Finite Group Databasebeta

← #36336198

Subgroups — #36336198

Order 21376.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923928C211yesmaximal, characteristic
2#24923631C211yesmaximal, characteristic, complemented
3#24923610C211yesmaximal, characteristic, complemented
4#24923593C211yesmaximal, characteristic, complemented
5#24923561C211yesmaximal, characteristic, complemented
6#24923517C211yesmaximal, characteristic, complemented
7#24923508C211yesmaximal, characteristic, complemented
8#24923491C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923424C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923277C211yesmaximal, characteristic, complemented
14#24923263C211yesmaximal, characteristic, complemented
15#556×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×Dic334C2211yescharacteristic
18C2×#14581940C2211yescharacteristic
19C2×#14581940C2211yescharacteristic
20C2×#14581937C2211yescharacteristic
21C2×#14581934C2211yescharacteristic
22C22×C167⋊C8C2211yescharacteristic
23#19128820C2211yescharacteristic
24#19128818—21no—
25#19128811C2211yescharacteristic
26#19128810—21no—
27#19128806C2211yescharacteristic
28#19128804—21no—
29C4×Dic334C2211yescharacteristic
30(C22⋊D167)⋊C4—21no—
31#19128765C2211yescharacteristic
32#19128765C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128755C2211yescharacteristic
37#19128755C2211yescharacteristic
38#19128755C2211yescharacteristic
39#19128755C2211yescharacteristic
40#19128755C2211yescharacteristic
41#19128755C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44#19128752C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48C4⋊Q8×C167C2211yescharacteristic
49#126×C167—21no—
50#126×C167—21no—
51#124×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53C22⋊Q8×C167C2211yescharacteristic
54C4×Q8×C167—21no—
55C4×Q8×C167C2211yescharacteristic
56C4×D4×C167C2211yescharacteristic
57#117×C167C2211yescharacteristic
58#117×C167C2211yescharacteristic
59#14581950—22no—
60#14581950—41no—
61C2×Dic334—22no—
62C2×Dic334—21no—
63C2×Dic334—41no—
64C2×Dic334C2311yescharacteristic
65C2×Dic334C2311yescharacteristic
66C2×D668C2311yescharacteristic
67C2×C22⋊D167—21no—
68C2×C22⋊D167—21no—
69C22×Dic167—21no—
70C2×C4×D167—21no—
71#14581940—22no—
72#14581940—41no—
73#14581937—41no—
74#14581934—41no—
75C2×C167⋊C8—41no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C2×C167⋊C8C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81Dic167⋊C4—21no—
82Dic167⋊C4—21no—
83C22⋊Dic167—21no—
84(C2×D167)⋊C4—21no—
85(C2×D167)⋊C4—21no—
86(C2×D167)⋊C4—21no—
87C4×Dic167—21no—
88C2×Q8×C167—21no—
89C2×Q8×C167C2311yescharacteristic
90C2×D4×C167D411yescharacteristic
91C22×C4×C167C2311yescharacteristic
92C22×C4×C167C2311yescharacteristic
93C4⋊C4×C167—21no—
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167D411yescharacteristic
98C4⋊C4×C167C2311yescharacteristic
99C4⋊C4×C167C2311yescharacteristic
100C4⋊C4×C167C2311yescharacteristic
101C22⋊C4×C167—21no—
102C22⋊C4×C167—21no—
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167D411yescharacteristic
106C22⋊C4×C167D411yescharacteristic
107C42×C167—21no—
108C42×C167—21no—
109C42×C167—21no—
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112Dic334—21no—
113Dic334—21no—
114Dic334—21no—
115Dic334—42no—
116Dic334#4112yes—
117D668—21no—
118D668—21no—
119C22⋊D167—42no—
120C22⋊D167—42no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C2×Dic167—21no—
124C2×Dic167—42no—
125C4×D167—42no—
126C167⋊C8—21no—
127C167⋊C8—21no—
128C167⋊C8—21no—
129C167⋊C8—21no—
130C23×C167C2×D411yescharacteristic
131C23×C167C2×D411yescharacteristic
132Q8×C167—21no—
133Q8×C167—21no—
134Q8×C167—42no—
135D4×C167—41no—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2411yescharacteristic
149C22×C167—21no—
150C22×C167—21no—
151C22×C167—21no—
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—21no—
155C22×C167C22×D411yescharacteristic
156C2×D167—41no—
157C2×D167—41no—
158C2×D167—41no—
159C4×C167—21no—
160C4×C167—21no—
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167—41no—
167C4×C167C2×#4111yescharacteristic
168C4×C167H(5,𝔽2)11yescharacteristic
169Dic167—22no—
170Dic167—41no—
171Dic167—41no—
172C2×C167—41no—
