Finite Group Databasebeta

← #36336114

Subgroups — #36336114

Order 21376.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923965C211yesmaximal, characteristic, complemented
2#24923637C211yesmaximal, characteristic, complemented
3#24923607C211yesmaximal, characteristic, complemented
4#24923597C211yesmaximal, characteristic, complemented
5#24923569C211yesmaximal, characteristic, complemented
6#24923517C211yesmaximal, characteristic, complemented
7#24923500C211yesmaximal, characteristic, complemented
8#24923471C211yesmaximal, characteristic, complemented
9#24923414C211yesmaximal, characteristic, complemented
10#24923414C211yesmaximal, characteristic, complemented
11#24923379C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923270C211yesmaximal, characteristic, complemented
14#24923265C211yesmaximal, characteristic, complemented
15#547×C167C211yesmaximal, characteristic, complemented
16C2×#14581937C2211yescharacteristic
17C2×#14581937C2211yescharacteristic
18C2×#14581935C2211yescharacteristic
19C2×#14581934C2211yescharacteristic
20C22×C167⋊C8C2211yescharacteristic
21#19128827—21no—
22#19128824C2211yescharacteristic
23#19128814—21no—
24#19128806C2211yescharacteristic
25#19128806C2211yescharacteristic
26#19128804—21no—
27C4×D668C2211yescharacteristic
28(C22⋊D167)⋊C4—21no—
29#19128791C2211yescharacteristic
30C2×C4⋊Dic167C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128756C2211yescharacteristic
39#19128756C2211yescharacteristic
40#19128756C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128752C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48C2×#41×C167C2211yescharacteristic
49C2×#41×C167C2211yescharacteristic
50C4⋊Q8×C167C2211yescharacteristic
51#123×C167—21no—
52#123×C167—21no—
53C22⋊Q8×C167C2211yescharacteristic
54C22⋊Q8×C167C2211yescharacteristic
55#121×C167—21no—
56C4×D4×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58C2×C4⋊C4×C167—21no—
59C2×D668C2311yescharacteristic
60C2×C22⋊D167—21no—
61C2×C22⋊D167—21no—
62C22×Dic167—21no—
63C2×C4×D167—21no—
64#14581937—41no—
65#14581937—41no—
66#14581935—22no—
67#14581934—41no—
68C2×C167⋊C8—41no—
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C2×C167⋊C8C2311yescharacteristic
73C4⋊Dic167—21no—
74C4⋊Dic167C2311yescharacteristic
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77Dic167⋊C4—21no—
78Dic167⋊C4—21no—
79C22⋊Dic167—21no—
80(C2×D167)⋊C4—21no—
81(C2×D167)⋊C4—21no—
82(C2×D167)⋊C4—21no—
83C4×Dic167—21no—
84#41×C167—22no—
85#41×C167—22no—
86#41×C167—41no—
87#41×C167—41no—
88C2×Q8×C167C2311yescharacteristic
89C2×Q8×C167C2311yescharacteristic
90C2×D4×C167—21no—
91C2×D4×C167—21no—
92C2×D4×C167D411yescharacteristic
93C2×D4×C167C2311yescharacteristic
94C22×C4×C167—21no—
95C22×C4×C167—21no—
96C22×C4×C167—21no—
97C22×C4×C167C2311yescharacteristic
98C22×C4×C167C2311yescharacteristic
99C4⋊C4×C167—21no—
100C4⋊C4×C167—21no—
101C4⋊C4×C167—21no—
102C4⋊C4×C167—21no—
103C4⋊C4×C167D411yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C22⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112D668—21no—
113D668—21no—
114C22⋊D167—42no—
115C22⋊D167—81no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—21no—
119C2×Dic167—81no—
120C4×D167—81no—
121C167⋊C8—21no—
122C167⋊C8—21no—
123C167⋊C8—21no—
124C167⋊C8—21no—
125C23×C167—21no—
126C23×C167C2×D411yescharacteristic
127C23×C167C2×D411yescharacteristic
128Q8×C167—21no—
129Q8×C167—21no—
130Q8×C167—21no—
131Q8×C167—21no—
132D4×C167—21no—
133D4×C167—41no—
134D4×C167—42no—
135D4×C167—42no—
136D4×C167#4112yes—
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×C167—42no—
145C2×C4×C167—42no—
146C2×C4×C167—42no—
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2411yescharacteristic
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—22no—
155C22×C167—21no—
156C22×C167—21no—
157C22×C167—21no—
158C22×C167—21no—
159C22×C167—41no—
160C22×C167—41no—
161C22×C167C22×D411yescharacteristic
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C4×C167—21no—
166C4×C167—21no—
167C4×C167—21no—
168C4×C167—21no—
169C4×C167—41no—
170C4×C167—41no—
171C4×C167—41no—
172C4×C167C2×#4111yescharacteristic
173C4×C167H(5,𝔽2)11yescharacteristic
174Dic167—41no—
175Dic167—41no—
176Dic167—41no—
177C2×C167—41no—
