Finite Group Databasebeta

← #35224453

Subgroups — #35224453

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782177C211yesmaximal, characteristic
2#117×D163C211yesmaximal, characteristic, complemented
3#23782009C211yesmaximal, characteristic, complemented
4#23782008C211yesmaximal, characteristic, complemented
5#23782002C211yesmaximal, characteristic, complemented
6#23781997C211yesmaximal, characteristic, complemented
7#23781992C211yesmaximal, characteristic, complemented
8#23781991C211yesmaximal, characteristic, complemented
9#23781985C211yesmaximal, characteristic, complemented
10#23781980C211yesmaximal, characteristic, complemented
11#23781701C211yesmaximal, characteristic, complemented
12#23781662C211yesmaximal, characteristic, complemented
13#23781659C211yesmaximal, characteristic, complemented
14C2×(C2×D163)⋊C8C211yesmaximal, characteristic, complemented
15#427×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C22×C163⋊C8C2211yescharacteristic
19#19104719—21no—
20#19104715C2211yescharacteristic
21#19104715C2211yescharacteristic
22#19104710C2211yescharacteristic
23#19104710C2211yescharacteristic
24#19104705—21no—
25C4×D652—21no—
26C4×D652—21no—
27#19104694C2211yescharacteristic
28#19104693C2211yescharacteristic
29#19104693C2211yescharacteristic
30#19104692—21no—
31#19104691—21no—
32C4⋊C4×D163C2211yescharacteristic
33C4⋊C4×D163C2211yescharacteristic
34C22⋊C4×D163—21no—
35C42×D163—21no—
36#19104668C2211yescharacteristic, complemented
37#19104668C2211yescharacteristic, complemented
38#19104666C2211yescharacteristic, complemented
39#19104666C2211yescharacteristic, complemented
40#19104660C2211yescharacteristic, complemented
41#19104660C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic, complemented
46#19104655C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic, complemented
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52C22×C8×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54#117×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic, complemented
56#107×C163C2211yescharacteristic, complemented
57#106×C163C2211yescharacteristic, complemented
58#106×C163C2211yescharacteristic, complemented
59C23×D163D411yescharacteristic
60#14579699—22no—
61#14579699—22no—
62C2×Dic326C2311yescharacteristic
63C2×D652—21no—
64C2×D652—21no—
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C22×Dic163D411yescharacteristic
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—22no—
72C2×C4×D163—22no—
73C2×C4×D163D411yescharacteristic
74C2×C4×D163D411yescharacteristic
75C2×C4×D163C2311yescharacteristic
76C2×C4×D163C2311yescharacteristic
77C2×C163⋊C8—22no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C22×C4×C163C2311yescharacteristic
97C2×C8×C163—22no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C42×C163—21no—
107C42×C163—21no—
108C22×D163—22no—
109C22×D163—22no—
110C22×D163—21no—
111C22×D163—41no—
112C22×D163C2×D411yescharacteristic
113C22×D163C2×D411yescharacteristic
114Dic326—21no—
115Dic326—21no—
116D652—21no—
117D652—21no—
118D652—42no—
119D652—42no—
120C22⋊D163—42no—
121C2×Dic163—22no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—22no—
128C4×D163—22no—
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—42no—
134C4×D163—42no—
135C4×D163—42no—
136C163⋊C8—21no—
137C163⋊C8—21no—
138C23×C163C2×D411yescharacteristic
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163#4112yes—
145C2×C4×C163#4112yes—
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163#12312yes—
152C22×C163C22×D411yescharacteristic
153C2×D163—21no—
154C2×D163—21no—
155C2×D163—21no—
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—44no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163#13512yes—
163C4×C163—41no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163#12312yes—
168C4×C163C2×#4111yescharacteristic
169C4×C163C2×#4111yescharacteristic
170Dic163—21no—
171Dic163—21no—
172Dic163—41no—
173Dic163—41no—
174Dic163—41no—
175C2×C163—22no—
176C2×C163C2×#12311yescharacteristic
177C2×C163C2×#13511yescharacteristic
178C2×C163#57511yescharacteristic
179D163—22no—
180D163—41no—
181D163—81no—
182C163#309111yescharacteristic, complemented, Sylow
183#3091—1631nomaximal, Sylow
184#525—1631no—
185C2×#117—1631no—
186#485—1631no—
187#485—1631no—
188#484—1631no—
189#483—1631no—
190#482—1631no—
191#482—1631no—
192#481—1631no—
