Finite Group Databasebeta

← #36334325

Subgroups — #36334325

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#24923743C211yesmaximal, characteristic, complemented
3#24923593C211yesmaximal, characteristic, complemented
4#24923558C211yesmaximal, characteristic, complemented
5#24923548C211yesmaximal, characteristic, complemented
6#24923487C211yesmaximal, characteristic, complemented
7#24923487C211yesmaximal, characteristic, complemented
8#24923419C211yesmaximal, characteristic, complemented
9#24923405C211yesmaximal, characteristic
10#24923380C211yesmaximal, characteristic, complemented
11#24923372C211yesmaximal, characteristic, complemented
12#24923354C211yesmaximal, characteristic, complemented
13#24923351C211yesmaximal, characteristic, complemented
14#24923327C211yesmaximal, characteristic, complemented
15#485×C167C211yesmaximal, characteristic, complemented
16C22×D668C2211yescharacteristic
17C2×#14581937C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×D1336C2211yescharacteristic
20C2×D1336C2211yescharacteristic
21#19128831C2211yescharacteristic, complemented
22#19128827—21no—
23#19128821C2211yescharacteristic, complemented
24#19128818—21no—
25#19128814—21no—
26#19128812—21no—
27#19128806C2211yescharacteristic
28Q8×Dic167C2211yescharacteristic, complemented
29C4×D668—21no—
30C4×D668—21no—
31#19128794C2211yescharacteristic
32#19128794C2211yescharacteristic
33#19128794C2211yescharacteristic
34#19128794C2211yescharacteristic
35#19128794C2211yescharacteristic, complemented
36#19128791C2211yescharacteristic
37C22⋊C4×D167—21no—
38C22⋊C4×D167—21no—
39#19128765C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128754C2211yescharacteristic
44#19128754C2211yescharacteristic
45#19128752C2211yescharacteristic
46#19128752C2211yescharacteristic
47#19128752C2211yescharacteristic, complemented
48#19128752C2211yescharacteristic, complemented
49C22⋊(C167⋊C8)C2211yescharacteristic
50#19128742C2211yescharacteristic, complemented
51C8×Dic167C2211yescharacteristic, complemented
52C22⋊Q8×C167C2211yescharacteristic
53#117×C167C2211yescharacteristic
54#107×C167C2211yescharacteristic, complemented
55#106×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#103×C167C2211yescharacteristic, complemented
58C22⋊C8×C167C2211yescharacteristic
59C23×D167—21no—
60C2×Dic334—21no—
61C2×D668—22no—
62C2×D668—22no—
63C2×D668—21no—
64C2×D668C2311yescharacteristic
65C2×D668C2311yescharacteristic
66C2×D668C2311yescharacteristic
67C2×C22⋊D167—21no—
68C2×C4×D167—21no—
69C2×C4×D167—21no—
70C2×C4×D167—21no—
71C2×C4×D167—21no—
72C2×C4×D167—21no—
73#14581937—22no—
74#14581937—22no—
75D1336—22no—
76D1336—22no—
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C4⋊Dic167—21no—
80C4⋊Dic167D411yescharacteristic
81C4⋊Dic167D411yescharacteristic
82C4⋊Dic167C2311yescharacteristic
83Dic167⋊C4—21no—
84C22⋊Dic167D411yescharacteristic
85C22⋊Dic167D411yescharacteristic
86(C2×D167)⋊C4—21no—
87(C2×D167)⋊C4—21no—
88(C2×D167)⋊C4—21no—
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94C4×Dic167—21no—
95C4×Dic167C2311yescharacteristic
96C4×Dic167C2311yescharacteristic
97C2×Q8×C167C2311yescharacteristic
98C22×C4×C167C2311yescharacteristic
99C2×C8×C167C2311yescharacteristic
100C2×C8×C167C2311yescharacteristic
101C4⋊C4×C167—21no—
102C4⋊C4×C167C2311yescharacteristic
103C4⋊C4×C167C2311yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C42×C167—21no—
108C22×D167—21no—
109C22×D167—21no—
110C22×D167—24no—
111C22×D167—21no—
112C22×D167—81no—
113C22×D167—81no—
114Dic334—81no—
115D668—24no—
116D668—21no—
117D668—21no—
118D668—21no—
119D668—21no—
120D668#4112yes—
121D668#4112yes—
122C22⋊D167—81no—
123C2×Dic167—21no—
124C2×Dic167C2×D411yescharacteristic
125C2×Dic167C2×D411yescharacteristic
126C2×Dic167C2×D411yescharacteristic
127C2×Dic167C2×D411yescharacteristic
128C4×D167—81no—
129C4×D167—81no—
130C4×D167—81no—
131C4×D167—81no—
132C4×D167—81no—
133C167⋊C8—21no—
134C167⋊C8—21no—
135C23×C167C2×D411yescharacteristic
136Q8×C167—21no—
137Q8×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×C167C2411yescharacteristic
146C8×C167—21no—
147C8×C167—21no—
148C22×C167—21no—
149C22×C167—21no—
150C22×C167—21no—
151C22×C167C22×D411yescharacteristic
152C2×D167—22no—
153C2×D167—22no—
154C2×D167—41no—
155C2×D167—41no—
156C2×D167—41no—
157C2×D167—41no—
158C2×D167—41no—
159C2×D167—41no—
160C2×D167—41no—
161C2×D167—84no—
