Finite Group Databasebeta

← #35226087

Subgroups — #35226087

Order 20864.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104765C211yesmaximal, characteristic, complemented
2C2×#19104755C211yesmaximal, characteristic, complemented
3#23782324C211yesmaximal, characteristic, complemented
4#23782057C211yesmaximal, characteristic, complemented
5#23781913C211yesmaximal, characteristic, complemented
6#23781904C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781872C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781825C211yesmaximal, characteristic, complemented
11#23781820C211yesmaximal, characteristic, complemented
12#23781810C211yesmaximal, characteristic, complemented
13#23781765C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15#547×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18#19104765—24no—
19#19104755—24no—
20C2×#14579689C2211yescharacteristic
21C2×#14579689C2211yescharacteristic
22C2×#14579689C2211yescharacteristic
23C2×#14579689C2211yescharacteristic
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic
26C2×#14579686C2211yescharacteristic
27C2×#14579686C2211yescharacteristic
28C2×#14579684C2211yescharacteristic
29C2×#14579683C2211yescharacteristic
30C22×C163⋊C8C2211yescharacteristic
31#19104723C2211yescharacteristic
32#19104721—21no—
33#19104714C2211yescharacteristic
34#19104713—21no—
35#19104709C2211yescharacteristic
36#19104707—21no—
37C4×Dic326C2211yescharacteristic
38(C22⋊D163)⋊C4—21no—
39#19104668C2211yescharacteristic
40C4⋊(C163⋊C8)C2211yescharacteristic
41#19104659C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104655C2211yescharacteristic
47C22⋊(C163⋊C8)C2211yescharacteristic
48C22⋊(C163⋊C8)C2211yescharacteristic
49#19104646C2211yescharacteristic
50C2×#41×C163C2211yescharacteristic
51C2×#41×C163C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53#123×C163—21no—
54#123×C163—21no—
55C22⋊Q8×C163C2211yescharacteristic
56C22⋊Q8×C163C2211yescharacteristic
57#121×C163—21no—
58C4×D4×C163C2211yescharacteristic
59C4×D4×C163C2211yescharacteristic
60C2×C4⋊C4×C163—21no—
61#14579699—22no—
62#14579699—41no—
63C2×Dic326—22no—
64C2×Dic326—21no—
65C2×Dic326—41no—
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73#14579689—22no—
74#14579689—22no—
75#14579689—22no—
76#14579689—41no—
77#14579687—22no—
78#14579687D414yescomplemented
79#14579686—22no—
80#14579686—41no—
81#14579684—22no—
82#14579683—22no—
83C2×C163⋊C8—22no—
84C2×C163⋊C8C2311yescharacteristic
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C4⋊Dic163C2311yescharacteristic
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95C4×Dic163—21no—
96#41×C163—22no—
97#41×C163—22no—
98#41×C163—41no—
99#41×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C2×D4×C163—21no—
103C2×D4×C163—21no—
104C2×D4×C163D411yescharacteristic
105C2×D4×C163C2311yescharacteristic
106C22×C4×C163—21no—
107C22×C4×C163—21no—
108C22×C4×C163—21no—
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C4⋊C4×C163—21no—
112C4⋊C4×C163—21no—
113C4⋊C4×C163—21no—
114C4⋊C4×C163—21no—
115C4⋊C4×C163D411yescharacteristic
116C4⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—42no—
128Dic326C2×D412yes—
129D652—21no—
130D652—21no—
131C22⋊D163—42no—
132C22⋊D163—42no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—42no—
137C4×D163—42no—
138C163⋊C8—21no—
139C163⋊C8—21no—
140C163⋊C8—21no—
141C163⋊C8C2×D412yes—
142C23×C163—21no—
143C23×C163C2×D411yescharacteristic
144C23×C163C2×D411yescharacteristic
145Q8×C163—21no—
146Q8×C163—21no—
147Q8×C163—21no—
148Q8×C163—21no—
149D4×C163—21no—
150D4×C163—41no—
151D4×C163—42no—
152D4×C163—42no—
153D4×C163C2×D412yes—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—42no—
162C2×C4×C163—42no—
163C2×C4×C163—42no—
164C2×C4×C163C2×D411yescharacteristic
165C2×C4×C163C2×D411yescharacteristic
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C22×C163—21no—
170C22×C163—21no—
171C22×C163—22no—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163C22×D411yescharacteristic
179C2×D163—41no—
180C2×D163—41no—
181C2×D163—41no—
182C4×C163—21no—
183C4×C163—21no—
184C4×C163—21no—
185C4×C163—21no—
186C4×C163—41no—
187C4×C163—41no—
188C4×C163—41no—
189C4×C163C22×D411yescharacteristic
190C4×C163H(5,𝔽2)11yescharacteristic
191Dic163—22no—
192Dic163—41no—
193Dic163—41no—
194C2×C163—41no—
195C2×C163—41no—
196C2×C163—42no—
197C2×C163D4211yescharacteristic
198C2×C163#57511yescharacteristic
199C2×C163#57811yescharacteristic
