Finite Group Databasebeta

← #36334400

Subgroups — #36334400

Order 21376.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923890C211yesmaximal, characteristic, complemented
2#24923604C211yesmaximal, characteristic, complemented
3#24923582C211yesmaximal, characteristic, complemented
4#24923564C211yesmaximal, characteristic, complemented
5#24923549C211yesmaximal, characteristic, complemented
6#24923491C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic, complemented
8#24923431C211yesmaximal, characteristic, complemented
9#24923414C211yesmaximal, characteristic
10#24923369C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12C4×#14581940C211yesmaximal, characteristic, complemented
13C4×#14581935C211yesmaximal, characteristic, complemented
14#24923330C211yesmaximal, characteristic, complemented
15#520×C167C211yesmaximal, characteristic, complemented
16C2×#14581940C2211yescharacteristic
17C2×#14581938C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20#19128832C2211yescharacteristic
21#19128827—21no—
22#19128820C2211yescharacteristic
23#19128818—21no—
24#19128814—21no—
25#19128811C2211yescharacteristic
26#19128810—21no—
27#19128806C2211yescharacteristic
28C4×Dic334C2211yescharacteristic
29C4×D668C2211yescharacteristic
30#19128765C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128765C2211yescharacteristic
33#19128765C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35C4⋊(C167⋊C8)C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128755C2211yescharacteristic
39#19128755C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44C22⋊(C167⋊C8)C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46#19128743C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48C2×#41×C167C2211yescharacteristic
49C4×Q8×C167C2211yescharacteristic
50C4×Q8×C167C2211yescharacteristic
51C4×D4×C167—21no—
52C4×D4×C167—21no—
53C4×D4×C167C2211yescharacteristic
54C4×D4×C167C2211yescharacteristic
55#117×C167—21no—
56#117×C167C2211yescharacteristic
57C2×C4⋊C4×C167—21no—
58C2×C4⋊C4×C167C2211yescharacteristic
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C4×D167—21no—
64#14581940—22no—
65#14581938—22no—
66#14581937—22no—
67#14581935—22no—
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C4⋊Dic167—21no—
73C4⋊Dic167C2311yescharacteristic
74C4⋊Dic167C2311yescharacteristic
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77C22⋊Dic167—21no—
78(C2×D167)⋊C4—21no—
79(C2×D167)⋊C4—21no—
80(C2×D167)⋊C4—21no—
81(C2×D167)⋊C4—21no—
82C4×Dic167—21no—
83#41×C167—22no—
84#41×C167—22no—
85C2×Q8×C167C2311yescharacteristic
86C2×D4×C167—21no—
87C2×D4×C167C2311yescharacteristic
88C22×C4×C167—21no—
89C22×C4×C167—21no—
90C22×C4×C167—21no—
91C22×C4×C167—21no—
92C22×C4×C167C2311yescharacteristic
93C4⋊C4×C167—21no—
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167D411yescharacteristic
97C4⋊C4×C167D411yescharacteristic
98C4⋊C4×C167C2311yescharacteristic
99C4⋊C4×C167C2311yescharacteristic
100C22⋊C4×C167—21no—
101C22⋊C4×C167—21no—
102C22⋊C4×C167D411yescharacteristic
103C22⋊C4×C167D411yescharacteristic
104C42×C167—21no—
105C42×C167—21no—
106C42×C167C2311yescharacteristic
107C42×C167C2311yescharacteristic
108C22×D167—21no—
109Dic334—21no—
110Dic334—21no—
111Dic334—81no—
112D668—21no—
113D668—21no—
114C22⋊D167—81no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C4×D167—81no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C167⋊C8—21no—
123C23×C167—21no—
124C23×C167C2×D411yescharacteristic
125Q8×C167—21no—
126Q8×C167#4112yes—
127D4×C167—24no—
128D4×C167—21no—
129D4×C167#4112yes—
130C2×C4×C167—24no—
131C2×C4×C167—21no—
132C2×C4×C167—21no—
133C2×C4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—81no—
137C2×C4×C167—81no—
138C2×C4×C167—81no—
139C2×C4×C167C2×D411yescharacteristic
140C2×C4×C167C2×D411yescharacteristic
141C2×C4×C167C2×D411yescharacteristic
142C2×C4×C167C2×D411yescharacteristic
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2411yescharacteristic
145C22×C167—21no—
146C22×C167—22no—
147C22×C167—21no—
148C22×C167—21no—
149C22×C167—41no—
150C22×C167—41no—
151C22×C167C22×D411yescharacteristic
152C2×D167—41no—
153C2×D167—41no—
154C2×D167—41no—
155C4×C167—21no—
156C4×C167—22no—
157C4×C167—21no—
158C4×C167—21no—
159C4×C167—21no—
160C4×C167—21no—
161C4×C167—41no—
162C4×C167—41no—
163C4×C167C2×#4111yescharacteristic
