Finite Group Databasebeta

← #14289035

Subgroups — #14289035

Order 1920.

456 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 98 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
1Q8×Dic30C211yesmaximal, characteristic, complemented
2C3×#65316C211yesmaximal, characteristic
3#11747083C211yesmaximal, characteristic
4#11747071C211yesmaximal, characteristic
5C5×#4535C211yesmaximal, characteristic, complemented
6#11745863C211yesmaximal, characteristic, complemented
7#11745844C211yesmaximal, characteristic, complemented
8#10599057—31nomaximal, Hall
9C3×C4×C5×Q8C2211yescharacteristic
10Q8×Dic15—21no—
11#91777—22no—
12#91777—21no—
13#91758—21no—
14#91758C2211yescharacteristic
15C4×Dic30—21no—
16C4×Dic30C2211yescharacteristic
17C2×C3×#688—21no—
18C3×#4020C2211yescharacteristic
19C2×#6657—21no—
20C2×#6657—22no—
21C5×C4⋊(C3⋊C8)C2211yescharacteristic
22C4×C5×C3⋊C8C2211yescharacteristic
23C3×#3941—21no—
24C3×C4⋊#168C2211yescharacteristic
25#90965—21no—
26#71033—51nomaximal, Hall
27Q8×Dic10—31no—
28#65316D311yescharacteristic
29#65137—31no—
30#64982—31no—
31C5×Q8⋊C8—31no—
32#64701—31no—
33#64676—31no—
34C2×C3×C5×Q8D411yescharacteristic
35C2×Dic30—22no—
36C2×Dic30D411yescharacteristic
37C2×Dic30D412yes—
38C2×C3×D20—21no—
39C2×C3×D20—21no—
40C2×C3×D20—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4D411yescharacteristic
43C3×C42×C5—21no—
44C3×C42×C5C2311yescharacteristic
45C4⋊Dic15—22no—
46C4⋊Dic15—21no—
47C4⋊Dic15—21no—
48C4⋊Dic15D411yescharacteristic
49#6712—22no—
50#6712—21no—
51C4×Dic15—22no—
52C4×Dic15—21no—
53C2×C5×C3⋊C8—21no—
54C2×C5×C3⋊C8—21no—
55C3×#688—42no—
56C2×C3×#168—21no—
57#6657—44no—
58#6657—42no—
59Q8×Dic6—51no—
60C3×#460—51no—
61#5076—51no—
62#5075—51no—
63#4535D511yescharacteristic, complemented
64#4484—51no—
65#4481—51no—
66C4×C5×Q8C2×D311yescharacteristic
67Q8×Dic5—61no—
68C2×#688—21no—
69C2×#677—61no—
70C2×#677—62no—
71#4034—62no—
72#4034—61no—
73#4020C2×D311yescharacteristic
74#4015—31no—
75#4015—61no—
76C4×Dic10—31no—
77C4×Dic10—61no—
78C5×C4⋊C8—31no—
79C4×C5×C8—31no—
80#3953—61no—
81#3941—21no—
82C4⋊#168C2×D311yescharacteristic
83#1623—151noSylow
84C22×C3×D5—41no—
85C3×C5×Q8—21no—
86C3×C5×Q8SD1612yescomplemented
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×Dic15—21no—
93C2×Dic15—22no—
94C2×Dic15—21no—
95Dic30—21no—
96Dic30—42no—
97Dic30—82no—
98Dic30SD1612yescomplemented
99C3×D20—42no—
100C3×D20—41no—
101C3×D20—41no—
102C3×D20—41no—
103C3×D20—41no—
104C3×#168—41no—
105C5×C3⋊C8—22no—
106C5×C3⋊C8—41no—
107C2×C3×SD16—101no—
108C3×#127—51no—
109C3×C4×Q8C2×D511yescharacteristic
110Q8×Dic3—101no—
111C2×#214—101no—
112C2×#214—102no—
113#888—102no—
114#888—101no—
115#869—51no—
116#869—101no—
117C4×Dic6—51no—
118C4×Dic6—101no—
119C3×C4⋊C8—51no—
120C3×#102—101no—
121#832—101no—
122C4⋊(C3⋊C8)C2×D511yescharacteristic
123C4×C3⋊C8C2×D511yescharacteristic
