Finite Group Databasebeta

← #10617776⋊C3 #14514041

Subgroups — #10617776⋊C3 #14514041

Order 1920.

508 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C5×#519⋊C3C211yesmaximal, characteristic
2#66146⋊C3C211yesmaximal, characteristic
3C2×Dic5×SL(2,𝔽3)C211yesmaximal, characteristic
4#65904⋊C3C211yesmaximal, characteristic, complemented
5#65903⋊C3C211yesmaximal, characteristic, complemented
6#65903⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic10C211yesmaximal, characteristic, complemented
8#10617776C311yesmaximal, characteristic, complemented, Hall
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10C3×#4024—41nomaximal
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13Dic5×SL(2,𝔽3)C2211yescharacteristic
14Dic5×SL(2,𝔽3)C2211yescharacteristic
15Dic5×SL(2,𝔽3)C2211yescharacteristic
16Dic5×SL(2,𝔽3)C2211yescharacteristic
17#3502⋊C3—51nomaximal, Hall
18C5×#519C2×C311yescharacteristic
19#66146C2×C311yescharacteristic
20C2×Q8×Dic5C2×C311yescharacteristic
21#65904—31no—
22#65904C2×C311yescharacteristic, complemented
23#65903—31no—
24#65903—31no—
25#65903C2×C311yescharacteristic, complemented
26#65903C2×C311yescharacteristic, complemented
27Q8×Dic10—31no—
28Q8×Dic10C2×C311yescharacteristic, complemented
29#65853—31no—
30#65853—31no—
31#65851—31no—
32#65851—31no—
33C22×C3×Dic5—41no—
34C2×C3×Dic10—41no—
35C2×C5×SL(2,𝔽3)Q812yescomplemented
36C2×C5×SL(2,𝔽3)C2311yescharacteristic
37C3×C5×C22⋊C4—41no—
38C3×C22⋊Dic5—41no—
39C3×C4⋊Dic5—41no—
40C3×#683—41no—
41C3×#683—41no—
42Q8×SL(2,𝔽3)—51no—
43#557⋊C3—51no—
44#557⋊C3—51no—
45#557⋊C3—51no—
46#519⋊C3—51no—
47#519⋊C3D511yescharacteristic
48C2×C4×SL(2,𝔽3)—51no—
49C22×C5×Q8C22×C311yescharacteristic
50C4×C5×Q8—31no—
51C4×C5×Q8—31no—
52C4×C5×Q8C22×C311yescharacteristic
53C4×C5×Q8C22×C311yescharacteristic
54C5×#117—31no—
55C5×#117—31no—
56Q8×Dic5—31no—
57Q8×Dic5—31no—
58Q8×Dic5—31no—
59Q8×Dic5—31no—
60Q8×Dic5C22×C311yescharacteristic
61Q8×Dic5C22×C311yescharacteristic
62Q8×Dic5C22×C311yescharacteristic
63Q8×Dic5C22×C311yescharacteristic
64#4072—31no—
65#4072—31no—
66C2×C4⋊Dic5—31no—
67C2×C4×Dic5—31no—
68#4036—32no—
69#4036—31no—
70#4036—31no—
71#4036—31no—
72#4036—31no—
73#4036—31no—
74#4036—31no—
75#4034—31no—
76#4034—31no—
77#4034—32no—
78#4034—31no—
79#4034—31no—
80#4034—31no—
81#4034—31no—
82#4024—32no—
83#4024—31no—
84#4024—31no—
85#4024—31no—
86#4024A411yescharacteristic
87#4016—32no—
88#4016—31no—
89#4016—31no—
90#4015—31no—
91#4015—32no—
92#4015—31no—
93C4×Dic10—32no—
94C4×Dic10—32no—
95C4×Dic10—31no—
96C4×Dic10—31no—
97C4×Dic10—31no—
98C4×Dic10—31no—
99#3502—51noSylow
100C23×C3×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×Dic5—41no—
104C2×C3×Dic5—41no—
105C2×C3×Dic5—41no—
106C2×C3×Dic5—41no—
107C2×C3×Dic5—82no—
108C3×Dic10—82no—
109C5×SL(2,𝔽3)C2×Q811yescharacteristic
110C22×SL(2,𝔽3)C2×D511yescharacteristic
111C3×C22⋊Q8—201no—
112C4×SL(2,𝔽3)—51no—
113C4×SL(2,𝔽3)—51no—
114C4×SL(2,𝔽3)—51no—
115C4×SL(2,𝔽3)—51no—
116C4×SL(2,𝔽3)C2×D511yescharacteristic
117C4×SL(2,𝔽3)C2×D511yescharacteristic
118C2×C5×Q8—31no—
119C2×C5×Q8—32no—
120C2×C5×Q8C3×Q812yescomplemented
