Finite Group Databasebeta

← #14419517

Subgroups — #14419517

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754410C211yesmaximal, characteristic
2C4×#6762C211yesmaximal, characteristic, complemented
3C3×#65757C211yesmaximal, characteristic, complemented
4#11752039C212yesmaximal, complemented
5#11751992C212yesmaximal, complemented
6#11751897C212yesmaximal, complemented
7#11751870C212yesmaximal, complemented
8#11751809C212yesmaximal, complemented
9#11751771C212yesmaximal, complemented
10#10617234—31nomaximal, Hall
11C2×#6762C2212yes—
12C2×#6739C2212yes—
13C2×D3×#170C2212yes—
14#91813C411yescharacteristic
15C5×#930C411yescharacteristic
16C2×C3×C4×Dic5C2211yescharacteristic
17Dic5×#67C2212yes—
18#91539C412yes—
19Dic5×D12C2211yescharacteristic
20#91398C2212yes—
21#91388C411yescharacteristic
22#91373C411yescharacteristic
23#91372C412yes—
24#91333C2211yescharacteristic
25#91231C2212yes—
26C4×C3⋊#170C2211yescharacteristic
27C4×C3⋊#170C2211yescharacteristic
28C3×C22⋊#170C2211yescharacteristic
29C3×C22⋊#170C2211yescharacteristic
30C3×#4004C2212yes—
31C3×#4002C2212yes—
32#91177C2212yes—
33Dic3⋊#170C2212yes—
34(C2×D3)⋊#170C2212yes—
35(C2×D3)⋊#170C2212yes—
36#91169C2212yes—
37Dic3×#170C2212yes—
38#79326—51nomaximal, Hall
39#65934—31no—
40#65774—32no—
41#65773—32no—
42#65773—32no—
43#65757D311yescharacteristic, complemented
44C4×#704—31no—
45#65707—32no—
46#65697—32no—
47#65691—32no—
48C22×C3×C4×C5C2×C411yescharacteristic
49C2×Dic30C2×C411yescharacteristic
50C2×C5×#67C2×C412yes—
51C2×C5×D12C2×C411yescharacteristic
52C22×C3×Dic5C2×C411yescharacteristic
53C22×C3×Dic5C2311yescharacteristic
54C2×D3×Dic5C2312yes—
55#6762—24no—
56C2×C3⋊#170C2312yes—
57C2×C3⋊#170C2312yes—
58C2×C3×#170C2312yes—
59C2×C3×#170C2312yes—
60#6739—24no—
61D3×#170—24no—
62#6719C2×C412yes—
63C4⋊Dic15C2×C411yescharacteristic
64#6712C2×C412yes—
65C5×(C2×D3)⋊C4C2×C412yes—
66C5×C4⋊Dic3C2×C411yescharacteristic
67C3×C4×Dic5C2×C411yescharacteristic
68C3×C4×Dic5C2×C411yescharacteristic
69C3×C4×Dic5C2311yescharacteristic
70C3×C4×Dic5C2311yescharacteristic
71Dic3⋊Dic5C2×C412yes—
72#6669C2×C412yes—
73#6669C2×C412yes—
74(C2×D3)⋊Dic5C2×C412yes—
75(C2×D3)⋊Dic5C2×C412yes—
76(C2×D3)⋊Dic5C2×C412yes—
77Dic3×Dic5C2312yes—
78#5479—51no—
79C3×#432—51no—
80C4×#208—51no—
81#4836—52no—
82#4828—52no—
83#4729—52no—
84#4728—52no—
85#4709—52no—
86#4708—52no—
87C2×#704—32no—
88C2×#704—32no—
89C22×#170—32no—
90C5×#121—31no—
91#4082—31no—
92#4081—32no—
93D4×Dic5—32no—
94D4×Dic5—31no—
95#4070—31no—
96C2×C4×Dic5C2×D311yescharacteristic
97#4036—31no—
98#4033—31no—
99#4025—32no—
100#4023—32no—
101C22⋊#170—32no—
102C22⋊#170—32no—
103C22⋊#170C2×D311yescharacteristic
104C22⋊#170C2×D311yescharacteristic
105#4004—32no—
106#4004C2×D312yes—
107#4002—32no—
108#4002—32no—
109#4002C2×D312yes—
110C4⋊#170—32no—
111C4×#170—32no—
112C4×#170—31no—
113C4×#170—31no—
114#2967—151noSylow
115C23×C3×C5C22×C411yescharacteristic
116C22×C5×D3C22×C412yes—
117C2×C3×C4×C5—41no—
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×Dic15C22×C412yes—
121C2×Dic15C22×C412yes—
122Dic30—41no—
123C5×#67—42no—
124C2×C5×Dic3C22×C412yes—
125C5×D12—41no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—41no—
128C2×C3×Dic5C22×C411yescharacteristic
129C2×C3×Dic5C22×C411yescharacteristic
130C2×C3×Dic5C22×C411yescharacteristic
131C2×C3×Dic5C2411yescharacteristic
132D3×Dic5—22no—
133D3×Dic5—22no—
134C3⋊#170—22no—
135C3⋊#170#4114yes—
