Finite Group Databasebeta

← #36336168

Subgroups — #36336168

Order 21376.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923915C211yesmaximal, characteristic
2#24923631C211yesmaximal, characteristic, complemented
3#24923610C211yesmaximal, characteristic, complemented
4#24923593C211yesmaximal, characteristic, complemented
5#24923556C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923500C211yesmaximal, characteristic, complemented
8#24923491C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923419C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923273C211yesmaximal, characteristic, complemented
14#24923263C211yesmaximal, characteristic, complemented
15#556×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×#14581940C2211yescharacteristic, complemented
19C2×#14581937C2211yescharacteristic
20C2×#14581937C2211yescharacteristic
21C2×#14581934C2211yescharacteristic
22C22×C167⋊C8C2211yescharacteristic, complemented
23#19128820C2211yescharacteristic
24#19128817—21no—
25#19128811C2211yescharacteristic
26#19128806C2211yescharacteristic
27#19128805—21no—
28#19128800—21no—
29C4×D668C2211yescharacteristic
30C22⋊C4×D167—21no—
31#19128765C2211yescharacteristic, complemented
32#19128763C2211yescharacteristic, complemented
33#19128763C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35#19128756C2211yescharacteristic, complemented
36#19128755C2211yescharacteristic, complemented
37#19128755C2211yescharacteristic, complemented
38#19128755C2211yescharacteristic, complemented
39#19128752C2211yescharacteristic
40#19128752C2211yescharacteristic
41#19128752C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44#19128752C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic, complemented
47#19128743C2211yescharacteristic, complemented
48C4⋊Q8×C167C2211yescharacteristic, complemented
49#126×C167—21no—
50#126×C167—21no—
51#124×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53C22⋊Q8×C167C2211yescharacteristic, complemented
54C4×Q8×C167—21no—
55C4×Q8×C167C2211yescharacteristic, complemented
56C4×D4×C167C2211yescharacteristic
57#117×C167C2211yescharacteristic
58#117×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60#14581950—22no—
61#14581950—41no—
62C2×Dic334C2311yescharacteristic
63C2×D668—22no—
64C2×D668—21no—
65C2×D668—41no—
66C2×D668C2311yescharacteristic
67C2×D668C2311yescharacteristic
68C2×C22⋊D167—21no—
69C2×C22⋊D167—21no—
70C2×C4×D167—21no—
71C2×C4×D167—21no—
72#14581940—41no—
73#14581937—22no—
74#14581937—41no—
75#14581934—41no—
76C2×C167⋊C8—41no—
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C2×C167⋊C8C2311yescharacteristic
80C2×C167⋊C8C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82Dic167⋊C4—21no—
83C22⋊Dic167—21no—
84(C2×D167)⋊C4—21no—
85(C2×D167)⋊C4—21no—
86(C2×D167)⋊C4—21no—
87(C2×D167)⋊C4—21no—
88C2×Q8×C167—21no—
89C2×Q8×C167C2311yescharacteristic
90C2×D4×C167D411yescharacteristic
91C22×C4×C167C2311yescharacteristic
92C22×C4×C167C2311yescharacteristic
93C4⋊C4×C167—21no—
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167D411yescharacteristic
98C4⋊C4×C167C2311yescharacteristic
99C4⋊C4×C167C2311yescharacteristic
100C4⋊C4×C167C2311yescharacteristic
101C22⋊C4×C167—21no—
102C22⋊C4×C167—21no—
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167D411yescharacteristic
106C22⋊C4×C167D411yescharacteristic
107C42×C167—21no—
108C42×C167—21no—
109C42×C167—21no—
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112C22×D167—21no—
113C22×D167—42no—
114C22×D167—42no—
115C22×D167—81no—
116Dic334—21no—
117Dic334—21no—
118D668—21no—
119D668—21no—
120D668—21no—
121D668—42no—
122D668#4112yes—
123C22⋊D167—42no—
124C22⋊D167—81no—
125C2×Dic167—21no—
126C2×Dic167—21no—
127C4×D167—42no—
128C4×D167—81no—
129C167⋊C8—21no—
130C167⋊C8—21no—
131C167⋊C8—21no—
132C167⋊C8—21no—
133C23×C167C2×D411yescharacteristic
134C23×C167C2×D411yescharacteristic
135Q8×C167—21no—
136Q8×C167—21no—
137Q8×C167—42no—
138D4×C167—41no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
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×C167C2×D411yescharacteristic
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2411yescharacteristic
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—21no—
155C22×C167—21no—
156C22×C167—21no—
157C22×C167—21no—
158C22×C167C22×D411yescharacteristic
159C2×D167—22no—
160C2×D167—41no—
161C2×D167—41no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—84no—
166C4×C167—21no—
167C4×C167—21no—
168C4×C167—21no—
169C4×C167—21no—
170C4×C167—41no—
171C4×C167—41no—
172C4×C167—41no—
173C4×C167—41no—
174C4×C167C2×#4111yescharacteristic
175C4×C167H(5,𝔽2)11yescharacteristic
176Dic167—41no—
177Dic167—41no—
178C2×C167—41no—
179C2×C167—41no—
180C2×C167#54611yescharacteristic
181C2×C167#57511yescharacteristic
