Finite Group Databasebeta

← #36362918

Subgroups — #36362918

Order 22144.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149725C211yesmaximal, characteristic, complemented
2#26012289C211yesmaximal, characteristic, complemented
3#26012055C211yesmaximal, characteristic, complemented
4#26011667C214yesmaximal, complemented
5#26011657C214yesmaximal, complemented
6#26011508C211yesmaximal, characteristic, complemented
7#26011507C211yesmaximal, characteristic, complemented
8C2×#19149595C211yesmaximal, characteristic, complemented
9#575×C173C211yesmaximal, characteristic, complemented
10#19149725—22no—
11#19149725—22no—
12#19149725—22no—
13#19149725C2214yes—
14C2×#14583827—21no—
15C2×#14583826—21no—
16C2×#14583826C2211yescharacteristic
17C22×D692—21no—
18C22×D692C2211yescharacteristic
19C2×D4×D173—21no—
20C2×D4×D173—22no—
21#19149710C2212yescomplemented
22#19149709—21no—
23#19149705C2212yescomplemented
24#19149704—21no—
25#19149699—22no—
26C2×#14583810C2211yescharacteristic, complemented
27#19149675C2212yescomplemented
28#19149668C2212yescomplemented
29#19149653—22no—
30#19149647—22no—
31#19149638C2211yescharacteristic, complemented
32#19149606C2214yescomplemented
33#19149605C2214yescomplemented
34#19149595C2214yescomplemented
35C2×#41×C173C2211yescharacteristic
36C2×#41×C173C2211yescharacteristic, complemented
37#137×C173—21no—
38#137×C173C2212yescomplemented
39AGL(1,ℤ/8)×C173—21no—
40AGL(1,ℤ/8)×C173C2212yescomplemented
41#135×C173—22no—
42C2×M16×C173C2211yescharacteristic, complemented
43C23×D173—22no—
44C23×D173D412yes—
45#14583827—24no—
46#14583827—24no—
47#14583826—24no—
48#14583826—24no—
49#14583826—21no—
50#14583826—21no—
51#14583826D412yescomplemented
52#14583826C2312yes—
53C2×Dic346D411yescharacteristic, complemented
54C2×D692—22no—
55C2×D692—24no—
56C2×D692—24no—
57C2×D692—21no—
58C2×D692—21no—
59C2×D692—21no—
60C2×D692—21no—
61C2×D692D412yes—
62C2×D692D411yescharacteristic, complemented
63C2×D692C2312yes—
64C2×C22⋊D173—22no—
65C2×C22⋊D173—24no—
66C2×C22⋊D173—21no—
67D4×D173—28no—
68D4×D173—24no—
69D4×D173—28no—
70D4×D173—24no—
71C2×C4×D173—22no—
72C2×C4×D173—21no—
73#14583816—22no—
74#14583814—22no—
75#14583813—22no—
76#14583811—22no—
77#14583810—21no—
78#14583810C2312yescomplemented
79C2×#13063436—21no—
80C4⋊Dic173—21no—
81C22⋊Dic173—21no—
82#14583803—24no—
83C4×Dic173C2312yescomplemented
84#41×C173—21no—
85#41×C173—21no—
86#41×C173—21no—
87#41×C173—21no—
88#41×C173—21no—
89#41×C173—21no—
90#41×C173D412yescomplemented
91#41×C173C2312yes—
92C2×Q8×C173D411yescharacteristic, complemented
93C2×Q8×C173C2311yescharacteristic, complemented
94C2×D4×C173—21no—
95C2×D4×C173—21no—
96C2×D4×C173D411yescharacteristic, complemented
97C2×D4×C173C2311yescharacteristic, complemented
98C22×C4×C173—21no—
99C22×C4×C173—21no—
100C22×C4×C173C2311yescharacteristic
101Q16×C173—22no—
102SD16×C173—22no—
103SD16×C173—22no—
104D8×C173—22no—
105M16×C173—21no—
106M16×C173C2312yescomplemented
107C2×C8×C173—21no—
108C22×D173—22no—
109C22×D173—21no—
110C22×D173—28no—
111C22×D173—22no—
112C22×D173—82no—
113C22×D173—84no—
114C22×D173C2×D414yes—
115Dic346—21no—
116Dic346C2×D412yes—
117D692—28no—
118D692—22no—
119D692—22no—
120D692—21no—
121D692—22no—
122D692—22no—