136C3×#170—22no—
137C3×#170—22no—
138C2×C3×C42—51no—
139#930AGL(1,𝔽5)11yescharacteristic
140#929—52no—
141C4×#67—52no—
142#924—51no—
143C2×#208—52no—
144C2×ΓL(1,𝔽25)—52no—
145C2×D3×C8—52no—
146#876—52no—
147#875—51no—
148C4×D12—51no—
149#872—52no—
150#870—51no—
151C4×Dic6—51no—
152C3×C4⋊C8—52no—
153C3×C22⋊C8—51no—
154C3×C22⋊C8—51no—
155C3×#97—52no—
156(C2×D3)⋊C8—52no—
157(C2×D3)⋊C8—52no—
158#815—52no—
159Dic3⋊C8—52no—
160Dic3⋊C8—52no—
161C8×Dic3—52no—
162#803—52no—
163C4×C3⋊C8—51no—
164C4×C3⋊C8—51no—
165C2×C5×D4—32no—
166C2×C5×D4—31no—
167C22×C4×C5C4×D311yescharacteristic
168C22×Dic5—32no—
169C22×Dic5C4×D311yescharacteristic
170C22×Dic5C22×D311yescharacteristic
171C2×Dic10—31no—
172#704—64no—
173#704—64no—
174C2×#170—32no—
175C2×#170—32no—
176C2×#170—64no—
177C2×#170C22×D312yes—
178C2×#170C22×D312yes—
179C5×C4⋊C4—31no—
180C5×C22⋊C4—32no—
181C22⋊Dic5—32no—
182C22⋊Dic5—32no—
183C22⋊Dic5—32no—
184C22⋊Dic5—32no—
185C4⋊Dic5—31no—
186C4⋊Dic5—32no—
187#683—32no—
188#683—32no—
189#683—32no—
190C4×Dic5—32no—
191C4×Dic5C4×D311yescharacteristic
192C4×Dic5C4×D311yescharacteristic
193C4×Dic5C22×D311yescharacteristic
194C4×Dic5C22×D311yescharacteristic
195#529—151no—
196#446—152no—
197#436—152no—
198#435—152no—
199#435—152no—
200#433—152no—
201#432—51no—
202#405—152no—
203C4×M16—151no—
204C22×C3×C5—21no—
205C22×C3×C5—21no—
206C22×C3×C5—21no—
207C22×C3×C5C23×C411yescharacteristic
208C2×C5×D3—22no—
209C2×C5×D3—22no—
210C2×C5×D3—22no—
211C3×C4×C5—21no—
212C3×C4×C5—21no—
213Dic15—22no—
214Dic15—22no—
215C3×Dic5—21no—
216C3×Dic5—21no—
217C3×Dic5—21no—
218C3×Dic5C2×#4112yes—
219C5×Dic3—22no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
223C2×#67C2×AGL(1,𝔽5)12yes—
224C2×D12C2×AGL(1,𝔽5)11yescharacteristic
225C2×C4×D3—52no—
226C2×Dic6—51no—
227C2×C3×C8—52no—
228C2×C3×C8—52no—
229C3×C42—51no—
230C3×C42—51no—
231C3×C42—51no—
232C3×C42—51no—
233#217—52no—
234(C2×D3)⋊C4—52no—
235(C2×D3)⋊C4—52no—
236(C2×D3)⋊C4—52no—
237(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
238C4⋊Dic3—51no—
239C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
240Dic3⋊C4—52no—
241Dic3⋊C4—52no—
242Dic3⋊C4—52no—
243Dic3⋊C4—52no—
244C4×Dic3—52no—
245#208—104no—
246C2×C3⋊C8—52no—
247C2×C3⋊C8—52no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5—32no—
251C23×C5C2×C4×D311yescharacteristic
252C5×D4—121no—
253C5×D4—122no—
254C2×C4×C5—32no—
255C2×C4×C5—41no—
256C2×C4×C5C2×C4×D311yescharacteristic
257C2×C4×C5C2×C4×D311yescharacteristic
258C2×Dic5—22no—
259C2×Dic5—32no—
260C2×Dic5—32no—
261C2×Dic5—41no—
262C2×Dic5—62no—
263C2×Dic5—62no—
264C2×Dic5C2×C4×D311yescharacteristic
265C2×Dic5C2×C4×D311yescharacteristic
266C2×Dic5C2×C4×D311yescharacteristic
267C2×Dic5C23×D311yescharacteristic
268Dic10—121no—
269#170—22no—
270#170—22no—
271#170—34no—
272#170—62no—
273C2×M16—152no—
274C2×M16—152no—
275C22×C8—152no—
276#126—152no—
277#125—151no—
278#124—151no—
279#123—152no—
280C22⋊Q8—151no—
281#121—31no—
282C4×Q8—151no—
283C4×D4—151no—
284C4×D4—152no—
285#117—152no—
286C2×C42—51no—
287C4⋊C8—52no—
288C4⋊C8—152no—
289C4⋊C8—152no—
290C22⋊C8—51no—
291C22⋊C8—51no—
292C22⋊C8—152no—
293C22⋊C8—152no—
294#97—52no—
295#97—152no—
296#97—152no—
297C4×C8—151no—
298C4×C8—151no—
299C4×C8—152no—
300C2×C3×C5—41no—
301C2×C3×C5C4×#4111yescharacteristic
302C2×C3×C5#56811yescharacteristic
303C2×C3×C5#56811yescharacteristic
304C5×D3—42no—
305C23×C3C22×AGL(1,𝔽5)11yescharacteristic
306C22×D3C22×AGL(1,𝔽5)12yes—
