Finite Group Databasebeta

← #14511813

Subgroups — #14511813

Order 1920.

530 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C22×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#91184C211yesmaximal, characteristic, complemented
3C2×C5×#859C211yesmaximal, characteristic, complemented
4#10601059—31nomaximal, Hall
5C22×C5×SL(2,𝔽3)C2211yescharacteristic
6C2×D5×SL(2,𝔽3)—21no—
7C2×D5×SL(2,𝔽3)—21no—
8C2×D5×SL(2,𝔽3)C412yescomplemented
9C2×#6665—41nomaximal
10#91184—21no—
11C5×#859—21no—
12#73148—51nomaximal, Hall
13C22×Q8×D5D311yescharacteristic
14C2×C5×#103—31no—
15C2×#3967—31no—
16C2×#3952—31no—
17C2×#3948—31no—
18#65276—31no—
19C2×#3942—31no—
20C23×C3×D5—41no—
21C22×Dic15—41no—
22C22×C5×Dic3—41no—
23C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
24C2×C5×SL(2,𝔽3)D411yescharacteristic
25C2×C5×SL(2,𝔽3)D411yescharacteristic
26D5×SL(2,𝔽3)—22no—
27#6665—44no—
28C23×SL(2,𝔽3)—51no—
29C2×#859—51no—
30C2×#859D511yescharacteristic, complemented
31C22×C5×Q8C2×D311yescharacteristic
32C2×Q8×D5—21no—
33C2×Q8×D5—21no—
34C2×Q8×D5—32no—
35C2×Q8×D5—61no—
36C2×Q8×D5—61no—
37C2×Q8×D5Dic312yescomplemented
38C22×C4×D5—31no—
39C22×Dic10—31no—
40C22×C5×C8—31no—
41C2×C5×C4⋊C4—31no—
42C2×#685—61no—
43C2×C4⋊Dic5—31no—
44C22×#168—31no—
45C5×#103—61no—
46C5×#103—61no—
47#3967—61no—
48#3967—61no—
49#3963—61no—
50#3952—62no—
51#3948—62no—
52#3942—62no—
53#1787—151noSylow
54C23×C3×C5—41no—
55C22×C3×D5—42no—
56C22×C3×D5—42no—
57C22×C3×D5—42no—
58C22×C3×D5—81no—
59C22×C3×D5—81no—
60C22×C3×D5—81no—
61C22×C3×D5—81no—
62C2×Dic15—42no—
63C2×Dic15—81no—
64C2×Dic15—81no—
65C2×C5×Dic3—42no—
66C2×C5×Dic3—81no—
67C2×C5×Dic3—81no—
68C5×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
69C22×SL(2,𝔽3)—52no—
70C22×SL(2,𝔽3)—101no—
71C22×SL(2,𝔽3)—101no—
72C22×SL(2,𝔽3)C2×D511yescharacteristic
73C2×#217—201no—
74#859—21no—
75#859—101no—
76C23×D5S411yescharacteristic
77C2×C5×Q8—31no—
78C2×C5×Q8—31no—
79C2×C5×Q8—31no—
80C2×C5×Q8C2×Dic311yescharacteristic
81C2×C5×Q8#6711yescharacteristic
82C2×C5×Q8#6711yescharacteristic
83C22×C4×C5—31no—
84C22×C4×C5—61no—
85C22×Dic5—31no—
86C22×Dic5—61no—
87Q8×D5—22no—
88Q8×D5—62no—
89Q8×D5—64no—
90Q8×D5—62no—
91Q8×D5—62no—
92C2×C4×D5—32no—
93C2×C4×D5—32no—
94C2×C4×D5—61no—
95C2×C4×D5—61no—
96C2×C4×D5—61no—
97C2×C4×D5—61no—
98C2×Dic10—32no—
99C2×Dic10—32no—
100C2×Dic10—32no—
101C2×Dic10—61no—
102C2×Dic10—61no—
103C2×Dic10—61no—
104C2×C5×C8—61no—
105C2×C5×C8—61no—
106C2×C5×C8—61no—
107C5×C4⋊C4—61no—
108C5×C4⋊C4—61no—
109#685—122no—
110C4⋊Dic5—61no—
111C4⋊Dic5—61no—
112C2×#168—61no—
113C2×#168—61no—
114C2×#168—61no—
115C23×Q8—51no—
116C2×#103—31no—
117C2×#103—151no—
118C2×#103—151no—
119C2×#103—151no—
120C2×C22⋊C8—151no—
121#380—151no—
122C22×C3×C5—41no—
123C22×C3×C5—41no—
124C22×C3×C5—41no—
125C22×C3×C5—41no—
126C22×C3×C5—41no—