124C2×C5×Q8#6711yescharacteristic
125C2×D20—21no—
126C2×D20—21no—
127C2×D20—21no—
128C2×Dic10—31no—
129C2×Dic10—32no—
130C2×Dic10—62no—
131C2×C5×C8—61no—
132C2×C5×C8—61no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4#6711yescharacteristic
135C42×C5—21no—
136C42×C5C22×D311yescharacteristic
137#688—42no—
138C4⋊Dic5—31no—
139C4⋊Dic5—62no—
140C4⋊Dic5—61no—
141C4⋊Dic5—61no—
142#683—62no—
143#683—61no—
144C4×Dic5—62no—
145C4×Dic5—61no—
146C2×#168—21no—
147#677—124no—
148#677—122no—
149Q82—151no—
150#492—151no—
151#460—51no—
152#460—151no—
153#333—151no—
154Q8⋊C8—31no—
155Q8⋊C8—151no—
156C22×C3×C5#12011yescharacteristic
157C2×C3×D5—81no—
158C2×C3×D5—81no—
159C2×C3×D5—81no—
160C3×C4×C5—22no—
161C3×C4×C5—21no—
162C3×C4×C5—41no—
163C3×C4×C5C2×SD1611yescharacteristic
164C3×C4×C5C2×SD1611yescharacteristic
165C3×C4×C5#13711yescharacteristic
166C3×C4×C5#13711yescharacteristic
167Dic15—22no—
168Dic15—42no—
169Dic15—41no—
170C2×C3×Q8D2011yescharacteristic
171C2×C3×D4—101no—
172C2×C3×D4—101no—
173C2×C3×D4—101no—
174C2×Dic6—51no—
175C2×Dic6—52no—
176C2×Dic6—102no—
177C3×SD16—202no—
178C2×C3×C8—101no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4D2011yescharacteristic
181C3×C42—21no—
182C3×C42C22×D511yescharacteristic
183#214—204no—
184#214—202no—
185C4⋊Dic3—51no—
186C4⋊Dic3—102no—
187C4⋊Dic3—101no—
188C4⋊Dic3—101no—
189Dic3⋊C4—102no—
190Dic3⋊C4—101no—
191C4×Dic3—102no—
192C4×Dic3—101no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C22×D5—41no—
196C5×Q8—21no—
197C5×Q8#21412yescomplemented
198C2×C4×C5—21no—
199C2×C4×C5—21no—
200C2×C4×C5D3×D411yescharacteristic
201C2×C4×C5D3×D411yescharacteristic
202C2×C4×C5C2×#6711yescharacteristic
203C2×Dic5—61no—
204C2×Dic5—61no—
205C2×Dic5—62no—
206D20—42no—
207D20—41no—
208D20—41no—
209D20—41no—
210D20—41no—
211Dic10—32no—
212Dic10—61no—
213Dic10—122no—
214Dic10—242no—
215C5×C8—62no—
216C5×C8—121no—
217#168—41no—
218C2×SD16—101no—
219C2×SD16—301no—
220C2×SD16—302no—
221C4⋊Q8—151no—
222C4⋊Q8—301no—
223C4⋊Q8—302no—
224C4⋊Q8—301no—
225#127—51no—
226C4×Q8—151no—
227C4×Q8—301no—
228C4×Q8—301no—
229C4×Q8D3×D511yescharacteristic
230C4⋊C8—31no—
231C4⋊C8—51no—
232#102—101no—
233#102—301no—
234C4×C8—31no—
235C2×C3×C5#44811yescharacteristic
236C2×C3×C5#44811yescharacteristic
237C2×C3×C5#45311yescharacteristic
238C3×D5—161no—
239C23×C3—201no—
240C3×Q8—21no—
241C3×Q8#67712yescomplemented
242C3×D4—202no—
243C3×D4—201no—
244C3×D4—201no—
245C3×D4—201no—
246C3×D4—201no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4C2×D2011yescharacteristic
250C2×C3×C4D4×D511yescharacteristic
251C2×C3×C4D4×D511yescharacteristic
252C2×Dic3—101no—
253C2×Dic3—101no—
254C2×Dic3—102no—
255Dic6—52no—
256Dic6—101no—
257Dic6—202no—
258Dic6—402no—
259C3×C8—201no—
260C3⋊C8—22no—
261C3⋊C8—41no—
262C22×C5#93711yescharacteristic
263C2×D5—81no—
264C2×D5—81no—
265C2×D5—81no—
266C4×C5—21no—
