Finite Group Databasebeta

← #14419333

Subgroups — #14419333

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754415C211yesmaximal, characteristic
2#11752418C211yesmaximal, characteristic, complemented
3C2×C3×C4×#170C211yesmaximal, characteristic, complemented
4#11752046C212yesmaximal, complemented
5#11752036C212yesmaximal, complemented
6#11751914C211yesmaximal, characteristic, complemented
7#11751893C211yesmaximal, characteristic, complemented
8#11751872C212yesmaximal, complemented
9#11751800C212yesmaximal, complemented
10#11751789C211yesmaximal, characteristic, complemented
11Dic6×#170C211yesmaximal, characteristic, complemented
12#10617265—31nomaximal, Hall
13C22×C3×#170C2211yescharacteristic
14C22×C3×#170C2211yescharacteristic
15C2×#6740C2212yes—
16C2×#6738C2212yes—
17#91819C411yescharacteristic
18C5×#924C411yescharacteristic
19C2×C3×C4×Dic5C2211yescharacteristic
20#91552C2212yes—
21#91546C412yes—
22#91418C2211yescharacteristic
23#91402C2212yes—
24#91388C411yescharacteristic
25#91387C412yes—
26#91375C411yescharacteristic
27Dic5×Dic6C2211yescharacteristic
28#91241C2212yes—
29#91231C2212yes—
30C4⋊(C3⋊#170)C2211yescharacteristic
31C4⋊(C3⋊#170)C2211yescharacteristic
32C3×C4×#170C2211yescharacteristic
33C3×C4×#170C2211yescharacteristic
34C3×C4×#170C2211yescharacteristic
35C3×C4×#170C2211yescharacteristic
36#91177C2212yes—
37Dic3⋊#170C2212yes—
38#91171C2212yes—
39#91171C2212yes—
40#91169C2212yes—
41Dic3×#170C2212yes—
42#78448—51nomaximal, Hall
43#65958—31no—
44#65788—31no—
45Q8×#170—31no—
46#65759—31no—
47C2×C4×#170D311yescharacteristic, complemented
48#65706—32no—
49#65705—31no—
50#65702—31no—
51#65701—32no—
52#65693—32no—
53#65692—32no—
54C22×C3×C4×C5C2×C411yescharacteristic
55C2×#1220C2×C412yes—
56C2×D60C2×C411yescharacteristic
57C2×C5×Dic6C2×C411yescharacteristic
58C22×C3×Dic5C2×C411yescharacteristic
59C22×C3×Dic5C2311yescharacteristic
60C2×#1200C2312yes—
61C2×C3⋊#170C2312yes—
62C2×C3⋊#170C2312yes—
63C2×C3×#170—22no—
64C2×C3×#170—22no—
65C2×C3×#170C2311yescharacteristic
66C2×C3×#170C2311yescharacteristic
67C2×C3×#170C2311yescharacteristic
68C2×C3×#170C2311yescharacteristic
69#6740—24no—
70#6738—24no—
71#6714C2×C412yes—
72C4⋊Dic15C2×C411yescharacteristic
73C5×#217C2×C412yes—
74C5×C4⋊Dic3C2×C411yescharacteristic
75C5×Dic3⋊C4C2×C412yes—
76C3×C4×Dic5C2×C411yescharacteristic
77C3×C4×Dic5C2×C411yescharacteristic
78C3×C4×Dic5C2311yescharacteristic
79C3×C4×Dic5C2311yescharacteristic
80Dic3⋊Dic5C2×C412yes—
81Dic3⋊Dic5C2×C412yes—
82#6669C2×C412yes—
83#6667C2×C412yes—
84#6667C2×C412yes—
85#6667C2×C412yes—
86Dic3×Dic5C2312yes—
87#5479—51no—
88C2×C3×C4×C8—51no—
89#5111—52no—
90C8×#67—52no—
91#4926—51no—
92#4690—52no—
93#4689—52no—
94#4688—51no—
95C8×D12—51no—
96#4679—51no—
97C8×Dic6—51no—
98C22×#170C2×D311yescharacteristic
99C22×#170C2×D311yescharacteristic
100C2×#700—32no—
101C2×D5⋊C8—32no—
102C5×C22⋊Q8—31no—