127C22×C3×C5—41no—
128C22×C3×C5—41no—
129C2×C3×D5—44no—
130C2×C3×D5—82no—
131C2×C3×D5—82no—
132C2×C3×D5—82no—
133C2×C3×D5—82no—
134C2×C3×D5—82no—
135C2×C3×D5—82no—
136Dic15—82no—
137C5×Dic3—82no—
138C24×C3—201no—
139C22×Dic3—41no—
140C22×Dic3—201no—
141C2×SL(2,𝔽3)—102no—
142C2×SL(2,𝔽3)C4×D511yescharacteristic
143C2×SL(2,𝔽3)D2011yescharacteristic
144C2×SL(2,𝔽3)C22⋊D511yescharacteristic
145#217—204no—
146C23×C5C2×S411yescharacteristic
147C22×D5—21no—
148C22×D5—21no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5#22612yes—
152C22×D5GL(2,𝔽3)12yes—
153C22×D5SL(2,ℤ/4)12yes—
154C5×Q8—31no—
155C5×Q8—31no—
156C5×Q8—31no—
157C5×Q8—61no—
158C5×Q8#21711yescharacteristic, complemented
159C2×C4×C5—31no—
160C2×C4×C5—31no—
161C2×C4×C5—31no—
162C2×C4×C5—31no—
163C2×C4×C5—31no—
164C2×C4×C5—31no—
165C2×C4×C5—121no—
166C2×C4×C5—121no—
167C2×C4×C5—121no—
168C2×Dic5—32no—
169C2×Dic5—61no—
170C2×Dic5—61no—
171C2×Dic5—121no—
172C2×Dic5—121no—
173C2×Dic5—121no—
174C4×D5—62no—
175C4×D5—62no—
176C4×D5—62no—
177C4×D5—62no—
178Dic10—34no—
179Dic10—62no—
180Dic10—62no—
181Dic10—62no—
182C5×C8—121no—
183#168—121no—
184C22×Q8—52no—
185C22×Q8—101no—
186C22×Q8—101no—
187C22×Q8—151no—
188C22×Q8—152no—
189C22×Q8—301no—
190C22×Q8—301no—
191C22×Q8D3×D511yescharacteristic
192C23×C4—151no—
193C22×C8—31no—
194C22×C8—151no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—151no—
197C2×C22⋊C4—301no—
198#103—61no—
199#103—61no—
200#103—302no—
201#103—302no—
202#103—301no—
203#103—301no—
204C22⋊C8—302no—
205#95—301no—
206C2×C3×C5—41no—
207C2×C3×C5—41no—
208C2×C3×C5—41no—
209C2×C3×C5—41no—
210C2×C3×C5—41no—
211C2×C3×C5—41no—
212C2×C3×C5—41no—
213C3×D5—84no—
214C23×C3—41no—
215C23×C3—202no—
216C23×C3—202no—
217C23×C3—202no—
218C23×C3—401no—
219C23×C3—401no—
220C23×C3—401no—
221C23×C3—401no—
222C2×Dic3—42no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—202no—
226C2×Dic3—401no—
227C2×Dic3—401no—
228SL(2,𝔽3)#68511yescharacteristic, complemented
229C22×C5C2×#22611yescharacteristic
230C22×C5C2×GL(2,𝔽3)11yescharacteristic
231C22×C5#98311yescharacteristic
232C22×C5#98311yescharacteristic
233C22×C5C2×SL(2,ℤ/4)11yescharacteristic
234C22×C5GL(2,ℤ/4)11yescharacteristic
235C22×C5GL(2,ℤ/4)11yescharacteristic
236C2×D5—22no—
237C2×D5—22no—
238C2×D5—22no—
239C2×D5—22no—
240C2×D5—22no—
241C2×D5—22no—
242C2×D5#85914yescomplemented
243C4×C5—31no—
244C4×C5—31no—
245C4×C5—31no—
246C4×C5—31no—
247C4×C5—241no—
248Dic5—62no—
249Dic5—241no—
250C24—51no—
251C2×Q8—31no—
252C2×Q8—31no—
253C2×Q8—31no—
254C2×Q8—102no—
255C2×Q8—152no—
256C2×Q8—152no—
257C2×Q8—152no—
258C2×Q8—301no—
259C2×Q8—301no—
260C2×Q8—301no—
261C2×Q8—304no—
262C2×Q8—302no—
263C2×Q8—302no—
264C2×Q8—302no—
265C2×Q8D5×Dic311yescharacteristic
266C2×Q8#120111yescharacteristic
267C2×Q8#120211yescharacteristic