123D692C2×D412yes—
124C22⋊D173—28no—
125C22⋊D173—22no—
126C22⋊D173—22no—
127C22⋊D173—84no—
128C2×Dic173—21no—
129C2×Dic173—22no—
130C4×D173—28no—
131C4×D173—22no—
132C4×D173—22no—
133#13063436—22no—
134C23×C173—21no—
135C23×C173—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173C2×D412yes—
141D4×C173—22no—
142D4×C173—22no—
143D4×C173—21no—
144D4×C173—21no—
145D4×C173—21no—
146D4×C173—41no—
147D4×C173—41no—
148D4×C173C2×D412yes—
149C2×C4×C173—21no—
150C2×C4×C173—22no—
151C2×C4×C173—22no—
152C2×C4×C173—21no—
153C2×C4×C173—41no—
154C2×C4×C173—41no—
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C8×C173—22no—
162C22×C173—22no—
163C22×C173—41no—
164C22×C173—42no—
165C22×C173—42no—
166C22×C173—41no—
167C22×C173#12011yescharacteristic
168C22×C173#12011yescharacteristic
169C22×C173C22×D411yescharacteristic
170C2×D173—24no—
171C2×D173—22no—
172C2×D173—44no—
173C2×D173—42no—
174C2×D173—44no—
175C2×D173—82no—
176C2×D173—88no—
177C4×C173—22no—
178C4×C173—41no—
179C4×C173#12011yescharacteristic
180C4×C173#12011yescharacteristic
181C4×C173C22×D411yescharacteristic
182C4×C173C22×D411yescharacteristic
183Dic173—22no—
184Dic173—42no—
185C2×C173—21no—
186C2×C173—21no—
187C2×C173—21no—
188C2×C173—42no—
189C2×C173—81no—
190C2×C173C2×#12011yescharacteristic
191D173—44no—
192D173—42no—
193C173#301911yescharacteristic, complemented, Sylow
194#3019—1731nomaximal, Sylow
195C2×H(5,𝔽2)—1731no—
196#575—1731no—
197#575D17311yescharacteristic, complemented
198#535—1731no—
199#455—1734no—
200#453—1734no—
201#421—1731no—
202#421—1731no—
203C2×#100—1731no—
204H(5,𝔽2)—3462no—
205H(5,𝔽2)—3462no—
206H(5,𝔽2)—3462no—
207H(5,𝔽2)—1734no—
208C2×#41—3461no—
209C2×#41—3461no—
210C2×#41—1731no—
211C2×#41C2×D17311yescharacteristic
212C2×#41C2×D17311yescharacteristic, complemented
213C22×D4—3461no—
214C22×D4—3462no—
215C22×D4—3461no—
216C22×D4—1731no—
217#137—21no—
218#137—3461no—
219#137—1732no—
220#137C2×D17312yescomplemented
221AGL(1,ℤ/8)—21no—
222AGL(1,ℤ/8)—3461no—
223AGL(1,ℤ/8)—1732no—
224AGL(1,ℤ/8)C2×D17312yescomplemented
225#135—22no—
226#135—3462no—
227C2×M16—1731no—
228C2×M16C2×D17311yescharacteristic, complemented
229#127—1732no—
230#124—1732no—
231#121—3462no—
232#120—3462no—
233#117—1731no—
234#104—1734no—
235#104—1734no—
236#100—1734no—
237C24—3462no—
238C24—1732no—
239#41—21no—
240#41—21no—
241#41—21no—
242#41—21no—
243#41—21no—
244#41—21no—
245#41—3464no—
246#41—3464no—
247#41—3464no—
248#41—3464no—
249#41—3461no—
250#41—3461no—
251#41—1732no—
252#41—1732no—
253#41C22⋊D17312yescomplemented
254#41C22×D17312yes—
255C2×Q8—1731no—
256C2×Q8C22⋊D17311yescharacteristic, complemented
257C2×Q8C22×D17311yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—3462no—
261C2×D4—3464no—
262C2×D4—3464no—
263C2×D4—3468no—
264C2×D4—3464no—
265C2×D4—3468no—
266C2×D4—3462no—
267C2×D4—3461no—
268C2×D4—3464no—
269C2×D4—3464no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—1732no—
275C2×D4—1731no—
276C2×D4—1732no—
277C2×D4C22⋊D17311yescharacteristic, complemented
278C2×D4C22×D17311yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3462no—
282C22×C4—3461no—
283C22×C4C22×D17311yescharacteristic
284Q16—22no—
285Q16—3462no—
286SD16—22no—
287SD16—22no—
288SD16—3462no—
289SD16—3462no—
