Finite Group Databasebeta

← #36336100

Subgroups — #36336100

Order 21376.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923920C211yesmaximal, characteristic, complemented
2#24923907C211yesmaximal, characteristic, complemented
3#24923630C211yesmaximal, characteristic, complemented
4#24923598C211yesmaximal, characteristic, complemented
5#24923587C211yesmaximal, characteristic, complemented
6#24923557C211yesmaximal, characteristic, complemented
7#24923522C211yesmaximal, characteristic, complemented
8#24923501C211yesmaximal, characteristic, complemented
9#24923487C211yesmaximal, characteristic, complemented
10#24923414C211yesmaximal, characteristic, complemented
11#24923404C211yesmaximal, characteristic, complemented
12#24923378C211yesmaximal, characteristic, complemented
13#24923338C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#417×C167C211yesmaximal, characteristic, complemented
16C2×#14581950—21no—
17C2×D4×D167C2211yescharacteristic
18C2×#14581937C2211yescharacteristic, complemented
19C2×D1336C2211yescharacteristic
20C2×#14581935C2211yescharacteristic
21C2×C8×D167C2211yescharacteristic
22#19128831C2211yescharacteristic, complemented
23#19128821C2211yescharacteristic
24#19128817—21no—
25#19128815—21no—
26#19128815—21no—
27#19128810—21no—
28#19128808C2211yescharacteristic
29#19128807C2211yescharacteristic
30#19128806C2211yescharacteristic
31#19128806C2211yescharacteristic, complemented
32#19128804—21no—
33#19128800—21no—
34D4×Dic167C2211yescharacteristic
35#19128794C2211yescharacteristic, complemented
36C4×C22⋊D167C2211yescharacteristic
37C4⋊C4×D167C2211yescharacteristic
38C2×(C2×D167)⋊C4—21no—
39#19128764C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41#19128763C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128756C2211yescharacteristic, complemented
44#19128754C2211yescharacteristic
45#19128754C2211yescharacteristic, complemented
46#19128753C2211yescharacteristic
47#19128752C2211yescharacteristic
48#19128752C2211yescharacteristic
49C22⋊(C167⋊C8)C2211yescharacteristic, complemented
50(C2×D167)⋊C8C2211yescharacteristic
51#19128742C2211yescharacteristic, complemented
52C2×#41×C167C2211yescharacteristic
53C2×M16×C167C2211yescharacteristic, complemented
54C2×C4⋊C4×C167C2211yescharacteristic, complemented
55#103×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic
58#102×C167C2211yescharacteristic
59C23×D167—21no—
60#14581950—44no—
61C2×Dic334—21no—
62C2×D668—21no—
63C2×D668C2311yescharacteristic
64C2×D668C2311yescharacteristic
65C2×C22⋊D167—21no—
66C2×C22⋊D167—21no—
67C2×C22⋊D167—21no—
68C2×C22⋊D167D411yescharacteristic
69D4×D167—22no—
70D4×D167—41no—
71C22×Dic167—21no—
72C2×C4×D167—21no—
73C2×C4×D167—21no—
74C2×C4×D167C2311yescharacteristic
75#14581937—41no—
76D1336—41no—
77#14581935—22no—
78C2×C167⋊C8C2311yescharacteristic
79C2×C167⋊C8C2311yescharacteristic
80C8×D167—41no—
81C4⋊Dic167C2311yescharacteristic
82C4⋊Dic167C2311yescharacteristic
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86Dic167⋊C4D411yescharacteristic
87C22⋊Dic167—21no—
88C22⋊Dic167D411yescharacteristic
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4—21no—
96(C2×D167)⋊C4D411yescharacteristic
97C4×Dic167C2311yescharacteristic
98#41×C167—22no—
99#41×C167—41no—
100C2×Q8×C167C2311yescharacteristic
101C2×D4×C167—21no—
102C2×D4×C167C2311yescharacteristic
103C22×C4×C167—21no—
104C22×C4×C167—21no—
105C22×C4×C167C2311yescharacteristic
106M16×C167—41no—
