—
179Dic167—41no—
180C2×C167—22no—
181C2×C167—41no—
182C2×C167—42no—
183C2×C167—81no—
184C2×C167#54611yescharacteristic
185C2×C167C2×#13711yescharacteristic
186C2×C167#57611yescharacteristic
187C167#331411yescharacteristic, complemented, Sylow
188#3314—1671nomaximal, Sylow
189#554—1671no—
190#546D16711yescharacteristic
191#489—1671no—
192#484—1671no—
193#480—1671no—
194#474—1671no—
195#468—1671no—
196#467—1671no—
197#464—1671no—
198#452—1671no—
199#450—1671no—
200#440—1671no—
201#435—1671no—
202#417—1671no—
203C2×#102—1671no—
204C2×#41C2×D16711yescharacteristic
205C22×D4C2×D16711yescharacteristic
206C2×Q16—1671no—
207C2×SD16—1671no—
208C2×SD16—1671no—
209C2×M16—1671no—
210C22×C8—1671no—
211C4⋊Q8—1671no—
212#125—3341no—
213#124C2×D16711yescharacteristic
214#123—21no—
215C22⋊Q8—3341no—
216C22⋊Q8—3341no—
217C22⋊Q8—1671no—
218C22⋊Q8—1671no—
219C22⋊Q8C2×D16711yescharacteristic
220#121—21no—
221#121C2×D16711yescharacteristic
222#120—21no—
223C4×Q8—1671no—
224C4×D4—3341no—
225C4×D4C2×D16711yescharacteristic
226C4×D4C2×D16711yescharacteristic
227C2×C4⋊C4—1671no—
228C2×C4⋊C4—1671no—
229C2×C22⋊C4—21no—
230#107—1671no—
231#107—1671no—
232#107—1671no—
233C4⋊C8—1671no—
234#103—1671no—
235#103—1671no—
236#103—1671no—
237#103—1671no—
238#102—1671no—
239#102—1671no—
240#102—1671no—
241#102—1671no—
242#102—1671no—
243#102—1671no—
244C22⋊C8—1671no—
245C22⋊C8—1671no—
246#97—1671no—
247C24—21no—
248#41—22no—
249#41—41no—
250C2×Q8—3341no—
251C2×Q8—1671no—
252C2×Q8C22×D16711yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4C22⋊D16711yescharacteristic
259C2×D4C22×D16711yescharacteristic
260C2×D4C22×D16711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—3341no—
264C22×C4—3341no—
265C22×C4C22×D16711yescharacteristic
266C22×C4C22×D16711yescharacteristic
267Q16—6681no—
268SD16—3342no—
269SD16—6681no—
270M16—6681no—
271C2×C8—6681no—
272C2×C8—1671no—
273C2×C8—1671no—
274C2×C8—1671no—
275C2×C8—1671no—
276C4⋊C4—21no—
277C4⋊C4—3341no—
278C4⋊C4—3341no—
279C4⋊C4—3341no—
280C4⋊C4—3341no—
281C4⋊C4—3341no—
282C4⋊C4—3341no—
283C4⋊C4—1671no—
284C4⋊C4—1671no—
285C4⋊C4—1671no—
286C4⋊C4C22⋊D16711yescharacteristic
287C4⋊C4C22×D16711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3341no—
294C22⋊C4—3341no—
295C22⋊C4C22⋊D16711yescharacteristic
296C22⋊C4C22⋊D16711yescharacteristic
297C42—3341no—
298C42C22×D16711yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C22⋊D16711yescharacteristic
305C23C2×C22⋊D16711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—13361no—
309Q8—3341no—
310Q8—3341no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#1458194912yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—13361no—
328C2×C4—13361no—
329C2×C4—3341no—
330C2×C4—3341no—
331C2×C4—3341no—
332C2×C4—3341no—
333C2×C4C2×C22⋊D16711yescharacteristic
334C2×C4C2×C22⋊D16711yescharacteristic
335C2×C4C2×C22⋊D16711yescharacteristic
336C2×C4C2×C22⋊D16711yescharacteristic
337C2×C4C23×D16711yescharacteristic
338C8—3341no—
339C8—3341no—
340C8—3341no—
341C8—3341no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22#1912886212yes—
355C22C22×C22⋊D16711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—6681no—
363C4—6681no—
364C4—6681no—
365C4—6681no—
366C4C2×#1458194911yescharacteristic
367C4#1912887711yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#2492391311yescharacteristic
373C2C2×#1912886211yescharacteristic
374C2#2492407211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radical#546×C167C2
socleC22×C167C22×D4
hypercenter#546D167
hypocenterC167#3314
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #24923976C2semidirect
2. #24923628C2semidirect
3. #24923602C2semidirect
4. #24923580C2semidirect
5. #24923549C2semidirect
6. #24923513C2semidirect
7. #24923508C2semidirect
8. #24923488C2semidirect
9. #24923431C2semidirect
10. #24923420C2semidirect
11. #24923369C2semidirect
12. #24923341C2semidirect
13. #24923270C2semidirect
14. C2×#19128753C2semidirect
16. C2×#14581940C22semidirect
19. C2×#14581934C22semidirect
21. #19128832C22semidirect
24. #19128811C22semidirect
27. C4×Dic334C22semidirect
30. C2×C4⋊Dic167C22semidirect
31. #19128765C22semidirect
32. #19128765C22semidirect
33. #19128765C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. #19128756C22semidirect
36. #19128756C22semidirect
37. #19128756C22semidirect
38. #19128755C22semidirect
46. C22⋊(C167⋊C8)C22semidirect
47. #19128743C22semidirect
187. C167#3314semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249236282
#249236022
#249235802
#249235492
#249235132
#249235082
#249234882
#249234312
#249234202
#249233692
#249233412
#249232702
C2×#191287532
#546×C1672
#3314167