Finite Group Databasebeta

← #14299898

Subgroups — #14299898

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754120C211yesmaximal, characteristic
2C3×#65294C211yesmaximal, characteristic
3#11748478C211yesmaximal, characteristic, complemented
4#11748440C211yesmaximal, characteristic, complemented
5C2×#90977C211yesmaximal, characteristic
6#11746354C211yesmaximal, characteristic, complemented
7C5×#4532C211yesmaximal, characteristic, complemented
8#10599566—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C2×#6710C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91284—22no—
17#91283—22no—
18D5×#208—22no—
19C3×#3937—22no—
20#90992—22no—
21Dic3⋊#168—22no—
22#90977C412yescomplemented
23#90977C412yescomplemented
24#70398—51nomaximal, Hall
25#66079—31no—
26#65413—31no—
27#65294D311yescharacteristic
28C2×#3937—31no—
29#65213—31no—
30C5×#339—31no—
31#64753—31no—
32C22×C3×C4×C5C2311yescharacteristic
33C22×C5×Dic3—21no—
34C2×C3×C22⋊D5—21no—
35C3×#709—21no—
36C3×#709—21no—
37C3×#709C2×C412yes—
38C3×#709D412yes—
39C2×C3×D20Q811yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×Dic10Q811yescharacteristic
42C3×C5×C4⋊C4—21no—
43C3×C5×C22⋊C4—21no—
44C3×C42×C5—21no—
45#6710—21no—
46#6710—21no—
47#6710—21no—
48#6710C2×C412yes—
49C2×C3⋊#168—21no—
50C2×C3⋊#168D412yes—
51C4×C5×Dic3—21no—
52C4×C5×Dic3C2×C411yescharacteristic
53C4×C5×Dic3C2×C411yescharacteristic
54C5×#208—21no—
55C5×#208D412yes—
56C2×C5×C3⋊C8—21no—
57C3×#681—21no—
58C3×#681—21no—
59C2×C3×#168—21no—
60#6649—24no—
61D5×C3⋊C8—24no—
62#5819—51no—
63C3×#421—51no—
64C2×#837—51no—
65#5124—51no—
66#5118—51no—
67#4533—51no—
68#4532D511yescharacteristic, complemented
69C2×#709C2×D311yescharacteristic
70C2×C5×M16—31no—
71C5×#117C2×D311yescharacteristic
72C2×C42×C5—31no—
73C2×#681—31no—
74C2×#681—31no—
75C2×#681C2×D311yescharacteristic
76#4053—62no—
77D5×M16—62no—
78#4047—62no—
79#3954—62no—
80#3937—22no—
81#3937—32no—
82#3937—32no—
83C4⋊#168—62no—
84#1861—151noSylow
85C23×C3×C5#4111yescharacteristic
86C22×C3×D5—21no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5C22×C411yescharacteristic
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×Q811yescharacteristic
93C2×C3×C4×C5#4111yescharacteristic
94C2×C5×Dic3—21no—
95C2×C5×Dic3—21no—
96C2×C5×Dic3—42no—
97C3×C22⋊D5—22no—
98C3×C22⋊D5—22no—
99C2×C3×Dic5—21no—
100C3×D20—21no—
101C3×D20C4⋊C412yes—
102C3×C4×D5—22no—
103C3×C4×D5—22no—
104C3×Dic10—21no—
105C3×Dic10C4⋊C412yes—
106C3⋊#168—22no—
107C3⋊#168—22no—
108C3×#168—41no—
109C5×C3⋊C8—22no—
110C2×C3×#41—51no—
111C2×C3×M16—51no—
112C3×#117C2×D511yescharacteristic
113#948—102no—
114#948—102no—
115C2×C4×Dic3C2×D511yescharacteristic
116C2×#208—51no—
117C2×#208—51no—
118C2×#208—102no—
119C2×#208C2×D511yescharacteristic
120C3×#104—102no—
121#837—52no—
122#837—52no—
123#822—102no—
124Dic3⋊C8—102no—
125C22×C4×C5—61no—
126C22×C4×C5C22×D311yescharacteristic
127C2×C22⋊D5—21no—
128#709—21no—
129#709—21no—
130#709C4×D312yes—
131#709#6712yes—
132C2×D20Dic611yescharacteristic
