Finite Group Databasebeta

← #14419354

Subgroups — #14419354

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#11754848C211yesmaximal, characteristic
2#11752433C211yesmaximal, characteristic, complemented
3C3×D4×#170C211yesmaximal, characteristic, complemented
4#11752039C212yesmaximal, complemented
5#67×#170C212yesmaximal, complemented
6#11751989C212yesmaximal, complemented
7#11751988C212yesmaximal, complemented
8C4×D3×#170C211yesmaximal, characteristic, complemented
9#11751826C211yesmaximal, characteristic, complemented
10#11751807C211yesmaximal, characteristic, complemented
11#11751768C211yesmaximal, characteristic, complemented
12#10617233—31nomaximal, Hall
13C2×#6762C2212yes—
14C22×C3×#170C2212yes—
15C2×D3×#170C2211yescharacteristic
16C2×D3×#170C2211yescharacteristic
17#91825C411yescharacteristic
18C5×#938C411yescharacteristic
19C3×D4×Dic5C2211yescharacteristic
20#91555C412yes—
21Dic5×#67C2212yes—
22#91531C412yes—
23#91528C2212yes—
24C4×D3×Dic5C2211yescharacteristic
25#91384C411yescharacteristic
26#91365C411yescharacteristic
27#91333C2211yescharacteristic
28#91241C2212yes—
29C4⋊(C3⋊#170)C2211yescharacteristic
30C4×C3⋊#170C2211yescharacteristic
31C3×C22⋊#170C2212yes—
32C3×C4⋊#170C2211yescharacteristic
33C3×C4×#170C2211yescharacteristic
34#91177C2212yes—
35Dic3⋊#170C2212yes—
36(C2×D3)⋊#170C2212yes—
37#91169C2212yes—
38#91169C2212yes—
39Dic3×#170C2211yescharacteristic
40Dic3×#170C2212yes—
41Dic3×#170C2211yescharacteristic
42#79401—51nomaximal, Hall
43#65934—31no—
44#65774—31no—
45#65773—32no—
46D4×#170—32no—
47D4×#170D311yescharacteristic, complemented
48#65759—31no—
49C2×C4×#170—31no—
50#65709—31no—
51#65703—31no—
52#65697—32no—
53#65691—32no—
54C2×C3×C5×D4C2×C411yescharacteristic
55C2×Dic30C2×C411yescharacteristic
56C2×C5×#67C2×C412yes—
57C2×C4×C5×D3C2×C411yescharacteristic
58C22×C3×Dic5C2312yes—
59C2×D3×Dic5C2×C411yescharacteristic
60C2×D3×Dic5C2311yescharacteristic
61#6762—24no—
62C2×C3⋊#170C2311yescharacteristic
63C2×C3⋊#170C2312yes—
64C2×C3⋊#170C2311yescharacteristic
65C2×C3×#170—24no—
66C2×C3×#170C2311yescharacteristic
67C2×C3×#170C2312yes—
68C2×C3×#170C2311yescharacteristic
69D3×#170—22no—
70D3×#170—22no—
71#6719C2×C412yes—
72#6719C2×C412yes—
73C4×Dic15C2×C411yescharacteristic
74C5×#217C2×C412yes—
75C5×C4⋊Dic3C2×C411yescharacteristic
76C3×C22⋊Dic5C2×C412yes—
77C3×C4⋊Dic5C2×C411yescharacteristic
78C3×C4×Dic5C2311yescharacteristic
79Dic3⋊Dic5C2×C412yes—
80#6669C2×C412yes—
81#6668C2×C412yes—
82#6668C2×C412yes—
83(C2×D3)⋊Dic5C2×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×#704—32no—
99C22×#170—31no—
100C22×#170—31no—
101C22×#170C2×D312yes—
102C5×#121—31no—
103#4082—31no—
104#4081—32no—
105D4×Dic5—32no—
106D4×Dic5C2×D311yescharacteristic
107#4070—31no—
108C2×C4×Dic5—31no—
109#4036—31no—
110#4033—31no—
111#4025—32no—
112#4023—32no—
113C22⋊#170—32no—
114C22⋊#170—32no—
115C22⋊#170C2×D312yes—
116#4004—32no—
117#4002—32no—
118#4002—32no—
119C4⋊#170—32no—
120C4⋊#170—31no—
121C4⋊#170C2×D311yescharacteristic
122C4×#170—31no—
123C4×#170—32no—
124C4×#170—31no—
125C4×#170—31no—
126C4×#170C2×D311yescharacteristic
127#2966—151noSylow
128C23×C3×C5C22×C412yes—
129C22×C5×D3C22×C411yescharacteristic
130C3×C5×D4—22no—
131C2×C3×C4×C5C22×C411yescharacteristic
132C2×Dic15C22×C411yescharacteristic
133C2×Dic15C22×C412yes—
