Finite Group Databasebeta

← #36334323

Subgroups — #36334323

Order 21376.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923892C211yesmaximal, characteristic, complemented
2#24923744C211yesmaximal, characteristic, complemented
3#24923607C211yesmaximal, characteristic, complemented
4#24923594C211yesmaximal, characteristic, complemented
5#24923557C211yesmaximal, characteristic, complemented
6#24923547C211yesmaximal, characteristic, complemented
7#24923486C211yesmaximal, characteristic, complemented
8#24923486C211yesmaximal, characteristic, complemented
9#24923403C211yesmaximal, characteristic
10#24923378C211yesmaximal, characteristic, complemented
11#24923370C211yesmaximal, characteristic, complemented
12SD16×Dic167C211yesmaximal, characteristic, complemented
13#24923349C211yesmaximal, characteristic, complemented
14#24923327C211yesmaximal, characteristic, complemented
15#450×C167C211yesmaximal, characteristic, complemented
16C2×#14581935C2211yescharacteristic
17C2×#14581935C2211yescharacteristic
18#19128831C2211yescharacteristic, complemented
19#19128827—21no—
20#19128821C2211yescharacteristic, complemented
21#19128818—21no—
22#19128814—21no—
23#19128812—21no—
24#19128806C2211yescharacteristic
25Q8×Dic167C2211yescharacteristic, complemented
26D4×Dic167—21no—
27D4×Dic167—21no—
28D4×Dic167C2211yescharacteristic
29D4×Dic167C2211yescharacteristic
30D4×Dic167C2211yescharacteristic
31D4×Dic167C2211yescharacteristic
32#19128794C2211yescharacteristic, complemented
33#19128791C2211yescharacteristic
34#19128791C2211yescharacteristic
35C2×C22⋊Dic167—21no—
36C2×C22⋊Dic167—21no—
37#19128766C2211yescharacteristic, complemented
38#19128765C2211yescharacteristic, complemented
39#19128764C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128756C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic, complemented
46#19128754C2211yescharacteristic, complemented
47#19128753C2211yescharacteristic
48#19128753C2211yescharacteristic
49C22⋊(C167⋊C8)C2211yescharacteristic
50#19128742C2211yescharacteristic, complemented
51C8×Dic167C2211yescharacteristic, complemented
52C22×D4×C167C2211yescharacteristic
53C2×SD16×C167C2211yescharacteristic
54C2×SD16×C167C2211yescharacteristic
55C22⋊Q8×C167C2211yescharacteristic
56#102×C167C2211yescharacteristic
57#102×C167C2211yescharacteristic
58C22⋊C8×C167C2211yescharacteristic
59C2×Dic334—21no—
60C2×D668C2311yescharacteristic
61C2×C22⋊D167—21no—
62C22×Dic167—21no—
63C22×Dic167—21no—
64C22×Dic167—21no—
65C22×Dic167—21no—
66C2×C4×D167—21no—
67#14581935—22no—
68#14581935—22no—
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C4⋊Dic167—21no—
72C4⋊Dic167D411yescharacteristic
73C4⋊Dic167D411yescharacteristic
74C4⋊Dic167C2311yescharacteristic
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77Dic167⋊C4—21no—
78C22⋊Dic167—21no—
79C22⋊Dic167—21no—
80C22⋊Dic167—21no—
81C22⋊Dic167—21no—
82C22⋊Dic167—21no—
83C22⋊Dic167D411yescharacteristic
84C22⋊Dic167D411yescharacteristic
85(C2×D167)⋊C4—21no—
86(C2×D167)⋊C4—21no—
87(C2×D167)⋊C4—21no—
88(C2×D167)⋊C4—21no—
89C4×Dic167—21no—
90C4×Dic167—21no—
91C4×Dic167C2311yescharacteristic
92C4×Dic167C2311yescharacteristic
93C24×C167—21no—
