Finite Group Databasebeta

← #46845739

Subgroups — #46845739

Order 34688.

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

normal subgroups only

This group has 704 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
1#31774732C211yesmaximal, characteristic
2#137×D271C212yesmaximal, complemented
3#31774467C211yesmaximal, characteristic, complemented
4#31774081C212yesmaximal, complemented
5#31774078C212yesmaximal, complemented
6#31774077C211yesmaximal, characteristic, complemented
7#31774075C211yesmaximal, characteristic, complemented
8#31774074C211yesmaximal, characteristic, complemented
9#31773912C212yesmaximal, complemented
10#31773878C211yesmaximal, characteristic, complemented
11#455×C271C211yesmaximal, characteristic, complemented
12#22576207—22no—
13#22576207C2212yes—
14#22576205—22no—
15#22576205—22no—
16#22576205C2211yescharacteristic
17C2×#16864653—22no—
18C2×#16864653—22no—
19C2×#16864653C2211yescharacteristic
20#41×D271—22no—
21#41×D271C2212yes—
22C2×Q8×D271C2211yescharacteristic, complemented
23#22576193C2212yescomplemented
24#22576192—22no—
25#22576191C2212yescomplemented
26#22576190C2212yescomplemented
27#22576189—22no—
28Q16×D271—22no—
29SD16×D271—22no—
30M16×D271C2212yescomplemented
31#22576160C2211yescharacteristic, complemented
32#22576158C2211yescharacteristic, complemented
33#22576147C2211yescharacteristic, complemented
34#22576142—22no—
35#22576137—22no—
36#22576117C2211yescharacteristic, complemented
37#22576088C2212yescomplemented
38#22576087C2212yescomplemented
39#22576086C2212yescomplemented
40#22576078C2212yescomplemented
41#22576077C2211yescharacteristic, complemented
42H(5,𝔽2)×C271C2211yescharacteristic
43#137×C271C2212yescomplemented
44#124×C271C2211yescharacteristic, complemented
45#104×C271C2212yescomplemented
46#100×C271C2211yescharacteristic, complemented
47#16864655D412yes—
48#16864654—24no—
49#16864654—22no—
50#16864654—22no—
51#16864654C2312yes—
52#16864653—24no—
53#16864653—24no—
54#16864653—24no—
55#16864653—22no—
56#16864653—22no—
57#16864653—24no—
58#16864653—22no—
59#16864653—22no—
60#16864653—22no—
61#16864653D412yes—
62C2×Dic542—22no—
63C2×Dic542—22no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—22no—
69Q8×D271—22no—
70Q8×D271D412yes—
71C2×C22⋊D271—24no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271D412yes—
75D4×D271—24no—
76D4×D271—22no—
77D4×D271—22no—
78D4×D271D412yes—
79C22×Dic271—22no—
80C22×Dic271D412yes—
81C2×C4×D271—22no—
82C2×C4×D271—22no—
83C2×C4×D271—21no—
84C2×C4×D271C2311yescharacteristic
85Dic1084—22no—
86#16864644—22no—
87#16864644—22no—
88#16864643—22no—
89#16864642—22no—
90#16864641—22no—
91#16864638C2312yes—
92#16864637—22no—
93C8×D271—22no—
94C4⋊Dic271—22no—
95Dic271⋊C4—21no—
96Dic271⋊C4—22no—
97C22⋊Dic271—22no—
98(C2×D271)⋊C4—21no—
99C4×Dic271C2311yescharacteristic
100#41×C271—22no—
101#41×C271C2312yes—
102C2×Q8×C271C2311yescharacteristic
103C2×D4×C271—22no—
104C2×D4×C271—22no—
105C2×D4×C271C2311yescharacteristic
106Q16×C271—22no—
107SD16×C271—22no—
108M16×C271C2312yes—
109C22⋊C4×C271—22no—
