Finite Group Databasebeta

← #36336220

Subgroups — #36336220

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#24923974C211yesmaximal, characteristic, complemented
2#24923907C211yesmaximal, characteristic
3#24923630C211yesmaximal, characteristic, complemented
4#24923603C211yesmaximal, characteristic, complemented
5#24923582C211yesmaximal, characteristic, complemented
6#24923547C211yesmaximal, characteristic, complemented
7#24923515C211yesmaximal, characteristic, complemented
8#24923509C211yesmaximal, characteristic, complemented
9#24923494C211yesmaximal, characteristic, complemented
10#24923431C211yesmaximal, characteristic, complemented
11#24923421C211yesmaximal, characteristic, complemented
12#24923366C211yesmaximal, characteristic, complemented
13#24923338C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#417×C167C211yesmaximal, characteristic, complemented
16C2×D4×D167C2211yescharacteristic
17C2×#14581940C2211yescharacteristic, complemented
18C2×#14581939C2211yescharacteristic
19C2×#14581938C2211yescharacteristic
20C2×C8×D167C2211yescharacteristic
21#19128831C2211yescharacteristic, complemented
22#19128822—21no—
23#19128821C2211yescharacteristic
24#19128815—21no—
25#19128812C2211yescharacteristic
26#19128811—21no—
27#19128811C2211yescharacteristic, complemented
28#19128810—21no—
29#19128808C2211yescharacteristic
30#19128806C2211yescharacteristic
31#19128804—21no—
32#19128800—21no—
33#19128797C2211yescharacteristic, complemented
34D4×Dic167C2211yescharacteristic
35C4×C22⋊D167—21no—
36C4×C22⋊D167C2211yescharacteristic
37C4⋊C4×D167C2211yescharacteristic
38C2×(C2×D167)⋊C4—21no—
39#19128766C2211yescharacteristic, complemented
40#19128765C2211yescharacteristic, complemented
41#19128765C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic
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—
60C2×Dic334—21no—
61C2×Dic334C2311yescharacteristic
62C2×D668C2311yescharacteristic
63C2×C22⋊D167—21no—
64C2×C22⋊D167—21no—
65C2×C22⋊D167D411yescharacteristic
66D4×D167—22no—
67D4×D167—41no—
68C22×Dic167—21no—
69C2×C4×D167—21no—
70C2×C4×D167C2311yescharacteristic
71#14581940—41no—
72#14581939—41no—
73#14581938—22no—
74C2×C167⋊C8C2311yescharacteristic
75C2×C167⋊C8C2311yescharacteristic
76C8×D167—41no—
77C4⋊Dic167—21no—
78C4⋊Dic167C2311yescharacteristic
79C4⋊Dic167C2311yescharacteristic
80Dic167⋊C4—21no—
81Dic167⋊C4—21no—
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4D411yescharacteristic
86C22⋊Dic167—21no—
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)⋊C4D411yescharacteristic
95C4×Dic167—21no—
96C4×Dic167C2311yescharacteristic
97#41×C167—22no—
98#41×C167—41no—
99C2×Q8×C167C2311yescharacteristic
100C2×D4×C167—21no—
101C2×D4×C167C2311yescharacteristic
102C22×C4×C167—21no—
103C22×C4×C167—21no—
104C22×C4×C167C2311yescharacteristic
105M16×C167—41no—
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167—21no—
109C4⋊C4×C167C2311yescharacteristic
110C4⋊C4×C167C2311yescharacteristic
111C22×D167—21no—
112C22×D167—42no—
113C22×D167—42no—
114C22×D167—81no—
115C22×D167C2×D411yescharacteristic
116Dic334—21no—
117Dic334—21no—
118Dic334—81no—
119D668—21no—
120D668—21no—
121C22⋊D167—22no—
122C22⋊D167—42no—
123C22⋊D167—81no—
124C2×Dic167—21no—
125C2×Dic167—21no—
126C2×Dic167—21no—
127C2×Dic167—81no—
128C2×Dic167C2×D411yescharacteristic
129C2×Dic167C2×D411yescharacteristic
130C2×Dic167C2×D411yescharacteristic
131C4×D167—21no—
132C4×D167—21no—
133C4×D167—81no—
134C167⋊C8—21no—
135C167⋊C8—21no—
136C23×C167—21no—
137C23×C167C2×D411yescharacteristic
138Q8×C167—21no—
139Q8×C167—21no—
140D4×C167—21no—
141D4×C167—42no—
142D4×C167#4112yes—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—42no—
149C2×C4×C167—81no—
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2411yescharacteristic
152C8×C167—21no—
153C8×C167—21no—
154C22×C167—21no—
155C22×C167—22no—
156C22×C167—21no—
157C22×C167—21no—
158C22×C167—41no—
159C22×C167—41no—
160C22×C167C22×D411yescharacteristic
161C2×D167—21no—
162C2×D167—21no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—41no—
166C2×D167—84no—
167C2×D167#13712yes—
168C4×C167—21no—
169C4×C167—41no—
170C4×C167—41no—
171C4×C167—41no—
172C4×C167C2×#4111yescharacteristic
173C4×C167H(5,𝔽2)11yescharacteristic
174Dic167—21no—
175Dic167—21no—
176Dic167—21no—
177Dic167—41no—
178Dic167—41no—
179Dic167—41no—
180C2×C167—42no—
181C2×C167—41no—
182C2×C167#54611yescharacteristic
