Finite Group Databasebeta

← #14282860

Subgroups — #14282860

Order 1920.

608 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 102 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#11754563C211yesmaximal, characteristic
2C4×#6763C211yesmaximal, characteristic, complemented
3C3×D5⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
4#11746500C212yesmaximal, complemented
5#11746498C212yesmaximal, complemented
6#10604989—31nomaximal, Hall
7C2×C5×#235C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×C4×ΓL(1,𝔽16)—21no—
11C2×C3×D5⋊C8—21no—
12#91815C411yescharacteristic, complemented
13#91563—21no—
14#91539—21no—
15#91449C411yescharacteristic, complemented
16#91449C2211yescharacteristic
17C4×#1201—21no—
18C4×#1201—21no—
19#91359C411yescharacteristic, complemented
20#91359C2211yescharacteristic
21#91235C2212yes—
22C3×C22⋊#170—21no—
23C3×#4003C2212yes—
24#69942—51nomaximal, Hall
25#66162—31no—
26C4×C22⋊AGL(1,𝔽5)—31no—
27D5⋊(C22⋊C8)D311yescharacteristic, complemented
28#64926—32no—
29#64920—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×C5×#67—21no—
34C5×#235—42no—
35C2×C5×D12C2×C411yescharacteristic
36C2×C4×C5×D3—21no—
37C2×C5×Dic6C2×C411yescharacteristic
38C22×C3×Dic5D411yescharacteristic
39C2×C3×C4×D5—42no—
40C2×C3×C4×D5C2×C411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5C2311yescharacteristic
44C2×#1201—21no—
45C2×#1201—21no—
46#6763—24no—
47C4×ΓL(1,𝔽16)—42no—
48C2×C3×#170—22no—
49C3×D5⋊C8—42no—
50#6719—21no—
51C4⋊Dic15C2×C411yescharacteristic
52C4⋊Dic15C2×C411yescharacteristic
53C4×Dic15—21no—
54#6669—21no—
55#6669—21no—
56(C2×D3)⋊Dic5—21no—
57(C2×D3)⋊Dic5—21no—
58#6665—21no—
59#6665—21no—
60#5791—51no—
61C2×C3×C22⋊C8—51no—
62C4×#217—51no—
63#4470—52no—
64#4467—52no—
65C2×C5×#41—31no—
66C22×C4×D5C2×D311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—61no—
68C2×C4×AGL(1,𝔽5)—61no—
69C2×D5⋊C8—21no—
70#4092—31no—
71#4092—31no—
72#4084—31no—
73#4084—31no—
74#4083—61no—
75#4081—61no—
76C4×C22⋊D5—61no—
77C4×C22⋊D5—61no—
78#4072—31no—
79C22⋊#170—21no—
80(C2×AGL(1,𝔽5))⋊C4—32no—
81#4003C2×D312yes—
82#1501—151noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C5×D3—21no—
85C22×C3×D5—22no—
86C22×C3×D5—22no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C411yescharacteristic
89C22×C3×D5C2×D411yescharacteristic
90C2×ΓL(1,𝔽16)—24no—
91C2×ΓL(1,𝔽16)—42no—
92C2×ΓL(1,𝔽16)—42no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×Dic15—21no—
97C2×Dic15—21no—
98C5×#67—42no—
99C2×C5×Dic3—21no—
100C5×D12—41no—
101C4×C5×D3—42no—
102C5×Dic6—41no—
103C2×C3×Dic5—41no—
104C2×C3×Dic5C22⋊C411yescharacteristic
105C2×C3×Dic5C2×D411yescharacteristic
106C3×C4×D5—22no—
107C3×C4×D5—22no—
108C3×C4×D5—41no—
109C3×C4×D5—41no—
110#1201—42no—
111#1201—42no—
112C3×#170—42no—
113C23×C3×C4—51no—
114C2×#235AGL(1,𝔽5)11yescharacteristic, complemented
115C22×C3×C8—101no—
116#953—101no—
117C2×#217—101no—
118#930—51no—
119#930—51no—
120#929—101no—
121C4×#67—101no—
122C4×#67—101no—
123#926—51no—
124#924—51no—
125#924—51no—
126C2×C4×Dic3—101no—
127C3×C22⋊C8—52no—
128C3×C22⋊C8—101no—
129C3⋊#95—52no—
130C23×D5D1211yescharacteristic
131C22×AGL(1,𝔽5)—62no—
132C5×#41—122no—
133C2×C5×Q8—31no—
134C2×C5×D4—31no—
135C2×C5×D4—61no—
136C22×C4×C5—61no—
137C22×C4×C5C4×D311yescharacteristic
138C2×C22⋊D5—61no—
139C2×C22⋊D5—61no—
140C22×Dic5#6711yescharacteristic
