Finite Group Databasebeta

← #14307269

Subgroups — #14307269

Order 1920.

520 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
1D5×#870C211yesmaximal, characteristic
2D5⋊(C4⋊(C3⋊C8))C211yesmaximal, characteristic
3C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#11746528C212yesmaximal
5#11746528C212yesmaximal
6#10604589—31nomaximal, Hall
7C2×D5⋊(C3⋊C8)C2212yes—
8C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
9C2×C3×C4×AGL(1,𝔽5)—21no—
10#91759C411yescharacteristic, complemented
11C5×#870C411yescharacteristic, complemented
12C3×C42×D5C2211yescharacteristic
13D5×C4⋊Dic3C2211yescharacteristic
14D5×C4⋊Dic3C2211yescharacteristic
15D5×Dic3⋊C4—21no—
16D5×Dic3⋊C4—21no—
17#91388C411yescharacteristic, complemented
18#91388C411yescharacteristic, complemented
19#91378—21no—
20#91378—21no—
21#91233—21no—
22C4⋊(C3⋊#170)—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4—21no—
24#69491—51nomaximal, Hall
25D5×#125—31no—
26C4×C4⋊AGL(1,𝔽5)D311yescharacteristic
27D5⋊(C4⋊C8)—31no—
28#64874—32no—
29#64874—32no—
30C22×C3×AGL(1,𝔽5)—22no—
31C2×C3×C4×D5D411yescharacteristic
32C2×C3×C4×D5D411yescharacteristic
33C2×C3×C4×D5C2311yescharacteristic
34C2×D5×Dic3—21no—
35C2×D5×Dic3—21no—
36C2×C3⋊#170—22no—
37D5⋊(C3⋊C8)—24no—
38C3×C4⋊AGL(1,𝔽5)—22no—
39C3×C4⋊AGL(1,𝔽5)Q814yes—
40C3×C4×AGL(1,𝔽5)—42no—
41C3×C42×C5C2×C411yescharacteristic
42C4⋊Dic15C2×C411yescharacteristic
43C4⋊Dic15C2×C411yescharacteristic
44#6712—21no—
45#6712—21no—
46C5×C4⋊Dic3C2×C411yescharacteristic
47C5×C4⋊Dic3C2×C411yescharacteristic
48C5×Dic3⋊C4—21no—
49C5×Dic3⋊C4—21no—
50C3×C4×Dic5C2×C411yescharacteristic
51C3×C4×Dic5D411yescharacteristic
52C3×C4×Dic5D411yescharacteristic
53Dic3⋊Dic5—21no—
54Dic3⋊Dic5—21no—
55#6669—21no—
56#6669—21no—
57#6668—21no—
58#6668—21no—
59C2×#870—51no—
60C3×C4×C4⋊C4—51no—
61C2×C4⋊(C3⋊C8)—51no—
62#4529—52no—
63#4529—52no—
64C2×C4⋊AGL(1,𝔽5)C2×D312yes—
65C2×C4×AGL(1,𝔽5)—21no—
66C2×D5⋊C8—32no—
67C5×#125—31no—
68D5×C4⋊C4—31no—
69D5×C4⋊C4—31no—
70D5×C4⋊C4—61no—
71D5×C4⋊C4—61no—
72#4036—31no—
73#4036—31no—
74#4035—61no—
75#4035—61no—
76C42×D5C2×D311yescharacteristic
77#4016—31no—
78(C2×AGL(1,𝔽5))⋊C4—21no—
79#4004—61no—
80C4⋊#170—61no—
81#1989—151noSylow
82C22×C3×D5C2×D411yescharacteristic
83C2×C3×AGL(1,𝔽5)—24no—
84C2×C3×AGL(1,𝔽5)—42no—
85C2×C3×AGL(1,𝔽5)—42no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22×C411yescharacteristic
89C2×Dic15—21no—
90C2×Dic15—21no—
91C2×C5×Dic3—21no—
92C2×C5×Dic3—21no—
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C2×D411yescharacteristic
96C3×C4×D5—22no—
97C3×C4×D5—21no—
98C3×C4×D5—21no—
99C3×C4×D5C2×Q812yes—
100C3×C4×D5#4112yes—
101D5×Dic3—42no—
102D5×Dic3—41no—
103D5×Dic3—41no—
104C3⋊#170—42no—
105C2×C3×C4⋊C4—52no—
106C2×C3×C42—51no—
107C2×C3×C42—101no—
108C2×C4⋊Dic3—51no—
109C2×C4⋊Dic3—51no—
110C2×Dic3⋊C4—101no—
111C2×Dic3⋊C4—101no—
112C22×C3⋊C8—52no—
113#870—51no—
114#870—51no—
115#870—51no—
116#870—101no—
117#870—101no—
118#870AGL(1,𝔽5)11yescharacteristic, complemented
119C3×#95—101no—
120C4⋊(C3⋊C8)—101no—
121C4⋊(C3⋊C8)—101no—
122C22×AGL(1,𝔽5)—22no—
123C2×C4×D5—61no—
124C2×C4×D5—61no—