307C2×C3×C4—41no—
308C2×C3×C4—51no—
309C2×C3×C4—51no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—102no—
313C2×C3×C4—201no—
314C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
315C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
316#67—42no—
317C2×Dic3—52no—
318C2×Dic3—52no—
319C2×Dic3C22×AGL(1,𝔽5)12yes—
320D12—41no—
321C4×D3—102no—
322C4×D3—102no—
323Dic6—201no—
324C3×C8—102no—
325C3×C8—102no—
326C3⋊C8—54no—
327C3⋊C8—102no—
328C22×C5—21no—
329C22×C5—21no—
330C22×C5—21no—
331C22×C5—62no—
332C22×C5—62no—
333C22×C5—62no—
334C22×C5C22×C4×D311yescharacteristic
335C4×C5—21no—
336C4×C5—21no—
337C4×C5—62no—
338Dic5—21no—
339Dic5—21no—
340Dic5—21no—
341Dic5—62no—
342Dic5—62no—
343Dic5D3×#4112yes—
344C2×Q8—151no—
345C2×D4—32no—
346C2×D4—31no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—152no—
350C22×C4D3×AGL(1,𝔽5)11yescharacteristic
351M16—304no—
352M16—304no—
353C2×C8—52no—
354C2×C8—52no—
355C2×C8—152no—
356C2×C8—152no—
357C2×C8—304no—
358C4⋊C4—31no—
359C4⋊C4—152no—
360C4⋊C4—152no—
361C4⋊C4—152no—
362C4⋊C4—152no—
363C4⋊C4—151no—
364C22⋊C4—32no—
365C22⋊C4—152no—
366C22⋊C4—152no—
367C22⋊C4—152no—
368C22⋊C4—152no—
369C42—51no—
370C42—51no—
371C42—51no—
372C42—51no—
373C42—152no—
374C3×C5#296711yescharacteristic, complemented, Hall
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3C23×AGL(1,𝔽5)11yescharacteristic
379C2×D3—22no—
380C2×D3—22no—
381C2×D3—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—52no—
385C3×C4—101no—
386C3×C4—101no—
387C3×C4—101no—
388Dic3—22no—
389Dic3—102no—
390Dic3—102no—
391C2×C5—41no—
392C2×C5—122no—
393C2×C5#552211yescharacteristic
394C2×C5#574511yescharacteristic
395C2×C5#574511yescharacteristic
396C23—32no—
397C23C2×D3×AGL(1,𝔽5)11yescharacteristic
398Q8—601no—
399D4—121no—
400D4—122no—
401C2×C4—32no—
402C2×C4—41no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—51no—
407C2×C4—102no—
408C2×C4—152no—
409C2×C4—152no—
410C2×C4—201no—
411C2×C4—302no—
412C2×C4—302no—
413C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
414C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
415C8—102no—
416C8—102no—
417C8—154no—
418C8—302no—
419C2×C3—41no—
420C2×C3#6625611yescharacteristic
421C2×C3#6625611yescharacteristic
422C2×C3#41×AGL(1,𝔽5)11yescharacteristic
423D3—42no—
424C5#7932611yescharacteristic, complemented, Sylow
425C22—21no—
426C22—21no—
427C22—21no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22C22×D3×AGL(1,𝔽5)11yescharacteristic
432C4—21no—
433C4—21no—
434C4—52no—
435C4—62no—
436C4—101no—
437C4—101no—
438C4—101no—
439C4—302no—
440C4—302no—
441C3#1061723411yescharacteristic, complemented, Sylow
442C2—41no—
443C2—122no—
444C2#1175682211yescharacteristic
445C2#1175683511yescharacteristic
446C2#1175686311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radicalC5×#930C4
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2967
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6762C2semidirect
3. C3×#65757C2semidirect
4. #11752039C2semidirect
5. #11751992C2semidirect
6. #11751897C2semidirect
7. #11751870C2semidirect
8. #11751809C2semidirect
9. #11751771C2semidirect
43. #65757D3semidirect
374. C3×C5#2967semidirect
424. C5#79326semidirect
441. C3#10617234semidirect

Maximal subgroups

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

subgroupindex
#117544102
C4×#67622
C3×#657572
#117520392
#117519922
#117518972
#117518702
#117518092
#117517712
#106172343
#793265