Finite Group Databasebeta

← #35224253

Subgroups — #35224253

Order 20864.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782463C211yesmaximal, characteristic, complemented
2C2×#19104755C211yesmaximal, characteristic, complemented
3C2×#19104742C211yesmaximal, characteristic, complemented
4#23781938C212yesmaximal, complemented
5#23781933C212yesmaximal, complemented
6#23781930C212yesmaximal, complemented
7#23781926C212yesmaximal, complemented
8C2×#19104670C211yesmaximal, characteristic, complemented
9#23781649C211yesmaximal, characteristic, complemented
10#23781647C211yesmaximal, characteristic, complemented
11#575×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13#19104766—21no—
14#19104765C2212yes—
15#19104761—21no—
16#19104760C2212yes—
17#19104755—21no—
18#19104755—21no—
19#19104755—21no—
20#19104755—21no—
21#19104755—22no—
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579687C2211yescharacteristic, complemented
24C2×#14579686C2211yescharacteristic, complemented
25C2×#14579684C2211yescharacteristic, complemented
26#19104742—21no—
27#19104742—21no—
28#19104742C2212yes—
29#19104742C2212yes—
30C2×#14579683—21no—
31C2×#14579683C2211yescharacteristic
32C2×#14579683C2211yescharacteristic
33C22×C163⋊C8—21no—
34C22×C163⋊C8C2211yescharacteristic
35#19104670C2212yescomplemented
36#19104670C2212yescomplemented
37#19104654C2212yescomplemented
38#19104653C2212yescomplemented
39#19104652C2212yescomplemented
40#19104651C2212yescomplemented
41C2×#41×C163C2211yescharacteristic
42C2×#41×C163C2211yescharacteristic
43#137×C163—21no—
44#137×C163C2212yes—
45AGL(1,ℤ/8)×C163—21no—
46AGL(1,ℤ/8)×C163C2212yes—
47#135×C163—22no—
48C2×M16×C163C2211yescharacteristic
49#14579699—21no—
50#14579699—21no—
51#14579699—21no—
52#14579699—21no—
53C2×Dic326C2311yescharacteristic
54C2×D652C2311yescharacteristic
55C2×C22⋊D163—21no—
56C2×C4×D163—21no—
57#14579689—21no—
58#14579689—21no—
59#14579689—22no—
60#14579687—21no—
61#14579687—21no—
62#14579687—22no—
63#14579686—21no—
64#14579686—21no—
65#14579686—22no—
66#14579684—21no—
67#14579684—21no—
68#14579684—22no—
69#14579683—22no—
70#14579683—22no—
71#14579683—21no—
72#14579683—21no—
73#14579683C2312yes—
74#14579683C2312yes—
75C2×C163⋊C8—22no—
76C2×C163⋊C8—22no—
77C2×C163⋊C8—21no—
78C2×C163⋊C8—21no—
79C2×C163⋊C8D411yescharacteristic
80C2×C163⋊C8D411yescharacteristic
81C2×C163⋊C8D411yescharacteristic
82C2×C163⋊C8D411yescharacteristic
83C2×C163⋊C8C2311yescharacteristic
84C2×C163⋊C8C2311yescharacteristic
85#41×C163—21no—
86#41×C163—21no—
87#41×C163—21no—
88#41×C163—21no—
89#41×C163—21no—
90#41×C163—21no—
91#41×C163D412yes—
92#41×C163C2312yes—
93C2×Q8×C163D411yescharacteristic
94C2×Q8×C163C2311yescharacteristic
95C2×D4×C163—21no—
96C2×D4×C163—21no—
97C2×D4×C163D411yescharacteristic
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163—21no—
100C22×C4×C163—21no—
101C22×C4×C163C2311yescharacteristic
102Q16×C163—22no—
103SD16×C163—22no—
104SD16×C163—22no—
105D8×C163—22no—
106M16×C163—21no—
107M16×C163C2312yes—
108C2×C8×C163—21no—
109C22×D163—21no—
110Dic326—21no—
111Dic326—21no—
112D652—21no—
113D652—21no—
114C22⋊D163—41no—
115C22⋊D163—41no—
116C2×Dic163—21no—
117C4×D163—41no—
118C4×D163—41no—
119C163⋊C8—22no—
120C163⋊C8—22no—
121C163⋊C8C2×D411yescharacteristic
122C163⋊C8C2×D411yescharacteristic
123C163⋊C8C2×D411yescharacteristic
124C163⋊C8C2×D411yescharacteristic
125C23×C163—21no—
126C23×C163—21no—
127C23×C163#4111yescharacteristic
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163—21no—
131Q8×C163C2×D412yes—
132D4×C163—22no—
133D4×C163—22no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—41no—
138D4×C163—41no—
139D4×C163C2×D412yes—
140C2×C4×C163—21no—
141C2×C4×C163—22no—
142C2×C4×C163—22no—
143C2×C4×C163—21no—
144C2×C4×C163—41no—
145C2×C4×C163—41no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163#4111yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—22no—
153C22×C163—22no—
154C22×C163—41no—
155C22×C163—42no—
156C22×C163—41no—
157C22×C163—81no—
158C22×C163#12111yescharacteristic
159C22×C163#12111yescharacteristic
160C22×C163C2×#4111yescharacteristic
161C2×D163—41no—
162C2×D163—81no—
163C4×C163—22no—
164C4×C163—41no—
165C4×C163#12111yescharacteristic
166C4×C163#12111yescharacteristic
167C4×C163C22×D411yescharacteristic
168C4×C163C22×D411yescharacteristic
169Dic163—41no—
170C2×C163—21no—