290D8—22no—
291D8—3462no—
292M16—21no—
293M16—3461no—
294M16—1732no—
295M16C22×D17312yescomplemented
296C2×C8—21no—
297C2×C8—3461no—
298C4⋊C4—3461no—
299C22⋊C4—3464no—
300C22⋊C4—3461no—
301C42—1732no—
302C23—21no—
303C23—21no—
304C23—3468no—
305C23—3462no—
306C23—3462no—
307C23—3461no—
308C23—13844no—
309C23—13842no—
310C23—1734no—
311C23D4×D17311yescharacteristic
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—3461no—
316Q8—1732no—
317Q8C2×C22⋊D17312yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—41no—
324D4—41no—
325D4—3468no—
326D4—3468no—
327D4—3462no—
328D4—3462no—
329D4—3462no—
330D4—3461no—
331D4—3462no—
332D4—3462no—
333D4—3462no—
334D4—13844no—
335D4—1732no—
336D4C2×C22⋊D17312yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—3468no—
344C2×C4—3461no—
345C2×C4—3462no—
346C2×C4—3462no—
347C2×C4—3462no—
348C2×C4D4×D17311yescharacteristic
349C2×C4D4×D17311yescharacteristic
350C2×C4D4×D17311yescharacteristic
351C2×C4C2×C22⋊D17311yescharacteristic
352C2×C4C2×C22⋊D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—22no—
355C8—3462no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—41no—
361C22—3464no—
362C22—3462no—
363C22—13842no—
364C22—13848no—
365C22—6924no—
366C22—6922no—
367C22—6924no—
368C22#1914964711yescharacteristic
369C22#1914964711yescharacteristic
370C22C2×D4×D17311yescharacteristic
371C4—22no—
372C4—41no—
373C4—3462no—
374C4—6922no—
375C4#1914964711yescharacteristic
376C4#1914964711yescharacteristic
377C4C2×D4×D17311yescharacteristic
378C4C22×C22⋊D17311yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—81no—
384C2—6922no—
385C2—6924no—
386C2C2×#1914964711yescharacteristic
subgroupG/H
centerC2C2×#19149647
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#575×C173C2
socleC2×C173C2×#120
hypercenter#575D173
hypocenterC173#3019
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2semidirect
2. #26012289C2semidirect
3. #26012055C2semidirect
4. #26011667C2semidirect
5. #26011657C2semidirect
6. #26011508C2semidirect
7. #26011507C2semidirect
8. C2×#19149595C2semidirect
9. #575×C173C2semidirect
21. #19149710C22semidirect
23. #19149705C22semidirect
26. C2×#14583810C22semidirect
27. #19149675C22semidirect
28. #19149668C22semidirect
31. #19149638C22semidirect
32. #19149606C22semidirect
33. #19149605C22semidirect
34. #19149595C22semidirect
36. C2×#41×C173C22semidirect
38. #137×C173C22semidirect
40. AGL(1,ℤ/8)×C173C22semidirect
42. C2×M16×C173C22semidirect
51. #14583826D4semidirect
53. C2×Dic346D4semidirect
62. C2×D692D4semidirect
78. #14583810C23semidirect
83. C4×Dic173C23semidirect
90. #41×C173D4semidirect
92. C2×Q8×C173D4semidirect
93. C2×Q8×C173C23semidirect
96. C2×D4×C173D4semidirect
97. C2×D4×C173C23semidirect
106. M16×C173C23semidirect
193. C173#3019semidirect
197. #575D173semidirect
212. C2×#41C2×D173semidirect
220. #137C2×D173semidirect
224. AGL(1,ℤ/8)C2×D173semidirect
228. C2×M16C2×D173semidirect
253. #41C22⋊D173semidirect
256. C2×Q8C22⋊D173semidirect
257. C2×Q8C22×D173semidirect
277. C2×D4C22⋊D173semidirect
278. C2×D4C22×D173semidirect
295. M16C22×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
#260122892
#260120552
#260116672
#260116572
#260115082
#260115072
C2×#191495952
#575×C1732
#3019173