173C2×C167—41no—
174C2×C167#54611yescharacteristic
175C2×C167#57511yescharacteristic
176C2×C167#57811yescharacteristic
177D167—81no—
178C167#331311yescharacteristic, complemented, Sylow
179#3313—1671nomaximal, Sylow
180#556D16711yescharacteristic, complemented
181#548—1671no—
182#489—1671no—
183#485—1671no—
184#482—1671no—
185#477—1671no—
186#468—1671no—
187#467—1671no—
188#464—1671no—
189#452—1671no—
190#451—1671no—
191#441—1671no—
192#435—1671no—
193#419—1671no—
194#416—1671no—
195C2×#41—1671no—
196C22×Q8—1671no—
197C2×Q16—1671no—
198C2×Q16—1671no—
199C2×SD16—1671no—
200C2×M16—1671no—
201C22×C8—1671no—
202C4⋊Q8C2×D16711yescharacteristic
203#126—21no—
204#126—21no—
205#124—21no—
206#124—3341no—
207#124—1671no—
208C22⋊Q8—3341no—
209C22⋊Q8—1671no—
210C22⋊Q8C2×D16711yescharacteristic
211C22⋊Q8C2×D16711yescharacteristic
212#121—3341no—
213#121—1671no—
214C4×Q8—21no—
215C4×Q8—1671no—
216C4×Q8C2×D16711yescharacteristic
217C4×D4—3341no—
218C4×D4C2×D16711yescharacteristic
219#117C2×D16711yescharacteristic
220#117C2×D16711yescharacteristic
221#107—1671no—
222#107—1671no—
223#106—1671no—
224C4⋊C8—1671no—
225#103—1671no—
226#103—1671no—
227#103—1671no—
228#103—1671no—
229#103—1671no—
230#103—1671no—
231#103—1671no—
232#102—1671no—
233#102—1671no—
234#102—1671no—
235C22⋊C8—1671no—
236C22⋊C8—1671no—
237#97—1671no—
238#41—3342no—
239#41—6681no—
240C2×Q8—21no—
241C2×Q8—3342no—
242C2×Q8—3341no—
243C2×Q8—6681no—
244C2×Q8—1671no—
245C2×Q8—1671no—
246C2×Q8C22×D16711yescharacteristic
247C2×D4—3341no—
248C2×D4—3341no—
249C2×D4—1671no—
250C2×D4C22⋊D16711yescharacteristic
251C22×C4—3341no—
252C22×C4—3341no—
253C22×C4C22×D16711yescharacteristic
254C22×C4C22×D16711yescharacteristic
255Q16—3342no—
256Q16—6681no—
257SD16—6681no—
258M16—6681no—
259C2×C8—6681no—
260C2×C8—1671no—
261C2×C8—1671no—
262C2×C8—1671no—
263C2×C8—1671no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3341no—
269C4⋊C4—3341no—
270C4⋊C4—1671no—
271C4⋊C4C22⋊D16711yescharacteristic
272C4⋊C4C22×D16711yescharacteristic
273C4⋊C4C22×D16711yescharacteristic
274C4⋊C4C22×D16711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4C22⋊D16711yescharacteristic
284C22⋊C4C22⋊D16711yescharacteristic
285C42—21no—
286C42—21no—
287C42—21no—
288C42—3341no—
289C42C22×D16711yescharacteristic
290C23—3341no—
291C23C2×C22⋊D16711yescharacteristic
292C23C2×C22⋊D16711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—3341no—
297Q8—3341no—
298Q8—3341no—
299Q8—6682no—
300Q8—1672no—
301D4—41no—
302D4—3341no—
303D4—3341no—
304D4—6682no—
305D4—6682no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—3341no—
315C2×C4—3341no—
316C2×C4—3341no—
317C2×C4—6682no—
318C2×C4—6682no—
319C2×C4C2×C22⋊D16711yescharacteristic
320C2×C4C2×C22⋊D16711yescharacteristic
321C2×C4C2×C22⋊D16711yescharacteristic
322C2×C4C2×C22⋊D16711yescharacteristic
323C2×C4C23×D16711yescharacteristic
324C8—3341no—
325C8—3341no—
326C8—3341no—
327C8—3341no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—6681no—
335C22—6681no—
336C22—6681no—
337C22C22×C22⋊D16711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3342no—
347C4—6681no—
348C4—6681no—
349C4#41×D16711yescharacteristic
350C4#1912887811yescharacteristic
351C2—41no—
352C2—41no—
353C2—13361no—
354C2#2492390711yescharacteristic
355C2#2492406711yescharacteristic
356C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#556×C167C2
socleC22×C167C22×D4
prime radical#24923928C2
hypercenter#556D167
hypocenterC167#3313
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #24923631C2semidirect
3. #24923610C2semidirect
4. #24923593C2semidirect
5. #24923561C2semidirect
6. #24923517C2semidirect
7. #24923508C2semidirect
8. #24923491C2semidirect
9. #24923429C2semidirect
10. #24923424C2semidirect
11. #24923374C2semidirect
12. #24923341C2semidirect
13. #24923277C2semidirect
14. #24923263C2semidirect
15. #556×C167C2semidirect
178. C167#3313semidirect
180. #556D167semidirect

Maximal subgroups

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

subgroupindex
#249239282
#249236312
#249236102
#249235932
#249235612
#249235172
#249235082
#249234912
#249234292
#249234242
#249233742
#249233412
#249232772
#249232632
#556×C1672
#3313167