133C2×C4×D5—21no—
134C2×Dic10Dic611yescharacteristic
135C5×M16—32no—
136C5×M16—61no—
137C2×C5×C8—61no—
138C5×C4⋊C4—21no—
139C5×C22⋊C4—21no—
140C42×C5—21no—
141C42×C5—31no—
142C42×C5—31no—
143C42×C5—61no—
144#681—21no—
145#681—21no—
146#681—32no—
147#681—61no—
148#681—61no—
149#681—61no—
150C2×#168—21no—
151C2×#168—32no—
152C2×#168—61no—
153#676—64no—
154C8×D5—64no—
155#568—151no—
156#430—151no—
157#423—151no—
158#421—51no—
159C2×#104—151no—
160#344—151no—
161#339—31no—
162C22×C3×C5—41no—
163C22×C3×C5C4×D411yescharacteristic
164C22×C3×C5C22⋊Q811yescharacteristic
165C22×C3×C5#12311yescharacteristic
166C2×C3×D5—22no—
167C2×C3×D5—81no—
168C3×C4×C5—81no—
169C3×C4×C5C2×C4⋊C411yescharacteristic
170C3×C4×C5C4×D411yescharacteristic
171C3×C4×C5#12011yescharacteristic
172C3×C4×C5C22⋊Q811yescharacteristic
173C3×Dic5—22no—
174C5×Dic3—22no—
175C5×Dic3—41no—
176C3×#41—52no—
177C3×#41—52no—
178C3×#41—101no—
179C3×#41—101no—
180C2×C3×Q8—51no—
181C2×C3×D4—51no—
182C2×C3×D4—101no—
183C22×C3×C4—101no—
184C22×C3×C4C22×D511yescharacteristic
185C22×Dic3—21no—
186C3×M16—101no—
187C3×M16—101no—
188C2×C3×C8—101no—
189C3×C4⋊C4—21no—
190C3×C22⋊C4—21no—
191C3×C42—21no—
192C4×Dic3—21no—
193C4×Dic3C4×D511yescharacteristic
194C4×Dic3C4×D511yescharacteristic
195#208—21no—
196#208—52no—
197#208—104no—
198#208—101no—
199#208—101no—
200#208—101no—
201#208D2012yes—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—52no—
204C2×C3⋊C8—104no—
205C2×C3⋊C8—101no—
206C23×C5#23711yescharacteristic
207C22×D5—21no—
208C2×C4×C5—41no—
209C2×C4×C5—61no—
210C2×C4×C5—61no—
211C2×C4×C5—122no—
212C2×C4×C5C2×Dic611yescharacteristic
213C2×C4×C5C2×C4×D311yescharacteristic
214C2×C4×C5D3×D411yescharacteristic
215C2×C4×C5D3×D411yescharacteristic
216C2×C4×C5#23711yescharacteristic
217C2×C4×C5C2×#6711yescharacteristic
218C22⋊D5—22no—
219C22⋊D5—22no—
220C2×Dic5—21no—
221D20—21no—
222D20Dic3⋊C412yes—
223C4×D5—22no—
224C4×D5—22no—
225Dic10—21no—
226Dic10Dic3⋊C412yes—
227C5×C8—62no—
228#168—41no—
229#168—62no—
230#168—62no—
231C2×#41—51no—
232#131—302no—
233#131—302no—
234C2×M16—31no—
235C2×M16—51no—
236C2×M16—151no—
237C2×M16—151no—
238C2×M16—302no—
239#117D3×D511yescharacteristic
240C2×C42—31no—
241#108—302no—
242C4⋊C8—302no—
243#104—102no—
244#104—152no—
245#104—152no—
246C2×C3×C5—21no—
247C2×C3×C5—21no—
248C2×C3×C5—21no—
249C2×C3×C5#38511yescharacteristic
250C3×D5—42no—
251C23×C3—101no—
252C23×C3#71111yescharacteristic
253C3×Q8—52no—
254C3×Q8—101no—
255C3×D4—52no—
256C3×D4—102no—
257C3×D4—102no—
258C3×D4—101no—
259C2×C3×C4—41no—
260C2×C3×C4—101no—
261C2×C3×C4—102no—
262C2×C3×C4—102no—
263C2×C3×C4C2×C4×D511yescharacteristic
264C2×C3×C4C2×D2011yescharacteristic
265C2×C3×C4#70911yescharacteristic
266C2×C3×C4D4×D511yescharacteristic
267C2×C3×C4D4×D511yescharacteristic
268C2×C3×C4Q8×D511yescharacteristic
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—42no—
272C3×C8—201no—
273C3⋊C8—22no—
274C3⋊C8—102no—
275C3⋊C8—102no—
276C22×C5—41no—
277C22×C5#87811yescharacteristic
