Finite Group Databasebeta

← #14439850

Subgroups — #14439850

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756487C211yesmaximal, characteristic
2#11752484C211yesmaximal, characteristic
3C5×#5080C211yesmaximal, characteristic, complemented
4C3×#65367C211yesmaximal, characteristic, complemented
5#11747439C214yesmaximal
6#11747151C214yesmaximal, complemented
7C4×#6648C211yesmaximal, characteristic
8#11746955C211yesmaximal, characteristic, complemented
9#11746948C211yesmaximal, characteristic, complemented
10#10613622—31nomaximal, Hall
11C3×C5×C4⋊Q8C2211yescharacteristic
12#91835—21no—
13Q8×Dic15—21no—
14#91786—21no—
15#91781C2212yes—
16#91779C2212yes—
17#91776—21no—
18C42×D15C2211yescharacteristic
19#91758C2211yescharacteristic
20C5×#869C2211yescharacteristic
21C3×#4015C2211yescharacteristic
22#91390—21no—
23#91384C2212yes—
24#91383—21no—
25#91347C2212yes—
26#91344—21no—
27#91331—21no—
28C2×#6648C2212yes—
29C5×#808C2214yes—
30C4×C5×C3⋊C8C2211yescharacteristic
31C3×#3939C2214yes—
32C3×C4×#168C2211yescharacteristic
33#90988C2214yes—
34#90987C2214yes—
35#90951C2212yes—
36#81938—51nomaximal, Hall
37#66062—31no—
38#66062—31no—
39C5×#498—31no—
40#65367—31no—
41#65367D311yescharacteristic, complemented
42#65149—34no—
43#65147—34no—
44C4×C8×D5—31no—
45#64969—31no—
46C2×C3×C5×Q8—21no—
47C2×C4×D15D411yescharacteristic
48C2×C4×D15D411yescharacteristic
49C2×C4×D15C2311yescharacteristic
50C2×Dic30—21no—
51C2×C5×Dic6—21no—
52C2×C3×Dic10—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4C2312yes—
55C3×C42×C5C2311yescharacteristic
56#6714—22no—
57C4⋊Dic15—21no—
58C4⋊Dic15—22no—
59C4⋊Dic15C2312yes—
60#6712—22no—
61C4×Dic15—22no—
62C4×Dic15D411yescharacteristic
63C4×Dic15D411yescharacteristic
64C4×Dic15C2311yescharacteristic
65C5×C4⋊Dic3—21no—
66C5×C4⋊Dic3C2312yes—
67C2×C5×C3⋊C8C2312yes—
68C3×C4⋊Dic5—21no—
69C3×C4⋊Dic5C2312yes—
70C2×C3×#168C2312yes—
71Dic3⋊Dic5—22no—
72#6668—22no—
73#6667—22no—
74Dic3×Dic5—22no—
75#6648Q818yescomplemented
76#5728—51no—
77#5728—51no—
78C3×#500—51no—
79#5080—51no—
80#5080D511yescharacteristic, complemented
81#4880—54no—
82#4878—54no—
83C4×D3×C8—51no—
84#4682—51no—
85C5×C4⋊Q8—31no—
86C5×C4⋊Q8C2×D311yescharacteristic
87#4092—61no—
88#4092—61no—
89Q8×Dic5—61no—
90Q8×Dic5—61no—
91C2×C8×D5—32no—
92#4043—61no—
93#4043—61no—
94#4038—32no—
95#4038—32no—
96#4036—32no—
97#4036—32no—
98#4033—61no—
99#4033—61no—
100C42×D5—31no—
101#4015—31no—
102#4015C2×D311yescharacteristic
103C5×#106—34no—
104C4×C5×C8—31no—
105#3949—34no—
106C8×Dic5—32no—
107#3939—34no—
108#3939C2×D314yes—
109C4×#168C2×D311yescharacteristic
110#3160—151noSylow
111C22×D15C2×D411yescharacteristic
112C3×C5×Q8—42no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2411yescharacteristic
117C2×Dic15—22no—
118C2×Dic15C2×D411yescharacteristic
119C2×Dic15C2×D411yescharacteristic
120C2×Dic15C2×D411yescharacteristic
121C4×D15—22no—
122C4×D15—22no—
123C4×D15C2×Q814yes—
124Dic30—42no—
125C2×C5×Dic3—22no—
126C5×Dic6—42no—
127C2×C3×Dic5—22no—
128C3×Dic10—42no—
129C3×#168C2×Q814yes—
130C5×C3⋊C8C2×Q814yes—