267C4×C5—22no—
268C4×C5—41no—
269C4×C5D3×SD1611yescharacteristic
270C4×C5#91511yescharacteristic
271C4×C5C2×#21411yescharacteristic
272C4×C5#95111yescharacteristic
273Dic5—62no—
274Dic5—122no—
275Dic5—121no—
276C2×Q8—151no—
277C2×Q8—152no—
278C2×Q8—302no—
279C2×Q8#120211yescharacteristic
280C2×D4—101no—
281C2×D4—101no—
282C2×D4—101no—
283SD16—202no—
284SD16—604no—
285SD16—602no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—101no—
289C4⋊C4—21no—
290C4⋊C4—151no—
291C4⋊C4—302no—
292C4⋊C4—302no—
293C4⋊C4—301no—
294C4⋊C4—301no—
295C4⋊C4—301no—
296C4⋊C4#120211yescharacteristic
297C42—21no—
298C42—302no—
299C42—301no—
300C42C2×D3×D511yescharacteristic
301C3×C5#162311yescharacteristic, complemented, Hall
302C22×C3—401no—
303C22×C3—401no—
304C22×C3—401no—
305C22×C3#402811yescharacteristic
306C3×C4—22no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4C2×#67711yescharacteristic
310C3×C4#405511yescharacteristic
311C3×C4D5×SD1611yescharacteristic
312C3×C4#406411yescharacteristic
313Dic3—102no—
314Dic3—202no—
315Dic3—201no—
316C2×C5#504011yescharacteristic
317C2×C5#517011yescharacteristic
318C2×C5#519811yescharacteristic
319D5—161no—
320C23—201no—
321Q8—21no—
322Q8—152no—
323Q8—301no—
324Q8—602no—
325Q8—1202no—
326Q8#665712yescomplemented
327D4—202no—
328D4—201no—
329D4—201no—
330D4—201no—
331D4—201no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—301no—
335C2×C4—301no—
336C2×C4—302no—
337C2×C4C2×#120211yescharacteristic
338C2×C4#679011yescharacteristic
339C2×C4#679011yescharacteristic
340C8—62no—
341C8—121no—
342C8—201no—
343C2×C3—801no—
344C2×C3#6502711yescharacteristic
345C2×C3#6509511yescharacteristic
346C2×C3#6511111yescharacteristic
347C5#7103311yescharacteristic, complemented, Sylow
348C22—401no—
349C22—401no—
350C22—401no—
351C22#9157211yescharacteristic
352C4—21no—
353C4—22no—
354C4—41no—
355C4—302no—
356C4—602no—
357C4—601no—
358C4C2×#665711yescharacteristic
359C4#9131311yescharacteristic
360C4#9150511yescharacteristic
361C4#9152211yescharacteristic
362C3#1059905711yescharacteristic, complemented, Sylow
363C2—801no—
364C2#1174854911yescharacteristic
365C2#1174928611yescharacteristic
366C2#1174942111yescharacteristic
subgroupG/H
centerC22#91572
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5#120
prime radicalC3×#4020C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#1623
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×Dic30C2semidirect
5. C5×#4535C2semidirect
6. #11745863C2semidirect
7. #11745844C2semidirect
63. #4535D5semidirect
86. C3×C5×Q8SD16semidirect
98. Dic30SD16semidirect
197. C5×Q8#214semidirect
241. C3×Q8#677semidirect
301. C3×C5#1623semidirect
326. Q8#6657semidirect
347. C5#71033semidirect
362. C3#10599057semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic302
C3×#653162
#117470832
#117470712
C5×#45352
#117458632
#117458442
#105990573
#710335