103#4093—31no—
104Q8×Dic5—31no—
105#4076—31no—
106C2×C4×Dic5C2×D311yescharacteristic
107#4044—32no—
108#4039—31no—
109#4038—32no—
110#4036—31no—
111#4032—32no—
112#4027—32no—
113C22⋊#170—32no—
114#4004—32no—
115#4003—32no—
116#4003—32no—
117#4002—32no—
118#4002—32no—
119C4⋊#170—32no—
120C4⋊#170—31no—
121C4⋊#170—31no—
122C4×#170—32no—
123C4×#170C2×D311yescharacteristic
124C4×#170C2×D311yescharacteristic
125C4×#170C2×D311yescharacteristic
126C4×#170C2×D311yescharacteristic
127#2966—151noSylow
128C23×C3×C5C22×C411yescharacteristic
129C22×D15C22×C412yes—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5C22×C411yescharacteristic
132C2×C3×C4×C5C22×C411yescharacteristic
133#1220—24no—
134C2×Dic15C22×C412yes—
135D60—22no—
136C2×C5×Dic3C22×C412yes—
137C2×C5×Dic3C22×C412yes—
138C5×Dic6—22no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—21no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5C22×C411yescharacteristic
143C2×C3×Dic5C22×C411yescharacteristic
144C2×C3×Dic5C22×C411yescharacteristic
145C2×C3×Dic5C2411yescharacteristic
146#1200—22no—
147#1200—22no—
148C3⋊#170—22no—
149C3⋊#170—22no—
150C3×#170—21no—
151C3×#170—21no—
152C3×#170#4112yes—
153C3×#170#4112yes—
154C22×C3×C8—51no—
155C22×C3×C8—51no—
156C2×C3×C42—51no—
157#930—51no—
158#929—52no—
159C4×#67—52no—
160#924AGL(1,𝔽5)11yescharacteristic
161C2×ΓL(1,𝔽25)—52no—
162C2×D3×C8—52no—
163#876—52no—
164#875—51no—
165C4×D12—51no—
166#872—52no—
167#870—51no—
168C4×Dic6—51no—
169C3×C4×C8—51no—
170C3×C4×C8—51no—
171C3×C4×C8—51no—
172C3×C4×C8—51no—
173#830—52no—
174(C2×D3)⋊C8—52no—
175(C2×D3)⋊C8—52no—
176#815—52no—
177Dic3⋊C8—52no—
178Dic3⋊C8—52no—
179C8×Dic3—52no—
180C4⋊(C3⋊C8)—51no—
181C4⋊(C3⋊C8)—51no—
182#803—52no—
183C2×C5×Q8—31no—
184C22×C4×C5C4×D311yescharacteristic
185C2×C22⋊D5—32no—
186C22×Dic5C4×D311yescharacteristic
187C22×Dic5C22×D311yescharacteristic
188C2×D20—31no—
189C2×C4×D5—32no—
190C2×#170—22no—
191C2×#170—22no—
192C2×#170—32no—
193C2×#170—32no—
194C2×#170C22×D311yescharacteristic
195C2×#170C22×D311yescharacteristic
196C2×#170C22×D311yescharacteristic
197C2×#170C22×D311yescharacteristic
198#700—64no—
199D5⋊C8—64no—
200C5×C4⋊C4—32no—
201C5×C4⋊C4—31no—
202C5×C22⋊C4—32no—
203#685—32no—
204#685—32no—
205#685—32no—
206#685—32no—
207C4⋊Dic5—31no—
208C4⋊Dic5—32no—
209C4⋊Dic5—32no—
210#683—32no—
211C4×Dic5—32no—
212C4×Dic5C4×D311yescharacteristic
213C4×Dic5C4×D311yescharacteristic
214C4×Dic5C22×D311yescharacteristic
215C4×Dic5C22×D311yescharacteristic
216#529—151no—
217#446—151no—
218C8×Q8—151no—
219#436—151no—
220#435—152no—
221C8×D4—151no—
222C8×D4—152no—
223#433—152no—
224#432—151no—
225#405—152no—
226C2×C4×C8—51no—
227C22×C3×C5—21no—
228C22×C3×C5—21no—
229C22×C3×C5#13112yes—
230C22×C3×C5C23×C411yescharacteristic
231C2×D15—22no—
232C2×D15—22no—
233C2×D15—22no—
234C3×C4×C5—21no—
235C3×C4×C5#13112yes—
236Dic15—22no—
237C3×Dic5—21no—
238C3×Dic5—21no—
239C3×Dic5—21no—