182C2×C167#57711yescharacteristic
183D167—42no—
184D167—81no—
185C167#331111yescharacteristic, complemented, Sylow
186#3311—1671nomaximal, Sylow
187#556D16711yescharacteristic, complemented
188#546—1671no—
189#489—1671no—
190#485—1671no—
191#482—1671no—
192#476—1671no—
193#470—1671no—
194#465—1671no—
195#464—1671no—
196#452—1671no—
197#450—1671no—
198#440—1671no—
199#435—1671no—
200#418—1671no—
201#416—1671no—
202C2×#41—1671no—
203C22×D4—1671no—
204C2×Q16—1671no—
205C2×SD16—1671no—
206C2×SD16—1671no—
207C2×M16—1671no—
208C22×C8—1671no—
209C4⋊Q8C2×D16711yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—1671no—
216C22⋊Q8C2×D16711yescharacteristic
217C22⋊Q8C2×D16711yescharacteristic, complemented
218#121—3341no—
219#121—1671no—
220#120—3341no—
221C4×Q8—21no—
222C4×Q8C2×D16711yescharacteristic, complemented
223C4×D4—1671no—
224C4×D4C2×D16711yescharacteristic
225#117C2×D16711yescharacteristic
226#117C2×D16711yescharacteristic, complemented
227C2×C22⋊C4—3341no—
228#107—1671no—
229#106—1671no—
230#106—1671no—
231C4⋊C8—1671no—
232#103—1671no—
233#103—1671no—
234#103—1671no—
235#103—1671no—
236#102—1671no—
237#102—1671no—
238#102—1671no—
239#102—1671no—
240#102—1671no—
241#102—1671no—
242C22⋊C8—1671no—
243C22⋊C8—1671no—
244#97—1671no—
245C24—3341no—
246#41—3342no—
247#41—6681no—
248C2×Q8—21no—
249C2×Q8—1671no—
250C2×Q8C22×D16711yescharacteristic
251C2×D4—3342no—
252C2×D4—3341no—
253C2×D4—3341no—
254C2×D4—3341no—
255C2×D4—6681no—
256C2×D4—1671no—
257C2×D4—1671no—
258C2×D4C22⋊D16711yescharacteristic
259C22×C4—3341no—
260C22×C4—3341no—
261C22×C4C22×D16711yescharacteristic
262C22×C4C22×D16711yescharacteristic
263Q16—6681no—
264SD16—3342no—
265SD16—6681no—
266M16—6681no—
267C2×C8—6681no—
268C2×C8—1671no—
269C2×C8—1671no—
270C2×C8—1671no—
271C2×C8—1671no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—3341no—
277C4⋊C4—1671no—
278C4⋊C4C22⋊D16711yescharacteristic
279C4⋊C4C22×D16711yescharacteristic
280C4⋊C4C22×D16711yescharacteristic
281C4⋊C4C22×D16711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—3341no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4C22⋊D16711yescharacteristic
292C22⋊C4C22⋊D16711yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D16711yescharacteristic
297C23—13361no—
298C23—3341no—
299C23—3341no—
300C23—6682no—
301C23—6682no—
302C23C2×C22⋊D16711yescharacteristic
303C23C2×C22⋊D16711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—3341no—
308Q8—3341no—
309D4—41no—
310D4—13361no—
311D4—3341no—
312D4—3341no—
313D4—3341no—
314D4—6682no—
315D4—6682no—
316D4—1672no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—13361no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—6682no—
329C2×C4C2×C22⋊D16711yescharacteristic
330C2×C4C2×C22⋊D16711yescharacteristic
331C2×C4C2×C22⋊D16711yescharacteristic
332C2×C4C2×C22⋊D16711yescharacteristic
333C2×C4C23×D16711yescharacteristic
334C8—3341no—
335C8—3341no—
336C8—3341no—
337C8—3341no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—13364no—
345C22—3342no—
346C22—6681no—
347C22—6681no—
348C22—6681no—
349C22—6681no—
350C22—6681no—
351C22C22×C22⋊D16711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6681no—
361C4—6681no—
362C4#41×D16711yescharacteristic
363C4#1912887811yescharacteristic
364C2—41no—
365C2—41no—
366C2—13361no—
367C2—6682no—
368C2#2492390711yescharacteristic
369C2#2492406711yescharacteristic
370C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#556×C167C2
socleC22×C167C22×D4
prime radical#24923915C2
hypercenter#556D167
hypocenterC167#3311
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #24923631C2semidirect
3. #24923610C2semidirect
4. #24923593C2semidirect
5. #24923556C2semidirect
6. #24923525C2semidirect
7. #24923500C2semidirect
8. #24923491C2semidirect
9. #24923429C2semidirect
10. #24923419C2semidirect
11. #24923367C2semidirect
12. #24923341C2semidirect
13. #24923273C2semidirect
14. #24923263C2semidirect
15. #556×C167C2semidirect
18. C2×#14581940C22semidirect
22. C22×C167⋊C8C22semidirect
31. #19128765C22semidirect
32. #19128763C22semidirect
33. #19128763C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. #19128756C22semidirect
36. #19128755C22semidirect
37. #19128755C22semidirect
38. #19128755C22semidirect
46. C22⋊(C167⋊C8)C22semidirect
47. #19128743C22semidirect
48. C4⋊Q8×C167C22semidirect
53. C22⋊Q8×C167C22semidirect
55. C4×Q8×C167C22semidirect
58. #117×C167C22semidirect
185. C167#3311semidirect
187. #556D167semidirect
209. C4⋊Q8C2×D167semidirect
217. C22⋊Q8C2×D167semidirect
222. C4×Q8C2×D167semidirect
226. #117C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239152
#249236312
#249236102
#249235932
#249235562
#249235252
#249235002
#249234912
#249234292
#249234192
#249233672
#249233412
#249232732
#249232632
#556×C1672
#3311167