110C42×C271C2311yescharacteristic
111C22×D271—22no—
112C22×D271—22no—
113C22×D271C2×D411yescharacteristic
114Dic542—24no—
115Dic542—22no—
116Dic542—22no—
117Dic542—22no—
118Dic542—41no—
119Dic542—41no—
120Dic542C2×D412yes—
121D1084—22no—
122D1084C2×D412yes—
123C22⋊D271—24no—
124C22⋊D271—24no—
125C22⋊D271—22no—
126C22⋊D271—22no—
127C22⋊D271—24no—
128C22⋊D271—22no—
129C2×Dic271—22no—
130C2×Dic271—24no—
131C2×Dic271—22no—
132C2×Dic271—22no—
133C2×Dic271—22no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271C2×D412yes—
137C2×Dic271C2×D411yescharacteristic
138C4×D271—24no—
139C4×D271—22no—
140C4×D271—22no—
141C4×D271—22no—
142C4×D271—42no—
143C4×D271C2×D412yes—
144C4×D271C2×D412yes—
145C271⋊C8—22no—
146C23×C271—22no—
147C23×C271C2×D412yes—
148Q8×C271—22no—
149Q8×C271C2×D412yes—
150D4×C271—24no—
151D4×C271—22no—
152D4×C271—22no—
153D4×C271C2×D412yes—
154C2×C4×C271—22no—
155C2×C4×C271—22no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271C2411yescharacteristic
159C8×C271—22no—
160C22×C271—22no—
161C22×C271—22no—
162C22×C271—42no—
163C22×C271—82no—
164C22×C271C22×D411yescharacteristic
165C2×D271—22no—
166C2×D271—41no—
167C2×D271—82no—
168C2×D271—82no—
169C2×D271#12012yes—
170C4×C271—22no—
171C4×C271—41no—
172C4×C271—41no—
173C4×C271C22×D412yes—
174Dic271—22no—
175Dic271—22no—
176Dic271—41no—
177Dic271—41no—
178Dic271#12011yescharacteristic
179Dic271#12011yescharacteristic
180C2×C271—21no—
181C2×C271—42no—
182C2×C271—42no—
183C2×C271C2×#12011yescharacteristic
184D271—22no—
185D271—42no—
186C271#302211yescharacteristic, complemented, Sylow
187#3022—2711nomaximal, Sylow
188#585—2711no—
189C2×#137—2712no—
190#536—2711no—
191#456—2712no—
192#456—2712no—
193#455—2711no—
194#455—2711no—
195#455—2711no—
196#455D27111yescharacteristic, complemented
197#421—2712no—
198#413—2711no—
1992−1+4—2712no—
2002−1+4—5422no—
201H(5,𝔽2)—2711no—
202H(5,𝔽2)—5422no—
203H(5,𝔽2)—5422no—
204H(5,𝔽2)C2×D27111yescharacteristic
205C2×#41—2711no—
206C2×#41—2712no—
207C2×#41—5422no—
208C2×#41—5422no—
209C2×#41—5422no—
210C22×Q8—2711no—
211#137—2712no—
212#137—2712no—
213#137—2712no—
214#137—5422no—
215#137—5422no—
216#137C2×D27112yescomplemented
217C2×Q16—5422no—
218C2×SD16—5422no—
219C2×M16—2712no—
220C4⋊Q8—2711no—
221C4⋊Q8—2711no—
222#124—2711no—
223#124C2×D27111yescharacteristic, complemented
224#123—5422no—
225C22⋊Q8—5422no—
226#117—2711no—
227#104—2712no—
228#104—2712no—
229#104—2712no—
230#104C2×D27112yescomplemented
231#101—2712no—
232#100—2711no—
233#100C2×D27111yescharacteristic, complemented
234#41—22no—
235#41—2712no—
236#41—2712no—
237#41—2712no—
238#41—5422no—
239#41—5422no—
240#41—5424no—
241#41—5422no—
242#41—5422no—
243#41—5422no—
244#41—5424no—
245#41—5422no—
246#41—5424no—
247#41—5424no—
248#41—5424no—
249#41—5422no—
250#41C22×D27112yes—
251C2×Q8—2711no—
252C2×Q8—2712no—
253C2×Q8—2711no—
254C2×Q8—5421no—
255C2×Q8—5422no—
256C2×Q8—5422no—
257C2×Q8—5422no—
258C2×Q8—5422no—
259C2×Q8—5422no—
260C2×Q8C22×D27111yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—2712no—
264C2×D4—2712no—
265C2×D4—5422no—