162C4×C167—21no—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167—41no—
167C4×C167C2×#4111yescharacteristic
168C4×C167C2×#4111yescharacteristic
169Dic167—21no—
170Dic167—21no—
171Dic167—21no—
172Dic167—21no—
173Dic167—41no—
174C2×C167—41no—
175C2×C167#52511yescharacteristic
176C2×C167#57611yescharacteristic
177C2×C167#57711yescharacteristic
178D167—42no—
179D167—42no—
180D167—81no—
181C167#311811yescharacteristic, complemented, Sylow
182#3118—1671nomaximal, Sylow
183#542—1671no—
184#518—1671no—
185#485D16711yescharacteristic, complemented
186#482—1671no—
187#476—1671no—
188#474—1671no—
189#463—1671no—
190#463—1671no—
191#450—1671no—
192#447—1671no—
193#442—1671no—
194#440—1671no—
195C4×SD16—1671no—
196C4×D8—1671no—
197#433—1671no—
198C22×D4—1671no—
199C2×SD16—1671no—
200C2×SD16—1671no—
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#117C2×D16711yescharacteristic
221C2×C22⋊C4—3341no—
222C2×C22⋊C4—3341no—
223#107—1671no—
224#107C2×D16711yescharacteristic, complemented
225#106—1671no—
226#106C2×D16711yescharacteristic, complemented
227C4⋊C8—1671no—
228C4⋊C8—1671no—
229#103C2×D16711yescharacteristic, complemented
230#103C2×D16711yescharacteristic, complemented
231#102—1671no—
232#102—1671no—
233#102—1671no—
234#102—1671no—
235#102—1671no—
236#102—1671no—
237C22⋊C8—1671no—
238C22⋊C8C2×D16711yescharacteristic
239#97—1671no—
240C4×C8—1671no—
241C24—3341no—
242C2×Q8—3341no—
243C2×Q8C22×D16711yescharacteristic
244C2×D4—3342no—
245C2×D4—3342no—
246C2×D4—3341no—
247C2×D4—3341no—
248C2×D4—1671no—
249C2×D4—1671no—
250C2×D4—1671no—
251C22×C4—3341no—
252C22×C4—3341no—
253C22×C4—3341no—
254C22×C4—3341no—
255C22×C4—3341no—
256C22×C4C22×D16711yescharacteristic
257SD16—3342no—
258SD16—3342no—
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⋊C4C22×D16711yescharacteristic
272C4⋊C4C22×D16711yescharacteristic
273C4⋊C4C22×D16711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
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—21no—
287C42—3341no—
288C42—1671no—
289C42—1671no—
290C23—13361no—
291C23—13361no—
292C23—3344no—
293C23—3341no—
294C23—3341no—
295C23—3341no—
296C23D4×D16711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—13361no—
300D4—13361no—
301D4—3344no—
302D4—3341no—
303D4—3341no—
304D4—3341no—
305D4—3341no—
306D4—1672no—
307D4—1672no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—13361no—
315C2×C4—13361no—
316C2×C4—13361no—
317C2×C4—13361no—
318C2×C4—13361no—
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—21no—
332C22—21no—
333C22—13364no—
334C22—3342no—
335C22—3342no—
336C22—6681no—
337C22—6681no—
338C22—6681no—
339C22—6681no—
340C22—6681no—
341C22—6681no—
342C22—6681no—
343C22C2×D4×D16711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3341no—
350C4—3341no—
351C4—3341no—
352C4—3341no—
353C4—6681no—
354C4#41×D16711yescharacteristic
355C4#41×D16711yescharacteristic
356C2—41no—
357C2—13361no—
358C2—6682no—
359C2—6682no—
360C2#2492377711yescharacteristic
361C2#2492407111yescharacteristic
362C2#2492407811yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#485×C167C2
socleC22×C167C22×D4
prime radical#24923405C2
hypercenter#485D167
hypocenterC167#3118
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #24923892C2semidirect
2. #24923743C2semidirect
3. #24923593C2semidirect
4. #24923558C2semidirect
5. #24923548C2semidirect
6. #24923487C2semidirect
7. #24923487C2semidirect
8. #24923419C2semidirect
10. #24923380C2semidirect
11. #24923372C2semidirect
12. #24923354C2semidirect
13. #24923351C2semidirect
14. #24923327C2semidirect
15. #485×C167C2semidirect
21. #19128831C22semidirect
23. #19128821C22semidirect
28. Q8×Dic167C22semidirect
35. #19128794C22semidirect
39. #19128765C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. Dic167⋊C8C22semidirect
47. #19128752C22semidirect
48. #19128752C22semidirect
50. #19128742C22semidirect
51. C8×Dic167C22semidirect
54. #107×C167C22semidirect
55. #106×C167C22semidirect
56. #103×C167C22semidirect
57. #103×C167C22semidirect
181. C167#3118semidirect
185. #485D167semidirect
224. #107C2×D167semidirect
226. #106C2×D167semidirect
229. #103C2×D167semidirect
230. #103C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249238922
#249237432
#249235932
#249235582
#249235482
#249234872
#249234192
#249234052
#249233802
#249233722
#249233542
#249233512
#249233272
#485×C1672
#3118167