Finite Group Databasebeta

← #14329880

Subgroups — #14329880

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#944C211yesmaximal, characteristic
2#11752449C211yesmaximal, characteristic
3C3×(C2×AGL(1,𝔽5))⋊Q8C211yesmaximal, characteristic
4#11751844C212yesmaximal
5#11751842C212yesmaximal
6#11751837C212yesmaximal
7#11751834C212yesmaximal
8#11751833C212yesmaximal
9AGL(1,𝔽5)×Dic3⋊C4C212yesmaximal
10#10617389—31nomaximal, Hall
11C2×C3×Q8×D5C2211yescharacteristic
12C2×D5×Dic6C2211yescharacteristic
13C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
14C2×C4×ΓL(1,𝔽16)C2211yescharacteristic
15C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
16C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
17C2×Dic3⋊AGL(1,𝔽5)C2212yes—
18C2×Dic3⋊AGL(1,𝔽5)C2212yes—
19C2×Dic3⋊AGL(1,𝔽5)C2212yes—
20C2×Dic3×AGL(1,𝔽5)C2212yes—
21C2×Dic3×AGL(1,𝔽5)C2212yes—
22C2×Dic3×AGL(1,𝔽5)C2212yes—
23#91833C411yescharacteristic, complemented
24C5×#944C411yescharacteristic, complemented
25D5×Dic3⋊C4C2212yes—
26D5×Dic3⋊C4C2212yes—
27C4×D5×Dic3C2211yescharacteristic
28#91389C411yescharacteristic, complemented
29#91385C411yescharacteristic, complemented
30#91327C412yescomplemented
31#91327C412yescomplemented
32#91235C2212yes—
33#91235C2212yes—
34C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
35C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
36(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
37(C2×AGL(1,𝔽5))⋊Dic3C2212yes—
38#76494—51nomaximal, Hall
39D5×C4⋊Q8—31no—
40(C2×AGL(1,𝔽5))⋊Q8—32no—
41(C2×AGL(1,𝔽5))⋊Q8—31no—
42(C2×AGL(1,𝔽5))⋊Q8D311yescharacteristic
43#65712—32no—
44#65712—32no—
45C4⋊C4×AGL(1,𝔽5)—32no—
46C4⋊C4×AGL(1,𝔽5)—32no—
47C4×C4⋊AGL(1,𝔽5)—32no—
48C22×ΓL(1,𝔽16)C2312yes—
49C22×ΓL(1,𝔽16)C2312yes—
50C22×C3×AGL(1,𝔽5)C2312yes—
51C22×C3×AGL(1,𝔽5)C2312yes—
52C2×C3×C5×Q8C2×C411yescharacteristic
53C2×Dic30C2×C411yescharacteristic
54C2×C5×Dic6C2×C411yescharacteristic
55C3×Q8×D5—41no—
56C3×Q8×D5—41no—
57C2×C3×C4×D5C2311yescharacteristic
58C2×C3×C4×D5C2311yescharacteristic
59C2×C3×C4×D5C2311yescharacteristic
60C2×C3×Dic10C2×C411yescharacteristic
61C2×C3×Dic10C2×C411yescharacteristic
62C2×C3×Dic10C2×C411yescharacteristic
63C2×#1204C2×C412yes—
64C2×D5×Dic3C2312yes—
65C2×D5×Dic3C2312yes—
66D5×Dic6—41no—
67D5×Dic6—41no—
68C4⋊ΓL(1,𝔽16)—41no—
69C4×ΓL(1,𝔽16)—41no—
70C3×C4⋊AGL(1,𝔽5)—41no—
71C3×C4×AGL(1,𝔽5)—41no—
72Dic3⋊AGL(1,𝔽5)—24no—
73Dic3⋊AGL(1,𝔽5)—42no—
74Dic3⋊AGL(1,𝔽5)Q818yescomplemented
75Dic3×AGL(1,𝔽5)—24no—
76Dic3×AGL(1,𝔽5)—24no—
77Dic3×AGL(1,𝔽5)—42no—
78#6712C2×C412yes—
79#6712C2×C412yes—
80C4×Dic15C2×C411yescharacteristic
81C5×Dic3⋊C4C2×C412yes—
82C5×Dic3⋊C4C2×C412yes—
83C4×C5×Dic3C2×C411yescharacteristic
