Finite Group Databasebeta

← #14304917

Subgroups — #14304917

Order 1920.

444 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 96 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#11752446C211yesmaximal, characteristic
2C2×C3×#4010C211yesmaximal, characteristic
3#11751825C211yesmaximal, characteristic, complemented
4C2×#91149C211yesmaximal, characteristic, complemented
5#11749207C211yesmaximal, characteristic
6#11746532C211yesmaximal, characteristic, complemented
7#11746526C211yesmaximal, characteristic, complemented
8#10603988—31nomaximal, Hall
9C2×C3×#713C2211yescharacteristic
10C2×#6758C2211yescharacteristic
11C2×D5⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×#700C2211yescharacteristic
14#91358C2211yescharacteristic
15C2×D5×C3⋊C8C2211yescharacteristic
16#91232—21no—
17#91232—21no—
18C3×#4010—22no—
19#91177—21no—
20Dic3⋊#170—21no—
21#91149—22no—
22#91119C411yescharacteristic, complemented
23C5×#835C411yescharacteristic, complemented
24#90965C411yescharacteristic, complemented
25#90964C411yescharacteristic, complemented
26#70372—51nomaximal, Hall
27#65787—31no—
28C2×#4010D311yescharacteristic
29C2×#3992—31no—
30#65708—31no—
31#65093—31no—
32#64896—31no—
33#64875—31no—
34C22×C3×AGL(1,𝔽5)—21no—
35C2×C3×C5×Q8C2×C411yescharacteristic
36C3×#713—21no—
37C3×#713—21no—
38C3×#713—21no—
39C3×#713—21no—
40C2×C3×D20—21no—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5C2311yescharacteristic
44C2×C3⋊#170—21no—
45C2×C3⋊#170D411yescharacteristic
46C2×C3⋊#170D411yescharacteristic
47#6758—21no—
48#6758—21no—
49D5⋊(C3⋊C8)—21no—
50D5⋊(C3⋊C8)—21no—
51C2×C3×#170—21no—
52C3×C4×AGL(1,𝔽5)—21no—
53C3×C4×AGL(1,𝔽5)D412yes—
54C3×#700—21no—
55C3×#700—21no—
56C4⋊Dic15C2×C411yescharacteristic
57C2×C3⋊#168C2×C411yescharacteristic
58C5×C4⋊Dic3C2×C411yescharacteristic
59C2×C5×C3⋊C8C2×C411yescharacteristic
60#6665—21no—
61Dic3×Dic5—21no—
62D5×C3⋊C8—21no—
63D5×C3⋊C8—21no—
64C2×C3×#104—51no—
65#5234—51no—
66#5233—51no—
67#5119—51no—
68C2×#803—51no—
69#4558—51no—
70#4523—51no—
71C2×#713C2×D311yescharacteristic
72C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
73C2×#700—31no—
74C2×#700C2×D311yescharacteristic
75C2×D5⋊C8—31no—
76C2×C8×D5—31no—
77#4038—31no—
78#4010—22no—
79#4004—61no—
80#4003—61no—
81#4003—61no—
82C4⋊#170—61no—
83#3992—62no—
84C5×#103—31no—
85#3967—31no—
86#3953—31no—
87#3941—31no—
88#1938—151noSylow
89C22×C3×D5—21no—
90C22×C3×D5#4111yescharacteristic
91C2×C3×AGL(1,𝔽5)—22no—
92C2×C3×AGL(1,𝔽5)—41no—
93C2×C3×AGL(1,𝔽5)—41no—
94C3×C5×Q8—21no—
95C3×C5×Q8—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5C22×C411yescharacteristic
98C2×Dic15—21no—
99C2×C5×Dic3—21no—
100C2×C3×Dic5C2×D411yescharacteristic
101C3×D20—21no—
102C3×D20—21no—
103C3×D20—41no—
104C3×D20—41no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107C3×C4×D5C2×D411yescharacteristic
108C3×C4×D5#4111yescharacteristic
109C3×C4×D5#4111yescharacteristic
110C3×C4×D5#4111yescharacteristic
111C3⋊#170—21no—
112C3⋊#170—21no—
113C3⋊#170—41no—
114C3×#170—41no—
115C3⋊#168—21no—
116C5×C3⋊C8—21no—
117C2×C3×#41—51no—
118C2×C3×M16—51no—
119C2×C3×C42—51no—
120#926—51no—
121C2×#208—51no—
122C22×C3⋊C8—51no—
123C22×C3⋊C8—51no—
124C3×#104—102no—
125#835—51no—
126#835AGL(1,𝔽5)11yescharacteristic, complemented
127#832—51no—
128#832—51no—