193#480—1631no—
194#427—1631no—
195#427D16311yescharacteristic, complemented
196#416—1631no—
197#416—1631no—
198C2×C22⋊C8—1631no—
199C2×#41—1631no—
200C23×C4—1631no—
201C22×C8—1631no—
202C22×C8C2×D16311yescharacteristic
203#123—3261no—
204C22⋊Q8—1631no—
205C22⋊Q8—1631no—
206#121—1631no—
207#121—1631no—
208#120—3261no—
209C4×D4—3261no—
210C4×D4—3261no—
211#117—3261no—
212#117—3261no—
213#117—1631no—
214#117—1631no—
215#117—1631no—
216#117C2×D16311yescharacteristic
217#117C2×D16311yescharacteristic
218C2×C4⋊C4—1631no—
219C2×C4⋊C4—1631no—
220C2×C22⋊C4—3261no—
221C2×C42—3261no—
222#107—1631no—
223#107—1631no—
224#107C2×D16311yescharacteristic, complemented
225#107C2×D16311yescharacteristic, complemented
226#106—1631no—
227#106—1631no—
228#106C2×D16311yescharacteristic, complemented
229#106C2×D16311yescharacteristic, complemented
230#103—1631no—
231#103—1631no—
232#103—1631no—
233#103—1631no—
234#102—1631no—
235#102—1631no—
236#102—1631no—
237#102—1631no—
238C22⋊C8—1631no—
239C22⋊C8—1631no—
240C22⋊C8—1631no—
241C22⋊C8—1631no—
242C24—1631no—
243#41—3262no—
244#41—3262no—
245C2×Q8—1631no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—3261no—
249C2×D4—1631no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3262no—
253C22×C4—3262no—
254C22×C4—3261no—
255C22×C4—1631no—
256C22×C4—1631no—
257C22×C4—1631no—
258C22×C4—1631no—
259C22×C4—1631no—
260C22×C4C22×D16311yescharacteristic
261C2×C8—22no—
262C2×C8—3262no—
263C2×C8—1631no—
264C2×C8—1631no—
265C2×C8C22×D16311yescharacteristic
266C2×C8C22×D16311yescharacteristic
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C42—21no—
287C42—21no—
288C42—3261no—
289C42—3261no—
290C42—3261no—
291C23—3262no—
292C23—3262no—
293C23—3261no—
294C23—6521no—
295C23—1631no—
296C23—1631no—
297C23D4×D16311yescharacteristic
298Q8—3261no—
299Q8—3261no—
300D4—3261no—
301D4—3261no—
302D4—6522no—
303D4—6522no—
304D4—6522no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3262no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3262no—
315C2×C4—3262no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—6522no—
320C2×C4—6522no—
321C2×C4—6522no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4D4×D16311yescharacteristic
325C2×C4#1457969912yes—
326C2×C4#1457970012yes—
327C2×C4C23×D16311yescharacteristic
328C8—21no—
329C8—21no—
330C8—3261no—
331C8—3261no—
332C22—21no—
333C22—21no—
334C22—3261no—
335C22—3261no—
336C22—3261no—
337C22—3261no—
338C22—3261no—
339C22—6524no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22—1632no—
344C22#1910472012yes—
345C22C2×D4×D16311yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3261no—
351C4—3261no—
352C4—6521no—
353C4—6521no—
354C4—6521no—
355C4#1910472012yes—
356C4C2×#1457969911yescharacteristic
357C4C2×#1457970011yescharacteristic
358C2—22no—
359C2—13041no—
360C2—3262no—
361C2—6521no—
362C2C2×#1910472011yescharacteristic
363C2#135×D16311yescharacteristic
364C2#2378246211yescharacteristic
subgroupG/H
centerC2×C4D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#427×C163C2
socleC22×C163C22×D4
prime radical#23782177C2
hypercenter#427D163
hypocenterC163#3091
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #117×D163C2semidirect
3. #23782009C2semidirect
4. #23782008C2semidirect
5. #23782002C2semidirect
6. #23781997C2semidirect
7. #23781992C2semidirect
8. #23781991C2semidirect
9. #23781985C2semidirect
10. #23781980C2semidirect
11. #23781701C2semidirect
12. #23781662C2semidirect
13. #23781659C2semidirect
14. C2×(C2×D163)⋊C8C2semidirect
15. #427×C163C2semidirect
36. #19104668C22semidirect
37. #19104668C22semidirect
38. #19104666C22semidirect
39. #19104666C22semidirect
40. #19104660C22semidirect
41. #19104660C22semidirect
42. #19104658C22semidirect
43. #19104658C22semidirect
44. #19104657C22semidirect
45. #19104657C22semidirect
46. #19104655C22semidirect
47. #19104655C22semidirect
55. #107×C163C22semidirect
56. #107×C163C22semidirect
57. #106×C163C22semidirect
58. #106×C163C22semidirect
182. C163#3091semidirect
195. #427D163semidirect
224. #107C2×D163semidirect
225. #107C2×D163semidirect
228. #106C2×D163semidirect
229. #106C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237821772
#117×D1632
#237820092
#237820082
#237820022
#237819972
#237819922
#237819912
#237819852
#237819802
#237817012
#237816622
#237816592
C2×(C2×D163)⋊C82
#427×C1632
#3091163