Finite Group Databasebeta

← #14281227

Subgroups — #14281227

Order 1920.

560 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 98 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×#91447C211yesmaximal, characteristic
2#11752415C211yesmaximal, characteristic, complemented
3C3×D5⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
4#11746568C212yesmaximal, complemented
5#11746566C212yesmaximal, complemented
6#10604206—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C22×#1201C411yescharacteristic
9C2×#6758—21no—
10C2×C3×D5⋊C8—21no—
11C2×#6712C411yescharacteristic
12C2×C5×(C2×D3)⋊C4C411yescharacteristic
13C2×Dic3⋊Dic5C2211yescharacteristic
14C2×(C2×D3)⋊Dic5C2211yescharacteristic
15C2×#6665C411yescharacteristic
16#91447—21no—
17#91447—21no—
18#91447—21no—
19#91447—21no—
20#91241C2212yes—
21#91232—21no—
22C3×C22⋊#170C2212yes—
23C3×#4003—21no—
24#69920—51nomaximal, Hall
25C2×#4029—31no—
26#65756—31no—
27D5⋊(C22⋊C8)D311yescharacteristic, complemented
28#64933—32no—
29#64913—32no—
30C23×C5×D3C2×C411yescharacteristic
31C23×C3×D5C2×C411yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C22×Dic15C2×C411yescharacteristic
34C22×Dic15C2×C411yescharacteristic
35C22×C5×Dic3C2×C411yescharacteristic
36C22×C3×Dic5C2311yescharacteristic
37C2×C3×C4×D5—42no—
38C2×C3×C4×D5D411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×#1201—21no—
43C2×#1201—21no—
44C2×#1201—42no—
45C2×C3⋊#170—22no—
46#6758—42no—
47C2×C3×#170—22no—
48C3×D5⋊C8—42no—
49#6712—21no—
50#6712—21no—
51C5×(C2×D3)⋊C4—21no—
52C5×(C2×D3)⋊C4—21no—
53Dic3⋊Dic5—21no—
54Dic3⋊Dic5—21no—
55(C2×D3)⋊Dic5—21no—
56(C2×D3)⋊Dic5—21no—
57#6665—21no—
58#6665—21no—
59C2×#929—51no—
60C2×C3×C22⋊C8—51no—
61#5207—51no—
62#4469—52no—
63#4468—52no—
64C22×C22⋊D5—31no—
65C22×C4×D5C2×D311yescharacteristic
66C2×#700—61no—
67C2×D5⋊C8—21no—
68C2×C5×C22⋊C4—31no—
69C2×C22⋊Dic5—31no—
70C2×#685—31no—
71C2×C4⋊Dic5—31no—
72C2×#683—31no—
73#4029—61no—
74#4029—61no—
75#4029—61no—
76#4029—61no—
77C22⋊#170—32no—
78C22⋊#170C2×D312yes—
79#4003—21no—
80#4003—61no—
81#1468—151noSylow
82C23×C3×C5C22×C411yescharacteristic
83C22×C5×D3—21no—
84C22×C5×D3—22no—
85C22×C5×D3—41no—
86C22×C5×D3—41no—
87C22×C3×D5—22no—
88C22×C3×D5—22no—
89C22×C3×D5—41no—
90C22×C3×D5C22⋊C411yescharacteristic
91C22×C3×D5C22⋊C411yescharacteristic
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5C22⋊C411yescharacteristic
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×Dic15—21no—
96C2×Dic15—21no—
97C2×Dic15—41no—
98C2×Dic15—41no—
99C2×C5×Dic3—21no—
100C2×C5×Dic3—22no—
101C2×C3×Dic5—41no—
102C2×C3×Dic5C2×D411yescharacteristic
103C2×C3×Dic5C2×D411yescharacteristic
104C3×C4×D5—22no—
105C3×C4×D5—22no—
106C3×C4×D5—41no—
107C3×C4×D5—41no—
108#1201—42no—
109#1201—42no—
110C3⋊#170—42no—
111C3×#170—42no—
112C23×C3×C4—51no—
113C22×#67—51no—
114C22×C3×C8—101no—
115C2×#217—51no—
116#929—101no—
117#929—101no—
118#929—101no—
119#929—101no—
120C2×(C2×D3)⋊C4—51no—
121C2×(C2×D3)⋊C4AGL(1,𝔽5)11yescharacteristic
122C2×Dic3⋊C4—51no—
123C2×Dic3⋊C4—51no—
124C2×#208—101no—
125C3×C22⋊C8—52no—
126C3×C22⋊C8—101no—
127#830—52no—
128#830—101no—
129C24×C5—31no—
130C23×D5C4×D311yescharacteristic
131C22×C4×C5—31no—
132C22×C4×C5C4×D311yescharacteristic
133C2×C22⋊D5—61no—
134C2×C22⋊D5—61no—
135C2×C22⋊D5—122no—
136C22×Dic5—31no—
137C22×Dic5—31no—
