Finite Group Databasebeta

← #14302542

Subgroups — #14302542

Order 1920.

620 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C3×#66262C211yesmaximal, characteristic, complemented
2C2×#91513C211yesmaximal, characteristic
3#11752441C211yesmaximal, characteristic, complemented
4#11751816C211yesmaximal, characteristic
5#11751719C211yesmaximal, characteristic
6#11746537C211yesmaximal, characteristic, complemented
7#11746519C211yesmaximal, characteristic, complemented
8#10604820—31nomaximal, Hall
9C2×C3×Q8×D5C2211yescharacteristic
10C2×#6772C2211yescharacteristic
11#91989C2211yescharacteristic
12D5⋊#208C2211yescharacteristic
13C3×Q8×AGL(1,𝔽5)—22no—
14C3×Q8×AGL(1,𝔽5)—22no—
15C3×#4130—22no—
16C3×#4130C2211yescharacteristic
17C3×#4130C2211yescharacteristic
18C2×#6718C411yescharacteristic, complemented
19C2×C5×#215C411yescharacteristic, complemented
20#91513—22no—
21#91513—22no—
22C2×#6662C411yescharacteristic, complemented
23C2×#6656C411yescharacteristic, complemented
24C2×#6649C2211yescharacteristic
25#91238—22no—
26#91172—22no—
27#91156—22no—
28#71662—51nomaximal, Hall
29#66262D311yescharacteristic, complemented
30C2×#4061—31no—
31#65782—31no—
32#65771—31no—
33#65769—31no—
34#64895—31no—
35#64894—31no—
36C2×C3×C5×Q8C2×C411yescharacteristic
37C2×Dic30C2×C411yescharacteristic
38C2×C5×D12C2×C411yescharacteristic
39C3×Q8×D5—21no—
40C3×Q8×D5—21no—
41C3×Q8×D5D412yes—
42C3×Q8×D5D412yes—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×Dic10—21no—
46C2×C3×Dic10C2×C411yescharacteristic
47C2×#1201—21no—
48C2×D3×Dic5—21no—
49#6772—21no—
50#6772—21no—
51#6772—21no—
52#6772—21no—
53#6763—21no—
54#6762—21no—
55C4⋊ΓL(1,𝔽16)—21no—
56C4×ΓL(1,𝔽16)—21no—
57#6758—21no—
58D5⋊(C3⋊C8)—21no—
59C3×C22⋊AGL(1,𝔽5)—21no—
60C3×C22⋊AGL(1,𝔽5)D412yes—
61C3×C4⋊AGL(1,𝔽5)—24no—
62C3×C4⋊AGL(1,𝔽5)—21no—
63C3×C4⋊AGL(1,𝔽5)—21no—
64C3×C4×AGL(1,𝔽5)—24no—
65C3×C4×AGL(1,𝔽5)—21no—
66C3×C4×AGL(1,𝔽5)D412yes—
67#6718—22no—
68C2×C3⋊#168C2×C411yescharacteristic
69C5×#215—22no—
70C2×C5×C3⋊C8C2×C411yescharacteristic
71#6662—22no—
72#6656—22no—
73#6649—21no—
74#6649—21no—
75C3×#519—51no—
76C2×#942—51no—
77#5225—51no—
78#5006—51no—
79#5005—51no—
80#4533—51no—
81#4531—51no—
82C2×Q8×D5C2×D311yescharacteristic
83C2×#711—31no—
84Q8×AGL(1,𝔽5)—22no—
85Q8×AGL(1,𝔽5)—22no—
86#4130—22no—
87#4130—31no—
88#4130C2×D311yescharacteristic
89#4130C2×D311yescharacteristic
90D5⋊M16—31no—
91C2×C5×SD16—31no—
92C2×#689—31no—
93C2×#687—31no—
94#4061—62no—
95#4061—62no—
96C2×Dic20—31no—
97C2×#676—31no—
98#4011—62no—
99#4009—62no—
100#4008—62no—
101#1905—151noSylow
102C22×C5×D3—21no—
103C22×C3×D5C2×D411yescharacteristic
104C2×ΓL(1,𝔽16)—41no—
105C2×C3×AGL(1,𝔽5)—22no—
106C2×C3×AGL(1,𝔽5)—22no—
107C3×C5×Q8—21no—
