Finite Group Databasebeta

← #44409088

Subgroups — #44409088

Order 29056.

762 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 133 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
1H(5,𝔽2)×D227C211yesmaximal, characteristic, complemented
2#28409828C212yesmaximal, complemented
3#28409595C211yesmaximal, characteristic, complemented
4#28409212C211yesmaximal, characteristic, complemented
5#28409211C212yesmaximal, complemented
6#28409210C211yesmaximal, characteristic, complemented
7#28409199C211yesmaximal, characteristic, complemented
8#28409198C212yesmaximal, complemented
9#28409047C212yesmaximal, complemented
10#100×D227C211yesmaximal, characteristic, complemented
11#453×C227C211yesmaximal, characteristic, complemented
12#20300334—22no—
13#20300334C2212yes—
14#20300333—22no—
15#20300333—22no—
16#20300333C2211yescharacteristic
17C2×#16813883C2211yescharacteristic, complemented
18#41×D227—22no—
19#41×D227C2212yes—
20C2×D4×D227—22no—
21C2×D4×D227—22no—
22C2×D4×D227C2211yescharacteristic
23#20300321C2212yescomplemented
24#20300318C2212yescomplemented
25#20300317—22no—
26#20300314C2212yescomplemented
27#20300312—22no—
28#20300309—22no—
29#20300307—22no—
30M16×D227C2212yescomplemented
31#20300284C2211yescharacteristic, complemented
32#20300276C2211yescharacteristic, complemented
33#20300275C2211yescharacteristic, complemented
34#20300260—22no—
35#20300256—22no—
36#20300245C2211yescharacteristic, complemented
37#20300216C2212yescomplemented
38#20300215C2212yescomplemented
39#20300214C2212yescomplemented
40#20300205C2211yescharacteristic, complemented
41#20300204C2212yescomplemented
42H(5,𝔽2)×C227C2211yescharacteristic
43AGL(1,ℤ/8)×C227C2212yescomplemented
44#127×C227C2211yescharacteristic, complemented
45#104×C227C2212yescomplemented
46#100×C227C2211yescharacteristic, complemented
47C23×D227—22no—
48C23×D227D412yes—
49#16813885—22no—
50#16813885D412yescomplemented
51#16813884—24no—
52#16813884—22no—
53#16813884—22no—
54#16813884C2312yes—
55#16813883—24no—
56#16813883—22no—
57#16813883—22no—
58#16813883—22no—
59#16813883—22no—
60#16813883D412yescomplemented
61C2×Dic454C2311yescharacteristic, complemented
62Q8×D227D412yescomplemented
63C2×D908—22no—
64C2×D908—22no—
65C2×D908C2311yescharacteristic
66C2×C22⋊D227—22no—
67C2×C22⋊D227—24no—
68C2×C22⋊D227—21no—
69C2×C22⋊D227—22no—
70C2×C22⋊D227—22no—
71C2×C22⋊D227D412yes—
72D4×D227—24no—
73D4×D227—24no—
74D4×D227—24no—
75D4×D227—22no—
76D4×D227—24no—
77D4×D227—22no—
78D4×D227—22no—
79D4×D227—22no—
80D4×D227—22no—
81D4×D227D412yescomplemented
82C22×Dic227—21no—
83C2×C4×D227—22no—
84C2×C4×D227—22no—
85C2×C4×D227C2311yescharacteristic
86Dic908—22no—
87#16813874—22no—
88#16813873—22no—
89#16813872—22no—
90#16813872—22no—
91#16813869—22no—
92#16813868C2312yescomplemented
93#16813867—22no—
94C8×D227—22no—
95Dic227⋊C4—21no—
96C22⋊Dic227—22no—
97(C2×D227)⋊C4—21no—
98(C2×D227)⋊C4—22no—
99C4×Dic227C2311yescharacteristic, complemented
100#41×C227—22no—
101#41×C227C2312yes—
102C2×D4×C227—22no—
103C2×D4×C227—22no—
104C2×D4×C227—22no—
105C2×D4×C227C2311yescharacteristic
106C2×D4×C227C2311yescharacteristic, complemented
107SD16×C227—22no—
108D8×C227—22no—
109M16×C227C2312yescomplemented
110C42×C227C2311yescharacteristic, complemented
111C22×D227—24no—
112C22×D227—22no—
113C22×D227—22no—
114C22×D227—22no—
115C22×D227—22no—
116C22×D227—82no—
117C22×D227—84no—
