Finite Group Databasebeta

← #14299884

Subgroups — #14299884

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#11756504C211yesmaximal, characteristic
2C3×#65413C211yesmaximal, characteristic, complemented
3#11748440C211yesmaximal, characteristic, complemented
4#11748428C211yesmaximal, characteristic
5C2×#90980C211yesmaximal, characteristic
6C5×#4555C211yesmaximal, characteristic, complemented
7#11745989C211yesmaximal, characteristic, complemented
8#10599566—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C2×#6710C2211yescharacteristic
12#91796—22no—
13M16×D15—22no—
14#91790—22no—
15C5×#926C2211yescharacteristic
16C2×C4×C5×Dic3C2211yescharacteristic
17C2×C3×#681C2211yescharacteristic
18C2×C3×#681C2211yescharacteristic
19C3×#3954—22no—
20Dic3⋊#168—22no—
21#90980—22no—
22#90980C412yescomplemented
23#90980C412yescomplemented
24#70735—51nomaximal, Hall
25#66079—31no—
26#65413D311yescharacteristic, complemented
27#65294—31no—
28C2×#3937—31no—
29#65213—31no—
30C5×#339—31no—
31#64753—31no—
32C22×C3×C4×C5C2311yescharacteristic
33C2×#1220—21no—
34#6817—21no—
35#6817—21no—
36#6817C2×C412yes—
37#6817D412yes—
38C2×D60Q811yescharacteristic
39C2×C4×D15—21no—
40C2×Dic30Q811yescharacteristic
41C22×C5×Dic3—21no—
42C3×C5×M16—21no—
43C3×C5×M16D412yes—
44C2×C3×C5×C8—21no—
45#6710—21no—
46#6710—21no—
47C2×C3⋊#168D412yes—
48#6705—24no—
49C8×D15—24no—
50C5×#217—21no—
51C5×C4⋊Dic3—21no—
52C4×C5×Dic3—21no—
53C4×C5×Dic3—21no—
54C4×C5×Dic3C2×C411yescharacteristic
55C4×C5×Dic3C2×C411yescharacteristic
56C3×#681—21no—
57C3×#681—21no—
58C3×#681—21no—
59C3×#681C2×C412yes—
60C2×C3×#168—21no—
61C2×C3×#168—21no—
62#5745—51no—
63C3×#430—51no—
64#5139—51no—
65C2×#825—51no—
66#5118—51no—
67#4559—51no—
68#4555D511yescharacteristic, complemented
69C2×#709—31no—
70C2×C5×M16C2×D311yescharacteristic
71C5×#117—31no—
72C2×C42×C5—31no—
73C2×#681—31no—
74C2×#681C2×D311yescharacteristic
75C2×#681C2×D311yescharacteristic
76#4053—62no—
77D5×M16—62no—
78#4047—62no—
79#3954—22no—
80#3937—32no—
81#3937—32no—
82#3937—62no—
83C4⋊#168—62no—
84#1861—151noSylow
85C23×C3×C5#4111yescharacteristic
86C22×D15—21no—
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×Q811yescharacteristic
92C2×C3×C4×C5#4111yescharacteristic
93#1220—22no—
94#1220—22no—
95C2×Dic15—21no—
96D60—21no—
97D60C4⋊C412yes—
98C4×D15—22no—
99C4×D15—22no—
100Dic30—21no—
101Dic30C4⋊C412yes—
102C2×C5×Dic3—21no—
103C2×C5×Dic3—21no—
104C2×C5×Dic3—42no—
105C2×C5×Dic3—41no—
106C3×C5×C8—22no—
107C3⋊#168—22no—
108C3×#168—22no—
109C3×#168—41no—
110C2×#235—51no—
111C2×C3×M16—51no—
112C2×C3×M16—51no—
113C2×C3×M16C2×D511yescharacteristic
114#926C2×D511yescharacteristic
115C2×C4×Dic3C2×D511yescharacteristic
116C2×#208—51no—
117#907—102no—
118D3×M16—102no—
119#901—102no—
120C3×#108—102no—
121#825—52no—
122#825—52no—
123#825—102no—
124Dic3⋊C8—102no—
125C22×C4×C5—61no—
126C22×C4×C5C22×D311yescharacteristic
127C2×C22⋊D5—61no—
128#709—32no—
129#709—32no—