164C4×C167C2×#4111yescharacteristic
165Dic167—41no—
166Dic167—41no—
167Dic167—41no—
168C2×C167—41no—
169C2×C167—42no—
170C2×C167#52511yescharacteristic
171C2×C167#57611yescharacteristic
172C2×C167#57811yescharacteristic
173D167—81no—
174C167#312411yescharacteristic, complemented, Sylow
175#3124—1671nomaximal, Sylow
176#542—1671no—
177#520D16711yescharacteristic, complemented
178#484—1671no—
179#480—1671no—
180#477—1671no—
181#474—1671no—
182#464—1671no—
183#463—1671no—
184#452—1671no—
185#449—1671no—
186#440—1671no—
187#440—1671no—
188C4×Q16—1671no—
189C4×D8—1671no—
190#433—1671no—
191C2×#41C2×D16711yescharacteristic
192C2×Q16—1671no—
193C2×SD16—1671no—
194C2×SD16—1671no—
195C2×D8—1671no—
196C4⋊Q8—1671no—
197#126—3341no—
198#124—3341no—
199#124—1671no—
200#123—3341no—
201C22⋊Q8—3341no—
202C22⋊Q8—1671no—
203#121—1671no—
204C4×Q8—1671no—
205C4×Q8C2×D16711yescharacteristic
206C4×Q8C2×D16711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—1671no—
210C4×D4C2×D16711yescharacteristic
211C4×D4C2×D16711yescharacteristic
212#117—21no—
213#117C2×D16711yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D16711yescharacteristic
216#107—1671no—
217#107—1671no—
218#107—1671no—
219#107—1671no—
220C4⋊C8—1671no—
221C4⋊C8—1671no—
222#103—1671no—
223#103—1671no—
224#103—1671no—
225#103—1671no—
226#102—1671no—
227#102—1671no—
228#102—1671no—
229#102—1671no—
230C22⋊C8—1671no—
231C22⋊C8—1671no—
232#97—1671no—
233C4×C8—1671no—
234#41—22no—
235#41—22no—
236C2×Q8—3341no—
237C2×Q8—1671no—
238C2×Q8C22×D16711yescharacteristic
239C2×D4—21no—
240C2×D4—3341no—
241C2×D4—1671no—
242C2×D4C22×D16711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3341no—
248C22×C4C22×D16711yescharacteristic
249Q16—3342no—
250SD16—3342no—
251SD16—3342no—
252D8—3342no—
253C2×C8—1671no—
254C2×C8—1671no—
255C2×C8—1671no—
256C2×C8—1671no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3341no—
261C4⋊C4—3341no—
262C4⋊C4—3341no—
263C4⋊C4—1671no—
264C4⋊C4—1671no—
265C4⋊C4C22⋊D16711yescharacteristic
266C4⋊C4C22⋊D16711yescharacteristic
267C4⋊C4C22×D16711yescharacteristic
268C4⋊C4C22×D16711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3341no—
272C22⋊C4—3341no—
273C22⋊C4—3341no—
274C22⋊C4—3341no—
275C22⋊C4—3341no—
276C22⋊C4C22⋊D16711yescharacteristic
277C22⋊C4C22⋊D16711yescharacteristic
278C42—21no—
279C42—21no—
280C42—3341no—
281C42C22×D16711yescharacteristic
282C42C22×D16711yescharacteristic
283C23—21no—
284C23—3341no—
285C23C2×C22⋊D16711yescharacteristic
286Q8—21no—
287Q8—13361no—
288Q8—3341no—
289Q8—3341no—
290Q8#1458195012yes—
291D4—24no—
292D4—21no—
293D4—13361no—
294D4—3341no—
295D4—3341no—
296D4#1458195012yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—13361no—
307C2×C4—3341no—
308C2×C4—3341no—
309C2×C4—3341no—
310C2×C4C2×C22⋊D16711yescharacteristic
311C2×C4C2×C22⋊D16711yescharacteristic
312C2×C4C2×C22⋊D16711yescharacteristic
313C2×C4C2×C22⋊D16711yescharacteristic
314C2×C4C2×C22⋊D16711yescharacteristic
315C2×C4C23×D16711yescharacteristic
316C8—3341no—
317C8—3341no—
318C8—3341no—
319C8—3341no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—6681no—
327C22—6681no—
328C22—6681no—
329C22C22×C22⋊D16711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—6681no—
339C4—6681no—
340C4—6681no—
341C4C2×#1458195011yescharacteristic
342C4C2×#1458195011yescharacteristic
343C2—41no—
344C2—42no—
345C2—13361no—
346C2#2492378011yescharacteristic
347C2#2492407211yescharacteristic
348C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#520×C167C2
socleC22×C167C22×D4
prime radical#24923414C2
hypercenter#520D167
hypocenterC167#3124
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #24923890C2semidirect
2. #24923604C2semidirect
3. #24923582C2semidirect
4. #24923564C2semidirect
5. #24923549C2semidirect
6. #24923491C2semidirect
7. #24923484C2semidirect
8. #24923431C2semidirect
10. #24923369C2semidirect
11. #24923367C2semidirect
12. C4×#14581940C2semidirect
13. C4×#14581935C2semidirect
14. #24923330C2semidirect
15. #520×C167C2semidirect
174. C167#3124semidirect
177. #520D167semidirect

Maximal subgroups

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

subgroupindex
#249238902
#249236042
#249235822
#249235642
#249235492
#249234912
#249234842
#249234312
#249234142
#249233692
#249233672
C4×#145819402
C4×#145819352
#249233302
#520×C1672
#3124167