Finite Group Databasebeta

← #14290028

Subgroups — #14290028

Order 1920.

500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756477C211yesmaximal, characteristic
2C3×#65359C211yesmaximal, characteristic, complemented
3#11747104C211yesmaximal, characteristic, complemented
4#11746971C211yesmaximal, characteristic, complemented
5C5×#4544C211yesmaximal, characteristic, complemented
6#11745845C211yesmaximal, characteristic, complemented
7#11745839C211yesmaximal, characteristic, complemented
8#10599200—31nomaximal, Hall
9C2×#6819—21no—
10C3×C5×#127C2211yescharacteristic
11#91829—21no—
12#91828—22no—
13#91827—21no—
14D4×Dic15—21no—
15C4×D60C2211yescharacteristic
16C4×Dic30C2211yescharacteristic
17C3×#4015C2211yescharacteristic
18C2×#6662—21no—
19C5×C4⋊(C3⋊C8)C2211yescharacteristic
20C5×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×#3964—22no—
22C3×C4×#168C2211yescharacteristic
23#90990—21no—
24#90971—21no—
25#90968—21no—
26#71207—51nomaximal, Hall
27#66052—31no—
28#65359D311yescharacteristic, complemented
29#65138—31no—
30#65135—31no—
31C5×#331—31no—
32#64704—31no—
33#64703—31no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4D412yes—
37C2×#1220—22no—
38C2×#1220—22no—
39C22×Dic15—22no—
40#6819—42no—
41#6819—42no—
42C2×D60D411yescharacteristic
43C2×C4×D15—21no—
44C2×Dic30D411yescharacteristic
45C2×C3×Dic10—21no—
46C3×C42×C5C2311yescharacteristic
47#6719—22no—
48#6714—21no—
49C4⋊Dic15D411yescharacteristic
50C4⋊Dic15D411yescharacteristic
51#6712—21no—
52C4×Dic15—21no—
53C2×C5×C3⋊C8—21no—
54C2×C5×C3⋊C8—21no—
55C3×C4⋊Dic5—21no—
56C3×C4⋊Dic5—21no—
57C2×C3×#168—21no—
58#6662—42no—
59#5718—51no—
60C3×#486—51no—
61#5090—51no—
62#5087—51no—
63#4544D511yescharacteristic, complemented
64#4459—51no—
65#4456—51no—
66C2×#711—61no—
67C5×#127C2×D311yescharacteristic
68#4086—61no—
69#4085—62no—
70#4084—61no—
71D4×Dic5—61no—
72C2×Dic20—61no—
73C4×D20—31no—
74#4015C2×D311yescharacteristic
75C4×Dic10—31no—
76C5×C4⋊C8—31no—
77C5×C4⋊C8—31no—
78#3964—22no—
79#3953—61no—
80#3950—61no—
81#3948—61no—
82C4×#168C2×D311yescharacteristic
83#1641—151noSylow
84C23×C3×C5—22no—
85C22×D15—21no—
86C3×C5×D4—42no—
87C3×C5×D4—41no—
88C3×C5×D4—41no—
89C3×C5×D4—41no—
90C3×C5×D4—41no—
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5C2×D411yescharacteristic
94#1220—44no—
95#1220—82no—
96C2×Dic15—21no—
97C2×Dic15—21no—
98C2×Dic15—21no—
99C2×Dic15—44no—
100D60—21no—
101D60—21no—
102C4×D15—41no—
103C4×D15—41no—
104Dic30—21no—
105Dic30D812yes—
106C2×C3×Dic5—41no—
107C3×Dic10—41no—
108C3×Dic10—41no—
109C3×#168—41no—
110C5×C3⋊C8—41no—
111C5×C3⋊C8—41no—
112C2×#237—101no—
113C3×C4⋊Q8—51no—
114C3×#127C2×D511yescharacteristic
115C2×#216—101no—
116#940—101no—
117#939—102no—
118#938—101no—
119D4×Dic3—101no—
120C4×D12—51no—
121C4×Dic6—51no—
122C3×#102—102no—
123C3×C4×C8—51no—
124#835—101no—
125#809—101no—
126#807—101no—
127C4⋊(C3⋊C8)C2×D511yescharacteristic
128C4⋊(C3⋊C8)C2×D511yescharacteristic
129C2×C5×D4—21no—
130C2×C5×D4—21no—