107C2×C8×C167C2311yescharacteristic
108C2×C8×C167C2311yescharacteristic
109C4⋊C4×C167—21no—
110C4⋊C4×C167C2311yescharacteristic
111C4⋊C4×C167C2311yescharacteristic
112C22×D167—21no—
113C22×D167—21no—
114C22×D167—42no—
115C22×D167—42no—
116C22×D167—81no—
117C22×D167C2×D411yescharacteristic
118Dic334—42no—
119D668—21no—
120D668—21no—
121D668—21no—
122D668—21no—
123D668—42no—
124C22⋊D167—22no—
125C22⋊D167—42no—
126C22⋊D167—42no—
127C22⋊D167—81no—
128C2×Dic167—21no—
129C2×Dic167—21no—
130C2×Dic167—81no—
131C2×Dic167C2×D411yescharacteristic
132C2×Dic167C2×D411yescharacteristic
133C2×Dic167C2×D411yescharacteristic
134C4×D167—21no—
135C4×D167—21no—
136C4×D167—42no—
137C4×D167—42no—
138C167⋊C8—21no—
139C167⋊C8—21no—
140C23×C167—21no—
141C23×C167C2×D411yescharacteristic
142Q8×C167—21no—
143Q8×C167—21no—
144D4×C167—21no—
145D4×C167—42no—
146D4×C167#4112yescomplemented
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167—21no—
150C2×C4×C167—21no—
151C2×C4×C167—21no—
152C2×C4×C167—42no—
153C2×C4×C167—42no—
154C2×C4×C167C2×D411yescharacteristic
155C2×C4×C167C2411yescharacteristic
156C8×C167—21no—
157C8×C167—21no—
158C22×C167—21no—
159C22×C167—22no—
160C22×C167—21no—
161C22×C167—21no—
162C22×C167—41no—
163C22×C167—41no—
164C22×C167C22×D411yescharacteristic
165C2×D167—21no—
166C2×D167—21no—
167C2×D167—41no—
168C2×D167—41no—
169C2×D167—41no—
170C2×D167—41no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—84no—
174C2×D167AGL(1,ℤ/8)12yes—
175C4×C167—21no—
176C4×C167—41no—
177C4×C167—41no—
178C4×C167—41no—
179C4×C167C2×#4111yescharacteristic
180C4×C167H(5,𝔽2)11yescharacteristic
181Dic167—21no—
182Dic167—21no—
183Dic167—21no—
184Dic167—41no—
185Dic167—41no—
186C2×C167—42no—
187C2×C167—41no—
188C2×C167#54611yescharacteristic
189C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C167#57711yescharacteristic
191D167—22no—
192D167—81no—
193D167—81no—
194C167#330611yescharacteristic, complemented, Sylow
195#3306—1671nomaximal, Sylow
196#547—1671no—
197#546—1671no—
198#489—1671no—
199#483—1671no—
200#481—1671no—
201#476—1671no—
202#469—1671no—
203#465—1671no—
204#463—1671no—
205#449—1671no—
206#447—1671no—
207#442—1671no—
208#435—1671no—
209#417D16711yescharacteristic, complemented
210C2×#102—1671no—
211C2×#41—3341no—
212C2×#41C2×D16711yescharacteristic
213C22×D4—1671no—
214C2×SD16—1671no—
215C2×D8—1671no—
216C2×D8—1671no—
217C2×M16C2×D16711yescharacteristic, complemented
218C22×C8—1671no—
219C4⋊Q8—1671no—
220#124—1671no—
221#123—3341no—
222#123—3341no—
223#123—3341no—
224C22⋊Q8—3341no—
225C22⋊Q8—1671no—
226#121—3341no—
227#121—1671no—
228#121—1671no—
229#121—1671no—
230#120—3341no—
231C4×D4—1671no—
232C4×D4—1671no—
233C4×D4—1671no—
234C2×C4⋊C4—1671no—
235C2×C4⋊C4C2×D16711yescharacteristic, complemented
236C2×C22⋊C4—3341no—
237#106—1671no—
238#106—1671no—
239#106—1671no—
240C4⋊C8—1671no—
241#103—1671no—
242#103C2×D16711yescharacteristic, complemented
243#103C2×D16711yescharacteristic, complemented
244#102—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102C2×D16711yescharacteristic
250#102C2×D16711yescharacteristic
251C22⋊C8—1671no—
252C22⋊C8—1671no—
253#97—1671no—
254C24—3341no—
255#41—22no—
256#41—41no—
257#41—6684no—
258C2×Q8—3341no—
259C2×Q8C22×D16711yescharacteristic
260C2×D4—21no—
261C2×D4—3341no—
262C2×D4—3341no—
263C2×D4—3342no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—6681no—
267C2×D4—1671no—
268C2×D4—1671no—
269C2×D4—1671no—