94C2×Q8×C167C2311yescharacteristic
95C2×D4×C167—22no—
96C2×D4×C167—22no—
97C2×D4×C167—21no—
98C2×D4×C167C2311yescharacteristic
99C2×D4×C167C2311yescharacteristic
100C22×C4×C167C2311yescharacteristic
101SD16×C167—22no—
102SD16×C167—22no—
103C2×C8×C167C2311yescharacteristic
104C2×C8×C167C2311yescharacteristic
105C4⋊C4×C167—21no—
106C4⋊C4×C167C2311yescharacteristic
107C22⋊C4×C167—21no—
108C22×D167—21no—
109Dic334—81no—
110D668—21no—
111D668—21no—
112C22⋊D167—81no—
113C2×Dic167—21no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—81no—
117C2×Dic167—81no—
118C2×Dic167—81no—
119C2×Dic167—81no—
120C2×Dic167C2×D411yescharacteristic
121C2×Dic167C2×D411yescharacteristic
122C2×Dic167C2×D411yescharacteristic
123C2×Dic167C2×D411yescharacteristic
124C4×D167—81no—
125C167⋊C8—21no—
126C167⋊C8—21no—
127C23×C167—24no—
128C23×C167—21no—
129C23×C167—21no—
130C23×C167—81no—
131C23×C167—81no—
132C23×C167C2×D411yescharacteristic
133Q8×C167—21no—
134Q8×C167—21no—
135D4×C167—24no—
136D4×C167—21no—
137D4×C167—21no—
138D4×C167#4112yes—
139D4×C167#4112yes—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2411yescharacteristic
146C8×C167—21no—
147C8×C167—21no—
148C22×C167—21no—
149C22×C167—22no—
150C22×C167—22no—
151C22×C167—21no—
152C22×C167—21no—
153C22×C167—41no—
154C22×C167—41no—
155C22×C167—41no—
156C22×C167—41no—
157C22×C167—84no—
158C22×C167C22×D411yescharacteristic
159C2×D167—41no—
160C2×D167—41no—
161C2×D167—41no—
162C4×C167—21no—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167C2×#4111yescharacteristic
166C4×C167C2×#4111yescharacteristic
167Dic167—21no—
168Dic167—21no—
169Dic167—21no—
170Dic167—21no—
171Dic167—41no—
172Dic167—41no—
173Dic167—41no—
174C2×C167—41no—
175C2×C167—42no—
176C2×C167—42no—
177C2×C167#52511yescharacteristic
178C2×C167#57611yescharacteristic
179C2×C167#57711yescharacteristic
180D167—81no—
181C167#311811yescharacteristic, complemented, Sylow
182#3118—1671nomaximal, Sylow
183#542—1671no—
184#518—1671no—
185#485—1671no—
186#482—1671no—
187#476—1671no—
188#474—1671no—
189#463—1671no—
190#463—1671no—
191#450D16711yescharacteristic, complemented
192#447—1671no—
193#442—1671no—
194#440—1671no—
195C4×SD16—1671no—
196C4×D8—1671no—
197#433—1671no—
198C22×D4C2×D16711yescharacteristic
199C2×SD16C2×D16711yescharacteristic
200C2×SD16C2×D16711yescharacteristic
201C2×D8—1671no—
202C2×D8—1671no—
203C4⋊Q8—1671no—
204#126—3341no—
205#124—3341no—
206#124—1671no—
207#123—3341no—
208C22⋊Q8—3341no—
209C22⋊Q8C2×D16711yescharacteristic
210#121—1671no—
211C4×Q8—1671no—
212C4×D4—3341no—
213C4×D4—3341no—
214C4×D4—1671no—
215C4×D4—1671no—
216C4×D4—1671no—
217C4×D4—1671no—
218C4×D4—1671no—
219#117—1671no—
220#117—1671no—
221C2×C22⋊C4—3341no—
222C2×C22⋊C4—3341no—
223#107—1671no—
224#107—1671no—
225#106—1671no—
226#106—1671no—
227C4⋊C8—1671no—
228C4⋊C8—1671no—
229#103—1671no—
230#103—1671no—
231#102—1671no—
232#102—1671no—
233#102—1671no—
234#102—1671no—
235#102C2×D16711yescharacteristic
236#102C2×D16711yescharacteristic
237C22⋊C8—1671no—
238C22⋊C8C2×D16711yescharacteristic
239#97—1671no—
240C4×C8—1671no—
241C24—21no—
242C2×Q8—3341no—
243C2×Q8C22×D16711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3341no—