134C2×Dic15C22×C411yescharacteristic
135Dic30—22no—
136C5×#67—24no—
137C2×C5×Dic3C22×C411yescharacteristic
138C2×C5×Dic3C22×C412yes—
139C4×C5×D3—22no—
140C2×C3×Dic5—22no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5C22×C411yescharacteristic
143C2×C3×Dic5C22×C412yes—
144C2×C3×Dic5C2411yescharacteristic
145D3×Dic5—22no—
146D3×Dic5—21no—
147D3×Dic5—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#938AGL(1,𝔽5)11yescharacteristic
157#936—51no—
158C4×#67—52no—
159#926—52no—
160C2×#208—52no—
161C2×D3×C8—51no—
162C2×D3×C8—51no—
163#897—51no—
164#890—51no—
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×C5×D4—32no—
184C2×C5×D4C4×D311yescharacteristic
185C22×C4×C5—31no—
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5C22×D312yes—
189C2×Dic10—31no—
190#704—64no—
191C2×#170—24no—
192C2×#170—31no—
193C2×#170—31no—
194C2×#170—32no—
195C2×#170—62no—
196C2×#170—62no—
197C2×#170C22×D311yescharacteristic
198C2×#170C22×D312yes—
199C2×#170C22×D311yescharacteristic
200C5×C4⋊C4—31no—
201C5×C22⋊C4—32no—
202C22⋊Dic5—32no—
203C22⋊Dic5—32no—
204C22⋊Dic5—32no—
205C22⋊Dic5C4×D312yes—
206C4⋊Dic5—32no—
207C4⋊Dic5C4×D311yescharacteristic
208#683—32no—
209#683—32no—
210#683—32no—
211C4×Dic5—31no—
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×C5—22no—
228C22×C3×C5—22no—
229C22×C3×C5—22no—
230C22×C3×C5C23×C411yescharacteristic
231C2×C5×D3—21no—
232C2×C5×D3—21no—
233C2×C5×D3#13112yes—
234C3×C4×C5#13112yes—
235Dic15—21no—
236Dic15—21no—
237Dic15—22no—
238C3×Dic5—22no—
239C3×Dic5—21no—
240C3×Dic5C2×#4111yescharacteristic
241C3×Dic5C2×#4111yescharacteristic
242C5×Dic3—21no—
243C5×Dic3—22no—
244C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
245C22×C3×C4—52no—
246C2×#67C2×AGL(1,𝔽5)12yes—
247C2×C4×D3—51no—
248C2×C4×D3—51no—
249C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
250C2×Dic6—51no—
251C2×C3×C8—51no—
252C2×C3×C8—51no—
253C2×C3×C8—52no—
254C2×C3×C8—104no—
255C3×C4⋊C4—51no—
256C3×C22⋊C4—52no—
257C3×C42—51no—
258#217—52no—
259#217—52no—
260#217C2×AGL(1,𝔽5)12yes—
261(C2×D3)⋊C4—52no—
262C4⋊Dic3—52no—
263C4⋊Dic3—52no—
264C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
265Dic3⋊C4—52no—
266Dic3⋊C4—52no—
267C4×Dic3—51no—
268C4×Dic3—51no—
269C4×Dic3—51no—
270C4×Dic3—52no—
271#208—104no—
272C2×C3⋊C8—51no—
273C2×C3⋊C8—51no—
274C2×C3⋊C8—52no—
275D3×C8—102no—
276D3×C8—102no—
277C23×C5—31no—
278C23×C5C2×C4×D312yes—
279C5×D4—22no—
280C5×D4—64no—
281C2×C4×C5—31no—
282C2×C4×C5—32no—
283C2×C4×C5—62no—
284C2×C4×C5C2×C4×D311yescharacteristic
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—31no—
288C2×Dic5—31no—
289C2×Dic5—32no—
290C2×Dic5—62no—
291C2×Dic5—61no—
292C2×Dic5—61no—
293C2×Dic5C2×C4×D311yescharacteristic
294C2×Dic5C2×C4×D312yes—
295C2×Dic5C23×D311yescharacteristic
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—151no—
309#124—151no—
310#123—152no—
311C22⋊Q8—151no—
312#121—31no—
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×C5—42no—
333C2×C3×C5C4×#4111yescharacteristic
334C2×C3×C5C2×#13111yescharacteristic
335C2×C3×C5C2×#13111yescharacteristic
336C5×D3—22no—
337C23×C3C22×AGL(1,𝔽5)12yes—
338C22×D3C22×AGL(1,𝔽5)11yescharacteristic
339C3×D4—22no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—52no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
346#67—24no—
347C2×Dic3—51no—
348C2×Dic3—51no—