171C2×C163—21no—
172C2×C163—21no—
173C2×C163—42no—
174C2×C163—81no—
175C2×C163C2×#12111yescharacteristic
176D163—81no—
177C163#306711yescharacteristic, complemented, Sylow
178#3067—1631nomaximal, Sylow
179#575—1631no—
180#575D16311yescharacteristic, complemented
181C2×#135—1631no—
182C2×#131—1631no—
183#473—1632no—
184#472—1632no—
185#471—1632no—
186#471—1632no—
187C2×#108—1631no—
188#413—1631no—
189#413—1631no—
190C2×#41—1631no—
191C2×#41C2×D16311yescharacteristic
192C2×#41C2×D16311yescharacteristic
193#137—21no—
194#137—3261no—
195#137—1632no—
196#137C2×D16312yes—
197AGL(1,ℤ/8)—21no—
198AGL(1,ℤ/8)—3261no—
199AGL(1,ℤ/8)—1632no—
200AGL(1,ℤ/8)C2×D16312yes—
201#135—22no—
202#135—3262no—
203#135—3261no—
204#135—3261no—
205#135—3261no—
206#135—3261no—
207C2×Q16—1631no—
208C2×SD16—1631no—
209C2×SD16—1631no—
210C2×D8—1631no—
211#131—3261no—
212#131—3261no—
213#131—1632no—
214#131—1632no—
215C2×M16—3261no—
216C2×M16—1631no—
217C2×M16—1631no—
218C2×M16C2×D16311yescharacteristic
219C22×C8—3261no—
220C22×C8—1631no—
221#108—1632no—
222#108—1632no—
223#101—1632no—
224#101—1632no—
225#100—1632no—
226#100—1632no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—3261no—
234#41—3261no—
235#41—3261no—
236#41—3261no—
237#41C22⋊D16312yes—
238#41C22×D16312yes—
239C2×Q8—1631no—
240C2×Q8C22⋊D16311yescharacteristic
241C2×Q8C22×D16311yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3261no—
245C2×D4—1631no—
246C2×D4C22⋊D16311yescharacteristic
247C2×D4C22×D16311yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3261no—
251C22×C4C22×D16311yescharacteristic
252Q16—22no—
253Q16—3262no—
254Q16—3261no—
255Q16—3261no—
256SD16—22no—
257SD16—22no—
258SD16—3262no—
259SD16—3262no—
260SD16—3261no—
261SD16—3261no—
262SD16—3261no—
263SD16—3261no—
264D8—22no—
265D8—3262no—
266D8—3261no—
267D8—3261no—
268M16—21no—
269M16—3261no—
270M16—3261no—
271M16—3262no—
272M16—3262no—
273M16—1632no—
274M16—1632no—
275M16C22×D16312yes—
276C2×C8—21no—
277C2×C8—3261no—
278C2×C8—3261no—
279C2×C8—3262no—
280C2×C8—3262no—
281C2×C8—1631no—
282C2×C8—1631no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8—1631no—
286C2×C8—1631no—
287C23—21no—
288C23—21no—
289C23—3261no—
290C23#1457969811yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—3261no—
295Q8—3261no—
296Q8C2×C22⋊D16312yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—3261no—
305D4—3261no—
306D4—6521no—
307D4—6521no—
308D4C2×C22⋊D16312yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—3261no—
316C2×C4—6521no—
317C2×C4—6521no—
318C2×C4D4×D16311yescharacteristic
319C2×C4D4×D16311yescharacteristic
320C2×C4C2×C22⋊D16311yescharacteristic
321C2×C4C2×C22⋊D16311yescharacteristic
322C2×C4#1457969811yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—22no—
325C8—3262no—
326C8—3262no—
327C8—1631no—
328C8—1631no—
329C8—1631no—
330C8—1631no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—13041no—
337C22—6521no—
338C22#1910470711yescharacteristic
339C22#1910470711yescharacteristic
340C22C2×#1457969811yescharacteristic
341C4—22no—
342C4—41no—
343C4—6521no—
344C4#1910470711yescharacteristic
345C4#1910470711yescharacteristic
346C4C2×D4×D16311yescharacteristic
347C4C22×C22⋊D16311yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—13041no—
354C2C2×#1910470711yescharacteristic
subgroupG/H
centerC4#19104707
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#575×C163C2
socleC2×C163C2×#121
hypercenter#575D163
hypocenterC163#3067
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #23782463C2semidirect
2. C2×#19104755C2semidirect
3. C2×#19104742C2semidirect
4. #23781938C2semidirect
5. #23781933C2semidirect
6. #23781930C2semidirect
7. #23781926C2semidirect
8. C2×#19104670C2semidirect
9. #23781649C2semidirect
10. #23781647C2semidirect
11. #575×C163C2semidirect
22. C2×#14579689C22semidirect
23. C2×#14579687C22semidirect
24. C2×#14579686C22semidirect
25. C2×#14579684C22semidirect
35. #19104670C22semidirect
36. #19104670C22semidirect
37. #19104654C22semidirect
38. #19104653C22semidirect
39. #19104652C22semidirect
40. #19104651C22semidirect
177. C163#3067semidirect
180. #575D163semidirect

Maximal subgroups

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

subgroupindex
#237824632
C2×#191047552
C2×#191047422
#237819382
#237819332
#237819302
#237819262
C2×#191046702
#237816492
#237816472
#575×C1632
#3067163