266C2×D4—5422no—
267C2×D4—5424no—
268C2×D4—5422no—
269C2×D4—5424no—
270C2×D4—5422no—
271C2×D4C22×D27111yescharacteristic
272C22×C4—2711no—
273C22×C4—2712no—
274C22×C4—5422no—
275C22×C4—5422no—
276C22×C4—5421no—
277C22×C4—5422no—
278Q16—22no—
279Q16—5422no—
280Q16—5422no—
281Q16—5422no—
282SD16—22no—
283SD16—5422no—
284SD16—5422no—
285SD16—5422no—
286M16—2712no—
287M16—5422no—
288M16C22×D27112yes—
289C2×C8—5422no—
290C4⋊C4—5421no—
291C4⋊C4—5422no—
292C4⋊C4—5422no—
293C22⋊C4—22no—
294C22⋊C4—5421no—
295C22⋊C4—5422no—
296C42—2711no—
297C42C22×D27111yescharacteristic
298C23—22no—
299C23—2711no—
300C23—5422no—
301C23—5422no—
302C23D4×D27112yes—
303Q8—22no—
304Q8—2712no—
305Q8—5422no—
306Q8—5424no—
307Q8—5422no—
308Q8—5422no—
309Q8—10841no—
310Q8—10841no—
311Q8D4×D27112yes—
312D4—24no—
313D4—22no—
314D4—22no—
315D4—2712no—
316D4—5422no—
317D4—5422no—
318D4—5424no—
319D4—5422no—
320D4—5424no—
321D4—5424no—
322D4—5422no—
323D4D4×D27112yes—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—2711no—
329C2×C4—2712no—
330C2×C4—2712no—
331C2×C4—2712no—
332C2×C4—5422no—
333C2×C4—5422no—
334C2×C4—5424no—
335C2×C4—5422no—
336C2×C4—5422no—
337C2×C4—5422no—
338C2×C4—5424no—
339C2×C4—5422no—
340C2×C4—5422no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—10842no—
344C2×C4C23×D27111yescharacteristic
345C8—22no—
346C8—5422no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—82no—
351C22—2712no—
352C22—5422no—
353C22—10841no—
354C22—21682no—
355C22—21682no—
356C22C2×D4×D27111yescharacteristic
357C4—22no—
358C4—41no—
359C4—41no—
360C4—2711no—
361C4—2711no—
362C4—5422no—
363C4—5422no—
364C4—10841no—
365C4—10841no—
366C4C2×D4×D27112yes—
367C2—21no—
368C2—42no—
369C2—42no—
370C2—5422no—
371C2—10842no—
372C2#120×D27111yescharacteristic
subgroupG/H
centerC2#120×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#455×C271C2
socleC2×C271C2×#120
prime radical#31774732C2
hypercenter#455D271
hypocenterC271#3022
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
2. #137×D271C2semidirect
3. #31774467C2semidirect
4. #31774081C2semidirect
5. #31774078C2semidirect
6. #31774077C2semidirect
7. #31774075C2semidirect
8. #31774074C2semidirect
9. #31773912C2semidirect
10. #31773878C2semidirect
11. #455×C271C2semidirect
22. C2×Q8×D271C22semidirect
23. #22576193C22semidirect
25. #22576191C22semidirect
26. #22576190C22semidirect
30. M16×D271C22semidirect
31. #22576160C22semidirect
32. #22576158C22semidirect
33. #22576147C22semidirect
36. #22576117C22semidirect
37. #22576088C22semidirect
38. #22576087C22semidirect
39. #22576086C22semidirect
40. #22576078C22semidirect
41. #22576077C22semidirect
43. #137×C271C22semidirect
44. #124×C271C22semidirect
45. #104×C271C22semidirect
46. #100×C271C22semidirect
186. C271#3022semidirect
196. #455D271semidirect
216. #137C2×D271semidirect
223. #124C2×D271semidirect
230. #104C2×D271semidirect
233. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747322
#137×D2712
#317744672
#317740812
#317740782
#317740772
#317740752
#317740742
#317739122
#317738782
#455×C2712
#3022271