131C2×C5×D4#6712yes—
132C2×C22⋊D5—62no—
133C2×C22⋊D5—62no—
134C22×Dic5—62no—
135#711—122no—
136#711—122no—
137C2×D20—31no—
138C2×C4×D5—61no—
139C2×Dic10—21no—
140C2×Dic10—31no—
141C2×C5×C8—61no—
142C2×C5×C8—61no—
143C42×C5C22×D311yescharacteristic
144C22⋊Dic5—62no—
145#685—61no—
146C4⋊Dic5—21no—
147C4⋊Dic5—21no—
148C4⋊Dic5—31no—
149C4⋊Dic5—31no—
150#683—61no—
151C4×Dic5—61no—
152C2×#168—21no—
153Dic20—122no—
154#550—151no—
155#486—51no—
156#474—151no—
157#462—151no—
158#331—31no—
159Q8⋊C8—151no—
160D4⋊C8—151no—
161C22×C3×C5—42no—
162C22×C3×C5—42no—
163C22×C3×C5—42no—
164C22×C3×C5#12011yescharacteristic
165C2×D15—41no—
166C2×D15—41no—
167C2×D15—41no—
168C3×C4×C5—21no—
169C3×C4×C5C2×D811yescharacteristic
170C3×C4×C5#13511yescharacteristic
171C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
172C3×C4×C5#13711yescharacteristic
173Dic15—22no—
174Dic15—41no—
175Dic15—41no—
176C3×Dic5—81no—
177C2×C3×Q8—101no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4D2012yes—
181C2×#67—102no—
182C2×#67—102no—
183C22×Dic3—102no—
184#237—202no—
185#237—202no—
186C2×D12—51no—
187C2×C4×D3—101no—
188C2×Dic6—51no—
189C2×C3×C8—101no—
190C3×C4⋊C4—101no—
191C3×C4⋊C4—101no—
192C3×C42C22×D511yescharacteristic
193#217—102no—
194#216—202no—
195(C2×D3)⋊C4—101no—
196C4⋊Dic3—51no—
197C4⋊Dic3—51no—
198Dic3⋊C4—101no—
199C4×Dic3—101no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C23×C5—22no—
203C22×D5—61no—
204C5×D4—42no—
205C5×D4—41no—
206C5×D4—41no—
207C5×D4—41no—
208C5×D4—41no—
209C2×C4×C5D3×D411yescharacteristic
210C2×C4×C5D3×D411yescharacteristic
211C2×C4×C5C2×#6711yescharacteristic
212C22⋊D5—124no—
213C22⋊D5—242no—
214C2×Dic5—41no—
215C2×Dic5—61no—
216C2×Dic5—61no—
217C2×Dic5—61no—
218C2×Dic5—124no—
219D20—61no—
220D20—61no—
221C4×D5—121no—
222C4×D5—121no—
223Dic10—32no—
224Dic10—41no—
225Dic10—41no—
226Dic10—61no—
227C5×C8—121no—
228C5×C8—121no—
229#168—41no—
230C2×#41—301no—
231C2×Q16—301no—
232C4⋊Q8—51no—
233#127—301no—
234#127D3×D511yescharacteristic
235#121—301no—
236#121—302no—
237C4×Q8—151no—
238C4×D4—151no—
239C4×D4—301no—
240#107—301no—
241C4⋊C8—31no—
242C4⋊C8—31no—
243#103—301no—
244#102—102no—
245#102—301no—
246C4×C8—51no—
247C2×C3×C5—82no—
248C2×C3×C5#44711yescharacteristic
249C2×C3×C5#45211yescharacteristic
250C2×C3×C5#45511yescharacteristic
251D15—81no—
252C23×C3—22no—
253C22×D3—101no—
254C3×Q8—201no—
255C3×Q8—201no—
256C3×D4—42no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C2×C3×C4—201no—
262C2×C3×C4C2×D2011yescharacteristic
263C2×C3×C4D4×D511yescharacteristic
264C2×C3×C4D4×D511yescharacteristic
265#67—204no—
266#67—402no—
267C2×Dic3—101no—
268C2×Dic3—101no—
269C2×Dic3—101no—
270C2×Dic3—204no—
271D12—101no—
272D12—101no—
273C4×D3—201no—
274C4×D3—201no—
275Dic6—52no—
276Dic6—101no—
277C3×C8—201no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C22×C5—42no—
281C22×C5—42no—
282C22×C5—42no—
283C22×C5#93711yescharacteristic
284C2×D5—121no—
285C2×D5—121no—
286C2×D5—121no—
