Finite Group Databasebeta

← #14290014

Subgroups — #14290014

Order 1920.

500 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 99 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#11756336C211yesmaximal, characteristic
2#11751132C211yesmaximal, characteristic, complemented
3#11750947C211yesmaximal, characteristic, complemented
4#11750795C211yesmaximal, characteristic, complemented
5C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
6#11746306C211yesmaximal, characteristic, complemented
7#11746280C211yesmaximal, characteristic, complemented
8#10599056—31nomaximal, Hall
9C2×#6821—21no—
10C3×C4×C5×Q8C2211yescharacteristic
11C2×#6718—21no—
12#91784—22no—
13#91784—21no—
14#91763—21no—
15#91763C2211yescharacteristic
16C4×D60—21no—
17C4×D60C2211yescharacteristic
18#91758C2211yescharacteristic
19C3×C5×C4⋊C8C2211yescharacteristic
20C3×C4×C5×C8C2211yescharacteristic
21#91105—21no—
22#91105—22no—
23#91102—21no—
24#91094—21no—
25C4⋊(C3⋊#168)C2211yescharacteristic
26#68263—51nomaximal, Hall
27#65911—31no—
28#65320—31no—
29#65135—31no—
30#64983—31no—
31C5×Q8⋊C8D311yescharacteristic, complemented
32#64705—31no—
33#64679—31no—
34C2×C3×C5×Q8D411yescharacteristic
35#6821—42no—
36#6821—42no—
37C2×D60—22no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60—22no—
41C2×D60D411yescharacteristic
42C2×D60D412yes—
43C2×C4×D15—22no—
44C2×C4×D15—21no—
45C2×Dic30—21no—
46C2×C3×C5×C8—21no—
47C2×C3×C5×C8—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C42×C5—21no—
51C3×C42×C5C2311yescharacteristic
52#6718—42no—
53#6714—22no—
54#6714—21no—
55C4⋊Dic15—21no—
56C4⋊Dic15—21no—
57C4⋊Dic15D411yescharacteristic
58C2×C3⋊#168—21no—
59#5577—51no—
60#5031—51no—
61#4846—51no—
62#4694—51no—
63C3×Q8⋊C8D511yescharacteristic, complemented
64#4485—51no—
65#4459—51no—
66C2×#713—61no—
67C4×C5×Q8C2×D311yescharacteristic
68C2×#689—61no—
69#4041—62no—
70#4041—61no—
71#4020—31no—
72#4020—61no—
73C4×D20—31no—
74C4×D20—61no—
75#4015—31no—
76C5×C4⋊C8C2×D311yescharacteristic
77C4×C5×C8C2×D311yescharacteristic
78#3953—61no—
79#3953—62no—
80#3950—61no—
81#3942—61no—
82C4⋊#168—31no—
83#1641—151noSylow
84C22×D15—21no—
85C22×D15—22no—
86C3×C5×Q8—21no—
87C3×C5×Q8D812yes—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×Dic15—21no—
94C2×Dic15—41no—
95D60—21no—
96D60—21no—
97D60—44no—
98D60—42no—
99D60—41no—
100D60—41no—
101D60—41no—
102D60—41no—
103D60—82no—
104C4×D15—44no—
105C4×D15—41no—
106C4×D15—41no—
107Dic30—41no—
108Dic30—41no—
109C3×C5×C8—41no—
110C3×C5×C8—41no—
111C3⋊#168—41no—
112C2×#239—101no—
113C3×C4×Q8C2×D511yescharacteristic
114C2×#216—101no—
115#895—102no—
116#895—101no—
117#874—51no—
118#874—101no—
119C4×D12—51no—
120C4×D12—101no—
121#869—51no—
122C3×C4⋊C8C2×D511yescharacteristic
123C3×C4×C8C2×D511yescharacteristic
124#821—102no—
125#821—101no—
126#818—101no—
127#810—101no—
128C4⋊(C3⋊C8)—51no—
129C2×C5×Q8D1211yescharacteristic
130#713—122no—
131#713—122no—
132C2×D20—31no—
133C2×D20—32no—
134C2×D20—62no—
135C2×D20—61no—