131C3×C4⋊Q8—51no—
132C3×C4⋊Q8C2×D511yescharacteristic
133#946—101no—
134#946—101no—
135Q8×Dic3—101no—
136Q8×Dic3—101no—
137C2×D3×C8—52no—
138#897—101no—
139#897—101no—
140#892—52no—
141#892—52no—
142#890—52no—
143#890—52no—
144#887—101no—
145#887—101no—
146C42×D3—51no—
147#869—51no—
148#869C2×D511yescharacteristic
149C3×#107—54no—
150C3×C4×C8—51no—
151#818—54no—
152C8×Dic3—52no—
153#808—54no—
154#808C2×D514yes—
155C4×C3⋊C8C2×D511yescharacteristic
156C2×C5×Q8—21no—
157C2×C5×Q8—61no—
158C2×C4×D5—31no—
159C2×C4×D5—31no—
160C2×C4×D5—31no—
161C2×Dic10—21no—
162C2×Dic10—61no—
163C2×C5×C8—32no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—32no—
166C5×C4⋊C4—61no—
167C5×C4⋊C4C22×D312yes—
168C42×C5C22×D311yescharacteristic
169#685—62no—
170#685—62no—
171C4⋊Dic5—21no—
172C4⋊Dic5—32no—
173C4⋊Dic5—62no—
174C4⋊Dic5—62no—
175C4⋊Dic5—61no—
176C4⋊Dic5C22×D312yes—
177#683—62no—
178#683—62no—
179C4×Dic5—31no—
180C4×Dic5—31no—
181C4×Dic5—31no—
182C4×Dic5—62no—
183C4×Dic5—62no—
184C2×#168C22×D312yes—
185C8×D5—38no—
186#533—151no—
187#533—151no—
188#500—51no—
189#500—151no—
190#499—154no—
191#498—31no—
192#498—151no—
193#427—154no—
194C2×C4×C8—151no—
195C22×C3×C5C22×D411yescharacteristic
196C2×D15—21no—
197C2×D15—21no—
198C2×D15C4⋊Q812yes—
199C3×C4×C5—42no—
200C3×C4×C5#13512yes—
201C3×C4×C5#13512yes—
202C3×C4×C5C22×Q812yes—
203Dic15—21no—
204Dic15—21no—
205Dic15—42no—
206Dic15C4⋊Q812yes—
207C3×Dic5—42no—
208C5×Dic3—42no—
209C2×C3×Q8—21no—
210C2×C3×Q8—101no—
211C2×C4×D3—51no—
212C2×C4×D3—51no—
213C2×C4×D3—51no—
214C2×Dic6—21no—
215C2×Dic6—101no—
216C2×C3×C8—52no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—52no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4C22×D512yes—
221C3×C42C22×D511yescharacteristic
222(C2×D3)⋊C4—102no—
223(C2×D3)⋊C4—102no—
224C4⋊Dic3—21no—
225C4⋊Dic3—52no—
226C4⋊Dic3—102no—
227C4⋊Dic3—102no—
228C4⋊Dic3—101no—
229C4⋊Dic3C22×D512yes—
230Dic3⋊C4—102no—
231Dic3⋊C4—102no—
232C4×Dic3—51no—
233C4×Dic3—51no—
234C4×Dic3—51no—
235C4×Dic3—102no—
236C4×Dic3—102no—
237C2×C3⋊C8C22×D512yes—
238D3×C8—58no—
239C22×D5—31no—
240C5×Q8—42no—
241C5×Q8—122no—
242C2×C4×C5—22no—
243C2×C4×C5—62no—
244C2×C4×C5D3×D411yescharacteristic
245C2×C4×C5D3×D411yescharacteristic
246C2×C4×C5C23×D311yescharacteristic
247C2×Dic5—22no—
248C2×Dic5—31no—
249C2×Dic5—31no—
250C2×Dic5—31no—
251C2×Dic5—62no—
252C4×D5—34no—
253C4×D5—62no—
254C4×D5—62no—
255Dic10—42no—
256Dic10—122no—
257C5×C8—34no—
258#168D3×Q814yes—
259C22×C8—152no—
260C4⋊Q8—31no—
261C4⋊Q8—51no—
262C4⋊Q8—151no—
263C4⋊Q8D3×D511yescharacteristic
264#125—152no—
265#125—152no—
266C22⋊Q8—301no—
267C22⋊Q8—301no—
268C22⋊Q8—301no—
269C22⋊Q8—301no—
270C4×Q8—301no—
271C4×Q8—301no—
272C4×Q8—301no—
273C4×Q8—301no—
274#117—152no—
275#117—152no—
276C2×C42—151no—
277#107—54no—
278#107—154no—
279#106—34no—
280#106—154no—
281C4×C8—31no—
282C4×C8—51no—
283C4×C8—152no—
284C2×C3×C5C2×C4⋊Q811yescharacteristic
285C2×C3×C5C2×#13511yescharacteristic
286C2×C3×C5C2×#13511yescharacteristic
287D15—22no—
288C22×D3—51no—
289C3×Q8—42no—
290C3×Q8—202no—
291C2×C3×C4—22no—
