Finite Group Databasebeta

← #35226532

Subgroups — #35226532

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782327C211yesmaximal, characteristic, complemented
2#23782320C211yesmaximal, characteristic, complemented
3#23782025C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23781980C211yesmaximal, characteristic, complemented
6#23781960C211yesmaximal, characteristic, complemented
7#23781910C211yesmaximal, characteristic, complemented
8#23781900C211yesmaximal, characteristic, complemented
9#23781880C211yesmaximal, characteristic, complemented
10#23781809C211yesmaximal, characteristic, complemented
11#23781802C211yesmaximal, characteristic, complemented
12#23781738C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15#440×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699—21no—
18C2×Q8×D163C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C2×#14579682C2211yescharacteristic
22C2×C8×D163C2211yescharacteristic
23#19104735C2211yescharacteristic
24#19104723—21no—
25#19104723C2211yescharacteristic
26#19104717—21no—
27#19104717—21no—
28#19104715—21no—
29#19104715C2211yescharacteristic
30#19104714—21no—
31#19104710—21no—
32#19104710C2211yescharacteristic
33#19104706C2211yescharacteristic
34#19104706C2211yescharacteristic
35C4×D652—21no—
36C4×D652C2211yescharacteristic
37C4⋊C4×D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59#14579700—22no—
60#14579700—41no—
61#14579699—44no—
62C2×Dic326—21no—
63C2×Dic326—21no—
64C2×Dic326C2311yescharacteristic
65Q8×D163—22no—
66Q8×D163—41no—
67C2×D652—21no—
68C2×D652—21no—
69C2×D652D411yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163—21no—
76C2×C4×D163—21no—
77C2×C4×D163C2311yescharacteristic
78C2×C4×D163C2311yescharacteristic
79#14579687—41no—
80#14579684—41no—
81#14579682—41no—
82C2×C163⋊C8C2311yescharacteristic
83C2×C163⋊C8C2311yescharacteristic
84C8×D163—41no—
85C4⋊Dic163—21no—
86C4⋊Dic163—21no—
87C4⋊Dic163D411yescharacteristic
88C4⋊Dic163C2311yescharacteristic
89C4⋊Dic163C2311yescharacteristic
90Dic163⋊C4—21no—
91Dic163⋊C4—21no—
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4D411yescharacteristic
99(C2×D163)⋊C4D411yescharacteristic
100C2×Q8×C163C2311yescharacteristic
101C2×D4×C163C2311yescharacteristic
102C22×C4×C163—21no—
103SD16×C163—22no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163—21no—
111C42×C163C2311yescharacteristic
112C22×D163—21no—
113C22×D163C2×D411yescharacteristic
114C22×D163C2×D411yescharacteristic
115Dic326—21no—
116Dic326—21no—
117Dic326—42no—
118Dic326—42no—
119D652—22no—
120D652—21no—
121D652—21no—
122D652—42no—
123D652—42no—
124C22⋊D163—42no—
125C22⋊D163—42no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—21no—
134C4×D163—21no—
135C4×D163—42no—
136C4×D163—42no—
137C4×D163—42no—
138C4×D163—42no—
139C163⋊C8—21no—
140C163⋊C8—21no—
141C23×C163—21no—
142Q8×C163—21no—
143Q8×C163#4112yescomplemented
144D4×C163—21no—
145D4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—42no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163C22×D411yescharacteristic
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163AGL(1,ℤ/8)12yes—
168C4×C163—22no—
169C4×C163—21no—
170C4×C163—21no—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163C2×#4111yescharacteristic
174C4×C163H(5,𝔽2)11yescharacteristic
175Dic163—21no—
176Dic163—21no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—81no—
181C2×C163#54611yescharacteristic
182C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C163#57811yescharacteristic
184D163—22no—
185D163—41no—
186D163—81no—
187C163#330811yescharacteristic, complemented, Sylow
188#3308—1631nomaximal, Sylow
189#548—1631no—
190#547—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#477—1631no—
195#468—1631no—
196#466—1631no—
197#463—1631no—