136C2×D20—61no—
137C2×D20—62no—
138C2×C4×D5—62no—
139C2×C4×D5—61no—
140C2×Dic10—61no—
141C2×C5×C8—21no—
142C2×C5×C8—21no—
143C5×C4⋊C4—21no—
144C5×C4⋊C4D1211yescharacteristic
145C42×C5—21no—
146C42×C5C22×D311yescharacteristic
147#689—122no—
148#685—62no—
149#685—61no—
150C4⋊Dic5—31no—
151C4⋊Dic5—61no—
152C4⋊Dic5—61no—
153C2×#168—61no—
154#550—151no—
155#486—151no—
156#474—151no—
157#462—151no—
158#331—151no—
159Q8⋊C8D1511yescharacteristic, complemented
160D4⋊C8—151no—
161C22×C3×C5#12011yescharacteristic
162C2×D15—42no—
163C2×D15—41no—
164C2×D15—41no—
165C2×D15—42no—
166C2×D15—42no—
167C2×D15—41no—
168C3×C4×C5—22no—
169C3×C4×C5—21no—
170C3×C4×C5—41no—
171C3×C4×C5C2×D811yescharacteristic
172C3×C4×C5#13511yescharacteristic
173C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
174C3×C4×C5#13711yescharacteristic
175Dic15—41no—
176Dic15—81no—
177C2×C3×Q8D2011yescharacteristic
178#239—202no—
179#239—202no—
180C2×D12—51no—
181C2×D12—52no—
182C2×D12—102no—
183C2×D12—101no—
184C2×D12—101no—
185C2×D12—102no—
186C2×C4×D3—102no—
187C2×C4×D3—101no—
188C2×Dic6—101no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4D2011yescharacteristic
193C3×C42—21no—
194C3×C42C22×D511yescharacteristic
195#216—202no—
196(C2×D3)⋊C4—102no—
197(C2×D3)⋊C4—101no—
198C4⋊Dic3—51no—
199C4⋊Dic3—101no—
200C4⋊Dic3—101no—
201C2×C3⋊C8—101no—
202C22×D5—61no—
203C22×D5—62no—
204C5×Q8—21no—
205C5×Q8D2412yes—
206C2×C4×C5—21no—
207C2×C4×C5—21no—
208C2×C4×C5C2×D1211yescharacteristic
209C2×C4×C5D3×D411yescharacteristic
210C2×C4×C5D3×D411yescharacteristic
211C2×Dic5—61no—
212C2×Dic5—121no—
213D20—61no—
214D20—61no—
215D20—122no—
216D20—124no—
217D20—121no—
218D20—121no—
219D20—121no—
220D20—121no—
221D20—242no—
222C4×D5—124no—
223C4×D5—121no—
224C4×D5—121no—
225Dic10—121no—
226Dic10—121no—
227C5×C8—41no—
228C5×C8—41no—
229#168—121no—
230C2×#41—301no—
231C2×Q16—301no—
232C4⋊Q8—151no—
233#127—151no—
234#127—301no—
235#121—302no—
236#121—301no—
237C4×Q8C2×D1511yescharacteristic
238C4×D4—151no—
239C4×D4—301no—
240#107—301no—
241C4⋊C8—151no—
242C4⋊C8C2×D1511yescharacteristic
243#103—301no—
244#102—301no—
245#102—302no—
246C4×C8C2×D1511yescharacteristic
247C2×C3×C5#44711yescharacteristic
248C2×C3×C5#45211yescharacteristic
249C2×C3×C5#45511yescharacteristic
250D15—81no—
251D15—82no—
252C22×D3—101no—
253C22×D3—102no—
254C3×Q8—21no—
255C3×Q8D4012yes—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4C2×D2011yescharacteristic
259C2×C3×C4D4×D511yescharacteristic
260C2×C3×C4D4×D511yescharacteristic
261C2×Dic3—101no—
262C2×Dic3—201no—
263D12—101no—
264D12—101no—
265D12—202no—
266D12—204no—
267D12—201no—
268D12—201no—
269D12—201no—
270D12—201no—
271D12—402no—
272C4×D3—204no—
273C4×D3—201no—
274C4×D3—201no—
275Dic6—201no—
276Dic6—201no—
277C3×C8—41no—
278C3×C8—41no—
279C3⋊C8—201no—
280C22×C5#88211yescharacteristic
281C2×D5—122no—
282C2×D5—121no—
283C2×D5—121no—
284C2×D5—122no—
285C2×D5—122no—
286C2×D5—121no—
287C4×C5—22no—
288C4×C5—21no—
289C4×C5—41no—
290C4×C5C2×D2411yescharacteristic