183C2×C167C2×#13711yescharacteristic
184C2×C167#57611yescharacteristic
185D167—22no—
186D167—81no—
187C167#331411yescharacteristic, complemented, Sylow
188#3314—1671nomaximal, Sylow
189#554—1671no—
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#417D16711yescharacteristic, complemented
203C2×#102—1671no—
204C2×#41C2×D16711yescharacteristic
205C22×D4—1671no—
206C2×Q16—1671no—
207C2×SD16—1671no—
208C2×SD16—1671no—
209C2×M16C2×D16711yescharacteristic, complemented
210C22×C8—1671no—
211C4⋊Q8—1671no—
212#125—3341no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—3341no—
216C22⋊Q8—3341no—
217C22⋊Q8—1671no—
218C22⋊Q8—1671no—
219C22⋊Q8—1671no—
220#121—3341no—
221#121—1671no—
222#120—3341no—
223C4×Q8—1671no—
224C4×D4—3341no—
225C4×D4—1671no—
226C4×D4—1671no—
227C2×C4⋊C4—1671no—
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#103C2×D16711yescharacteristic, complemented
237#103C2×D16711yescharacteristic, complemented
238#102—1671no—
239#102—1671no—
240#102—1671no—
241#102—1671no—
242#102C2×D16711yescharacteristic
243#102C2×D16711yescharacteristic
244C22⋊C8—1671no—
245C22⋊C8—1671no—
246#97—1671no—
247C24—3341no—
248#41—22no—
249#41—41no—
250C2×Q8—3341no—
251C2×Q8—1671no—
252C2×Q8C22×D16711yescharacteristic
253C2×D4—21no—
254C2×D4—3341no—
255C2×D4—3342no—
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×C4—1671no—
266C22×C4C22×D16711yescharacteristic
267Q16—6681no—
268SD16—3342no—
269SD16—6681no—
270M16—41no—
271C2×C8—6681no—
272C2×C8—1671no—
273C2×C8—1671no—
274C2×C8C22×D16711yescharacteristic
275C2×C8C22×D16711yescharacteristic
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—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—3341no—
292C22⋊C4—3341no—
293C22⋊C4—3341no—
294C22⋊C4—3341no—
295C22⋊C4—1671no—
296C22⋊C4—1671no—
297C42—3341no—
298C42—1671no—
299C23—21no—
300C23—13361no—
301C23—3341no—
302C23—6682no—
303C23—6682no—
304C23—1671no—
305C23D4×D16711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—13361no—
309Q8—3341no—
310Q8—3341no—
311D4—21no—
312D4—42no—
313D4—13361no—
314D4—3342no—
315D4—3341no—
316D4—3341no—
317D4—6682no—
318D4#1458195012yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—13361no—
327C2×C4—13361no—
328C2×C4—3341no—
329C2×C4—3341no—
330C2×C4—3341no—
331C2×C4—3341no—
332C2×C4—3341no—
333C2×C4—1671no—
334C2×C4—1671no—
335C2×C4—1671no—
336C2×C4D4×D16711yescharacteristic
337C2×C4C23×D16711yescharacteristic
338C8—21no—
339C8—21no—
340C8—3341no—
341C8—3341no—
342C22—21no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—13364no—
349C22—3341no—
350C22—3341no—
351C22—6681no—
352C22—6681no—
353C22—6681no—
354C22—1672no—
355C22C2×D4×D16711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3341no—
361C4—3341no—
362C4—3341no—
363C4—6681no—
364C4—6681no—
365C4—6681no—
366C4C2×#1458195011yescharacteristic
367C4#1912887711yescharacteristic
368C2—41no—
369C2—42no—
370C2—13361no—
371C2—3342no—
372C2#2492390911yescharacteristic
373C2#137×D16711yescharacteristic
374C2#2492407311yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#417×C167C2
socleC22×C167C22×D4
prime radical#24923907C2
hypercenter#417D167
hypocenterC167#3314
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. #24923974C2semidirect
3. #24923630C2semidirect
4. #24923603C2semidirect
5. #24923582C2semidirect
6. #24923547C2semidirect
7. #24923515C2semidirect
8. #24923509C2semidirect
9. #24923494C2semidirect
10. #24923431C2semidirect
11. #24923421C2semidirect
12. #24923366C2semidirect
13. #24923338C2semidirect
14. #102×D167C2semidirect
15. #417×C167C2semidirect
17. C2×#14581940C22semidirect
21. #19128831C22semidirect
27. #19128811C22semidirect
33. #19128797C22semidirect
39. #19128766C22semidirect
40. #19128765C22semidirect
41. #19128765C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128757C22semidirect
44. #19128756C22semidirect
49. C22⋊(C167⋊C8)C22semidirect
51. #19128742C22semidirect
53. C2×M16×C167C22semidirect
54. C2×C4⋊C4×C167C22semidirect
55. #103×C167C22semidirect
56. #103×C167C22semidirect
187. C167#3314semidirect
202. #417D167semidirect
209. C2×M16C2×D167semidirect
228. C2×C4⋊C4C2×D167semidirect
236. #103C2×D167semidirect
237. #103C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239742
#249239072
#249236302
#249236032
#249235822
#249235472
#249235152
#249235092
#249234942
#249234312
#249234212
#249233662
#249233382
#102×D1672
#417×C1672
#3314167