Finite Group Databasebeta

← #36336169

Subgroups — #36336169

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#24923990C211yesmaximal, characteristic, complemented
2#24923910C211yesmaximal, characteristic
3#24923630C211yesmaximal, characteristic, complemented
4#24923607C211yesmaximal, characteristic, complemented
5#24923595C211yesmaximal, characteristic, complemented
6#24923558C211yesmaximal, characteristic, complemented
7#24923501C211yesmaximal, characteristic, complemented
8#24923493C211yesmaximal, characteristic, complemented
9#24923430C211yesmaximal, characteristic, complemented
10#24923419C211yesmaximal, characteristic, complemented
11#24923372C211yesmaximal, characteristic, complemented
12#24923338C211yesmaximal, characteristic, complemented
13#24923275C211yesmaximal, characteristic, complemented
14#24923264C211yesmaximal, characteristic, complemented
15#470×C167C211yesmaximal, characteristic, complemented
16C2×#14581951C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×#14581937C2211yescharacteristic
20C2×C8×D167C2211yescharacteristic, complemented
21#19128831C2211yescharacteristic, complemented
22#19128827—21no—
23#19128825—21no—
24#19128821C2211yescharacteristic
25#19128818—21no—
26#19128817—21no—
27#19128808C2211yescharacteristic, complemented
28#19128807—21no—
29#19128807C2211yescharacteristic
30#19128802—21no—
31Q8×Dic167C2211yescharacteristic, complemented
32#19128797—21no—
33#19128794C2211yescharacteristic
34C4×C22⋊D167C2211yescharacteristic
35#19128791C2211yescharacteristic
36#19128790C2211yescharacteristic, complemented
37C22⋊C4×D167—21no—
38#19128765C2211yescharacteristic, complemented
39#19128763C2211yescharacteristic, complemented
40Dic167⋊C8C2211yescharacteristic, complemented
41#19128756C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128754C2211yescharacteristic
44#19128754C2211yescharacteristic
45#19128754C2211yescharacteristic
46#19128754C2211yescharacteristic
47#19128752C2211yescharacteristic
48#19128752C2211yescharacteristic
49C22⋊(C167⋊C8)C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51#19128742C2211yescharacteristic, complemented
52C2×Q16×C167C2211yescharacteristic, complemented
53C2×M16×C167C2211yescharacteristic
54C22⋊Q8×C167C2211yescharacteristic
55C22⋊Q8×C167C2211yescharacteristic
56#106×C167C2211yescharacteristic, complemented
57#103×C167C2211yescharacteristic, complemented
58#103×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60#14581951—22no—
61#14581951—41no—
62C2×Dic334—21no—
63C2×D668—22no—
64C2×D668—21no—
65C2×D668—21no—
66C2×D668—21no—
67C2×D668—41no—
68C2×D668C2311yescharacteristic
69C2×D668C2311yescharacteristic
70C2×C22⋊D167D411yescharacteristic
71C2×C4×D167—21no—
72C2×C4×D167—21no—
73C2×C4×D167C2311yescharacteristic
74#14581937—22no—
75#14581937—41no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C8×D167—41no—
79C4⋊Dic167—21no—
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4D411yescharacteristic
85C22⋊Dic167—21no—
86C22⋊Dic167D411yescharacteristic
87(C2×D167)⋊C4—21no—
88(C2×D167)⋊C4—21no—
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4D411yescharacteristic
94C4×Dic167—21no—
95C4×Dic167—21no—
96C4×Dic167—21no—
97C4×Dic167C2311yescharacteristic
98C2×Q8×C167C2311yescharacteristic
99C2×Q8×C167C2311yescharacteristic
100C22×C4×C167C2311yescharacteristic
101Q16×C167—41no—
102M16×C167—41no—
103C2×C8×C167C2311yescharacteristic
104C2×C8×C167C2311yescharacteristic
105C4⋊C4×C167—21no—
106C4⋊C4×C167—21no—
107C4⋊C4×C167C2311yescharacteristic
108C4⋊C4×C167C2311yescharacteristic
109C22⋊C4×C167—21no—
110C22⋊C4×C167—21no—
111C22×D167—21no—
112C22×D167—21no—
113C22×D167—42no—
114C22×D167—42no—
115C22×D167—81no—
116C22×D167C2×D411yescharacteristic
117Dic334—42no—
118D668—21no—
119D668—21no—
120D668—21no—
121D668—42no—
122D668—42no—
123D668—81no—
124D668#4112yes—
125C22⋊D167—41no—
126C2×Dic167—21no—
127C2×Dic167—21no—
128C2×Dic167C2×D411yescharacteristic
129C2×Dic167C2×D411yescharacteristic
130C2×Dic167C2×D411yescharacteristic
131C4×D167—21no—
132C4×D167—21no—
133C4×D167—42no—
134C4×D167—81no—
135C167⋊C8—21no—
136C167⋊C8—21no—
137C23×C167C2×D411yescharacteristic
138Q8×C167—21no—
139Q8×C167—21no—
140Q8×C167—21no—
141Q8×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—21no—
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2411yescharacteristic
150C8×C167—21no—
151C8×C167—21no—
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—21no—
155C22×C167C22×D411yescharacteristic
156C2×D167—21no—
157C2×D167—22no—
158C2×D167—21no—
159C2×D167—21no—
160C2×D167—41no—
161C2×D167—41no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—84no—
166C4×C167—21no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167—41no—