291C4×C5#90811yescharacteristic
292C4×C5#91611yescharacteristic
293C4×C5#91811yescharacteristic
294Dic5—121no—
295Dic5—241no—
296#41—602no—
297#41—602no—
298C2×Q8—301no—
299C2×Q8D6011yescharacteristic
300C2×D4—151no—
301C2×D4—152no—
302C2×D4—302no—
303C2×D4—302no—
304C2×D4—301no—
305C2×D4—301no—
306C22×C4—302no—
307C22×C4—301no—
308Q16—602no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—301no—
312C4⋊C4—21no—
313C4⋊C4—151no—
314C4⋊C4—301no—
315C4⋊C4—301no—
316C4⋊C4D6011yescharacteristic
317C22⋊C4—302no—
318C22⋊C4—301no—
319C42—21no—
320C42C22×D1511yescharacteristic
321C3×C5#164111yescharacteristic, complemented, Hall
322C22×C3#402811yescharacteristic
323C2×D3—201no—
324C2×D3—202no—
325C2×D3—202no—
326C2×D3—201no—
327C2×D3—202no—
328C2×D3—201no—
329C3×C4—21no—
330C3×C4—22no—
331C3×C4—41no—
332C3×C4C2×D4011yescharacteristic
333C3×C4#405411yescharacteristic
334C3×C4#406211yescharacteristic
335C3×C4#406411yescharacteristic
336Dic3—201no—
337Dic3—401no—
338C2×C5#473211yescharacteristic
339C2×C5#480811yescharacteristic
340C2×C5#482511yescharacteristic
341D5—241no—
342D5—242no—
343C23—302no—
344C23—301no—
345Q8—21no—
346Q8—601no—
347Q8—601no—
348Q8D12012yes—
349D4—301no—
350D4—301no—
351D4—602no—
352D4—604no—
353D4—601no—
354D4—601no—
355D4—601no—
356D4—601no—
357D4—1202no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—301no—
361C2×C4—604no—
362C2×C4—601no—
363C2×C4—601no—
364C2×C4—601no—
365C2×C4C2×D6011yescharacteristic
366C2×C4D4×D1511yescharacteristic
367C2×C4D4×D1511yescharacteristic
368C8—41no—
369C8—41no—
370C8—601no—
371C2×C3#6502111yescharacteristic
372C2×C3#6509911yescharacteristic
373C2×C3#6511411yescharacteristic
374D3—401no—
375D3—402no—
376C5#6826311yescharacteristic, complemented, Sylow
377C22—602no—
378C22—601no—
379C22—602no—
380C22—601no—
381C22—601no—
382C22—602no—
383C22#9177111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—601no—
388C4—1201no—
389C4C2×D12011yescharacteristic
390C4#9179711yescharacteristic
391C4#9180511yescharacteristic
392C4#9180711yescharacteristic
393C3#1059905611yescharacteristic, complemented, Sylow
394C2—1202no—
395C2—1201no—
396C2#1175083311yescharacteristic
397C2#1175090911yescharacteristic
398C2#1175092611yescharacteristic
subgroupG/H
centerC22#91771
commutator, cosocleC3×C42×C5C23
FrattiniC42C22×D15
FittingC3×C5×Q8⋊C8C2
socleC22×C3×C5#120
prime radical#11756336C2
hypercenterQ8⋊C8D15
hypocenterC3×C5#1641
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #11751132C2semidirect
3. #11750947C2semidirect
4. #11750795C2semidirect
5. C3×C5×Q8⋊C8C2semidirect
6. #11746306C2semidirect
7. #11746280C2semidirect
31. C5×Q8⋊C8D3semidirect
63. C3×Q8⋊C8D5semidirect
159. Q8⋊C8D15semidirect
321. C3×C5#1641semidirect
376. C5#68263semidirect
393. C3#10599056semidirect

Maximal subgroups

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

subgroupindex
#117563362
#117511322
#117509472
#117507952
C3×C5×Q8⋊C82
#117463062
#117462802
#105990563
#682635