Finite Group Databasebeta

← #14419345

Subgroups — #14419345

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#11753521C211yesmaximal, characteristic
2#11752364C211yesmaximal, characteristic, complemented
3C3×#65702C211yesmaximal, characteristic, complemented
4#11751903C212yesmaximal, complemented
5#11751881C212yesmaximal, complemented
6C4×#6738C211yesmaximal, characteristic, complemented
7#11751826C211yesmaximal, characteristic, complemented
8#11751791C211yesmaximal, characteristic, complemented
9#11751770C212yesmaximal, complemented
10#11751768C211yesmaximal, characteristic, complemented
11#11751767C212yesmaximal, complemented
12#10617332—31nomaximal, Hall
13C2×#6758C2212yes—
14C2×C3×D5⋊C8C2212yes—
15C2×#6738C2211yescharacteristic
16C2×#6738C2211yescharacteristic
17#91786C411yescharacteristic
18C5×#890C411yescharacteristic
19C3×#4039C2211yescharacteristic
20#91465C411yescharacteristic
21#91444C2212yes—
22C4×#1200C2211yescharacteristic
23#91364C411yescharacteristic
24#91362C412yes—
25#91358C2212yes—
26#91356C412yes—
27#91333C2211yescharacteristic
28#91232C2212yes—
29C4⋊(C3⋊#170)C2211yescharacteristic
30C4×C3⋊#170C2211yescharacteristic
31C3×#4003C2212yes—
32C3×C4⋊#170C2211yescharacteristic
33C3×C4×#170C2211yescharacteristic
34#91177C2212yes—
35Dic3⋊#170C2212yes—
36#91171C2212yes—
37#91169C2212yes—
38#91169C2212yes—
39Dic3×#170C2211yescharacteristic
40Dic3×#170C2212yes—
41Dic3×#170C2211yescharacteristic
42#79401—51nomaximal, Hall
43#66025—31no—
44#65709—31no—
45#65706—32no—
46#65705—31no—
47#65704—32no—
48#65703—31no—
49#65702—32no—
50#65702D311yescharacteristic, complemented
51#65701—32no—
52#65683—31no—
53C4×D5⋊C8—31no—
54C2×C4×D15C2×C411yescharacteristic
55C2×Dic30C2×C411yescharacteristic
56C2×C3×D20C2×C411yescharacteristic
57C2×C3×C4×D5C2312yes—
58C2×#1202C2×C412yes—
59C2×#1200C2×C411yescharacteristic
60C2×#1200C2311yescharacteristic
61C2×C3⋊#170C2311yescharacteristic
62C2×C3⋊#170C2312yes—
63C2×C3⋊#170C2311yescharacteristic
64#6758—24no—
65C2×C3×#170C2311yescharacteristic
66C2×C3×#170C2312yes—
67C2×C3×#170C2311yescharacteristic
68C3×D5⋊C8—24no—
69#6738—22no—
70#6738—22no—
71C3×C5×C4⋊C4C2×C411yescharacteristic
72#6714C2×C412yes—
73C4⋊Dic15C2×C412yes—
74C5×C4⋊Dic3C2×C411yescharacteristic
75C5×C4⋊Dic3C2×C412yes—
76C5×Dic3⋊C4C2×C412yes—
77C4×C5×Dic3C2×C411yescharacteristic
78C3×#685C2×C412yes—
79C3×C4×Dic5C2311yescharacteristic
80#6669C2×C412yes—
81#6665C2×C412yes—
82#6665C2×C412yes—
83#6665C2×C412yes—
84Dic3×Dic5C2×C411yescharacteristic
85Dic3×Dic5C2311yescharacteristic
86Dic3×Dic5C2312yes—
87#5557—51no—
88C3×C8×D4—51no—
89C8×#67—52no—
90#5101—52no—
91#5012—51no—
92#4835—51no—
93#4831—51no—
94#4728—52no—
95#4720—52no—
96C4×D3×C8—51no—
97C8×Dic6—51no—
98C2×#700—32no—
99C2×D5⋊C8—31no—
100C2×D5⋊C8—31no—
101C2×D5⋊C8C2×D312yes—