130#709—61no—
131#709—61no—
132C2×D20—31no—
133C2×C4×D5—61no—
134C2×Dic10—31no—
135C5×M16—21no—
136C5×M16#6712yes—
137C2×C5×C8—21no—
138C5×C4⋊C4—61no—
139C5×C22⋊C4—61no—
140C42×C5—31no—
141C42×C5—31no—
142C42×C5—61no—
143C42×C5—61no—
144#681—21no—
145#681—21no—
146#681—21no—
147#681—61no—
148#681—61no—
149#681C4×D312yes—
150C2×#168—21no—
151C2×#168—21no—
152C2×#168—32no—
153#676—64no—
154C8×D5—64no—
155#568—151no—
156#430—51no—
157#423—151no—
158#421—151no—
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×D15—22no—
167C2×D15—81no—
168C3×C4×C5C2×C4⋊C411yescharacteristic
169C3×C4×C5C4×D411yescharacteristic
170C3×C4×C5#12011yescharacteristic
171C3×C4×C5C22⋊Q811yescharacteristic
172Dic15—22no—
173C5×Dic3—22no—
174C5×Dic3—41no—
175C5×Dic3—81no—
176C22×C3×C4C22×D511yescharacteristic
177C2×#67—101no—
178C22×Dic3—21no—
179#235—52no—
180#235—52no—
181#235—101no—
182#235—101no—
183C2×D12—51no—
184C2×C4×D3—101no—
185C2×Dic6—51no—
186C3×M16—21no—
187C3×M16—52no—
188C3×M16—101no—
189C3×M16—101no—
190C3×M16—101no—
191C3×M16D2012yes—
192C2×C3×C8—21no—
193C2×C3×C8—101no—
194C2×C3×C8—101no—
195#217—21no—
196C4⋊Dic3—21no—
197C4×Dic3—21no—
198C4×Dic3—21no—
199C4×Dic3C4×D511yescharacteristic
200C4×Dic3C4×D511yescharacteristic
201#208—101no—
202#208—101no—
203C2×C3⋊C8—52no—
204ΓL(1,𝔽25)—104no—
205D3×C8—104no—
206C23×C5#23911yescharacteristic
207C22×D5—61no—
208C2×C4×C5—61no—
209C2×C4×C5—61no—
210C2×C4×C5—122no—
211C2×C4×C5—121no—
212C2×C4×C5C2×C4×D311yescharacteristic
213C2×C4×C5#23511yescharacteristic
214C2×C4×C5D3×D411yescharacteristic
215C2×C4×C5D3×D411yescharacteristic
216C2×C4×C5D3×Q811yescharacteristic
217C2×C4×C5C2×#6711yescharacteristic
218C22⋊D5—62no—
219C22⋊D5—62no—
220C2×Dic5—61no—
221D20—32no—
222D20—61no—
223C4×D5—62no—
224C4×D5—62no—
225Dic10—32no—
226Dic10—61no—
227C5×C8—22no—
228#168—22no—
229#168—41no—
230#168—62no—
231C2×#41—151no—
232#131—302no—
233#131—302no—
234C2×M16—51no—
235C2×M16—51no—
236C2×M16—151no—
237C2×M16—302no—
238C2×M16D3×D511yescharacteristic
239#117—31no—
240C2×C42—31no—
241#108—102no—
242C4⋊C8—302no—
243#104—152no—
244#104—152no—
245#104—302no—
246C2×C3×C5—21no—
247C2×C3×C5—21no—
248C2×C3×C5—21no—
249C2×C3×C5#38511yescharacteristic
250D15—42no—
251C23×C3#71111yescharacteristic
252C22×D3—101no—
253C2×C3×C4C2×C4×D511yescharacteristic
254C2×C3×C4C2×D2011yescharacteristic
255C2×C3×C4#70911yescharacteristic
256C2×C3×C4D4×D511yescharacteristic
257C2×C3×C4D4×D511yescharacteristic
258C2×C3×C4Q8×D511yescharacteristic
259#67—102no—
260#67—102no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—42no—
264C2×Dic3—41no—
265C2×Dic3—101no—
266D12—52no—
267D12—101no—
268C4×D3—102no—
269C4×D3—102no—
270Dic6—52no—
271Dic6—101no—
272C3×C8—22no—
273C3×C8—102no—
274C3×C8—201no—
275C3⋊C8—102no—
276C22×C5—41no—
277C22×C5#89311yescharacteristic
278C22×C5#89411yescharacteristic
279C22×C5#94611yescharacteristic
280C2×D5—62no—
281C2×D5—241no—
282C4×C5—62no—