141C2×C4×D5—42no—
142C2×C4×D5C4×D311yescharacteristic
143C2×C4×D5D1211yescharacteristic
144C2×C4×D5#6711yescharacteristic
145C2×C4×D5C22×D311yescharacteristic
146C22⋊AGL(1,𝔽5)—64no—
147C2×#170—22no—
148C4×AGL(1,𝔽5)—122no—
149D5⋊C8—42no—
150C22⋊Dic5—61no—
151C22⋊Dic5—61no—
152C22⋊Dic5—61no—
153#685—61no—
154#685—61no—
155C4⋊Dic5—31no—
156C4⋊Dic5—31no—
157#683—61no—
158#683—61no—
159C4×Dic5—61no—
160#525—151no—
161C2×C22⋊C8—51no—
162C4×C22⋊C4—151no—
163#328—152no—
164#327—152no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5—21no—
168C22×C3×C5C2×C22⋊C411yescharacteristic
169C2×C5×D3—41no—
170C2×C5×D3—41no—
171C2×C5×D3—41no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—44no—
177C2×C3×D5#10412yes—
178C2×C3×D5#13512yes—
179ΓL(1,𝔽16)—44no—
180C3×C4×C5—21no—
181C3×C4×C5#9912yes—
182Dic15—41no—
183Dic15—41no—
184C3×Dic5—21no—
185C3×Dic5—21no—
186C5×Dic3—41no—
187C24×C3—51no—
188C22×C3×C4—51no—
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—51no—
192C22×C3×C4—51no—
193C22×C3×C4—202no—
194C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
195C2×#67—21no—
196C2×#67—101no—
197C2×#67—101no—
198C22×Dic3—102no—
199#235—42no—
200C2×D12C2×AGL(1,𝔽5)11yescharacteristic
201C2×C4×D3—21no—
202C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
203C2×C3×C8—102no—
204C2×C3×C8—202no—
205#217—101no—
206#217—101no—
207#217—104no—
208#217—101no—
209(C2×D3)⋊C4—101no—
210(C2×D3)⋊C4—101no—
211C4⋊Dic3—51no—
212C4⋊Dic3—51no—
213Dic3⋊C4—101no—
214Dic3⋊C4—101no—
215C4×Dic3—101no—
216C4×Dic3—202no—
217C23×C5—61no—
218C23×C5(C2×D3)⋊C411yescharacteristic
219C22×D5—22no—
220C22×D5—22no—
221C22×D5—41no—
222C22×D5(C2×D3)⋊C411yescharacteristic
223C22×D5C2×D1211yescharacteristic
224C2×AGL(1,𝔽5)—64no—
225C2×AGL(1,𝔽5)—122no—
226C2×AGL(1,𝔽5)—122no—
227C5×Q8—121no—
228C5×D4—121no—
229C5×D4—122no—
230C2×C4×C5—22no—
231C2×C4×C5—61no—
232C2×C4×C5—122no—
233C2×C4×C5(C2×D3)⋊C411yescharacteristic
234C2×C4×C5C2×C4×D311yescharacteristic
235C22⋊D5—122no—
236C22⋊D5—122no—
237C2×Dic5—41no—
238C2×Dic5—61no—
239C2×Dic5—61no—
240C2×Dic5(C2×D3)⋊C411yescharacteristic
241C2×Dic5C2×#6711yescharacteristic
242C4×D5—22no—
243C4×D5—22no—
244C4×D5—41no—
245C4×D5—41no—
246#170—42no—
247C2×#41—31no—
248C23×C4—51no—
249C22×C8—101no—
250#123—301no—
251C22⋊Q8—151no—
252C22⋊Q8—151no—
253#121—151no—
254#121—151no—
255#120—301no—
256C4×D4—301no—
257C4×D4—301no—
258#117—151no—
259C2×C22⋊C4—301no—
260C2×C42—301no—
261C22⋊C8—52no—
262C22⋊C8—101no—
263#95—152no—
264C2×C3×C5—41no—
265C2×C3×C5C2×#9911yescharacteristic
266C2×C3×C5#41611yescharacteristic
267C2×C3×C5C2×#10411yescharacteristic
268C3×D5—22no—
269C3×D5—22no—
270C5×D3—81no—
271C23×C3—51no—
272C23×C3—51no—
273C23×C3—102no—
274C23×C3—102no—
275C23×C3—201no—
276C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
277C22×D3—21no—
278C2×C3×C4—22no—
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—42no—
289#67—202no—
290#67—202no—
291C2×Dic3—21no—
292C2×Dic3—104no—
293C2×Dic3—101no—
294C2×Dic3—101no—
295C2×Dic3—202no—
296C2×Dic3—202no—
297D12—41no—
298C4×D3—42no—
299Dic6—41no—
300C3×C8—202no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—121no—
305C22×C5—121no—
306C22×C5—121no—
307C22×C5C2×(C2×D3)⋊C411yescharacteristic
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—21no—
311C2×D5—21no—
312C2×D5—44no—