268C22×C4—31no—
269C22×C4—61no—
270C22×C4—152no—
271C22×C4—152no—
272C22×C4—151no—
273C22×C4—301no—
274C22×C4—301no—
275C22×C4—301no—
276C22×C4—301no—
277C22×C4—301no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—301no—
282C2×C8—301no—
283C2×C8—301no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—301no—
287C4⋊C4—301no—
288C22⋊C4—602no—
289C3×C5—41noHall
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—204no—
298C22×C3—402no—
299C22×C3—402no—
300C22×C3—402no—
301C22×C3—402no—
302C22×C3—402no—
303C22×C3—402no—
304Dic3—82no—
305Dic3—402no—
306C2×C5C2×#85911yescharacteristic
307C2×C5#541911yescharacteristic
308C2×C5#541911yescharacteristic
309C2×C5#542011yescharacteristic
310C2×C5#542111yescharacteristic
311C2×C5#542111yescharacteristic
312C2×C5#543211yescharacteristic
313D5—24no—
314C23—52no—
315C23—52no—
316C23—52no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23D5×S411yescharacteristic
322Q8—31no—
323Q8—31no—
324Q8—31no—
325Q8—61no—
326Q8—154no—
327Q8—302no—
328Q8—302no—
329Q8—302no—
330Q8#666511yescharacteristic, complemented
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—152no—
341C2×C4—302no—
342C2×C4—302no—
343C2×C4—302no—
344C2×C4—302no—
345C2×C4—301no—
346C2×C4—301no—
347C2×C4—601no—
348C2×C4—601no—
349C2×C4—601no—
350C8—121no—
351C8—601no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—404no—
360C5#7314811yescharacteristic, complemented, Sylow
361C22—54no—
362C22—102no—
363C22—102no—
364C22—102no—
365C22—102no—
366C22—102no—
367C22—102no—
368C22D5×#22611yescharacteristic
369C22#9189611yescharacteristic
370C22#9189711yescharacteristic
371C22D5×GL(2,𝔽3)11yescharacteristic
372C22D5×SL(2,ℤ/4)11yescharacteristic
373C22#9190411yescharacteristic
374C22#9190511yescharacteristic
375C4—31no—
376C4—31no—
377C4—31no—
378C4—31no—
379C4—241no—
380C4—302no—
381C4—1201no—
382C3—41noSylow
383C2—104no—
384C2#1175160511yescharacteristic
385C2#1175160611yescharacteristic
386C2#1175160711yescharacteristic
387C2D5×#85911yescharacteristic
388C2#1175161311yescharacteristic
389C2#1175161411yescharacteristic
390C2#1175163911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D5×S4
commutatorC2×C5×SL(2,𝔽3)C2×C4
FittingC22×C5×Q8C2×D3
socleC23×C5C2×S4
prime radicalC22×D5×SL(2,𝔽3)C2
cosocleC22×C5×SL(2,𝔽3)C22
hypocenterC5×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#91184C2semidirect
3. C2×C5×#859C2semidirect
8. C2×D5×SL(2,𝔽3)C4semidirect
30. C2×#859D5semidirect
37. C2×Q8×D5Dic3semidirect
68. C5×SL(2,𝔽3)C22⋊C4semidirect
158. C5×Q8#217semidirect
228. SL(2,𝔽3)#685semidirect
242. C2×D5#859semidirect
330. Q8#6665semidirect
360. C5#73148semidirect

Maximal subgroups

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

subgroupindex
C22×D5×SL(2,𝔽3)2
C2×#911842
C2×C5×#8592
#106010593
C2×#66654
#731485