84Dic3⋊Dic5C2×C412yes—
85Dic3⋊Dic5C2×C412yes—
86#6669C2×C412yes—
87#6669C2×C412yes—
88Dic3×Dic5C2×C412yes—
89C2×#944—51no—
90C3×#391—51no—
91#5229—51no—
92#4976—52no—
93C4×Dic3⋊C4—52no—
94#4649—52no—
95#4649—52no—
96#4647—52no—
97#4646—52no—
98C2×Q8×D5—31no—
99C2×Q8×D5C2×D311yescharacteristic
100C2×C4⋊AGL(1,𝔽5)—32no—
101C2×C4⋊AGL(1,𝔽5)—32no—
102C2×C4⋊AGL(1,𝔽5)—32no—
103C2×C4⋊AGL(1,𝔽5)—31no—
104C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
105C2×C4×AGL(1,𝔽5)—32no—
106C2×C4×AGL(1,𝔽5)—32no—
107C2×C4×AGL(1,𝔽5)—32no—
108C2×C4×AGL(1,𝔽5)—31no—
109C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
110C5×C4⋊Q8—31no—
111#4090—31no—
112#4090—31no—
113D5×C4⋊C4—32no—
114D5×C4⋊C4—32no—
115#4034—32no—
116#4034—32no—
117C42×D5—31no—
118#4015—31no—
119(C2×AGL(1,𝔽5))⋊C4—32no—
120(C2×AGL(1,𝔽5))⋊C4—32no—
121(C2×AGL(1,𝔽5))⋊C4—32no—
122(C2×AGL(1,𝔽5))⋊C4—32no—
123(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
124(C2×AGL(1,𝔽5))⋊C4C2×D312yes—
125#2369—151noSylow
126C22×C3×D5C2411yescharacteristic
127C2×ΓL(1,𝔽16)—22no—
128C2×ΓL(1,𝔽16)—22no—
129C2×ΓL(1,𝔽16)—22no—
130C2×ΓL(1,𝔽16)—22no—
131C2×ΓL(1,𝔽16)—22no—
132C2×ΓL(1,𝔽16)C2×Q814yes—
133C2×C3×AGL(1,𝔽5)—22no—
134C2×C3×AGL(1,𝔽5)—22no—
135C2×C3×AGL(1,𝔽5)—22no—
136C2×C3×AGL(1,𝔽5)—22no—
137C2×C3×AGL(1,𝔽5)—22no—
138C2×C3×AGL(1,𝔽5)C2×Q814yes—
139C3×C5×Q8—41no—
140C2×C3×C4×C5C22×C411yescharacteristic
141C2×C3×C4×C5C22×C411yescharacteristic
142C2×C3×C4×C5C22×C411yescharacteristic
143C2×Dic15C22×C412yes—
144C2×Dic15C22×C412yes—
145Dic30—41no—
146C2×C5×Dic3C22×C412yes—
147C2×C5×Dic3C22×C412yes—
148C5×Dic6—41no—
149C2×C3×Dic5C22×C411yescharacteristic
150C2×C3×Dic5C22×C411yescharacteristic
151C2×C3×Dic5C22×C411yescharacteristic
152C3×C4×D5—21no—
153C3×C4×D5—21no—
154C3×C4×D5—21no—
155C3×C4×D5—21no—
156C3×C4×D5—21no—
157C3×C4×D5—21no—
158C3×Dic10—41no—
159C3×Dic10—41no—
160C3×Dic10—41no—
161#1204—42no—
162D5×Dic3—22no—
163D5×Dic3—22no—
164D5×Dic3C2×Q814yes—
165D5×Dic3#4114yes—
166C22×C3×Q8—51no—
167C22×Dic6—51no—
168C2×C3×C4⋊C4—51no—
169C2×C3×C42—51no—
170#944—51no—
171#944—51no—
172#944—52no—
173#944—52no—
174#944—51no—
175#944AGL(1,𝔽5)11yescharacteristic, complemented
176C2×C4⋊Dic3—51no—
177C2×Dic3⋊C4—52no—
178C2×Dic3⋊C4—52no—
179C2×Dic3⋊C4—52no—
180C2×Dic3⋊C4—52no—
181C2×Dic3⋊C4—52no—
182C2×C4×Dic3—51no—
183C2×C4×Dic3—52no—
184C2×C4×Dic3—52no—
185C2×C4×Dic3—52no—
186C2×C4×Dic3—51no—
187C3×#95—52no—
188C3×#95—52no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C3⋊#95—52no—
192C3⋊#95—52no—
193C22×AGL(1,𝔽5)—32no—
194C22×AGL(1,𝔽5)—32no—
195C22×AGL(1,𝔽5)C22×D312yes—
196C22×AGL(1,𝔽5)C22×D312yes—
197C2×C5×Q8—31no—
198C2×C5×Q8C4×D311yescharacteristic
199Q8×D5—41no—
