Finite Group Databasebeta

← #36336213

Subgroups — #36336213

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#24923915C211yesmaximal, characteristic
2#24923631C211yesmaximal, characteristic, complemented
3#24923604C211yesmaximal, characteristic, complemented
4#24923582C211yesmaximal, characteristic, complemented
5#24923546C211yesmaximal, characteristic, complemented
6#24923517C211yesmaximal, characteristic, complemented
7#24923508C211yesmaximal, characteristic, complemented
8#24923491C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923419C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923268C211yesmaximal, characteristic, complemented
14C2×#19128752C211yesmaximal, characteristic, complemented
15#554×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×#14581940C2211yescharacteristic, complemented
19C2×#14581937C2211yescharacteristic
20C2×#14581937C2211yescharacteristic
21C2×#14581934C2211yescharacteristic, complemented
22C22×C167⋊C8C2211yescharacteristic
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#19128765C2211yescharacteristic, complemented
33#19128765C2211yescharacteristic, 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#125×C167—21no—
50C22⋊Q8×C167—21no—
51C22⋊Q8×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53C22⋊Q8×C167C2211yescharacteristic, complemented
54C4×Q8×C167C2211yescharacteristic, complemented
55C4×D4×C167—21no—
56C4×D4×C167C2211yescharacteristic
57C2×C4⋊C4×C167C2211yescharacteristic
58C2×C4⋊C4×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×C167—21no—
92C22×C4×C167—21no—
93C22×C4×C167C2311yescharacteristic
94C22×C4×C167C2311yescharacteristic
95C4⋊C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167—21no—
98C4⋊C4×C167—21no—
99C4⋊C4×C167—21no—
100C4⋊C4×C167—21no—
101C4⋊C4×C167D411yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C4⋊C4×C167C2311yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167D411yescharacteristic
108C22⋊C4×C167D411yescharacteristic
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—81no—
138D4×C167—22no—
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×C167—81no—
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—21no—
158C22×C167—21no—
159C22×C167#13712yes—
160C22×C167C22×D411yescharacteristic
161C2×D167—22no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—41no—
166C2×D167—41no—
167C2×D167—84no—
168C4×C167—21no—
169C4×C167—21no—
170C4×C167—21no—
171C4×C167—21no—
172C4×C167—41no—
173C4×C167—41no—
174C4×C167—41no—
175C4×C167—41no—
176C4×C167C2×#4111yescharacteristic
177C4×C167H(5,𝔽2)11yescharacteristic
178Dic167—41no—
179Dic167—41no—
180C2×C167—22no—
181C2×C167—41no—
182C2×C167#54611yescharacteristic
183C2×C167C2×#13711yescharacteristic
184C2×C167#57611yescharacteristic
185D167—42no—
186D167—81no—
187C167#331411yescharacteristic, complemented, Sylow
188#3314—1671nomaximal, Sylow
189#554D16711yescharacteristic, complemented
190#546—1671no—
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×#41—1671no—
205C22×D4—1671no—
206C2×Q16—1671no—
207C2×SD16—1671no—
208C2×SD16—1671no—
209C2×M16—1671no—
210C22×C8—1671no—
211C4⋊Q8C2×D16711yescharacteristic, complemented
212#125—21no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—1671no—
218C22⋊Q8C2×D16711yescharacteristic
219C22⋊Q8C2×D16711yescharacteristic, complemented
220#121—3341no—
221#121—1671no—
222#120—3341no—
223C4×Q8C2×D16711yescharacteristic, complemented
224C4×D4—21no—
225C4×D4—1671no—
226C4×D4C2×D16711yescharacteristic
227C2×C4⋊C4C2×D16711yescharacteristic
228C2×C4⋊C4C2×D16711yescharacteristic, complemented
229C2×C22⋊C4—3341no—
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—3341no—
248#41—3342no—
249#41—6681no—
250C2×Q8—21no—
251C2×Q8—1671no—
252C2×Q8C22×D16711yescharacteristic
253C2×D4—3342no—
254C2×D4—3341no—
255C2×D4—3341no—
256C2×D4—3341no—
257C2×D4—6681no—
258C2×D4—1671no—
259C2×D4—1671no—
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—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—3341no—
283C4⋊C4—1671no—
284C4⋊C4C22⋊D16711yescharacteristic
285C4⋊C4C22×D16711yescharacteristic
286C4⋊C4C22×D16711yescharacteristic
287C4⋊C4C22×D16711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—3341no—
291C22⋊C4—3341no—
292C22⋊C4—3341no—
293C22⋊C4—3341no—
294C22⋊C4—3341no—
295C22⋊C4C22⋊D16711yescharacteristic
296C22⋊C4C22⋊D16711yescharacteristic
297C42—21no—
298C42C22×D16711yescharacteristic
299C23—13361no—
300C23—3341no—
301C23—3341no—
302C23—6682no—
303C23—6682no—
304C23C2×C22⋊D16711yescharacteristic
305C23C2×C22⋊D16711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—81no—
309Q8—3341no—
310Q8—3341no—
311D4—22no—
312D4—13361no—
313D4—3341no—
314D4—3341no—
315D4—3341no—
316D4—6682no—
317D4—6682no—
318D4—1672no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—13361no—
330C2×C4—3341no—
331C2×C4—3341no—
332C2×C4—6682no—
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—21no—
346C22—21no—
347C22—13364no—
348C22—3342no—
349C22—6681no—
350C22—6681no—
351C22—6681no—
352C22—6681no—
353C22—6681no—
354C22#1912886312yes—
355C22C22×C22⋊D16711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—6681no—
365C4—6681no—
366C4#41×D16711yescharacteristic
367C4#1912887811yescharacteristic
368C2—22no—
369C2—41no—
370C2—13361no—
371C2—6682no—
372C2#2492390711yescharacteristic
373C2C2×#1912886311yescharacteristic
374C2#2492407211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#554×C167C2
socleC22×C167C22×D4
prime radical#24923915C2
hypercenter#554D167
hypocenterC167#3314
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #24923631C2semidirect
3. #24923604C2semidirect
4. #24923582C2semidirect
5. #24923546C2semidirect
6. #24923517C2semidirect
7. #24923508C2semidirect
8. #24923491C2semidirect
9. #24923429C2semidirect
10. #24923419C2semidirect
11. #24923367C2semidirect
12. #24923341C2semidirect
13. #24923268C2semidirect
14. C2×#19128752C2semidirect
15. #554×C167C2semidirect
18. C2×#14581940C22semidirect
21. C2×#14581934C22semidirect
31. #19128765C22semidirect
32. #19128765C22semidirect
33. #19128765C22semidirect
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
54. C4×Q8×C167C22semidirect
58. C2×C4⋊C4×C167C22semidirect
187. C167#3314semidirect
189. #554D167semidirect
211. C4⋊Q8C2×D167semidirect
219. C22⋊Q8C2×D167semidirect
223. C4×Q8C2×D167semidirect
228. C2×C4⋊C4C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239152
#249236312
#249236042
#249235822
#249235462
#249235172
#249235082
#249234912
#249234292
#249234192
#249233672
#249233412
#249232682
C2×#191287522
#554×C1672
#3314167