287C4×C5—21no—
288C4×C5D3×D811yescharacteristic
289C4×C5#91111yescharacteristic
290C4×C5#91511yescharacteristic
291C4×C5#91611yescharacteristic
292Dic5—62no—
293Dic5—81no—
294Dic5—121no—
295Dic5—121no—
296#41—602no—
297#41—602no—
298C2×Q8—101no—
299C2×Q8—151no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—151no—
303C2×D4—302no—
304C2×D4—302no—
305C2×D4#120212yes—
306C22×C4—302no—
307C22×C4—301no—
308Q16—602no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—101no—
312C4⋊C4—101no—
313C4⋊C4—101no—
314C4⋊C4—151no—
315C4⋊C4—151no—
316C4⋊C4—301no—
317C22⋊C4—302no—
318C22⋊C4—301no—
319C42—301no—
320C42C2×D3×D511yescharacteristic
321C3×C5#164111yescharacteristic, complemented, Hall
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3#402811yescharacteristic
326C2×D3—201no—
327C2×D3—201no—
328C2×D3—201no—
329C3×C4—21no—
330C3×C4—401no—
331C3×C4D5×D811yescharacteristic
332C3×C4#405811yescharacteristic
333C3×C4#406011yescharacteristic
334C3×C4#406111yescharacteristic
335Dic3—102no—
336Dic3—201no—
337Dic3—201no—
338C2×C5—82no—
339C2×C5#506111yescharacteristic
340C2×C5#515611yescharacteristic
341C2×C5#517311yescharacteristic
342D5—241no—
343C23—22no—
344C23—301no—
345Q8—152no—
346Q8—201no—
347Q8—201no—
348Q8—301no—
349D4—42no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—301no—
355D4—301no—
356D4—604no—
357D4—1202no—
358C2×C4—201no—
359C2×C4—301no—
360C2×C4—301no—
361C2×C4—301no—
362C2×C4—604no—
363C2×C4—601no—
364C2×C4—601no—
365C2×C4C2×#120211yescharacteristic
366C2×C4#679011yescharacteristic
367C2×C4#679011yescharacteristic
368C8—121no—
369C8—121no—
370C8—201no—
371C2×C3—82no—
372C2×C3#6503511yescharacteristic
373C2×C3#6506211yescharacteristic
374C2×C3#6507211yescharacteristic
375D3—401no—
376C5#7120711yescharacteristic, complemented, Sylow
377C22—42no—
378C22—42no—
379C22—42no—
380C22—601no—
381C22—601no—
382C22—601no—
383C22#9157211yescharacteristic
384C4—21no—
385C4—302no—
386C4—401no—
387C4—601no—
388C4—601no—
389C4#9148111yescharacteristic
390C4#9148811yescharacteristic
391C4#9149811yescharacteristic
392C4#9149911yescharacteristic
393C3#1059920011yescharacteristic, complemented, Sylow
394C2—82no—
395C2—1201no—
396C2#1174904011yescharacteristic
397C2#1174906311yescharacteristic
398C2#1174914911yescharacteristic
subgroupG/H
centerC22#91572
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C5×#127C22
socleC22×C3×C5#120
prime radical#11756477C2
hypercenter#127D3×D5
hypocenterC3×C5#1641
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. C3×#65359C2semidirect
3. #11747104C2semidirect
4. #11746971C2semidirect
5. C5×#4544C2semidirect
6. #11745845C2semidirect
7. #11745839C2semidirect
28. #65359D3semidirect
63. #4544D5semidirect
321. C3×C5#1641semidirect
376. C5#71207semidirect
393. C3#10599200semidirect

Maximal subgroups

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

subgroupindex
#117564772
C3×#653592
#117471042
#117469712
C5×#45442
#117458452
#117458392
#105992003
#712075