292C2×C3×C4—102no—
293C2×C3×C4D4×D511yescharacteristic
294C2×C3×C4D4×D511yescharacteristic
295C2×C3×C4C23×D511yescharacteristic
296C2×Dic3—22no—
297C2×Dic3—51no—
298C2×Dic3—51no—
299C2×Dic3—51no—
300C2×Dic3—102no—
301C4×D3—54no—
302C4×D3—102no—
303C4×D3—102no—
304Dic6—42no—
305Dic6—202no—
306C3×C8—54no—
307C3⋊C8Q8×D514yes—
308C22×C5C2×D3×D411yescharacteristic
309C2×D5—32no—
310C2×D5—61no—
311C2×D5—61no—
312C4×C5—42no—
313C4×C5—122no—
314C4×C5#91212yes—
315C4×C5#91912yes—
316C4×C5C2×D3×Q812yes—
317Dic5—32no—
318Dic5—42no—
319Dic5—61no—
320Dic5—61no—
321Dic5—122no—
322C2×Q8—21no—
323C2×Q8—61no—
324C2×Q8—101no—
325C2×Q8—301no—
326C22×C4—151no—
327C22×C4—151no—
328C22×C4—151no—
329C2×C8—32no—
330C2×C8—52no—
331C2×C8—158no—
332C4⋊C4—21no—
333C4⋊C4—32no—
334C4⋊C4—52no—
335C4⋊C4—61no—
336C4⋊C4—101no—
337C4⋊C4—152no—
338C4⋊C4—302no—
339C4⋊C4—302no—
340C4⋊C4—302no—
341C4⋊C4—302no—
342C4⋊C4—301no—
343C4⋊C4C2×D3×D512yes—
344C22⋊C4—302no—
345C22⋊C4—302no—
346C42—151no—
347C42—151no—
348C42—151no—
349C42—302no—
350C42—302no—
351C42C2×D3×D511yescharacteristic
352C3×C5#316011yescharacteristic, complemented, Hall
353C22×C3C2×D4×D511yescharacteristic
354C2×D3—52no—
355C2×D3—101no—
356C2×D3—101no—
357C3×C4—42no—
358C3×C4—202no—
359C3×C4#406212yes—
360C3×C4#406212yes—
361C3×C4C2×Q8×D512yes—
362Dic3—42no—
363Dic3—52no—
364Dic3—101no—
365Dic3—101no—
366Dic3—202no—
367C2×C5D3×C4⋊Q811yescharacteristic
368C2×C5C2×#91211yescharacteristic
369C2×C5C2×#91911yescharacteristic
370D5—62no—
371C23—151no—
372Q8—42no—
373Q8—122no—
374Q8—202no—
375Q8—602no—
376C2×C4—22no—
377C2×C4—62no—
378C2×C4—102no—
379C2×C4—151no—
380C2×C4—154no—
381C2×C4—151no—
382C2×C4—151no—
383C2×C4—302no—
384C2×C4—302no—
385C2×C4—302no—
386C2×C4#679011yescharacteristic
387C2×C4#679011yescharacteristic
388C2×C4C22×D3×D511yescharacteristic
389C8—34no—
390C8—54no—
391C2×C3D5×C4⋊Q811yescharacteristic
392C2×C3C2×#406211yescharacteristic
393C2×C3C2×#406211yescharacteristic
394D3—102no—
395C5#8193811yescharacteristic, complemented, Sylow
396C22—152no—
397C22—301no—
398C22—301no—
399C22C2×#679011yescharacteristic
400C4—42no—
401C4—122no—
402C4—152no—
403C4—202no—
404C4—301no—
405C4—301no—
406C4—602no—
407C4#9149712yes—
408C4#9152312yes—
409C4D3×Q8×D512yes—
410C3#1061362211yescharacteristic, complemented, Sylow
411C2—302no—
412C2C2×#9149711yescharacteristic
413C2C2×#9152311yescharacteristic
414C2#1175515311yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radicalC22×D15C2×D4
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3160
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C5×#5080C2semidirect
4. C3×#65367C2semidirect
6. #11747151C2semidirect
8. #11746955C2semidirect
9. #11746948C2semidirect
41. #65367D3semidirect
75. #6648Q8semidirect
80. #5080D5semidirect
352. C3×C5#3160semidirect
395. C5#81938semidirect
410. C3#10613622semidirect

Maximal subgroups

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

subgroupindex
#117564872
#117524842
C5×#50802
C3×#653672
#117474392
#117471512
C4×#66482
#117469552
#117469482
#106136223
#819385