240C3×Dic5C2×#4111yescharacteristic
241C3×Dic5C2×#4111yescharacteristic
242C5×Dic3—22no—
243C5×Dic3—22no—
244C22×C3×C4—51no—
245C22×C3×C4—51no—
246C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
247C2×#67—52no—
248C2×D12—51no—
249C2×C4×D3—52no—
250C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
251C2×C3×C8—51no—
252C2×C3×C8—51no—
253C2×C3×C8—51no—
254C2×C3×C8—51no—
255C2×C3×C8—102no—
256C2×C3×C8—102no—
257C3×C42—51no—
258C3×C42—51no—
259C3×C42—51no—
260C3×C42—51no—
261#217C2×AGL(1,𝔽5)12yes—
262(C2×D3)⋊C4—52no—
263(C2×D3)⋊C4—52no—
264(C2×D3)⋊C4—52no—
265(C2×D3)⋊C4—52no—
266C4⋊Dic3—51no—
267C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
268Dic3⋊C4—52no—
269Dic3⋊C4—52no—
270Dic3⋊C4—52no—
271Dic3⋊C4C2×AGL(1,𝔽5)12yes—
272C4×Dic3—52no—
273C2×C3⋊C8—52no—
274C2×C3⋊C8—52no—
275ΓL(1,𝔽25)—104no—
276D3×C8—104no—
277C23×C5C2×C4×D311yescharacteristic
278C22×D5—32no—
279C5×Q8—62no—
280C2×C4×C5—22no—
281C2×C4×C5—32no—
282C2×C4×C5—32no—
283C2×C4×C5C2×C4×D311yescharacteristic
284C2×C4×C5C2×C4×D311yescharacteristic
285C22⋊D5—64no—
286C2×Dic5—22no—
287C2×Dic5—21no—
288C2×Dic5—21no—
289C2×Dic5—32no—
290C2×Dic5C2×C4×D311yescharacteristic
291C2×Dic5C2×C4×D311yescharacteristic
292C2×Dic5C2×C4×D311yescharacteristic
293C2×Dic5C23×D311yescharacteristic
294D20—62no—
295C4×D5—62no—
296C4×D5—62no—
297#170—21no—
298#170—21no—
299#170—62no—
300#170—62no—
301#170#23512yes—
302#170#23512yes—
303C2×M16—152no—
304C22×C8—51no—
305C22×C8—51no—
306C22×C8—152no—
307#126—152no—
308#125—151no—
309#124—151no—
310#123—152no—
311C22⋊Q8—31no—
312#121—151no—
313C4×Q8—151no—
314C4×D4—151no—
315C4×D4—152no—
316#117—152no—
317C2×C42—51no—
318C4⋊C8—151no—
319C4⋊C8—151no—
320C4⋊C8—152no—
321C4⋊C8—152no—
322C22⋊C8—152no—
323C22⋊C8—152no—
324C22⋊C8—152no—
325#97—152no—
326#97—152no—
327C4×C8—51no—
328C4×C8—51no—
329C4×C8—51no—
330C4×C8—51no—
331C4×C8—152no—
332C2×C3×C5—22no—
333C2×C3×C5C4×#4111yescharacteristic
334C2×C3×C5C2×#13111yescharacteristic
335C2×C3×C5C2×#13111yescharacteristic
336D15—42no—
337C23×C3C22×AGL(1,𝔽5)11yescharacteristic
338C22×D3—52no—
339C2×C3×C4—22no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—51no—
343C2×C3×C4—51no—
344C2×C3×C4—101no—
345C2×C3×C4—101no—
346C2×C3×C4—102no—
347C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
348C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
349#67—104no—
350C2×Dic3—52no—
351C2×Dic3C22×AGL(1,𝔽5)12yes—
352C2×Dic3C22×AGL(1,𝔽5)12yes—
353D12—102no—
354C4×D3—102no—
355C4×D3—102no—
356Dic6—22no—
357C3×C8—52no—
358C3×C8—52no—
359C3×C8—101no—
360C3×C8—101no—
361C3⋊C8—102no—
362C3⋊C8—102no—
363C22×C5—21no—
364C22×C5—21no—
365C22×C5#90112yes—
366C22×C5C22×C4×D311yescharacteristic
367C2×D5—62no—
368C2×D5—62no—
369C2×D5—62no—
370C4×C5—21no—
371C4×C5—62no—
372C4×C5—62no—
373C4×C5#90112yes—
374Dic5—21no—
375Dic5—21no—
376Dic5—21no—
377Dic5—62no—
378Dic5C2×#23511yescharacteristic