121C2×C5×Q8C23×C311yescharacteristic
122C22×C4×C5—31no—
123C22×Dic5—31no—
124C22×Dic5C2×A411yescharacteristic
125C2×Dic10—31no—
126C2×Dic10—32no—
127C2×Dic10—31no—
128C2×Dic10—31no—
129C2×Dic10C2×A411yescharacteristic
130C5×C4⋊C4—31no—
131C5×C4⋊C4—31no—
132C5×C4⋊C4—31no—
133C5×C4⋊C4—31no—
134C5×C22⋊C4—31no—
135C5×C22⋊C4C2×A411yescharacteristic
136C42×C5—32no—
137C42×C5—31no—
138C42×C5—31no—
139C22⋊Dic5—31no—
140C22⋊Dic5C2×A411yescharacteristic
141C4⋊Dic5—32no—
142C4⋊Dic5—32no—
143C4⋊Dic5—31no—
144C4⋊Dic5—31no—
145C4⋊Dic5—31no—
146C4⋊Dic5—31no—
147C4⋊Dic5—31no—
148C4⋊Dic5—31no—
149C4⋊Dic5—31no—
150C4⋊Dic5—31no—
151C4⋊Dic5—31no—
152C4⋊Dic5C2×A411yescharacteristic
153#683—32no—
154#683—32no—
155#683—31no—
156#683—31no—
157#683—31no—
158#683—31no—
159#683—31no—
160#683—31no—
161#683C2×A411yescharacteristic
162#683C2×A411yescharacteristic
163C4×Dic5—32no—
164C4×Dic5—31no—
165C4×Dic5—31no—
166C4×Dic5—31no—
167C4×Dic5—31no—
168C4×Dic5—31no—
169C4×Dic5—31no—
170Q82—51no—
171Q82—151no—
172#557—51no—
173#557—51no—
174#557—51no—
175#557—151no—
176#557—151no—
177#557—151no—
178#544—151no—
179#544—151no—
180#533—151no—
181#533—151no—
182#519—51no—
183#519C3×D511yescharacteristic
184C2×C4×Q8—51no—
185C22×C3×C5—42no—
186C22×C3×C5—41no—
187C22×C3×C5—81no—
188C22×C3×C5—81no—
189C3×C4×C5—81no—
190C3×C4×C5—81no—
191C3×Dic5—42no—
192C3×Dic5—81no—
193C3×Dic5—81no—
194C3×Dic5—81no—
195C2×C3×Q8—201no—
196C22×C3×C4—201no—
197C2×SL(2,𝔽3)Dic1012yescomplemented
198C2×SL(2,𝔽3)C22×D511yescharacteristic
199C3×C4⋊C4—201no—
200C3×C4⋊C4—201no—
201C3×C4⋊C4—201no—
202C3×C22⋊C4—41no—
203C3×C22⋊C4—201no—
204C23×C5C22×A411yescharacteristic
205C5×Q8—32no—
206C5×Q8—31no—
207C5×Q8—32no—
208C5×Q8C2×C3×Q811yescharacteristic
209C2×C4×C5—32no—
210C2×C4×C5—32no—
211C2×C4×C5—31no—
212C2×C4×C5—31no—
213C2×C4×C5—31no—
214C2×C4×C5—31no—
215C2×C4×C5C22×A411yescharacteristic
216C2×C4×C5C22×A411yescharacteristic
217C2×Dic5—22no—
218C2×Dic5—31no—
219C2×Dic5—31no—
220C2×Dic5—31no—
221C2×Dic5—31no—
222C2×Dic5—121no—
223C2×Dic5C22×A411yescharacteristic
224C2×Dic5C22×A411yescharacteristic
225C2×Dic5C22×A411yescharacteristic
226C2×Dic5C22×A411yescharacteristic
227Dic10—22no—
228Dic10—62no—
229Dic10—121no—
230Dic10—121no—
231Dic10—122no—
232C22×Q8C2×C3×D511yescharacteristic
233C4⋊Q8—151no—
234C4⋊Q8—152no—
235C4⋊Q8—151no—
236C4⋊Q8—151no—
237C4⋊Q8—151no—
238C4⋊Q8—151no—
239C4⋊Q8—151no—
240C4⋊Q8—152no—
241C4⋊Q8—151no—
242C4⋊Q8—151no—
243#125—151no—
244#125—151no—
245#125—152no—
246#125—151no—
247#125—151no—
248#125—151no—
249#125—151no—
250#125—151no—
251#125—151no—
252#125—152no—
253C22⋊Q8—51no—
254C22⋊Q8—151no—
255C22⋊Q8—151no—
256C22⋊Q8—152no—
257C22⋊Q8—151no—
258C4×Q8—31no—
259C4×Q8—31no—
260C4×Q8—51no—
261C4×Q8—51no—
262C4×Q8—51no—
263C4×Q8—51no—
264C4×Q8—151no—
265C4×Q8—151no—
266C4×Q8—151no—
267C4×Q8—151no—
268C4×Q8—152no—
269C4×Q8—152no—
270C4×Q8—151no—
271C4×Q8—151no—
272C4×Q8—151no—
273C4×Q8—151no—
274C4×Q8C2×C3×D511yescharacteristic
275C4×Q8C2×C3×D511yescharacteristic
276#117—31no—
277#117—31no—
278#117—151no—
279#117—151no—
280C2×C4⋊C4—151no—