125C2×C4×D5#6711yescharacteristic
126C2×C4×D5#6711yescharacteristic
127C2×C4×D5C22×D311yescharacteristic
128C2×#170—62no—
129C4⋊AGL(1,𝔽5)—22no—
130C4⋊AGL(1,𝔽5)Dic614yes—
131C4×AGL(1,𝔽5)—42no—
132D5⋊C8—64no—
133C5×C4⋊C4—31no—
134C5×C4⋊C4—31no—
135C5×C4⋊C4—61no—
136C5×C4⋊C4—61no—
137C42×C5C4×D311yescharacteristic
138C4⋊Dic5—31no—
139C4⋊Dic5—31no—
140C4⋊Dic5—61no—
141C4⋊Dic5—61no—
142#683—61no—
143#683—61no—
144#683—61no—
145#683—61no—
146#683—61no—
147#683—61no—
148C4×Dic5C4×D311yescharacteristic
149C4×Dic5D1211yescharacteristic
150C4×Dic5D1211yescharacteristic
151C2×#125—151no—
152C2×C4⋊C8—151no—
153C4×C4⋊C4—51no—
154#340—152no—
155#340—152no—
156C22×C3×C5C2×C22⋊C411yescharacteristic
157C2×C3×D5C22⋊Q812yes—
158C2×C3×D5#13512yes—
159C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
160C2×C3×D5#13711yescharacteristic
161C3×AGL(1,𝔽5)—44no—
162C3×C4×C5—21no—
163C3×C4×C5—21no—
164C3×C4×C5C4×Q812yes—
165Dic15—41no—
166Dic15—41no—
167C3×Dic5—21no—
168C3×Dic5—21no—
169C3×Dic5#12411yescharacteristic
170C3×Dic5C4⋊Q811yescharacteristic
171C5×Dic3—41no—
172C5×Dic3—41no—
173C22×C3×C4—51no—
174C22×C3×C4—51no—
175C22×C3×C4—51no—
176C22×C3×C4—102no—
177C22×Dic3—101no—
178C22×Dic3—101no—
179C3×C4⋊C4—54no—
180C3×C4⋊C4—102no—
181C3×C42—51no—
182C3×C42—51no—
183C3×C42—51no—
184C3×C42—202no—
185C3×C42C2×AGL(1,𝔽5)11yescharacteristic
186C4⋊Dic3—51no—
187C4⋊Dic3—51no—
188C4⋊Dic3—101no—
189C4⋊Dic3—101no—
190C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
191C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
192Dic3⋊C4—21no—
193Dic3⋊C4—21no—
194Dic3⋊C4—101no—
195Dic3⋊C4—101no—
196Dic3⋊C4—101no—
197Dic3⋊C4—101no—
198Dic3⋊C4—101no—
199Dic3⋊C4—101no—
200C2×C3⋊C8—102no—
201C2×C3⋊C8—104no—
202C22×D5C2×#6711yescharacteristic
203C2×AGL(1,𝔽5)—24no—
204C2×AGL(1,𝔽5)—42no—
205C2×AGL(1,𝔽5)—42no—
206C2×C4×C5—61no—
207C2×C4×C5—61no—
208C2×C4×C5(C2×D3)⋊C411yescharacteristic
209C2×C4×C5(C2×D3)⋊C411yescharacteristic
210C2×C4×C5C2×C4×D311yescharacteristic
211C2×Dic5—61no—
212C2×Dic5—61no—
213C2×Dic5(C2×D3)⋊C411yescharacteristic
214C2×Dic5(C2×D3)⋊C411yescharacteristic
215C2×Dic5C2×D1211yescharacteristic
216C4×D5—21no—
217C4×D5—21no—
218C4×D5—22no—
219C4×D5—121no—
220C4×D5—121no—
221C4×D5—122no—
222C4×D5C2×Dic612yes—
223C4×D5#23512yes—
224#170—122no—
225C22×C8—152no—
226#125—31no—
227#125—151no—
228#125—151no—
229#125—151no—
230#125—301no—
231#125—301no—
232C2×C4⋊C4—52no—
233C2×C4⋊C4—151no—
234C2×C4⋊C4—151no—
235C2×C4⋊C4—301no—
236C2×C4⋊C4—301no—
237C2×C42—51no—
238C2×C42—101no—
239C4⋊C8—301no—
240C4⋊C8—301no—
241#95—101no—
242C2×C3×C5#39111yescharacteristic
243C2×C3×C5#41611yescharacteristic
244C2×C3×C5#41711yescharacteristic
245C3×D5#47812yes—
246C3×D5#47912yes—
247C23×C3—51no—
248C2×C3×C4—51no—
249C2×C3×C4—52no—
250C2×C3×C4—52no—
251C2×C3×C4—51no—
252C2×C3×C4—51no—
253C2×C3×C4—101no—
254C2×C3×C4—101no—
255C2×C3×C4—102no—
256C2×C3×C4—104no—
257C2×C3×C4—202no—
258C2×C3×C4—202no—
259C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
260C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
261C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—21no—