313C2×D5#82512yes—
314C2×D5#90412yes—
315AGL(1,𝔽5)—124no—
316C4×C5—21no—
317C4×C5—121no—
318C4×C5#80612yes—
319Dic5—21no—
320Dic5—21no—
321Dic5—121no—
322Dic5—121no—
323C24—51no—
324#41—122no—
325C2×Q8—31no—
326C2×D4—31no—
327C2×D4—61no—
328C2×D4—301no—
329C2×D4—301no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—51no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—61no—
336C22×C4—202no—
337C22×C4—302no—
338C22×C4D3×AGL(1,𝔽5)11yescharacteristic
339C2×C8—102no—
340C2×C8—202no—
341C4⋊C4—151no—
342C4⋊C4—151no—
343C4⋊C4—301no—
344C4⋊C4—301no—
345C22⋊C4—301no—
346C22⋊C4—301no—
347C22⋊C4—301no—
348C22⋊C4—301no—
349C22⋊C4—304no—
350C22⋊C4—301no—
351C42—301no—
352C42—602no—
353C3×C5#150111yescharacteristic, complemented, Hall
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—52no—
358C22×C3—52no—
359C22×C3—101no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3—204no—
364C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
365C2×D3—41no—
366C2×D3—41no—
367C2×D3—41no—
368C3×C4—21no—
369C3×C4—101no—
370C3×C4—101no—
371C3×C4#401112yes—
372Dic3—41no—
373Dic3—204no—
374Dic3—201no—
375Dic3—201no—
376C2×C5—41no—
377C2×C5—241no—
378C2×C5C2×#80611yescharacteristic
379C2×C5#511311yescharacteristic
380C2×C5C2×#82511yescharacteristic
381D5—22no—
382D5—22no—
383C23—51no—
384C23—51no—
385C23—61no—
386C23—102no—
387C23—102no—
388C23—201no—
389C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
390Q8—121no—
391D4—121no—
392D4—122no—
393D4—602no—
394D4—602no—
395C2×C4—22no—
396C2×C4—51no—
397C2×C4—51no—
398C2×C4—61no—
399C2×C4—102no—
400C2×C4—102no—
401C2×C4—122no—
402C2×C4—201no—
403C2×C4—201no—
404C2×C4—201no—
405C2×C4—304no—
406C2×C4—301no—
407C2×C4—301no—
408C2×C4—602no—
409C2×C4—602no—
410C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
411C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
412C8—202no—
413C2×C3—41no—
414C2×C3—102no—
415C2×C3—102no—
416C2×C3D5⋊#10411yescharacteristic
417C2×C3#6579411yescharacteristic
418C2×C3C2×#401111yescharacteristic
419D3—81no—
420C5#6994211yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—21no—
424C22—52no—
425C22—52no—
426C22—101no—
427C22—101no—
428C22—101no—
429C22—101no—
430C22—121no—
431C22—121no—
432C22—121no—
433C22—204no—
434C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
435C4—21no—
436C4—101no—
437C4—101no—
438C4—121no—
439C4—604no—
440C4—601no—
441C4—601no—
442C4#9117212yes—
443C3#1060498911yescharacteristic, complemented, Sylow
444C2—41no—
445C2—102no—
446C2—102no—
447C2—241no—
448C2#1175170411yescharacteristic
449C2D5⋊#82511yescharacteristic
450C2C2×#9117211yescharacteristic
subgroupG/H
centerC2×C4(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11754563C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1501
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6763C2semidirect
3. C3×D5⋊(C22⋊C8)C2semidirect
4. #11746500C2semidirect
5. #11746498C2semidirect
7. C2×C5×#235C4semidirect
12. #91815C4semidirect
15. #91449C4semidirect
19. #91359C4semidirect
27. D5⋊(C22⋊C8)D3semidirect
114. C2×#235AGL(1,𝔽5)semidirect
353. C3×C5#1501semidirect
420. C5#69942semidirect
443. C3#10604989semidirect

Maximal subgroups

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

subgroupindex
#117545632
C4×#67632
C3×D5⋊(C22⋊C8)2
#117465002
#117464982
#106049893
#699425