118C22×D227C2×D412yes—
119C22×D227C2×D411yescharacteristic
120Dic454—22no—
121Dic454—22no—
122Dic454C2×D412yes—
123D908—24no—
124D908—22no—
125D908—22no—
126D908C2×D412yes—
127C22⋊D227—24no—
128C22⋊D227—24no—
129C22⋊D227—22no—
130C22⋊D227—22no—
131C22⋊D227—24no—
132C22⋊D227—22no—
133C22⋊D227—41no—
134C22⋊D227—41no—
135C22⋊D227—82no—
136C2×Dic227—22no—
137C2×Dic227—22no—
138C2×Dic227—21no—
139C2×Dic227—21no—
140C2×Dic227—42no—
141C2×Dic227C2×D411yescharacteristic
142C4×D227—24no—
143C4×D227—22no—
144C4×D227—22no—
145C4×D227—22no—
146C4×D227C2×D412yes—
147C4×D227C2×D412yes—
148C227⋊C8—22no—
149C23×C227—22no—
150C23×C227—21no—
151C23×C227C2×D412yes—
152Q8×C227C2×D412yes—
153D4×C227—24no—
154D4×C227—22no—
155D4×C227—22no—
156D4×C227—22no—
157D4×C227—82no—
158D4×C227C2×D412yes—
159C2×C4×C227—22no—
160C2×C4×C227—22no—
161C2×C4×C227—21no—
162C2×C4×C227C2411yescharacteristic
163C8×C227—22no—
164C22×C227—22no—
165C22×C227—22no—
166C22×C227—42no—
167C22×C227—42no—
168C22×C227—41no—
169C22×C227—82no—
170C22×C227C22×D411yescharacteristic
171C2×D227—22no—
172C2×D227—22no—
173C2×D227—41no—
174C2×D227—44no—
175C2×D227—44no—
176C2×D227—42no—
177C2×D227—82no—
178C2×D227—84no—
179C2×D227—82no—
180C2×D227#12011yescharacteristic
181C2×D227#12011yescharacteristic
182C4×C227—22no—
183C4×C227—41no—
184C4×C227C22×D412yes—
185Dic227—22no—
186Dic227—41no—
187Dic227—41no—
188Dic227#12012yes—
189C2×C227—21no—
190C2×C227—42no—
191C2×C227—42no—
192C2×C227—81no—
193C2×C227C2×#12011yescharacteristic
194D227—21no—
195D227—21no—
196D227—42no—
197D227—42no—
198C227#301911yescharacteristic, complemented, Sylow
199#3019—2271nomaximal, Sylow
200C2×H(5,𝔽2)—2271no—
201#575—2272no—
202#535—2271no—
203#455—2271no—
204#455—2272no—
205#455—2271no—
206#453—2272no—
207#453—2271no—
208#453D22711yescharacteristic, complemented
209#421—2272no—
210C2×#100—2271no—
211H(5,𝔽2)—4542no—
212H(5,𝔽2)—4542no—
213H(5,𝔽2)—4542no—
214H(5,𝔽2)—2272no—
215H(5,𝔽2)—2271no—
216H(5,𝔽2)C2×D22711yescharacteristic
217C2×#41—4542no—
218C2×#41—2272no—
219C2×#41—2271no—
220C22×D4—4542no—
221C22×D4—4542no—
222C22×D4—2271no—
223#137—4542no—
224#137—2272no—
225#137—2272no—
226AGL(1,ℤ/8)—4542no—
227AGL(1,ℤ/8)—2272no—
228AGL(1,ℤ/8)C2×D22712yescomplemented
229#135—4542no—
230#135—4542no—
231C2×M16—2272no—
232#127—2271no—
233#127C2×D22711yescharacteristic, complemented
234#124—2271no—
235#124—2271no—
236#121—4542no—
237#120—4542no—
238#117—2271no—
239#104—2272no—
240#104—2272no—
241#104—2272no—
242#104C2×D22712yescomplemented
243#100—2271no—
244#100—2272no—
245#100C2×D22711yescharacteristic, complemented
246C24—4542no—
247C24—2272no—
248#41—22no—
249#41—4544no—
250#41—4542no—
251#41—4542no—
252#41—4542no—
253#41—4542no—
254#41—4544no—
255#41—4542no—
256#41—4542no—
257#41—4542no—
258#41—2272no—
259#41—2272no—
260#41—2272no—
261#41C22×D22712yes—
262C2×Q8—2272no—
263C2×Q8—2271no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—4542no—
268C2×D4—4542no—
269C2×D4—4544no—
270C2×D4—4544no—
271C2×D4—4544no—
272C2×D4—4542no—
273C2×D4—4544no—
274C2×D4—4542no—
275C2×D4—4542no—
276C2×D4—4541no—
277C2×D4—4542no—
278C2×D4—4544no—
279C2×D4—4542no—
280C2×D4—4542no—
281C2×D4—4542no—
282C2×D4—4542no—
283C2×D4—2272no—
284C2×D4—2272no—
285C2×D4—2271no—
286C2×D4C22×D22711yescharacteristic
287C2×D4C22×D22711yescharacteristic, complemented
288C22×C4—4542no—
289C22×C4—4542no—
290C22×C4—4541no—