264C2×Dic3—101no—
265C2×Dic3—101no—
266C2×Dic3—201no—
267C2×Dic3—201no—
268C2×Dic3—202no—
269C3⋊C8—202no—
270C22×C5C2×(C2×D3)⋊C411yescharacteristic
271C2×D5#92412yes—
272C2×D5#93211yescharacteristic
273C2×D5#94211yescharacteristic
274C2×D5#95012yes—
275AGL(1,𝔽5)—44no—
276C4×C5—21no—
277C4×C5—21no—
278C4×C5—121no—
279C4×C5—121no—
280C4×C5C4×Dic612yes—
281Dic5—21no—
282Dic5—21no—
283Dic5—121no—
284Dic5—121no—
285Dic5#86911yescharacteristic
286Dic5#87511yescharacteristic
287C22×C4—51no—
288C22×C4—51no—
289C22×C4—51no—
290C22×C4—102no—
291C22×C4—301no—
292C22×C4—301no—
293C2×C8—302no—
294C2×C8—304no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—54no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—102no—
301C4⋊C4—151no—
302C4⋊C4—151no—
303C4⋊C4—301no—
304C4⋊C4—301no—
305C4⋊C4—301no—
306C4⋊C4—301no—
307C4⋊C4—301no—
308C4⋊C4—301no—
309C4⋊C4—301no—
310C4⋊C4—301no—
311C42—51no—
312C42—51no—
313C42—51no—
314C42—202no—
315C42D3×AGL(1,𝔽5)11yescharacteristic
316C3×C5#198911yescharacteristic, complemented, Hall
317C22×C3—51no—
318C22×C3—51no—
319C22×C3—52no—
320C22×C3—52no—
321C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—51no—
325C3×C4—51no—
326C3×C4—101no—
327C3×C4—101no—
328C3×C4—204no—
329C3×C4Q8×AGL(1,𝔽5)12yes—
330Dic3—41no—
331Dic3—41no—
332Dic3—201no—
333Dic3—201no—
334C2×C5#492911yescharacteristic
335C2×C5#496611yescharacteristic
336C2×C5#500211yescharacteristic
337D5#500912yes—
338D5#502212yes—
339C23—51no—
340C2×C4—51no—
341C2×C4—52no—
342C2×C4—52no—
343C2×C4—51no—
344C2×C4—51no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—101no—
348C2×C4—102no—
349C2×C4—101no—
350C2×C4—104no—
351C2×C4—202no—
352C2×C4—202no—
353C2×C4—301no—
354C2×C4—301no—
355C2×C4—602no—
356C2×C4—601no—
357C2×C4—601no—
358C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
359C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
360C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
361C8—602no—
362C2×C3—52no—
363C2×C3—52no—
364C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
365C2×C3#6579311yescharacteristic
366C2×C3#6579411yescharacteristic
367C5#6949111yescharacteristic, complemented, Sylow
368C22—52no—
369C22—51no—
370C22—51no—
371C22—52no—
372C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
373C4—21no—
374C4—21no—
375C4—51no—
376C4—51no—
377C4—101no—
378C4—101no—
379C4—121no—
380C4—121no—
381C4—204no—
382C4—601no—
383C4—601no—
384C4AGL(1,𝔽5)×Dic612yes—
385C3#1060458911yescharacteristic, complemented, Sylow
386C2—52no—
387C2—52no—
388C2#1175170311yescharacteristic
389C2#1175170511yescharacteristic
390C2#1175174911yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1989
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
10. #91759C4semidirect
11. C5×#870C4semidirect
17. #91388C4semidirect
18. #91388C4semidirect
118. #870AGL(1,𝔽5)semidirect
316. C3×C5#1989semidirect
367. C5#69491semidirect
385. C3#10604589semidirect

Maximal subgroups

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

subgroupindex
D5×#8702
D5⋊(C4⋊(C3⋊C8))2
C3×C4×C4⋊AGL(1,𝔽5)2
#117465282
#106045893
#694915