102C5×#125—31no—
103#4044—32no—
104#4043—31no—
105#4041—31no—
106#4040—32no—
107#4039—32no—
108#4039C2×D311yescharacteristic
109#4038—32no—
110#4033—31no—
111#4021—31no—
112C42×D5—31no—
113#4004—32no—
114#4003—32no—
115#4003—32no—
116#4003C2×D312yes—
117#4002—32no—
118#4002—32no—
119C4⋊#170—32no—
120C4⋊#170—31no—
121C4⋊#170C2×D311yescharacteristic
122C4×#170—31no—
123C4×#170—31no—
124C4×#170—32no—
125C4×#170—31no—
126C4×#170C2×D311yescharacteristic
127#2966—151noSylow
128C22×D15C22×C411yescharacteristic
129C22×C3×D5C22×C412yes—
130C2×C3×C4×C5C22×C412yes—
131C2×C3×C4×C5C22×C411yescharacteristic
132C2×Dic15C22×C411yescharacteristic
133C2×Dic15C22×C412yes—
134C4×D15—22no—
135Dic30—22no—
136C2×C5×Dic3C22×C411yescharacteristic
137C2×C5×Dic3C22×C411yescharacteristic
138C2×C5×Dic3C22×C412yes—
139C2×C3×Dic5C22×C411yescharacteristic
140C2×C3×Dic5C2411yescharacteristic
141C3×D20—22no—
142C3×C4×D5—22no—
143C3×C4×D5—22no—
144#1202—24no—
145#1200—22no—
146#1200—21no—
147#1200—21no—
148C3⋊#170—21no—
149C3⋊#170—22no—
150C3⋊#170#4112yes—
151C3×#170—21no—
152C3×#170—22no—
153C3×#170#4112yes—
154C22×C3×C8—52no—
155C3×C4×D4—51no—
156#938—51no—
157#936—51no—
158C4×#67—52no—
159#926—52no—
160C2×#208—52no—
161C2×D3×C8—51no—
162C2×D3×C8—51no—
163#897—51no—
164#890AGL(1,𝔽5)11yescharacteristic
165#883—52no—
166#879—52no—
167C42×D3—51no—
168C4×Dic6—51no—
169C3×C4⋊C8—51no—
170C3×C22⋊C8—52no—
171C3×C4×C8—51no—
172#830—52no—
173(C2×D3)⋊C8—52no—
174#815—52no—
175#815—52no—
176Dic3⋊C8—52no—
177Dic3⋊C8—52no—
178C8×Dic3—52no—
179C8×Dic3—51no—
180C8×Dic3—51no—
181C4⋊(C3⋊C8)—51no—
182C4×C3⋊C8—51no—
183C2×D20—32no—
184C2×D20C4×D311yescharacteristic
185C2×C4×D5—31no—
186C2×C4×D5—31no—
187C2×C4×D5—31no—
188C2×C4×D5C22×D312yes—
189C2×Dic10—31no—
190C2×#170—31no—
191C2×#170—31no—
192C2×#170—32no—
193C2×#170C22×D311yescharacteristic
194C2×#170C22×D312yes—
195C2×#170C22×D311yescharacteristic
196#700—64no—
197D5⋊C8—24no—
198D5⋊C8—62no—
199D5⋊C8—62no—
200C5×C4⋊C4—32no—
201C5×C4⋊C4—31no—
202C5×C4⋊C4—32no—
203C5×C4⋊C4C4×D311yescharacteristic
204C42×C5—31no—
205#685—32no—
206#685—32no—
207#685—32no—
208#685—32no—
209#685C4×D312yes—
210C4⋊Dic5—32no—
211#683—32no—
212C4×Dic5—31no—
213C4×Dic5—31no—
214C4×Dic5—32no—
215C4×Dic5C22×D311yescharacteristic
216#529—151no—
217#446—151no—
218C8×Q8—151no—
219#436—151no—
220#435—152no—
221C8×D4—51no—
222C8×D4—152no—
223#433—152no—
224#432—151no—
225#405—152no—
226C2×C4×C8—151no—
227C22×C3×C5C23×C411yescharacteristic
228C2×D15—21no—
229C2×D15—21no—
230C2×D15#13112yes—
231C2×C3×D5—22no—
232C2×C3×D5—22no—
233C2×C3×D5—22no—
234C3×C4×C5—22no—
235C3×C4×C5#13112yes—
236Dic15—22no—
237Dic15—21no—
238C3×Dic5—21no—
239C3×Dic5C2×#4111yescharacteristic
240C3×Dic5C2×#4111yescharacteristic
241C5×Dic3—21no—
242C5×Dic3—21no—
243C5×Dic3—22no—