200Q8×D5—41no—
201Q8×D5—121no—
202Q8×D5—121no—
203C2×C4×D5—32no—
204C2×C4×D5—32no—
205C2×C4×D5C22×D311yescharacteristic
206C2×C4×D5C22×D311yescharacteristic
207C2×C4×D5C22×D311yescharacteristic
208C2×Dic10—31no—
209C2×Dic10—32no—
210C2×Dic10C4×D311yescharacteristic
211C2×Dic10C4×D311yescharacteristic
212C2×Dic10C4×D311yescharacteristic
213C4⋊AGL(1,𝔽5)—38no—
214C4⋊AGL(1,𝔽5)—41no—
215C4⋊AGL(1,𝔽5)—64no—
216C4⋊AGL(1,𝔽5)—121no—
217C4⋊AGL(1,𝔽5)—122no—
218C4×AGL(1,𝔽5)—41no—
219C4×AGL(1,𝔽5)—64no—
220C4×AGL(1,𝔽5)—64no—
221C4×AGL(1,𝔽5)—121no—
222C4×AGL(1,𝔽5)—122no—
223C5×C4⋊C4—32no—
224C5×C4⋊C4—32no—
225C42×C5—31no—
226C4⋊Dic5—32no—
227C4⋊Dic5—32no—
228#683—32no—
229#683—32no—
230#683—32no—
231#683—32no—
232C4×Dic5—31no—
233C4×Dic5—32no—
234C2×C4⋊Q8—151no—
235#391—51no—
236#391—152no—
237#391—151no—
238#389—152no—
239#389—152no—
240C4×C4⋊C4—152no—
241C4×C4⋊C4—152no—
242C4×C4⋊C4—152no—
243C22×C3×C5C23×C411yescharacteristic
244C2×C3×D5C22×Q812yes—
245C2×C3×D5C2×#4112yes—
246C2×C3×D5H(5,𝔽2)11yescharacteristic
247C2×C3×D5H(5,𝔽2)11yescharacteristic
248ΓL(1,𝔽16)—24no—
249ΓL(1,𝔽16)—42no—
250C3×AGL(1,𝔽5)—24no—
251C3×AGL(1,𝔽5)—42no—
252C3×C4×C5—21no—
253C3×C4×C5—21no—
254C3×C4×C5—21no—
255Dic15—22no—
256Dic15C4×Q814yes—
257C3×Dic5—21no—
258C3×Dic5—21no—
259C3×Dic5—21no—
260C5×Dic3—22no—
261C5×Dic3C4×Q814yes—
262C2×C3×Q8—51no—
263C2×C3×Q8—51no—
264C2×C3×Q8—51no—
265C2×C3×Q8—201no—
266C2×C3×Q8—201no—
267C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
268C22×C3×C4—52no—
269C22×C3×C4—51no—
270C22×C3×C4—51no—
271C22×C3×C4—51no—
272C22×C3×C4—52no—
273C22×Dic3—52no—
274C22×Dic3—52no—
275C22×Dic3—52no—
276C22×Dic3—52no—
277C2×Dic6—51no—
278C2×Dic6—52no—
279C2×Dic6—201no—
280C2×Dic6—201no—
281C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
282C3×C4⋊C4—201no—
283C3×C42—201no—
284C4⋊Dic3—201no—
285Dic3⋊C4—58no—
286Dic3⋊C4—52no—
287Dic3⋊C4—52no—
288Dic3⋊C4—52no—
289Dic3⋊C4—52no—
290Dic3⋊C4—52no—
291Dic3⋊C4—52no—
292Dic3⋊C4—104no—
293Dic3⋊C4—202no—
294Dic3⋊C4C2×AGL(1,𝔽5)12yes—
295Dic3⋊C4C2×AGL(1,𝔽5)12yes—
296C4×Dic3—51no—
297C4×Dic3—52no—
298C4×Dic3—104no—
299C4×Dic3—104no—
300C4×Dic3—202no—
301C4×Dic3—201no—
302C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
303C22×D5C23×D311yescharacteristic
304C2×AGL(1,𝔽5)—22no—
305C2×AGL(1,𝔽5)—22no—
306C2×AGL(1,𝔽5)—22no—
307C2×AGL(1,𝔽5)—22no—
308C2×AGL(1,𝔽5)—22no—
309C2×AGL(1,𝔽5)—34no—
310C2×AGL(1,𝔽5)—62no—
311C2×AGL(1,𝔽5)—62no—
312C2×AGL(1,𝔽5)—62no—
313C2×AGL(1,𝔽5)—62no—
314C2×AGL(1,𝔽5)—62no—
315C2×AGL(1,𝔽5)D3×Q814yes—
316C5×Q8—41no—
317C5×Q8—121no—
318C2×C4×C5—32no—
319C2×C4×C5—32no—
320C2×C4×C5C2×C4×D311yescharacteristic
321C2×C4×C5C2×C4×D311yescharacteristic
