Finite Group Databasebeta

← #36334247

Subgroups — #36334247

Order 21376.

392 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#24923803C211yesmaximal, characteristic, complemented
2#24923610C211yesmaximal, characteristic, complemented
3#24923593C211yesmaximal, characteristic, complemented
4#24923576C211yesmaximal, characteristic, complemented
5#24923569C211yesmaximal, characteristic, complemented
6#24923491C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic, complemented
8#24923431C211yesmaximal, characteristic, complemented
9#24923414C211yesmaximal, characteristic
10#24923379C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12#24923369C211yesmaximal, characteristic, complemented
13#24923367C211yesmaximal, characteristic, complemented
14#24923326C211yesmaximal, characteristic, complemented
15C4×#41×C167C211yesmaximal, characteristic, complemented
16C2×#14581940C2211yescharacteristic
17C2×#14581938C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20#19128826—21no—
21#19128824C2211yescharacteristic
22#19128820C2211yescharacteristic
23#19128815—21no—
24#19128811C2211yescharacteristic
25#19128806C2211yescharacteristic
26C4×Dic334C2211yescharacteristic
27C4×D668C2211yescharacteristic
28C4×C22⋊D167—21no—
29#19128789—21no—
30#19128765C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
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
47#19128743C2211yescharacteristic
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×C42×C167—21no—
58C2×C42×C167C2211yescharacteristic
59C2×Dic334C2311yescharacteristic
60C2×D668C2311yescharacteristic
61C2×C22⋊D167—21no—
62C2×C4×D167—21no—
63#14581940—22no—
64#14581938—22no—
65#14581937—22no—
66#14581935—22no—
67C2×C167⋊C8C2311yescharacteristic
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C4⋊Dic167C2311yescharacteristic
72C4⋊Dic167C2311yescharacteristic
73Dic167⋊C4—21no—
74Dic167⋊C4—21no—
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×C167D411yescharacteristic
92C22×C4×C167D411yescharacteristic
93C22×C4×C167C2311yescharacteristic
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167C2311yescharacteristic
97C4⋊C4×C167C2311yescharacteristic
98C22⋊C4×C167—21no—
99C22⋊C4×C167—21no—
100C22⋊C4×C167—21no—
101C42×C167—21no—
102C42×C167—21no—
103C42×C167—21no—
104C42×C167D411yescharacteristic
105C42×C167D411yescharacteristic
106C42×C167C2311yescharacteristic
107C42×C167C2311yescharacteristic
108C22×D167—21no—
109Dic334—21no—
110Dic334—21no—
111D668—21no—
112D668—21no—
113C22⋊D167—42no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C4×D167—42no—
118C167⋊C8—21no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C23×C167—21no—
123C23×C167C2×D411yescharacteristic
124Q8×C167—21no—
125Q8×C167#4112yes—
126D4×C167—24no—
127D4×C167—21no—
128D4×C167#4112yes—
129C2×C4×C167—24no—
130C2×C4×C167—21no—
131C2×C4×C167—21no—
132C2×C4×C167—21no—
133C2×C4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—41no—
136C2×C4×C167—41no—
137C2×C4×C167—42no—
138C2×C4×C167—42no—
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#57511yescharacteristic
172C2×C167#57511yescharacteristic
173D167—81no—
174C167#311111yescharacteristic, complemented, Sylow
175#3111—1671nomaximal, Sylow
176#529—1671no—
177C4×#41D16711yescharacteristic, complemented
178#485—1671no—
179#482—1671no—
180#479—1671no—
181#478—1671no—
182#464—1671no—
183#463—1671no—
184#452—1671no—
185#449—1671no—
186#442—1671no—
187#441—1671no—
188#440—1671no—
189#440—1671no—
190#432—1671no—
191C2×#41C2×D16711yescharacteristic
192C2×Q16—1671no—
193C2×SD16—1671no—
194C2×SD16—1671no—
195C2×D8—1671no—
196#126—3341no—
197#125—1671no—
198#124—1671no—
199#123—3341no—
200C22⋊Q8—1671no—
201#121—1671no—
202C4×Q8—1671no—
203C4×Q8C2×D16711yescharacteristic
204C4×Q8C2×D16711yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—3341no—
208C4×D4—1671no—
209C4×D4C2×D16711yescharacteristic
210C4×D4C2×D16711yescharacteristic
211#117—21no—
212#117—3341no—
213#117C2×D16711yescharacteristic
214C2×C42—21no—
215C2×C42C2×D16711yescharacteristic
216#107—1671no—
217#107—1671no—
218#106—1671no—
219#106—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—
233#97—1671no—
234#41—22no—
235#41—22no—
236C2×Q8—1671no—
237C2×Q8C22×D16711yescharacteristic
238C2×D4—21no—
239C2×D4—3341no—
240C2×D4—1671no—
241C2×D4C22×D16711yescharacteristic
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—3341no—
246C22×C4C22⋊D16711yescharacteristic
247C22×C4C22⋊D16711yescharacteristic
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—3341no—
260C4⋊C4—3341no—
261C4⋊C4—3341no—
262C4⋊C4—3341no—
263C4⋊C4—1671no—
264C4⋊C4—1671no—
265C4⋊C4C22×D16711yescharacteristic
266C4⋊C4C22×D16711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3341no—
271C22⋊C4—3341no—
272C22⋊C4—3341no—
273C22⋊C4—3341no—
274C22⋊C4—3341no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—3341no—
279C42C22⋊D16711yescharacteristic
280C42C22⋊D16711yescharacteristic
281C42C22×D16711yescharacteristic
282C42C22×D16711yescharacteristic
283C23—21no—
284C23—3341no—
285C23C2×C22⋊D16711yescharacteristic
286Q8—21no—
287Q8—3341no—
288Q8—3341no—
289Q8#1458195012yes—
290D4—24no—
291D4—21no—
292D4—3341no—
293D4—3341no—
294D4—6682no—
295D4#1458195012yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—3341no—
307C2×C4—3341no—
308C2×C4—3341no—
309C2×C4—6682no—
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#2492406711yescharacteristic
348C2#2492406711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×#41×C167C2
socleC22×C167C22×D4
prime radical#24923414C2
hypercenterC4×#41D167
hypocenterC167#3111
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #24923803C2semidirect
2. #24923610C2semidirect
3. #24923593C2semidirect
4. #24923576C2semidirect
5. #24923569C2semidirect
6. #24923491C2semidirect
7. #24923484C2semidirect
8. #24923431C2semidirect
10. #24923379C2semidirect
11. #24923374C2semidirect
12. #24923369C2semidirect
13. #24923367C2semidirect
14. #24923326C2semidirect
15. C4×#41×C167C2semidirect
174. C167#3111semidirect
177. C4×#41D167semidirect

Maximal subgroups

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

subgroupindex
#249238032
#249236102
#249235932
#249235762
#249235692
#249234912
#249234842
#249234312
#249234142
#249233792
#249233742
#249233692
#249233672
#249233262
C4×#41×C1672
#3111167