170C4×C167—41no—
171C4×C167C2×#4111yescharacteristic
172C4×C167H(5,𝔽2)11yescharacteristic
173Dic167—21no—
174Dic167—21no—
175Dic167—21no—
176Dic167—41no—
177Dic167—41no—
178C2×C167—41no—
179C2×C167#54611yescharacteristic
180C2×C167#57511yescharacteristic
181C2×C167#57711yescharacteristic
182D167—41no—
183D167—42no—
184D167—81no—
185C167#331111yescharacteristic, complemented, Sylow
186#3311—1671nomaximal, Sylow
187#556—1671no—
188#546—1671no—
189#489—1671no—
190#485—1671no—
191#482—1671no—
192#476—1671no—
193#470D16711yescharacteristic, complemented
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×Q16C2×D16711yescharacteristic, complemented
205C2×SD16—1671no—
206C2×SD16—1671no—
207C2×M16C2×D16711yescharacteristic
208C22×C8—1671no—
209C4⋊Q8—1671no—
210#126—3341no—
211#126—3341no—
212#124—3341no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—1671no—
216C22⋊Q8C2×D16711yescharacteristic
217C22⋊Q8C2×D16711yescharacteristic
218#121—3341no—
219#121—1671no—
220#120—3341no—
221C4×Q8—3341no—
222C4×Q8—1671no—
223C4×D4—1671no—
224C4×D4—1671no—
225#117—1671no—
226#117—1671no—
227C2×C22⋊C4—3341no—
228#107—1671no—
229#106—1671no—
230#106C2×D16711yescharacteristic, complemented
231C4⋊C8—1671no—
232#103—1671no—
233#103—1671no—
234#103C2×D16711yescharacteristic, complemented
235#103C2×D16711yescharacteristic, complemented
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—3341no—
249C2×Q8C22×D16711yescharacteristic
250C2×Q8C22×D16711yescharacteristic
251C2×D4—3341no—
252C2×D4—3342no—
253C2×D4—3341no—
254C2×D4—3341no—
255C2×D4—6681no—
256C2×D4—1671no—
257C2×D4—1671no—
258C2×D4—1671no—
259C22×C4—3341no—
260C22×C4—3341no—
261C22×C4—1671no—
262C22×C4C22×D16711yescharacteristic
263Q16—41no—
264SD16—3342no—
265SD16—6681no—
266M16—41no—
267C2×C8—6681no—
268C2×C8—1671no—
269C2×C8—1671no—
270C2×C8C22×D16711yescharacteristic
271C2×C8C22×D16711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—3341no—
277C4⋊C4—1671no—
278C4⋊C4—1671no—
279C4⋊C4—1671no—
280C4⋊C4C22×D16711yescharacteristic
281C4⋊C4C22×D16711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—1671no—
292C22⋊C4—1671no—
293C42—3341no—
294C42—3341no—
295C42—3341no—
296C42—1671no—
297C23—13361no—
298C23—3341no—
299C23—3341no—
300C23—6682no—
301C23—6682no—
302C23—1671no—
303C23D4×D16711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—6682no—
309D4—13361no—
310D4—3341no—
311D4—3341no—
312D4—3341no—
313D4—6682no—
314D4—6682no—
315D4—6681no—
316D4—1672no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—13361no—
324C2×C4—3341no—
325C2×C4—3341no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—6682no—
329C2×C4—1671no—
330C2×C4—1671no—
331C2×C4—1671no—
332C2×C4D4×D16711yescharacteristic
333C2×C4C23×D16711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3341no—
337C8—3341no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—13364no—
342C22—3341no—
343C22—3342no—
344C22—3341no—
345C22—3341no—
346C22—6681no—
347C22—6681no—
348C22—6681no—
349C22—6681no—
350C22—6681no—
351C22C2×D4×D16711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3341no—
358C4—3341no—
359C4—3341no—
360C4—6681no—
361C4—6681no—
362C4#41×D16711yescharacteristic
363C4#1912887711yescharacteristic
364C2—41no—
365C2—13361no—
366C2—6682no—
367C2—6681no—
368C2#2492390811yescharacteristic
369C2#2492406811yescharacteristic
370C2#2492407811yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#470×C167C2
socleC22×C167C22×D4
prime radical#24923910C2
hypercenter#470D167
hypocenterC167#3311
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #24923990C2semidirect
3. #24923630C2semidirect
4. #24923607C2semidirect
5. #24923595C2semidirect
6. #24923558C2semidirect
7. #24923501C2semidirect
8. #24923493C2semidirect
9. #24923430C2semidirect
10. #24923419C2semidirect
11. #24923372C2semidirect
12. #24923338C2semidirect
13. #24923275C2semidirect
14. #24923264C2semidirect
15. #470×C167C2semidirect
20. C2×C8×D167C22semidirect
21. #19128831C22semidirect
27. #19128808C22semidirect
31. Q8×Dic167C22semidirect
36. #19128790C22semidirect
38. #19128765C22semidirect
39. #19128763C22semidirect
40. Dic167⋊C8C22semidirect
41. #19128756C22semidirect
42. #19128756C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. #19128742C22semidirect
52. C2×Q16×C167C22semidirect
56. #106×C167C22semidirect
57. #103×C167C22semidirect
58. #103×C167C22semidirect
185. C167#3311semidirect
193. #470D167semidirect
204. C2×Q16C2×D167semidirect
230. #106C2×D167semidirect
234. #103C2×D167semidirect
235. #103C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239902
#249239102
#249236302
#249236072
#249235952
#249235582
#249235012
#249234932
#249234302
#249234192
#249233722
#249233382
#249232752
#249232642
#470×C1672
#3311167