283C4×C5—121no—
284C4×C5—241no—
285C4×C5D3×C4⋊C411yescharacteristic
286C4×C5#89611yescharacteristic
287C4×C5C4×#6711yescharacteristic
288C4×C5#93711yescharacteristic
289Dic5—62no—
290#41—152no—
291#41—152no—
292#41—301no—
293#41—301no—
294C2×Q8—151no—
295C2×D4—151no—
296C2×D4—301no—
297C22×C4—61no—
298C22×C4—301no—
299C22×C4C2×D3×D511yescharacteristic
300M16—21no—
301M16—52no—
302M16—101no—
303M16—101no—
304M16—101no—
305M16—304no—
306M16—301no—
307M16—301no—
308M16#120212yes—
309C2×C8—21no—
310C2×C8—101no—
311C2×C8—101no—
312C2×C8—152no—
313C2×C8—304no—
314C4⋊C4—61no—
315C22⋊C4—61no—
316C42—31no—
317C42—31no—
318C42—61no—
319C42—61no—
320C3×C5#186111yescharacteristic, complemented, Hall
321C22×C3—41no—
322C22×C3#402711yescharacteristic
323C22×C3#402911yescharacteristic
324C22×C3#404311yescharacteristic
325C2×D3—102no—
326C2×D3—401no—
327C3×C4C4×D2011yescharacteristic
328C3×C4#402811yescharacteristic
329C3×C4D5×C4⋊C411yescharacteristic
330C3×C4#404211yescharacteristic
331Dic3—22no—
332Dic3—41no—
333Dic3—81no—
334Dic3—102no—
335C2×C5—21no—
336C2×C5—21no—
337C2×C5—21no—
338C2×C5#498911yescharacteristic
339D5—122no—
340C23—301no—
341C23#676811yescharacteristic
342Q8—152no—
343Q8—301no—
344D4—152no—
345D4—302no—
346D4—302no—
347D4—301no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—122no—
351C2×C4—121no—
352C2×C4—301no—
353C2×C4—302no—
354C2×C4—302no—
355C2×C4D15⋊Q811yescharacteristic
356C2×C4#677111yescharacteristic
357C2×C4C4×D3×D511yescharacteristic
358C2×C4C2×#120211yescharacteristic
359C2×C4#679011yescharacteristic
360C2×C4#679011yescharacteristic
361C8—22no—
362C8—102no—
363C8—201no—
364C8—302no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3#6495711yescharacteristic
369D3—202no—
370C5#7073511yescharacteristic, complemented, Sylow
371C22—41no—
372C22—302no—
373C22—1201no—
374C22#9135511yescharacteristic
375C22#9139011yescharacteristic
376C22#9140611yescharacteristic
377C4—62no—
378C4—121no—
379C4—241no—
380C4—302no—
381C4#9137711yescharacteristic
382C4#9140411yescharacteristic
383C4C4×#120211yescharacteristic
384C4#9157211yescharacteristic
385C3#1059956611yescharacteristic, complemented, Sylow
386C2—21no—
387C2—21no—
388C2—21no—
389C2—602no—
390C2#1174872911yescharacteristic
subgroupG/H
centerC4#91572
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#385
prime radicalC2×#6817C22
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1861
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C3×#65413C2semidirect
3. #11748440C2semidirect
6. C5×#4555C2semidirect
7. #11745989C2semidirect
22. #90980C4semidirect
23. #90980C4semidirect
26. #65413D3semidirect
68. #4555D5semidirect
320. C3×C5#1861semidirect
370. C5#70735semidirect
385. C3#10599566semidirect

Maximal subgroups

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

subgroupindex
#117565042
C3×#654132
#117484402
#117484282
C2×#909802
C5×#45552
#117459892
#105995663
#707355