138C22×Dic5C22×D311yescharacteristic
139C2×C4×D5—42no—
140C2×C4×D5D1211yescharacteristic
141C2×C4×D5D1211yescharacteristic
142C2×C4×D5#6711yescharacteristic
143C2×C4×D5#6711yescharacteristic
144C2×#170—22no—
145C2×#170—62no—
146#700—122no—
147D5⋊C8—42no—
148C5×C22⋊C4—61no—
149C5×C22⋊C4—61no—
150C22⋊Dic5—61no—
151C22⋊Dic5—61no—
152#685—61no—
153#685—61no—
154C4⋊Dic5—61no—
155C4⋊Dic5—61no—
156#683—61no—
157#683—61no—
158C2×#123—151no—
159#407—151no—
160C2×C22⋊C8—51no—
161#324—152no—
162#323—152no—
163C22×C3×C5—21no—
164C22×C3×C5—21no—
165C22×C3×C5C2×C22⋊C411yescharacteristic
166C22×C3×C5#13112yes—
167C2×C5×D3—41no—
168C2×C5×D3—41no—
169C2×C5×D3—41no—
170C2×C5×D3—82no—
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—44no—
177C2×C3×D5#9912yes—
178C3×C4×C5—21no—
179C3×C4×C5—21no—
180Dic15—41no—
181Dic15—41no—
182C3×Dic5—21no—
183C3×Dic5—21no—
184C5×Dic3—41no—
185C24×C3—51no—
186C23×D3C2×AGL(1,𝔽5)11yescharacteristic
187C22×C3×C4—51no—
188C22×C3×C4—51no—
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—51no—
192C22×C3×C4—202no—
193C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
194C2×#67—101no—
195C2×#67—101no—
196C2×#67—202no—
197C22×Dic3—51no—
198C22×Dic3—51no—
199C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
200C2×C3×C8—102no—
201C2×C3×C8—202no—
202#217—101no—
203#217—101no—
204(C2×D3)⋊C4—21no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—101no—
207(C2×D3)⋊C4—101no—
208Dic3⋊C4—101no—
209Dic3⋊C4—101no—
210Dic3⋊C4—101no—
211Dic3⋊C4—101no—
212#208—202no—
213C2×C3⋊C8—102no—
214C23×C5—61no—
215C23×C5—62no—
216C23×C5—121no—
217C23×C5—121no—
218C23×C5C2×C4×D311yescharacteristic
219C22×D5—22no—
220C22×D5—22no—
221C22×D5—41no—
222C22×D5(C2×D3)⋊C411yescharacteristic
223C22×D5(C2×D3)⋊C411yescharacteristic
224C2×C4×C5—41no—
225C2×C4×C5—62no—
226C2×C4×C5—61no—
227C2×C4×C5(C2×D3)⋊C411yescharacteristic
228C2×C4×C5(C2×D3)⋊C411yescharacteristic
229C22⋊D5—122no—
230C22⋊D5—122no—
231C2×Dic5—41no—
232C2×Dic5—61no—
233C2×Dic5—61no—
234C2×Dic5—121no—
235C2×Dic5—121no—
236C2×Dic5C2×D1211yescharacteristic
237C2×Dic5C2×#6711yescharacteristic
238C4×D5—22no—
239C4×D5—22no—
240C4×D5—41no—
241C4×D5—41no—
242#170—42no—
243#170—122no—
244C22×D4—151no—
245C23×C4—51no—
246C2×M16—301no—
247C22×C8—101no—
248#123—301no—
249#123—301no—
250#123—301no—
251#123—301no—
252C2×C4⋊C4—151no—
253C2×C4⋊C4—151no—
254C2×C22⋊C4—31no—
255C2×C22⋊C4—151no—
256C2×C22⋊C4—151no—
257C22⋊C8—52no—
258C22⋊C8—101no—
259C22⋊C8—152no—
260C22⋊C8—301no—
261C2×C3×C5—22no—
262C2×C3×C5#40811yescharacteristic
263C2×C3×C5C2×#9911yescharacteristic
264C2×C3×C5#41311yescharacteristic
265C3×D5—22no—
266C3×D5—41no—
267C5×D3—81no—
268C23×C3—51no—
269C23×C3—51no—
270C23×C3—102no—
271C23×C3—102no—
272C23×C3—201no—
273C23×C3C22×AGL(1,𝔽5)11yescharacteristic
274C22×D3—21no—
275C22×D3—22no—
276C22×D3—41no—
277C22×D3—41no—
278C2×C3×C4—41no—
279C2×C3×C4—51no—
280C2×C3×C4—51no—
281C2×C3×C4—102no—
282C2×C3×C4—102no—
283C2×C3×C4—201no—
284C2×C3×C4—201no—
285C2×C3×C4—201no—
286C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
287C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
288#67—202no—
289#67—202no—
290C2×Dic3—22no—
291C2×Dic3—21no—
292C2×Dic3—101no—
293C2×Dic3—101no—
294C2×Dic3—201no—
295C2×Dic3—201no—
296C3×C8—202no—
297C3⋊C8—202no—
298C22×C5—21no—
299C22×C5—21no—