244C2×C3×D4—51no—
245C22×C3×C4—52no—
246C2×#67—52no—
247C2×C4×D3—51no—
248C2×C4×D3—51no—
249C2×C4×D3—51no—
250C2×Dic6—51no—
251C2×C3×C8—51no—
252C2×C3×C8—51no—
253C2×C3×C8—52no—
254C2×C3×C8—104no—
255C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
256C3×C22⋊C4—52no—
257C3×C42—51no—
258#217—52no—
259#217—52no—
260#217—52no—
261(C2×D3)⋊C4—52no—
262C4⋊Dic3—52no—
263C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
264C4⋊Dic3C2×AGL(1,𝔽5)12yes—
265Dic3⋊C4—52no—
266Dic3⋊C4C2×AGL(1,𝔽5)12yes—
267C4×Dic3—51no—
268C4×Dic3—51no—
269C4×Dic3—52no—
270C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
271#208—104no—
272C2×C3⋊C8—51no—
273C2×C3⋊C8—51no—
274C2×C3⋊C8—52no—
275D3×C8—102no—
276D3×C8—102no—
277C22×D5—31no—
278C22×D5C2×C4×D312yes—
279C2×C4×C5—32no—
280C2×C4×C5—31no—
281C2×C4×C5—31no—
282C2×C4×C5C2×C4×D312yes—
283C2×C4×C5C2×C4×D311yescharacteristic
284C2×Dic5—31no—
285C2×Dic5—32no—
286C2×Dic5C2×C4×D311yescharacteristic
287C2×Dic5C23×D311yescharacteristic
288D20—22no—
289D20—64no—
290C4×D5—22no—
291C4×D5—22no—
292C4×D5—62no—
293C4×D5—62no—
294C4×D5—61no—
295C4×D5—61no—
296Dic10—62no—
297#170—22no—
298#170—21no—
299#170—32no—
300#170—61no—
301#170—62no—
302#170#23712yes—
303C2×M16—152no—
304C22×C8—52no—
305C22×C8—151no—
306C22×C8—151no—
307#126—152no—
308#125—31no—
309#124—151no—
310#123—152no—
311C22⋊Q8—151no—
312#121—151no—
313C4×Q8—151no—
314C4×D4—51no—
315C4×D4—152no—
316#117—152no—
317C2×C42—151no—
318C4⋊C8—51no—
319C4⋊C8—151no—
320C4⋊C8—152no—
321C4⋊C8—152no—
322C22⋊C8—52no—
323C22⋊C8—152no—
324C22⋊C8—152no—
325#97—152no—
326#97—152no—
327C4×C8—51no—
328C4×C8—151no—
329C4×C8—151no—
330C4×C8—151no—
331C4×C8—152no—
332C2×C3×C5C4×#4111yescharacteristic
333C2×C3×C5C2×#13111yescharacteristic
334C2×C3×C5C2×#13111yescharacteristic
335D15—22no—
336C3×D5—42no—
337C23×C3—52no—
338C22×D3—51no—
339C3×D4—102no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4C22×AGL(1,𝔽5)12yes—
345C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
346#67—104no—
347C2×Dic3—51no—
348C2×Dic3—52no—
349C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
350C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
351C2×Dic3C22×AGL(1,𝔽5)12yes—
352C4×D3—101no—
353C4×D3—101no—
354C4×D3—102no—
355C4×D3—102no—
356Dic6—102no—
357C3×C8—52no—
358C3×C8—101no—
359C3×C8—102no—
360C3⋊C8—52no—
361C3⋊C8—101no—
362C3⋊C8—102no—
363C22×C5C22×C4×D311yescharacteristic
364C2×D5—22no—
365C2×D5—22no—
366C2×D5—22no—
367C2×D5—32no—
368C2×D5—61no—
369C2×D5—61no—
370C4×C5—22no—
371C4×C5—61no—
372C4×C5—62no—
373C4×C5—61no—
374C4×C5#90112yes—
375Dic5—21no—
376Dic5—62no—
377Dic5—61no—
378Dic5C2×#23711yescharacteristic
379Dic5D3×#4111yescharacteristic
380C2×Q8—151no—
381C2×D4—51no—
382C2×D4—152no—
383C22×C4—52no—
384C22×C4—151no—
385C22×C4—151no—
386C22×C4—151no—