270C2×D4C22×D16711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4—3341no—
276C22×C4—1671no—
277C22×C4C22×D16711yescharacteristic
278SD16—6681no—
279D8—3342no—
280D8—6681no—
281M16—41no—
282C2×C8—6681no—
283C2×C8—1671no—
284C2×C8—1671no—
285C2×C8C22×D16711yescharacteristic
286C2×C8C22×D16711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3341no—
289C4⋊C4—3341no—
290C4⋊C4—3341no—
291C4⋊C4—1671no—
292C4⋊C4—1671no—
293C4⋊C4—1671no—
294C4⋊C4C22×D16711yescharacteristic
295C4⋊C4C22×D16711yescharacteristic
296C22⋊C4—3341no—
297C22⋊C4—3341no—
298C22⋊C4—3341no—
299C22⋊C4—3341no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4—3341no—
304C22⋊C4—1671no—
305C22⋊C4—1671no—
306C42—1671no—
307C23—21no—
308C23—13361no—
309C23—3341no—
310C23—3341no—
311C23—6682no—
312C23—6682no—
313C23—1671no—
314C23D4×D16711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6682no—
318D4—21no—
319D4—42no—
320D4—13361no—
321D4—3342no—
322D4—3341no—
323D4—3341no—
324D4—3341no—
325D4—3341no—
326D4—6682no—
327D4—6682no—
328D4—6682no—
329D4#1458195012yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—13361no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—3341no—
341C2×C4—3341no—
342C2×C4—6682no—
343C2×C4—6682no—
344C2×C4—1671no—
345C2×C4—1671no—
346C2×C4—1671no—
347C2×C4D4×D16711yescharacteristic
348C2×C4C23×D16711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3341no—
352C8—3341no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—13364no—
360C22—3341no—
361C22—3341no—
362C22—6681no—
363C22—6681no—
364C22—6681no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22—1672no—
369C22C2×D4×D16711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3341no—
375C4—3341no—
376C4—3341no—
377C4—6681no—
378C4—6681no—
379C4C2×#1458195011yescharacteristic
380C4#1912887711yescharacteristic
381C2—41no—
382C2—42no—
383C2—13361no—
384C2—13361no—
385C2—3342no—
386C2#2492390911yescharacteristic
387C2AGL(1,ℤ/8)×D16711yescharacteristic
388C2#2492407811yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#417×C167C2
socleC22×C167C22×D4
hypercenter#417D167
hypocenterC167#3306
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #24923920C2semidirect
2. #24923907C2semidirect
3. #24923630C2semidirect
4. #24923598C2semidirect
5. #24923587C2semidirect
6. #24923557C2semidirect
7. #24923522C2semidirect
8. #24923501C2semidirect
9. #24923487C2semidirect
10. #24923414C2semidirect
11. #24923404C2semidirect
12. #24923378C2semidirect
13. #24923338C2semidirect
14. #102×D167C2semidirect
15. #417×C167C2semidirect
18. C2×#14581937C22semidirect
22. #19128831C22semidirect
31. #19128806C22semidirect
35. #19128794C22semidirect
39. #19128764C22semidirect
40. #19128763C22semidirect
41. #19128763C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128756C22semidirect
45. #19128754C22semidirect
49. C22⋊(C167⋊C8)C22semidirect
51. #19128742C22semidirect
53. C2×M16×C167C22semidirect
54. C2×C4⋊C4×C167C22semidirect
55. #103×C167C22semidirect
56. #103×C167C22semidirect
146. D4×C167#41semidirect
194. C167#3306semidirect
209. #417D167semidirect
217. C2×M16C2×D167semidirect
235. C2×C4⋊C4C2×D167semidirect
242. #103C2×D167semidirect
243. #103C2×D167semidirect
329. D4#14581950semidirect

Maximal subgroups

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

subgroupindex
#249239202
#249239072
#249236302
#249235982
#249235872
#249235572
#249235222
#249235012
#249234872
#249234142
#249234042
#249233782
#249233382
#102×D1672
#417×C1672
#3306167