108C3×C5×Q8C22⋊C412yes—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×Dic15—21no—
112Dic30—21no—
113Dic30—21no—
114C5×D12—21no—
115C5×D12—21no—
116C2×C3×Dic5—21no—
117C2×C3×Dic5#4111yescharacteristic
118C3×C4×D5—22no—
119C3×C4×D5—22no—
120C3×C4×D5C2×D411yescharacteristic
121C3×C4×D5C2×D411yescharacteristic
122C3×C4×D5C2×D411yescharacteristic
123C3×C4×D5#4111yescharacteristic
124C3×Dic10—21no—
125C3×Dic10—22no—
126C3×Dic10—22no—
127C3×Dic10C22⋊C412yes—
128#1201—41no—
129#1201—41no—
130D3×Dic5—41no—
131D3×Dic5—41no—
132C3⋊#170—41no—
133C3⋊#168—21no—
134C5×C3⋊C8—21no—
135C22×C3×Q8—51no—
136C2×#235—51no—
137C3×C4×Q8—102no—
138C3×C4×Q8—102no—
139C3×#117—51no—
140C3×#117—51no—
141C3×#117—102no—
142C2×#216—51no—
143C2×#216—51no—
144#942—102no—
145#942—102no—
146C2×#215—51no—
147C2×#215AGL(1,𝔽5)11yescharacteristic, complemented
148#926—51no—
149C2×#208—51no—
150C2×#208—51no—
151#835—102no—
152#806—102no—
153#805—102no—
154C2×C5×Q8C4×D311yescharacteristic
155C2×C5×D4—31no—
156C2×C22⋊D5—61no—
157C22×Dic5—61no—
158Q8×D5—21no—
159Q8×D5—21no—
160Q8×D5D1212yes—
161Q8×D5#6712yes—
162#711—61no—
163#711—61no—
164#711—61no—
165#711—61no—
166C2×C4×D5—21no—
167C2×C4×D5C22×D311yescharacteristic
168C2×Dic10—21no—
169C2×Dic10—31no—
170C2×Dic10C4×D311yescharacteristic
171C22⋊AGL(1,𝔽5)—21no—
172C22⋊AGL(1,𝔽5)—61no—
173C22⋊AGL(1,𝔽5)#6712yes—
174#704—61no—
175C4⋊AGL(1,𝔽5)—24no—
176C4⋊AGL(1,𝔽5)—21no—
177C4⋊AGL(1,𝔽5)—21no—
178C4⋊AGL(1,𝔽5)—61no—
179C4×AGL(1,𝔽5)—24no—
180C4×AGL(1,𝔽5)—21no—
181C4×AGL(1,𝔽5)—61no—
182C4×AGL(1,𝔽5)#6712yes—
183#700—61no—
184D5⋊C8—61no—
185C5×SD16—62no—
186C2×C5×C8—31no—
187#689—62no—
188#687—62no—
189C2×#168—31no—
190Dic20—62no—
191#676—61no—
192#676—61no—
193C2×#137—151no—
194#519—51no—
195#421—151no—
196#419—151no—
197#410—151no—
198#344—151no—
199H(3,ℤ/4)—151no—
200C22×C3×C5C4×D411yescharacteristic
201C2×C5×D3—41no—
202C2×C5×D3—81no—
203C2×C3×D5—41no—
204C2×C3×D5#12011yescharacteristic
205C2×C3×D5#12111yescharacteristic
206ΓL(1,𝔽16)—81no—
207C3×AGL(1,𝔽5)—42no—
208C3×AGL(1,𝔽5)—42no—
209C3×C4×C5—22no—
210C3×C4×C5C2×C22⋊C411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212Dic15—41no—
213C3×Dic5—22no—
214C3×Dic5#12111yescharacteristic
215C3×Dic5#12311yescharacteristic
216C2×C3×Q8—52no—
217C2×C3×Q8—52no—
218C2×C3×Q8—51no—
219C2×C3×Q8—101no—
220C2×C3×Q8—101no—
221C2×C3×Q8—101no—
222C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
223C22×C3×C4—51no—
224C22×C3×C4—101no—
225C2×#67—101no—
226#235—101no—
227#235—101no—
228#235—101no—
229#235—101no—
230C2×D12C2×AGL(1,𝔽5)11yescharacteristic
231C2×C4×D3—101no—
232C2×Dic6—51no—
233C3×C4⋊C4—101no—
234C3×C4⋊C4—101no—
235C3×C4⋊C4—104no—
236C3×C22⋊C4—52no—
237C3×C22⋊C4—101no—