278C22×C5#67⋊C411yescharacteristic
279C22×C5#93511yescharacteristic
280C2×D5—22no—
281C2×D5—81no—
282C4×C5—62no—
283C4×C5—81no—
284C4×C5—121no—
285C4×C5#87811yescharacteristic
286C4×C5#67⋊C411yescharacteristic
287C4×C5C2×Dic3⋊C411yescharacteristic
288C4×C5#93711yescharacteristic
289Dic5—22no—
290#41—52no—
291#41—52no—
292#41—101no—
293#41—101no—
294C2×Q8—51no—
295C2×D4—51no—
296C2×D4—101no—
297C22×C4—61no—
298C22×C4—101no—
299C22×C4C2×D3×D511yescharacteristic
300M16—32no—
301M16—61no—
302M16—101no—
303M16—101no—
304M16—152no—
305M16—304no—
306M16—301no—
307M16—301no—
308M16—301no—
309C2×C8—61no—
310C2×C8—101no—
311C2×C8—152no—
312C2×C8—304no—
313C2×C8—301no—
314C4⋊C4—21no—
315C22⋊C4—21no—
316C42—21no—
317C42—31no—
318C42—31no—
319C42—61no—
320C3×C5#186111yescharacteristic, complemented, Hall
321C22×C3—41no—
322C22×C3—102no—
323C22×C3—401no—
324C22×C3#402711yescharacteristic
325C22×C3#402911yescharacteristic
326C22×C3#404311yescharacteristic
327C3×C4—81no—
328C3×C4—102no—
329C3×C4C4×D2011yescharacteristic
330C3×C4#402811yescharacteristic
331C3×C4D5×C4⋊C411yescharacteristic
332C3×C4#404211yescharacteristic
333Dic3—22no—
334Dic3—41no—
335C2×C5—21no—
336C2×C5—21no—
337C2×C5—21no—
338C2×C5#494611yescharacteristic
339D5—42no—
340C23—101no—
341C23#678011yescharacteristic
342Q8—52no—
343Q8—101no—
344D4—52no—
345D4—102no—
346D4—102no—
347D4—101no—
348C2×C4—41no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—101no—
352C2×C4—102no—
353C2×C4—102no—
354C2×C4—122no—
355C2×C4D5×Dic611yescharacteristic
356C2×C4C4×D3×D511yescharacteristic
357C2×C4D3×D2011yescharacteristic
358C2×C4#677711yescharacteristic
359C2×C4#677911yescharacteristic
360C2×C4D5×#6711yescharacteristic
361C8—62no—
362C8—201no—
363C8—302no—
364C8—302no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—202no—
369C2×C3#6495711yescharacteristic
370C5#7039811yescharacteristic, complemented, Sylow
371C22—41no—
372C22—102no—
373C22—401no—
374C22#9142211yescharacteristic
375C22#9144211yescharacteristic
376C22#9154811yescharacteristic
377C4—62no—
378C4—81no—
379C4—102no—
380C4—121no—
381C4D5×Dic3⋊C411yescharacteristic
382C4#9139511yescharacteristic
383C4#9143011yescharacteristic
384C4#9147411yescharacteristic
385C3#1059956611yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—21no—
389C2—202no—
390C2#1174889011yescharacteristic
subgroupG/H
centerC4#91474
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#385
prime radicalC2×C3×#709C22
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1861
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #11748478C2semidirect
4. #11748440C2semidirect
6. #11746354C2semidirect
7. C5×#4532C2semidirect
22. #90977C4semidirect
23. #90977C4semidirect
68. #4532D5semidirect
320. C3×C5#1861semidirect
370. C5#70398semidirect
385. C3#10599566semidirect

Maximal subgroups

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

subgroupindex
#117541202
C3×#652942
#117484782
#117484402
C2×#909772
#117463542
C5×#45322
#105995663
#703985