248C2×D4—1671no—
249C2×D4C22×D16711yescharacteristic
250C2×D4C22×D16711yescharacteristic
251C22×C4—3341no—
252C22×C4—3341no—
253C22×C4—3341no—
254C22×C4—3341no—
255C22×C4—3341no—
256C22×C4C22×D16711yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—3342no—
260D8—3342no—
261C2×C8—1671no—
262C2×C8—1671no—
263C2×C8C22×D16711yescharacteristic
264C2×C8C22×D16711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3341no—
267C4⋊C4—3341no—
268C4⋊C4—1671no—
269C4⋊C4—1671no—
270C4⋊C4—1671no—
271C4⋊C4—1671no—
272C4⋊C4—1671no—
273C4⋊C4C22×D16711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—3341no—
276C22⋊C4—3341no—
277C22⋊C4—3341no—
278C22⋊C4—3341no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C22⋊C4—1671no—
285C22⋊C4—1671no—
286C42—3341no—
287C42—3341no—
288C42—1671no—
289C42—1671no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3341no—
296C23D4×D16711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—13361no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—13361no—
304D4—3341no—
305D4—3341no—
306D4#1458194912yes—
307D4#1458194912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—13361no—
313C2×C4—13361no—
314C2×C4—13361no—
315C2×C4—13361no—
316C2×C4—13361no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4—1671no—
321C2×C4—1671no—
322C2×C4—1671no—
323C2×C4—1671no—
324C2×C4D4×D16711yescharacteristic
325C2×C4C23×D16711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3341no—
329C8—3341no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—6681no—
341C22—6681no—
342C22—6681no—
343C22C2×D4×D16711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3341no—
348C4—3341no—
349C4—3341no—
350C4—3341no—
351C4—6681no—
352C4—6681no—
353C4—6681no—
354C4C2×#1458194911yescharacteristic
355C4C2×#1458194911yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—13361no—
360C2#2492377511yescharacteristic
361C2#2492407311yescharacteristic
362C2#2492407511yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#450×C167C2
socleC22×C167C22×D4
prime radical#24923403C2
hypercenter#450D167
hypocenterC167#3118
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #24923892C2semidirect
2. #24923744C2semidirect
3. #24923607C2semidirect
4. #24923594C2semidirect
5. #24923557C2semidirect
6. #24923547C2semidirect
7. #24923486C2semidirect
8. #24923486C2semidirect
10. #24923378C2semidirect
11. #24923370C2semidirect
12. SD16×Dic167C2semidirect
13. #24923349C2semidirect
14. #24923327C2semidirect
15. #450×C167C2semidirect
18. #19128831C22semidirect
20. #19128821C22semidirect
25. Q8×Dic167C22semidirect
32. #19128794C22semidirect
37. #19128766C22semidirect
38. #19128765C22semidirect
39. #19128764C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128756C22semidirect
44. #19128756C22semidirect
45. #19128754C22semidirect
46. #19128754C22semidirect
50. #19128742C22semidirect
51. C8×Dic167C22semidirect
181. C167#3118semidirect
191. #450D167semidirect

Maximal subgroups

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

subgroupindex
#249238922
#249237442
#249236072
#249235942
#249235572
#249235472
#249234862
#249234032
#249233782
#249233702
SD16×Dic1672
#249233492
#249233272
#450×C1672
#3118167