178C2×C167—41no—
179C2×C167—42no—
180C2×C167#54611yescharacteristic
181C2×C167#57511yescharacteristic
182C2×C167#57711yescharacteristic
183D167—81no—
184C167#330711yescharacteristic, complemented, Sylow
185#3307—1671nomaximal, Sylow
186#553—1671no—
187#547D16711yescharacteristic, complemented
188#491—1671no—
189#485—1671no—
190#483—1671no—
191#478—1671no—
192#468—1671no—
193#465—1671no—
194#461—1671no—
195#449—1671no—
196#449—1671no—
197#442—1671no—
198#435—1671no—
199#417—1671no—
200#416—1671no—
201C2×#41C2×D16711yescharacteristic
202C2×#41C2×D16711yescharacteristic
203C2×SD16—1671no—
204C2×SD16—1671no—
205C2×D8—1671no—
206C2×M16—1671no—
207C22×C8—1671no—
208C4⋊Q8C2×D16711yescharacteristic
209#126—3341no—
210#125—1671no—
211#123—21no—
212#123—21no—
213#123—3341no—
214C22⋊Q8C2×D16711yescharacteristic
215C22⋊Q8C2×D16711yescharacteristic
216#121—21no—
217#121—3341no—
218#121—1671no—
219#121—1671no—
220C4×D4—3341no—
221C4×D4—1671no—
222C4×D4C2×D16711yescharacteristic
223C4×D4C2×D16711yescharacteristic
224#117—1671no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1671no—
227#107—1671no—
228#106—1671no—
229#106—1671no—
230C4⋊C8—1671no—
231#103—1671no—
232#103—1671no—
233#103—1671no—
234#103—1671no—
235#103—1671no—
236#103—1671no—
237#102—1671no—
238#102—1671no—
239#102—1671no—
240#102—1671no—
241C22⋊C8—1671no—
242C22⋊C8—1671no—
243#97—1671no—
244#41—22no—
245#41—22no—
246#41—41no—
247#41—41no—
248C2×Q8C22×D16711yescharacteristic
249C2×Q8C22×D16711yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3341no—
253C2×D4—3341no—
254C2×D4—1671no—
255C2×D4C22⋊D16711yescharacteristic
256C2×D4C22×D16711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3341no—
261C22×C4—3341no—
262C22×C4C22×D16711yescharacteristic
263C22×C4C22×D16711yescharacteristic
264SD16—6681no—
265SD16—6681no—
266D8—3342no—
267M16—6681no—
268C2×C8—6681no—
269C2×C8—1671no—
270C2×C8—1671no—
271C2×C8—1671no—
272C2×C8—1671no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—3341no—
278C4⋊C4—3341no—
279C4⋊C4—3341no—
280C4⋊C4—1671no—
281C4⋊C4—1671no—
282C4⋊C4—1671no—
283C4⋊C4C22⋊D16711yescharacteristic
284C4⋊C4C22×D16711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—3341no—
292C22⋊C4C22⋊D16711yescharacteristic
293C22⋊C4C22⋊D16711yescharacteristic
294C42—3341no—
295C42C22×D16711yescharacteristic
296C23—21no—
297C23—3341no—
298C23C2×C22⋊D16711yescharacteristic
299C23C2×C22⋊D16711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—41no—
306D4—42no—
307D4—42no—
308D4—13361no—
309D4—3341no—
310D4—3341no—
311D4—6682no—
312D4#1458194912yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—13361no—
324C2×C4—13361no—
325C2×C4—3341no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4C2×C22⋊D16711yescharacteristic
329C2×C4C2×C22⋊D16711yescharacteristic
330C2×C4C2×C22⋊D16711yescharacteristic
331C2×C4C2×C22⋊D16711yescharacteristic
332C2×C4C23×D16711yescharacteristic
333C8—3341no—
334C8—3341no—
335C8—3341no—
336C8—3341no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—6681no—
347C22—6681no—
348C22—6681no—
349C22C22×C22⋊D16711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—6681no—
358C4—6681no—
359C4—6681no—
360C4C2×#1458194911yescharacteristic
361C4#1912887711yescharacteristic
362C2—41no—
363C2—41no—
364C2—42no—
365C2—13361no—
366C2#2492391311yescharacteristic
367C2#2492406711yescharacteristic
368C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#547×C167C2
socleC22×C167C22×D4
hypercenter#547D167
hypocenterC167#3307
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #24923965C2semidirect
2. #24923637C2semidirect
3. #24923607C2semidirect
4. #24923597C2semidirect
5. #24923569C2semidirect
6. #24923517C2semidirect
7. #24923500C2semidirect
8. #24923471C2semidirect
9. #24923414C2semidirect
10. #24923414C2semidirect
11. #24923379C2semidirect
12. #24923341C2semidirect
13. #24923270C2semidirect
14. #24923265C2semidirect
15. #547×C167C2semidirect
184. C167#3307semidirect
187. #547D167semidirect

Maximal subgroups

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

subgroupindex
#249239652
#249236372
#249236072
#249235972
#249235692
#249235172
#249235002
#249234712
#249234142
#249233792
#249233412
#249232702
#249232652
#547×C1672
#3307167