198#449—1631no—
199#448—1631no—
200#440D16311yescharacteristic, complemented
201#435—1631no—
202#417—1631no—
203C2×#103—1631no—
204C2×#41—3261no—
205C2×#41—1631no—
206C22×Q8—1631no—
207C2×SD16—1631no—
208C2×SD16C2×D16311yescharacteristic
209C2×D8—1631no—
210C2×M16—1631no—
211C22×C8—1631no—
212C4⋊Q8—1631no—
213#124—3261no—
214#124—1631no—
215#123—3261no—
216#123—3261no—
217C22⋊Q8—3261no—
218C22⋊Q8—3261no—
219C22⋊Q8—1631no—
220#121—3261no—
221#121—1631no—
222#121—1631no—
223#121—1631no—
224C4×Q8C2×D16311yescharacteristic
225C4×D4—3261no—
226C4×D4—1631no—
227C4×D4C2×D16311yescharacteristic
228C2×C4⋊C4—1631no—
229C2×C4⋊C4—1631no—
230#107—1631no—
231#107—1631no—
232#107C2×D16311yescharacteristic
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103—1631no—
238#103—1631no—
239#103C2×D16311yescharacteristic
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102C2×D16311yescharacteristic
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97C2×D16311yescharacteristic
247#41—3262no—
248#41—6524no—
249#41—6521no—
250C2×Q8—3261no—
251C2×Q8—3261no—
252C2×Q8—3262no—
253C2×Q8—6521no—
254C2×Q8—1631no—
255C2×Q8C22×D16311yescharacteristic
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—1631no—
261C2×D4—1631no—
262C2×D4C22×D16311yescharacteristic
263C22×C4—21no—
264C22×C4—3261no—
265C22×C4—3261no—
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—1631no—
269C22×C4—1631no—
270SD16—22no—
271SD16—6521no—
272D8—6521no—
273M16—6521no—
274C2×C8—6521no—
275C2×C8—1631no—
276C2×C8—1631no—
277C2×C8C22×D16311yescharacteristic
278C2×C8C22×D16311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4C22×D16311yescharacteristic
288C4⋊C4C22×D16311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—1631no—
297C22⋊C4—1631no—
298C42—21no—
299C42C22×D16311yescharacteristic
300C23—21no—
301C23—3261no—
302C23—1631no—
303C23—1631no—
304Q8—21no—
305Q8—3261no—
306Q8—3261no—
307Q8—6522no—
308Q8—6522no—
309Q8#1457969912yescomplemented
310D4—21no—
311D4—21no—
312D4—3262no—
313D4—3261no—
314D4—3261no—
315D4—6522no—
316D4—6522no—
317D4—6522no—
318D4—6522no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—6522no—
331C2×C4—6522no—
332C2×C4—6522no—
333C2×C4—6522no—
334C2×C4—1631no—
335C2×C4—1631no—
336C2×C4D4×D16311yescharacteristic
337C2×C4D4×D16311yescharacteristic
338C2×C4C23×D16311yescharacteristic
339C8—21no—
340C8—21no—
341C8—3261no—
342C8—3261no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3261no—
347C22—3261no—
348C22—3261no—
349C22—3261no—
350C22—3261no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22—1632no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—3261no—
362C4—3261no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4C2×#1457969911yescharacteristic
367C4#1910478111yescharacteristic
368C2—81no—
369C2—13041no—
370C2—3262no—
371C2—6521no—
372C2#2378231111yescharacteristic
373C2AGL(1,ℤ/8)×D16311yescharacteristic
374C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#440×C163C2
socleC22×C163C22×D4
hypercenter#440D163
hypocenterC163#3308
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #23782327C2semidirect
2. #23782320C2semidirect
3. #23782025C2semidirect
4. #23782002C2semidirect
5. #23781980C2semidirect
6. #23781960C2semidirect
7. #23781910C2semidirect
8. #23781900C2semidirect
9. #23781880C2semidirect
10. #23781809C2semidirect
11. #23781802C2semidirect
12. #23781738C2semidirect
13. #23781665C2semidirect
14. #103×D163C2semidirect
15. #440×C163C2semidirect
143. Q8×C163#41semidirect
187. C163#3308semidirect
200. #440D163semidirect
309. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823272
#237823202
#237820252
#237820022
#237819802
#237819602
#237819102
#237819002
#237818802
#237818092
#237818022
#237817382
#237816652
#103×D1632
#440×C1632
#3308163