238C3×C42—52no—
239C3×C42—101no—
240C3×C42—104no—
241#217—101no—
242#216—102no—
243#216—102no—
244#215—22no—
245#215—102no—
246C4⋊Dic3—101no—
247C4×Dic3—101no—
248#208—101no—
249#208—101no—
250#208—101no—
251#208—101no—
252C2×C3⋊C8—51no—
253C2×C3⋊C8—101no—
254C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
255C23×C5—61no—
256C22×D5C2×#6711yescharacteristic
257C2×AGL(1,𝔽5)—22no—
258C2×AGL(1,𝔽5)—22no—
259C2×AGL(1,𝔽5)—121no—
260C5×Q8—21no—
261C5×Q8(C2×D3)⋊C412yes—
262C5×D4—61no—
263C5×D4—61no—
264C2×C4×C5—21no—
265C2×C4×C5C2×C4×D311yescharacteristic
266C22⋊D5—121no—
267C22⋊D5—121no—
268C2×Dic5—21no—
269C2×Dic5—61no—
270C2×Dic5—121no—
271C2×Dic5—121no—
272C2×Dic5#23511yescharacteristic
273C4×D5—22no—
274C4×D5—22no—
275C4×D5C2×D1211yescharacteristic
276C4×D5#23511yescharacteristic
277C4×D5C2×#6711yescharacteristic
278C4×D5C2×#6711yescharacteristic
279Dic10—21no—
280Dic10—22no—
281Dic10—22no—
282Dic10—61no—
283Dic10—61no—
284Dic10(C2×D3)⋊C412yes—
285#170—121no—
286C5×C8—61no—
287#168—61no—
288C2×#41—151no—
289C22×Q8—51no—
290#137—302no—
291#137—302no—
292C2×Q16—151no—
293C2×Q16—151no—
294C2×SD16—31no—
295C2×SD16—151no—
296C2×M16—151no—
297C2×M16—151no—
298C4×Q8—102no—
299C4×Q8—102no—
300#117—51no—
301#117—51no—
302#117—102no—
303#117—151no—
304#104—302no—
305#103—302no—
306#99—302no—
307C2×C3×C5—21no—
308C2×C3×C5#38611yescharacteristic
309C3×D5—21no—
310C3×D5—21no—
311C5×D3—81no—
312C23×C3—51no—
313C22×D3—21no—
314C3×Q8—21no—
315C3×Q8—52no—
316C3×Q8—102no—
317C3×Q8—102no—
318C3×Q8—101no—
319C3×Q8C22⋊AGL(1,𝔽5)12yes—
320C2×C3×C4—21no—
321C2×C3×C4—51no—
322C2×C3×C4—51no—
323C2×C3×C4—51no—
324C2×C3×C4—51no—
325C2×C3×C4—51no—
326C2×C3×C4—102no—
327C2×C3×C4—102no—
328C2×C3×C4—101no—
329C2×C3×C4—102no—
330C2×C3×C4—102no—
331C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
332#67—201no—
333#67—201no—
334C2×Dic3—101no—
335C2×Dic3—201no—
336D12—21no—
337D12—21no—
338C4×D3—201no—
339C4×D3—201no—
340Dic6—101no—
341Dic6—101no—
342C3⋊C8—21no—
343C3⋊C8—101no—
344C3⋊C8—201no—
345C22×C5—121no—
346C22×C5—241no—
347C22×C5C4×#6711yescharacteristic
348C2×D5—41no—
349C2×D5#93011yescharacteristic
350C2×D5#95311yescharacteristic
351AGL(1,𝔽5)—42no—
352AGL(1,𝔽5)—42no—
353AGL(1,𝔽5)—241no—
354C4×C5—22no—
355C4×C5C2×(C2×D3)⋊C411yescharacteristic
356C4×C5C4×#6711yescharacteristic
357Dic5—22no—
358Dic5—121no—
359Dic5#92911yescharacteristic
360Dic5#93011yescharacteristic
361#41—301no—
362#41—301no—
363#41—301no—
364#41—301no—
365C2×Q8—52no—
366C2×Q8—52no—
367C2×Q8—51no—
368C2×Q8—101no—
369C2×Q8—101no—
370C2×Q8—101no—
371C2×Q8—151no—
372C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
373C2×D4—31no—
374C2×D4—301no—
375C22×C4—51no—
376C22×C4—101no—