322C2×C4×C5C2×C4×D311yescharacteristic
323C2×Dic5—32no—
324C2×Dic5—32no—
325C2×Dic5C2×C4×D311yescharacteristic
326C2×Dic5C2×C4×D311yescharacteristic
327C2×Dic5C2×C4×D311yescharacteristic
328C4×D5—21no—
329C4×D5—21no—
330C4×D5—21no—
331C4×D5—21no—
332C4×D5—21no—
333C4×D5—21no—
334C4×D5—34no—
335C4×D5—34no—
336C4×D5—62no—
337C4×D5—62no—
338Dic10—41no—
339Dic10—41no—
340Dic10—41no—
341Dic10—122no—
342Dic10—121no—
343C22×Q8—51no—
344C22×Q8—151no—
345C4⋊Q8—31no—
346C4⋊Q8—151no—
347C4⋊Q8—152no—
348C4⋊Q8—152no—
349C4⋊Q8—151no—
350C4⋊Q8—151no—
351C2×C4⋊C4—51no—
352C2×C4⋊C4—152no—
353C2×C4⋊C4—152no—
354C2×C4⋊C4—152no—
355C2×C4⋊C4—152no—
356C2×C4⋊C4—152no—
357C2×C4⋊C4—151no—
358C2×C42—51no—
359C2×C42—151no—
360C2×C42—152no—
361C2×C42—152no—
362C2×C42—152no—
363C2×C42—151no—
364#95—52no—
365#95—52no—
366#95—152no—
367#95—152no—
368#95—152no—
369#95—152no—
370C2×C3×C5C2×C4×Q812yes—
371C2×C3×C5#51811yescharacteristic
372C3×D5#55412yes—
373C3×D5#55511yescharacteristic
374C3×D5Q8211yescharacteristic
375C23×C3—51no—
376C3×Q8—41no—
377C3×Q8—201no—
378C3×Q8—201no—
379C3×Q8—201no—
380C2×C3×C4—54no—
381C2×C3×C4—51no—
382C2×C3×C4—51no—
383C2×C3×C4—51no—
384C2×C3×C4—101no—
385C2×C3×C4—101no—
386C2×C3×C4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—101no—
389C2×C3×C4—101no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4—102no—
394C2×C3×C4—102no—
395C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
396C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
397C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
398C2×Dic3—52no—
399C2×Dic3—54no—
400C2×Dic3—54no—
401C2×Dic3—54no—
402C2×Dic3—52no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3—102no—
407C2×Dic3—102no—
408C2×Dic3—102no—
409C2×Dic3—102no—
410C2×Dic3C22×AGL(1,𝔽5)12yes—
411C2×Dic3C22×AGL(1,𝔽5)12yes—
412Dic6—41no—
413Dic6—202no—
414Dic6—201no—
415C22×C5C22×C4×D311yescharacteristic
416C2×D5#100411yescharacteristic
417C2×D5#100411yescharacteristic
418C2×D5C2×D3×Q812yes—
419C2×D5D3×#4112yes—
420AGL(1,𝔽5)—24no—
421AGL(1,𝔽5)—42no—
422AGL(1,𝔽5)—64no—
423AGL(1,𝔽5)—122no—
424C4×C5—21no—
425C4×C5—21no—
426C4×C5—21no—
427C4×C5—34no—
428C4×C5—62no—
429Dic5—21no—
430Dic5—21no—
431Dic5—21no—
432Dic5—34no—
433Dic5—62no—
434C2×Q8—31no—
435C2×Q8—51no—
436C2×Q8—51no—
437C2×Q8—51no—
438C2×Q8—152no—
439C2×Q8—151no—
440C2×Q8—201no—
441C2×Q8—201no—
442C2×Q8—601no—
443C2×Q8—601no—
444C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
445C22×C4—52no—
446C22×C4—51no—
447C22×C4—51no—
448C22×C4—51no—
449C22×C4—52no—
450C22×C4—152no—
451C22×C4—152no—
452C22×C4—152no—
453C22×C4—152no—
454C4⋊C4—32no—
455C4⋊C4—32no—
456C4⋊C4—152no—
457C4⋊C4—152no—
458C4⋊C4—152no—
459C4⋊C4—152no—