291C22×C4—2271no—
292Q16—4542no—
293Q16—4542no—
294SD16—22no—
295SD16—4542no—
296SD16—4542no—
297SD16—4542no—
298D8—22no—
299D8—4542no—
300M16—4542no—
301M16—2272no—
302M16C22×D22712yescomplemented
303C2×C8—4542no—
304C4⋊C4—4541no—
305C22⋊C4—4541no—
306C22⋊C4—4542no—
307C22⋊C4—4542no—
308C42—2271no—
309C42C22×D22711yescharacteristic, complemented
310C23—22no—
311C23—21no—
312C23—18162no—
313C23—18164no—
314C23—4542no—
315C23—4544no—
316C23—4542no—
317C23—4542no—
318C23—4542no—
319C23—2272no—
320C23—2271no—
321C23D4×D22712yes—
322Q8—4542no—
323Q8—4542no—
324Q8—2272no—
325Q8D4×D22712yes—
326D4—24no—
327D4—22no—
328D4—22no—
329D4—22no—
330D4—82no—
331D4—18162no—
332D4—9081no—
333D4—9081no—
334D4—4544no—
335D4—4542no—
336D4—4544no—
337D4—4544no—
338D4—4542no—
339D4—4542no—
340D4—4544no—
341D4—4542no—
342D4—4542no—
343D4—2272no—
344D4D4×D22712yes—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—9082no—
349C2×C4—4544no—
350C2×C4—4542no—
351C2×C4—4542no—
352C2×C4—4541no—
353C2×C4—4541no—
354C2×C4—4542no—
355C2×C4—4542no—
356C2×C4—4542no—
357C2×C4—2271no—
358C2×C4—2272no—
359C2×C4—2272no—
360C2×C4C23×D22711yescharacteristic
361C8—22no—
362C8—4542no—
363C22—22no—
364C22—22no—
365C22—42no—
366C22—42no—
367C22—41no—
368C22—82no—
369C22—18162no—
370C22—18164no—
371C22—18162no—
372C22—9082no—
373C22—9081no—
374C22—9084no—
375C22—9084no—
376C22—4542no—
377C22—4542no—
378C22—2271no—
379C22—2271no—
380C22C2×D4×D22711yescharacteristic
381C4—22no—
382C4—41no—
383C4—9081no—
384C4—9081no—
385C4—4542no—
386C4—2272no—
387C4C2×D4×D22712yes—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—81no—
392C2—9082no—
393C2—9082no—
394C2—4541no—
395C2—4541no—
396C2#120×D22711yescharacteristic
subgroupG/H
centerC2#120×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#453×C227C2
socleC2×C227C2×#120
hypercenter#453D227
hypocenterC227#3019
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. H(5,𝔽2)×D227C2semidirect
2. #28409828C2semidirect
3. #28409595C2semidirect
4. #28409212C2semidirect
5. #28409211C2semidirect
6. #28409210C2semidirect
7. #28409199C2semidirect
8. #28409198C2semidirect
9. #28409047C2semidirect
10. #100×D227C2semidirect
11. #453×C227C2semidirect
17. C2×#16813883C22semidirect
23. #20300321C22semidirect
24. #20300318C22semidirect
26. #20300314C22semidirect
30. M16×D227C22semidirect
31. #20300284C22semidirect
32. #20300276C22semidirect
33. #20300275C22semidirect
36. #20300245C22semidirect
37. #20300216C22semidirect
38. #20300215C22semidirect
39. #20300214C22semidirect
40. #20300205C22semidirect
41. #20300204C22semidirect
43. AGL(1,ℤ/8)×C227C22semidirect
44. #127×C227C22semidirect
45. #104×C227C22semidirect
46. #100×C227C22semidirect
50. #16813885D4semidirect
60. #16813883D4semidirect
61. C2×Dic454C23semidirect
62. Q8×D227D4semidirect
81. D4×D227D4semidirect
92. #16813868C23semidirect
99. C4×Dic227C23semidirect
106. C2×D4×C227C23semidirect
109. M16×C227C23semidirect
110. C42×C227C23semidirect
198. C227#3019semidirect
208. #453D227semidirect
228. AGL(1,ℤ/8)C2×D227semidirect
233. #127C2×D227semidirect
242. #104C2×D227semidirect
245. #100C2×D227semidirect
287. C2×D4C22×D227semidirect
302. M16C22×D227semidirect
309. C42C22×D227semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D2272
#284098282
#284095952
#284092122
#284092112
#284092102
#284091992
#284091982
#284090472
#100×D2272
#453×C2272
#3019227