349C2×Dic3—52no—
350C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
351C2×Dic3C22×AGL(1,𝔽5)12yes—
352C4×D3—22no—
353C4×D3—101no—
354C4×D3—101no—
355C4×D3—102no—
356Dic6—102no—
357C3×C8—52no—
358C3×C8—101no—
359C3×C8—102no—
360C3⋊C8—52no—
361C3⋊C8—101no—
362C3⋊C8—102no—
363C22×C5—22no—
364C22×C5—22no—
365C22×C5—22no—
366C22×C5—32no—
367C22×C5—61no—
368C22×C5—61no—
369C22×C5C22×C4×D311yescharacteristic
370C4×C5—61no—
371C4×C5—62no—
372C4×C5#90112yes—
373Dic5—21no—
374Dic5—22no—
375Dic5—62no—
376Dic5—61no—
377Dic5—61no—
378Dic5C2×#23711yescharacteristic
379Dic5D3×#4111yescharacteristic
380C2×Q8—151no—
381C2×D4—32no—
382C2×D4D3×AGL(1,𝔽5)11yescharacteristic
383C22×C4—31no—
384C22×C4—52no—
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—31no—
398C4⋊C4—51no—
399C4⋊C4—152no—
400C4⋊C4—152no—
401C4⋊C4—152no—
402C4⋊C4—152no—
403C22⋊C4—32no—
404C22⋊C4—52no—
405C22⋊C4—152no—
406C22⋊C4—152no—
407C22⋊C4—152no—
408C42—51no—
409C42—151no—
410C42—152no—
411C42—151no—
412C42—151no—
413C3×C5#296611yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3C23×AGL(1,𝔽5)11yescharacteristic
418C2×D3—21no—
419C2×D3—21no—
420C2×D3#413112yes—
421C3×C4—51no—
422C3×C4—51no—
423C3×C4—101no—
424C3×C4—102no—
425C3×C4#413112yes—
426Dic3—21no—
427Dic3—22no—
428Dic3—102no—
429Dic3—101no—
430Dic3—101no—
431C2×C5—42no—
432C2×C5—62no—
433C2×C5C4×#23711yescharacteristic
434C2×C5C2×#90111yescharacteristic
435C2×C5D3×#13111yescharacteristic
436C23—31no—
437C23C2×D3×AGL(1,𝔽5)12yes—
438Q8—302no—
439D4—22no—
440D4—64no—
441C2×C4—31no—
442C2×C4—32no—
443C2×C4—51no—
444C2×C4—51no—
445C2×C4—52no—
446C2×C4—62no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—151no—
450C2×C4—151no—
451C2×C4—152no—
452C2×C4—301no—
453C2×C4—301no—
454C2×C4—302no—
455C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
456C8—52no—
457C8—102no—
458C8—101no—
459C8—152no—
460C8—301no—
461C8—302no—
462C2×C3—42no—
463C2×C3C2×#413111yescharacteristic
464C2×C3C2×#413111yescharacteristic
465C2×C3#41×AGL(1,𝔽5)11yescharacteristic
466D3—22no—
467C5#7940111yescharacteristic, complemented, Sylow
468C22—22no—
469C22—22no—
470C22—22no—
471C22—32no—
472C22—61no—
473C22—61no—
474C22C22×D3×AGL(1,𝔽5)11yescharacteristic
475C4—51no—
476C4—51no—
477C4—62no—
478C4—61no—
479C4—101no—
480C4—102no—
481C4—301no—
482C4—301no—
483C4—302no—
484C4#9192712yes—
485C3#1061723311yescharacteristic, complemented, Sylow
486C2—42no—
487C2—62no—
488C2D3×#413111yescharacteristic
489C2AGL(1,𝔽5)×#23711yescharacteristic
490C2C2×#9192711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC2×C3×C5×D4C2×C4
prime radicalC5×#938C4
cosocleC2×C3×Dic5C24
hypercenterC2×D4D3×AGL(1,𝔽5)
hypocenterC3×C5#2966
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11752433C2semidirect
3. C3×D4×#170C2semidirect
4. #11752039C2semidirect
5. #67×#170C2semidirect
6. #11751989C2semidirect
7. #11751988C2semidirect
8. C4×D3×#170C2semidirect
9. #11751826C2semidirect
10. #11751807C2semidirect
11. #11751768C2semidirect
47. D4×#170D3semidirect
413. C3×C5#2966semidirect
467. C5#79401semidirect
485. C3#10617233semidirect

Maximal subgroups

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

subgroupindex
#117548482
#117524332
C3×D4×#1702
#117520392
#67×#1702
#117519892
#117519882
C4×D3×#1702
#117518262
#117518072
#117517682
#106172333
#794015