129#830—101no—
130#830—101no—
131Dic3⋊C8—101no—
132Dic3⋊C8—101no—
133#803—102no—
134C22×AGL(1,𝔽5)—21no—
135C2×C5×Q8C4×D311yescharacteristic
136#713—21no—
137#713—21no—
138#713—21no—
139#713—21no—
140C2×D20—21no—
141C2×D20C4×D311yescharacteristic
142C2×C4×D5—21no—
143C2×C4×D5C22×D311yescharacteristic
144C2×#170—21no—
145C2×#170—31no—
146C2×#170—31no—
147C2×#170—61no—
148C4×AGL(1,𝔽5)—21no—
149C4×AGL(1,𝔽5)#6712yes—
150#700—21no—
151#700—21no—
152#700—61no—
153#700—61no—
154D5⋊C8—61no—
155D5⋊C8—61no—
156C2×C5×C8—31no—
157C5×C4⋊C4—31no—
158#685—61no—
159C4⋊Dic5—31no—
160C4×Dic5—61no—
161C2×#168—31no—
162C8×D5—61no—
163C8×D5—61no—
164#424—151no—
165C2×#104—51no—
166#416—151no—
167#408—151no—
168C2×#97—151no—
169#341—151no—
170#339—151no—
171C22×C3×C5C4×D411yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—41no—
175C2×C3×D5—41no—
176C2×C3×D5—41no—
177C2×C3×D5#12311yescharacteristic
178C2×C3×D5#12611yescharacteristic
179C3×AGL(1,𝔽5)—42no—
180C3×C4×C5—41no—
181C3×C4×C5#11711yescharacteristic
182C3×C4×C5C4×D411yescharacteristic
183Dic15—41no—
184C3×Dic5#12111yescharacteristic
185C3×Dic5#12411yescharacteristic
186C5×Dic3—41no—
187C3×#41—101no—
188C3×#41—101no—
189C3×#41—101no—
190C3×#41—101no—
191C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
192C2×C3×D4—51no—
193C2×C3×D4—101no—
194C22×C3×C4—51no—
195C22×C3×C4—101no—
196C22×C3×C4—101no—
197C3×M16—101no—
198C3×M16—101no—
199C2×C3×C8—101no—
200C3×C42—52no—
201C3×C42—101no—
202#217—101no—
203C4⋊Dic3—51no—
204C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
205C4×Dic3—101no—
206#208—101no—
207#208—101no—
208C2×C3⋊C8—51no—
209C2×C3⋊C8—51no—
210C2×C3⋊C8—51no—
211C2×C3⋊C8—101no—
212C2×C3⋊C8—101no—
213C2×C3⋊C8—101no—
214C2×C3⋊C8—101no—
215C2×C3⋊C8—101no—
216C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic
217C22×D5—21no—
218C22×D5#23711yescharacteristic
219C2×AGL(1,𝔽5)—22no—
220C2×AGL(1,𝔽5)—41no—
221C2×AGL(1,𝔽5)—41no—
222C5×Q8—21no—
223C5×Q8—21no—
224C2×C4×C5—21no—
225C2×C4×C5—61no—
226C2×C4×C5C2×C4×D311yescharacteristic
227C2×Dic5—61no—
228C2×Dic5D3×D411yescharacteristic
229D20—21no—
230D20—21no—
231D20—41no—
232D20—41no—
233C4×D5—41no—
234C4×D5—41no—
235C4×D5#23511yescharacteristic
236C4×D5#23711yescharacteristic
237C4×D5#23711yescharacteristic
238C4×D5C2×#6711yescharacteristic
239#170—41no—
240#170—61no—
241#170—61no—
242#170—121no—
243C5×C8—61no—
244#168—61no—
245C2×#41—51no—
246C2×M16—51no—
247C2×M16—151no—
248C22×C8—151no—
249C22×C8—151no—
250#117—151no—
251C2×C42—51no—
252C4⋊C8—301no—
253C4⋊C8—301no—
254#104—102no—
255#103—31no—
256#103—151no—
257#102—151no—
258#102—151no—
259C22⋊C8—301no—
260C22⋊C8—301no—
261#97—302no—
262C2×C3×C5#38811yescharacteristic
263C2×C3×C5#44411yescharacteristic
264C2×C3×C5#44411yescharacteristic
265C3×D5—21no—
266C3×D5—21no—
267C3×D5—81no—
268C23×C3—51no—
269C23×C3—101no—
270C3×Q8—21no—
271C3×Q8—21no—
272C3×D4—101no—
273C3×D4—101no—
274C3×D4—201no—
275C3×D4—201no—
276C2×C3×C4—21no—
277C2×C3×C4—51no—
278C2×C3×C4—51no—
279C2×C3×C4—51no—
280C2×C3×C4—51no—
281C2×C3×C4—51no—
282C2×C3×C4—102no—
283C2×C3×C4—201no—
284C2×C3×C4—201no—
285C2×C3×C4—201no—
286C2×C3×C4—201no—
287C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
288C2×Dic3—21no—
289C2×Dic3—101no—
290C3×C8—201no—
291C3⋊C8—21no—
292C3⋊C8—101no—
293C3⋊C8—101no—