200D163—81no—
201C163#326911yescharacteristic, complemented, Sylow
202#3269—1631nomaximal, Sylow
203C2×#137—1631no—
204C2×#135—1631no—
205#548—1631no—
206#547D16311yescharacteristic, complemented
207#497—1631no—
208#468—1631no—
209#467—1631no—
210#463—1631no—
211#462—1631no—
212#452—1631no—
213#452—1631no—
214#451—1631no—
215#449—1631no—
216#440—1631no—
217#435—1631no—
218C2×#41—1631no—
219C2×#41C2×D16311yescharacteristic
220C2×#41C2×D16311yescharacteristic
221C22×Q8—1631no—
222#137—3264no—
223#135—3264no—
224C2×Q16—1631no—
225C2×Q16—1631no—
226C2×Q16—1631no—
227C2×Q16—1631no—
228C2×SD16—1631no—
229C2×SD16—1631no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×D8—1631no—
233C2×M16—1631no—
234C22×C8—1631no—
235C4⋊Q8C2×D16311yescharacteristic
236#124—3261no—
237#124—1631no—
238#123—21no—
239#123—21no—
240C22⋊Q8—3261no—
241C22⋊Q8—1631no—
242C22⋊Q8C2×D16311yescharacteristic
243C22⋊Q8C2×D16311yescharacteristic
244#121—21no—
245#121—3261no—
246#121—1631no—
247C4×Q8—1631no—
248C4×D4—3261no—
249C4×D4C2×D16311yescharacteristic
250C4×D4C2×D16311yescharacteristic
251C2×C4⋊C4—21no—
252#107—1631no—
253C4⋊C8—1631no—
254#103—1631no—
255#103—1631no—
256#103—1631no—
257#103—1631no—
258#102—1631no—
259#102—1631no—
260C22⋊C8—1631no—
261C22⋊C8—1631no—
262#97—1631no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—3262no—
268#41—6521no—
269C2×Q8—3262no—
270C2×Q8—3261no—
271C2×Q8—6521no—
272C2×Q8—1631no—
273C2×Q8—1631no—
274C2×Q8C22×D16311yescharacteristic
275C2×Q8C22×D16311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3261no—
279C2×D4—3261no—
280C2×D4—1631no—
281C2×D4C22⋊D16311yescharacteristic
282C2×D4C22×D16311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—3261no—
287C22×C4—3261no—
288C22×C4C22×D16311yescharacteristic
289C22×C4C22×D16311yescharacteristic
290Q16—3262no—
291Q16—3262no—
292Q16—3262no—
293Q16—6521no—
294SD16—3262no—
295SD16—3262no—
296SD16—6521no—
297SD16—1634no—
298D8—3262no—
299M16—3262no—
300C2×C8—3262no—
301C2×C8—1631no—
302C2×C8—1631no—
303C2×C8—1631no—
304C2×C8—1631no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—3261no—
310C4⋊C4—3261no—
311C4⋊C4—1631no—
312C4⋊C4C22⋊D16311yescharacteristic
313C4⋊C4C22×D16311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—3261no—
318C22⋊C4—3261no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4C22⋊D16311yescharacteristic
322C22⋊C4C22⋊D16311yescharacteristic
323C42—3261no—
324C42C22×D16311yescharacteristic
325C23—21no—
326C23—3261no—
327C23C2×C22⋊D16311yescharacteristic
328C23C2×C22⋊D16311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—3261no—
334Q8—3261no—
335Q8—3261no—
336Q8—6522no—
337Q8—1632no—
338D4—21no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—3261no—
343D4—3261no—
344D4—6522no—
345D4—6522no—
346D4D4×D16312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—3261no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—6522no—
361C2×C4—6522no—
362C2×C4C2×C22⋊D16311yescharacteristic
363C2×C4C2×C22⋊D16311yescharacteristic
364C2×C4C2×C22⋊D16311yescharacteristic
365C2×C4C2×C22⋊D16311yescharacteristic
366C2×C4C23×D16311yescharacteristic
367C8—3261no—
368C8—3261no—
369C8—3261no—
370C8—1632no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—6521no—
381C22—6521no—
382C22—6521no—
383C22C22×C22⋊D16311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—3262no—
392C4—6521no—
393C4—6521no—
394C4C2×D4×D16311yescharacteristic
395C4#1910478011yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—13041no—
400C2D4×C22⋊D16311yescharacteristic
401C2#2378246311yescharacteristic
402C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#547×C163C2
socleC22×C163C22×D4
hypercenter#547D163
hypocenterC163#3269
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19104765C2semidirect
2. C2×#19104755C2semidirect
3. #23782324C2semidirect
4. #23782057C2semidirect
5. #23781913C2semidirect
6. #23781904C2semidirect
7. #23781880C2semidirect
8. #23781872C2semidirect
9. #23781827C2semidirect
10. #23781825C2semidirect
11. #23781820C2semidirect
12. #23781810C2semidirect
13. #23781765C2semidirect
14. #23781737C2semidirect
15. #547×C163C2semidirect
78. #14579687D4semidirect
201. C163#3269semidirect
206. #547D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047652
C2×#191047552
#237823242
#237820572
#237819132
#237819042
#237818802
#237818722
#237818272
#237818252
#237818202
#237818102
#237817652
#237817372
#547×C1632
#3269163