387M16—304no—
388C2×C8—51no—
389C2×C8—51no—
390C2×C8—52no—
391C2×C8—104no—
392C2×C8—151no—
393C2×C8—151no—
394C2×C8—152no—
395C2×C8—302no—
396C2×C8—302no—
397C4⋊C4—32no—
398C4⋊C4—32no—
399C4⋊C4—31no—
400C4⋊C4—152no—
401C4⋊C4—152no—
402C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
403C22⋊C4—52no—
404C22⋊C4—152no—
405C22⋊C4—152no—
406C22⋊C4—152no—
407C22⋊C4—152no—
408C42—31no—
409C42—51no—
410C42—151no—
411C42—152no—
412C42—151no—
413C3×C5#296611yescharacteristic, complemented, Hall
414C22×C3—102no—
415C22×C3—102no—
416C22×C3—102no—
417C22×C3C23×AGL(1,𝔽5)11yescharacteristic
418C2×D3—52no—
419C2×D3—101no—
420C2×D3—101no—
421C3×C4—22no—
422C3×C4—51no—
423C3×C4—51no—
424C3×C4—101no—
425C3×C4#413312yes—
426Dic3—21no—
427Dic3—22no—
428Dic3—21no—
429Dic3—102no—
430Dic3—101no—
431C2×C5C4×#23711yescharacteristic
432C2×C5C2×#90111yescharacteristic
433C2×C5D3×#13111yescharacteristic
434D5—42no—
435D5—62no—
436C23—52no—
437C23—151no—
438Q8—302no—
439D4—102no—
440D4—304no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—32no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—151no—
449C2×C4—152no—
450C2×C4—301no—
451C2×C4—301no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4C2×D3×AGL(1,𝔽5)12yes—
455C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
456C8—52no—
457C8—102no—
458C8—101no—
459C8—152no—
460C8—301no—
461C8—302no—
462C2×C3—202no—
463C2×C3C2×#413311yescharacteristic
464C2×C3D5⋊#13111yescharacteristic
465C2×C3#41×AGL(1,𝔽5)11yescharacteristic
466D3—102no—
467C5#7940111yescharacteristic, complemented, Sylow
468C22—102no—
469C22—102no—
470C22—102no—
471C22—152no—
472C22—301no—
473C22—301no—
474C22C22×D3×AGL(1,𝔽5)11yescharacteristic
475C4—22no—
476C4—51no—
477C4—51no—
478C4—61no—
479C4—61no—
480C4—62no—
481C4—101no—
482C4—301no—
483C4—302no—
484C4#9191412yes—
485C3#1061733211yescharacteristic, complemented, Sylow
486C2—202no—
487C2—302no—
488C2C2×#9191411yescharacteristic
489C2AGL(1,𝔽5)×#23711yescharacteristic
490C2#1175684911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C4⋊C4C2×C4
prime radical#91465C4
cosocleC2×C3×Dic5C24
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2966
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11752364C2semidirect
3. C3×#65702C2semidirect
4. #11751903C2semidirect
5. #11751881C2semidirect
6. C4×#6738C2semidirect
7. #11751826C2semidirect
8. #11751791C2semidirect
9. #11751770C2semidirect
10. #11751768C2semidirect
11. #11751767C2semidirect
50. #65702D3semidirect
413. C3×C5#2966semidirect
467. C5#79401semidirect
485. C3#10617332semidirect

Maximal subgroups

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

subgroupindex
#117535212
#117523642
C3×#657022
#117519032
#117518812
C4×#67382
#117518262
#117517912
#117517702
#117517682
#117517672
#106173323
#794015