460C4⋊C4—158no—
461C4⋊C4—152no—
462C4⋊C4—152no—
463C4⋊C4—201no—
464C4⋊C4—304no—
465C4⋊C4—602no—
466C4⋊C4—601no—
467C42—31no—
468C42—152no—
469C42—151no—
470C42—201no—
471C42—304no—
472C42—304no—
473C42—601no—
474C42—602no—
475C3×C5#236911yescharacteristic, complemented, Hall
476C22×C3—51no—
477C22×C3—51no—
478C22×C3—52no—
479C22×C3—52no—
480C22×C3C23×AGL(1,𝔽5)11yescharacteristic
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—104no—
485C3×C4—101no—
486C3×C4—101no—
487C3×C4—101no—
488C3×C4—202no—
489Dic3—22no—
490Dic3—54no—
491Dic3—104no—
492Dic3—102no—
493Dic3—202no—
494Dic3Q8×AGL(1,𝔽5)14yes—
495C2×C5#548611yescharacteristic
496C2×C5C4×D3×Q812yes—
497D5#563612yes—
498D5#567011yescharacteristic
499D5#572211yescharacteristic
500C23—51no—
501Q8—41no—
502Q8—121no—
503Q8—201no—
504Q8—201no—
505Q8—201no—
506Q8—601no—
507Q8—602no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—54no—
511C2×C4—51no—
512C2×C4—51no—
513C2×C4—51no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—102no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—154no—
526C2×C4—154no—
527C2×C4—152no—
528C2×C4—154no—
529C2×C4—152no—
530C2×C4—302no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—302no—
537C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
538C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
539C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
540C2×C3—52no—
541C2×C3—51no—
542C2×C3—51no—
543C2×C3#6625811yescharacteristic
544C2×C3C2×Q8×AGL(1,𝔽5)12yes—
545C5#7649411yescharacteristic, complemented, Sylow
546C22—52no—
547C22—52no—
548C22—51no—
549C22—51no—
550C22C22×D3×AGL(1,𝔽5)11yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—34no—
555C4—62no—
556C4—104no—
557C4—101no—
558C4—101no—
559C4—101no—
560C4—154no—
561C4—202no—
562C4—304no—
563C4—302no—
564C4—602no—
565C3#1061738911yescharacteristic, complemented, Sylow
566C2—52no—
567C2—51no—
568C2—51no—
569C2D3×Q8×AGL(1,𝔽5)12yes—
570C2#1175687111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×Q8C2×C4
prime radical, cosocleC22×C3×D5C24
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#2369
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
23. #91833C4semidirect
24. C5×#944C4semidirect
28. #91389C4semidirect
29. #91385C4semidirect
30. #91327C4semidirect
31. #91327C4semidirect
74. Dic3⋊AGL(1,𝔽5)Q8semidirect
175. #944AGL(1,𝔽5)semidirect
475. C3×C5#2369semidirect
545. C5#76494semidirect
565. C3#10617389semidirect

Maximal subgroups

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

subgroupindex
D5×#9442
#117524492
C3×(C2×AGL(1,𝔽5))⋊Q82
#117518442
#117518422
#117518372
#117518342
#117518332
AGL(1,𝔽5)×Dic3⋊C42
#106173893
#764945