377C22×C4—301no—
378Q16—302no—
379Q16—302no—
380SD16—62no—
381SD16—302no—
382M16—301no—
383M16—301no—
384M16—301no—
385M16—301no—
386C2×C8—31no—
387C2×C8—151no—
388C2×C8—301no—
389C4⋊C4—101no—
390C4⋊C4—101no—
391C4⋊C4—104no—
392C4⋊C4—301no—
393C22⋊C4—52no—
394C22⋊C4—101no—
395C22⋊C4—301no—
396C42—52no—
397C42—101no—
398C42—104no—
399C42—301no—
400C3×C5#190511yescharacteristic, complemented, Hall
401C22×C3—51no—
402C22×C3—51no—
403C22×C3—201no—
404C22×C3D4×AGL(1,𝔽5)11yescharacteristic
405C2×D3—41no—
406C2×D3—81no—
407C3×C4—22no—
408C3×C4—51no—
409C3×C4—51no—
410C3×C4—102no—
411C3×C4—202no—
412C3×C4—202no—
413C3×C4D4×AGL(1,𝔽5)11yescharacteristic
414C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
415Dic3—201no—
416Dic3—401no—
417C2×C5—21no—
418C2×C5—241no—
419C2×C5#521311yescharacteristic
420D5—21no—
421D5—21no—
422C23—51no—
423C23—61no—
424Q8—21no—
425Q8—52no—
426Q8—102no—
427Q8—102no—
428Q8—101no—
429Q8—301no—
430Q8—301no—
431Q8(C2×D3)⋊AGL(1,𝔽5)12yes—
432D4—61no—
433D4—61no—
434D4—601no—
435D4—601no—
436C2×C4—21no—
437C2×C4—51no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—51no—
442C2×C4—102no—
443C2×C4—102no—
444C2×C4—101no—
445C2×C4—102no—
446C2×C4—102no—
447C2×C4—301no—
448C2×C4—601no—
449C2×C4—601no—
450C2×C4—601no—
451C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
452C8—61no—
453C8—301no—
454C8—601no—
455C2×C3—21no—
456C2×C3—101no—
457C2×C3—101no—
458C2×C3#6577911yescharacteristic
459D3—81no—
460C5#7166211yescharacteristic, complemented, Sylow
461C22—51no—
462C22—51no—
463C22—121no—
464C22—201no—
465C22—241no—
466C22AGL(1,𝔽5)×#6711yescharacteristic
467C4—22no—
468C4—51no—
469C4—51no—
470C4—102no—
471C4—202no—
472C4—202no—
473C4—601no—
474C4—1201no—
475C4C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
476C4AGL(1,𝔽5)×#6711yescharacteristic
477C3#1060482011yescharacteristic, complemented, Sylow
478C2—21no—
479C2—101no—
480C2—101no—
481C2—241no—
482C2#1175202211yescharacteristic
subgroupG/H
centerC2#11752022
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC2×C3×C5#386
prime radicalC2×#6772C22
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1905
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#66262C2semidirect
3. #11752441C2semidirect
6. #11746537C2semidirect
7. #11746519C2semidirect
18. C2×#6718C4semidirect
19. C2×C5×#215C4semidirect
22. C2×#6662C4semidirect
23. C2×#6656C4semidirect
29. #66262D3semidirect
147. C2×#215AGL(1,𝔽5)semidirect
400. C3×C5#1905semidirect
460. C5#71662semidirect
477. C3#10604820semidirect

Maximal subgroups

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

subgroupindex
C3×#662622
C2×#915132
#117524412
#117518162
#117517192
#117465372
#117465192
#106048203
#716625