294C3⋊C8—101no—
295C3⋊C8—201no—
296C22×C5#67⋊C411yescharacteristic
297C2×D5—21no—
298C2×D5—21no—
299C2×D5—41no—
300C2×D5—41no—
301C2×D5—41no—
302C2×D5#87911yescharacteristic
303C2×D5#93511yescharacteristic
304AGL(1,𝔽5)—42no—
305C4×C5—41no—
306C4×C5—121no—
307C4×C5#87711yescharacteristic
308C4×C5C4×#6711yescharacteristic
309Dic5—121no—
310Dic5#88511yescharacteristic
311Dic5#93811yescharacteristic
312#41—101no—
313#41—101no—
314#41—101no—
315#41—101no—
316C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
317C2×D4—51no—
318C2×D4—101no—
319C22×C4—51no—
320C22×C4—101no—
321C22×C4—101no—
322M16—101no—
323M16—101no—
324M16—301no—
325M16—301no—
326C2×C8—31no—
327C2×C8—101no—
328C2×C8—151no—
329C2×C8—151no—
330C2×C8—151no—
331C2×C8—301no—
332C2×C8—301no—
333C2×C8—301no—
334C2×C8—301no—
335C2×C8—301no—
336C4⋊C4—31no—
337C4⋊C4—151no—
338C22⋊C4—301no—
339C42—52no—
340C42—101no—
341C42—301no—
342C3×C5#193811yescharacteristic, complemented, Hall
343C22×C3—51no—
344C22×C3—51no—
345C22×C3—101no—
346C22×C3—101no—
347C22×C3—201no—
348C22×C3—201no—
349C22×C3—201no—
350C22×C3D4×AGL(1,𝔽5)11yescharacteristic
351C3×C4—41no—
352C3×C4—51no—
353C3×C4—51no—
354C3×C4—202no—
355C3×C4#413011yescharacteristic
356C3×C4D4×AGL(1,𝔽5)11yescharacteristic
357Dic3—41no—
358Dic3—201no—
359C2×C5#482311yescharacteristic
360C2×C5#482311yescharacteristic
361C2×C5#494411yescharacteristic
362D5—21no—
363D5—21no—
364D5—81no—
365C23—51no—
366C23—101no—
367Q8—21no—
368Q8—21no—
369D4—101no—
370D4—101no—
371D4—201no—
372D4—201no—
373C2×C4—21no—
374C2×C4—51no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—51no—
378C2×C4—51no—
379C2×C4—61no—
380C2×C4—102no—
381C2×C4—201no—
382C2×C4—201no—
383C2×C4—201no—
384C2×C4—201no—
385C2×C4—301no—
386C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
387C8—61no—
388C8—201no—
389C8—301no—
390C8—301no—
391C8—301no—
392C8—601no—
393C2×C3—101no—
394C2×C3—101no—
395C2×C3—401no—
396C2×C3#6570011yescharacteristic
397C2×C3#6573711yescharacteristic
398C2×C3#6574111yescharacteristic
399C5#7037211yescharacteristic, complemented, Sylow
400C22—51no—
401C22—51no—
402C22—101no—
403C22—101no—
404C22—201no—
405C22—201no—
406C22—201no—
407C22#67⋊AGL(1,𝔽5)11yescharacteristic
408C4—41no—
409C4—51no—
410C4—51no—
411C4—121no—
412C4—202no—
413C4—601no—
414C4#9192311yescharacteristic
415C4AGL(1,𝔽5)×#6711yescharacteristic
416C3#1060398811yescharacteristic, complemented, Sylow
417C2—101no—
418C2—101no—
419C2—401no—
420C2#1175198011yescharacteristic
421C2#1175198211yescharacteristic
422C2#1175201711yescharacteristic
subgroupG/H
centerC22#67⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C4×D4
prime radicalC2×C3×#713C22
cosocleC2×C3×C4×D5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1938
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11751825C2semidirect
4. C2×#91149C2semidirect
6. #11746532C2semidirect
7. #11746526C2semidirect
22. #91119C4semidirect
23. C5×#835C4semidirect
24. #90965C4semidirect
25. #90964C4semidirect
126. #835AGL(1,𝔽5)semidirect
342. C3×C5#1938semidirect
399. C5#70372semidirect
416. C3#10603988semidirect

Maximal subgroups

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

subgroupindex
#117524462
C2×C3×#40102
#117518252
C2×#911492
#117492072
#117465322
#117465262
#106039883
#703725