379Dic5C2×#23511yescharacteristic
380C2×Q8—31no—
381C2×D4—152no—
382C2×D4—151no—
383C22×C4—51no—
384C22×C4—51no—
385C22×C4—152no—
386C22×C4D3×AGL(1,𝔽5)11yescharacteristic
387M16—304no—
388C2×C8—51no—
389C2×C8—51no—
390C2×C8—51no—
391C2×C8—51no—
392C2×C8—102no—
393C2×C8—102no—
394C2×C8—152no—
395C2×C8—152no—
396C2×C8—304no—
397C4⋊C4—32no—
398C4⋊C4—31no—
399C4⋊C4—152no—
400C4⋊C4—152no—
401C4⋊C4—152no—
402C4⋊C4—151no—
403C22⋊C4—32no—
404C22⋊C4—152no—
405C22⋊C4—152no—
406C22⋊C4—152no—
407C22⋊C4—152no—
408C42—51no—
409C42—51no—
410C42—51no—
411C42—51no—
412C42—152no—
413C3×C5#296611yescharacteristic, complemented, Hall
414C22×C3—21no—
415C22×C3—21no—
416C22×C3#413312yes—
417C22×C3C23×AGL(1,𝔽5)11yescharacteristic
418C2×D3—102no—
419C2×D3—102no—
420C2×D3—102no—
421C3×C4—21no—
422C3×C4—51no—
423C3×C4—51no—
424C3×C4—101no—
425C3×C4—101no—
426C3×C4—101no—
427C3×C4#413112yes—
428Dic3—22no—
429Dic3—22no—
430Dic3—102no—
431C2×C5—22no—
432C2×C5C4×#23511yescharacteristic
433C2×C5C2×#90111yescharacteristic
434C2×C5C2×#90111yescharacteristic
435D5—122no—
436C23—152no—
437C23C2×D3×AGL(1,𝔽5)11yescharacteristic
438Q8—62no—
439D4—304no—
440D4—302no—
441C2×C4—22no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—51no—
448C2×C4—101no—
449C2×C4—101no—
450C2×C4—102no—
451C2×C4—152no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
455C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
456C8—52no—
457C8—52no—
458C8—101no—
459C8—101no—
460C8—302no—
461C8—302no—
462C2×C3—22no—
463C2×C3C2×#413111yescharacteristic
464C2×C3C2×#413311yescharacteristic
465C2×C3#41×AGL(1,𝔽5)11yescharacteristic
466D3—202no—
467C5#7844811yescharacteristic, complemented, Sylow
468C22—21no—
469C22—21no—
470C22—302no—
471C22—302no—
472C22—302no—
473C22#9191412yes—
474C22C22×D3×AGL(1,𝔽5)11yescharacteristic
475C4—21no—
476C4—51no—
477C4—51no—
478C4—62no—
479C4—62no—
480C4—101no—
481C4—101no—
482C4—101no—
483C4—302no—
484C4#9192712yes—
485C3#1061726511yescharacteristic, complemented, Sylow
486C2—22no—
487C2—602no—
488C2C2×#9191411yescharacteristic
489C2AGL(1,𝔽5)×#23511yescharacteristic
490C2C2×#9192711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radical#91819C4
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2966
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. #11752418C2semidirect
3. C2×C3×C4×#170C2semidirect
4. #11752046C2semidirect
5. #11752036C2semidirect
6. #11751914C2semidirect
7. #11751893C2semidirect
8. #11751872C2semidirect
9. #11751800C2semidirect
10. #11751789C2semidirect
11. Dic6×#170C2semidirect
47. C2×C4×#170D3semidirect
413. C3×C5#2966semidirect
467. C5#78448semidirect
485. C3#10617265semidirect

Maximal subgroups

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

subgroupindex
#117544152
#117524182
C2×C3×C4×#1702
#117520462
#117520362
#117519142
#117518932
#117518722
#117518002
#117517892
Dic6×#1702
#106172653
#784485