300C22×C5—121no—
301C22×C5—121no—
302C22×C5—121no—
303C22×C5—242no—
304C22×C5#90112yes—
305C22×C5C2×(C2×D3)⋊C411yescharacteristic
306C2×D5—21no—
307C2×D5—21no—
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—21no—
311C2×D5—44no—
312C2×D5#80612yes—
313C4×C5—21no—
314C4×C5—21no—
315C4×C5—121no—
316Dic5—21no—
317Dic5—21no—
318Dic5—121no—
319Dic5—121no—
320C24—31no—
321C24—51no—
322C2×D4—301no—
323C2×D4—301no—
324C2×D4—602no—
325C22×C4—31no—
326C22×C4—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—51no—
330C22×C4—51no—
331C22×C4—151no—
332C22×C4—151no—
333C22×C4—202no—
334C22×C4D3×AGL(1,𝔽5)11yescharacteristic
335M16—602no—
336C2×C8—102no—
337C2×C8—202no—
338C2×C8—302no—
339C4⋊C4—301no—
340C4⋊C4—301no—
341C4⋊C4—301no—
342C4⋊C4—301no—
343C22⋊C4—61no—
344C22⋊C4—61no—
345C22⋊C4—301no—
346C22⋊C4—301no—
347C22⋊C4—301no—
348C22⋊C4—301no—
349C3×C5#146811yescharacteristic, complemented, Hall
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—52no—
353C22×C3—101no—
354C22×C3—101no—
355C22×C3—101no—
356C22×C3—101no—
357C22×C3—101no—
358C22×C3—204no—
359C22×C3#413112yes—
360C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—82no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—101no—
368C3×C4—101no—
369Dic3—41no—
370Dic3—201no—
371Dic3—201no—
372C2×C5—22no—
373C2×C5—241no—
374C2×C5C2×#80611yescharacteristic
375C2×C5#511011yescharacteristic
376C2×C5#513211yescharacteristic
377D5—22no—
378D5—41no—
379C23—51no—
380C23—51no—
381C23—61no—
382C23—62no—
383C23—102no—
384C23—102no—
385C23—121no—
386C23—121no—
387C23—201no—
388C23C2×D3×AGL(1,𝔽5)11yescharacteristic
389D4—602no—
390D4—602no—
391C2×C4—41no—
392C2×C4—51no—
393C2×C4—51no—
394C2×C4—62no—
395C2×C4—61no—
396C2×C4—102no—
397C2×C4—102no—
398C2×C4—201no—
399C2×C4—201no—
400C2×C4—201no—
401C2×C4—301no—
402C2×C4—301no—
403C2×C4—601no—
404C2×C4—601no—
405C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
406C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
407C8—202no—
408C8—602no—
409C2×C3—22no—
410C2×C3—102no—
411C2×C3—201no—
412C2×C3#6576111yescharacteristic
413C2×C3D5⋊#9911yescharacteristic
414C2×C3#6577611yescharacteristic
415D3—81no—
416C5#6992011yescharacteristic, complemented, Sylow
417C22—21no—
418C22—21no—
419C22—52no—
420C22—101no—
421C22—101no—
422C22—101no—
423C22—101no—
424C22—101no—
425C22—121no—
426C22—121no—
427C22—121no—
428C22—204no—
429C22—242no—
430C22#9191112yes—
431C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
432C4—21no—
433C4—21no—
434C4—101no—
435C4—101no—
436C4—121no—
437C4—601no—
438C4—601no—
439C3#1060420611yescharacteristic, complemented, Sylow
440C2—22no—
441C2—102no—
442C2—201no—
443C2—241no—
444C2#1175180111yescharacteristic
445C2#1175181811yescharacteristic
446C2D5⋊#80611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#1201C4
cosocleC22×C3×Dic5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1468
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752415C2semidirect
3. C3×D5⋊(C22⋊C8)C2semidirect
4. #11746568C2semidirect
5. #11746566C2semidirect
27. D5⋊(C22⋊C8)D3semidirect
349. C3×C5#1468semidirect
416. C5#69920semidirect
439. C3#10604206semidirect

Maximal subgroups

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

subgroupindex
C2×#914472
#117524152
C3×D5⋊(C22⋊C8)2
#117465682
#117465662
#106042063
#699205