281C2×C42—151no—
282C2×C3×C5—41no—
283C2×C3×C5—41no—
284C2×C3×C5—41no—
285C2×C3×C5—82no—
286C23×C3—41no—
287C3×Q8—402no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—201no—
291C2×C3×C4—201no—
292C2×C3×C4—201no—
293C2×C3×C4—201no—
294C2×C3×C4—402no—
295SL(2,𝔽3)C2×Dic1011yescharacteristic
296C22×C5—21no—
297C22×C5—21no—
298C22×C5Q8×A412yes—
299C22×C5C23×A411yescharacteristic
300C4×C5—21no—
301C4×C5—21no—
302C4×C5—31no—
303C4×C5—32no—
304C4×C5—31no—
305C4×C5—61no—
306C4×C5—61no—
307Dic5—21no—
308Dic5—21no—
309Dic5—21no—
310Dic5—61no—
311Dic5—61no—
312Dic5—61no—
313Dic5—61no—
314Dic52−1+4⋊C312yes—
315C2×Q8—31no—
316C2×Q8—32no—
317C2×Q8—51no—
318C2×Q8—151no—
319C2×Q8—151no—
320C2×Q8—152no—
321C2×Q8—151no—
322C2×Q8C3×Dic1012yescomplemented
323C2×Q8C22×C3×D511yescharacteristic
324C22×C4—31no—
325C22×C4—51no—
326C22×C4—151no—
327C4⋊C4—31no—
328C4⋊C4—31no—
329C4⋊C4—31no—
330C4⋊C4—31no—
331C4⋊C4—51no—
332C4⋊C4—51no—
333C4⋊C4—51no—
334C4⋊C4—151no—
335C4⋊C4—151no—
336C4⋊C4—151no—
337C4⋊C4—151no—
338C4⋊C4—151no—
339C4⋊C4—151no—
340C4⋊C4—151no—
341C4⋊C4—151no—
342C4⋊C4—151no—
343C4⋊C4—151no—
344C4⋊C4—151no—
345C4⋊C4—151no—
346C4⋊C4—152no—
347C4⋊C4—152no—
348C4⋊C4—152no—
349C4⋊C4—152no—
350C4⋊C4—151no—
351C4⋊C4—151no—
352C4⋊C4—151no—
353C22⋊C4—31no—
354C22⋊C4—51no—
355C22⋊C4—151no—
356C22⋊C4D5×A411yescharacteristic
357C42—32no—
358C42—31no—
359C42—31no—
360C42—151no—
361C42—151no—
362C42—151no—
363C42—151no—
364C42—151no—
365C42—151no—
366C42—152no—
367C3×C5—41noHall
368C22×C3—42no—
369C22×C3—41no—
370C22×C3—81no—
371C22×C3—81no—
372C3×C4—81no—
373C3×C4—81no—
374C3×C4—202no—
375C3×C4—401no—
376C3×C4—401no—
377C3×C4—401no—
378C2×C5—22no—
379C2×C5C2×Q8×A411yescharacteristic
380C2×C5C2×2−1+4⋊C311yescharacteristic
381C2×C5#594511yescharacteristic
382C23C2×D5×A411yescharacteristic
383Q8—32no—
384Q8—31no—
385Q8—32no—
386Q8—102no—
387Q8—302no—
388Q8—601no—
389Q8—601no—
390Q8—602no—
391Q8C2×C3×Dic1011yescharacteristic
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—31no—
395C2×C4—31no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—51no—
399C2×C4—51no—
400C2×C4—51no—
401C2×C4—51no—
402C2×C4—102no—
403C2×C4—151no—
404C2×C4—151no—
405C2×C4—151no—
406C2×C4—151no—
407C2×C4—601no—
408C2×C4C2×D5×A411yescharacteristic
409C2×C4C2×D5×A411yescharacteristic
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—82no—
414C5#3502⋊C311yescharacteristic, complemented, Sylow
415C22—21no—
416C22—21no—
417C22A4×Dic1012yes—
418C22C22×D5×A411yescharacteristic
419C4—21no—
420C4—21no—
421C4—31no—
422C4—32no—
423C4—31no—
424C4—52no—
425C4—61no—
426C4—61no—
427C4—101no—
428C4—101no—
429C4—101no—
430C4—301no—
431C4—301no—
432C4—301no—
433C4—301no—
434C3—41noSylow
435C2—22no—
436C2C2×A4×Dic1011yescharacteristic
437C2D5×2−1+4⋊C311yescharacteristic
438C2#66287⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC5×#519C2×C3
socleC22×C5C23×A4
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC22⋊C4D5×A4
hypocenterC5×Q8C2×C3×Q8
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #65904⋊C3C2semidirect
5. #65903⋊C3C2semidirect
6. #65903⋊C3C2semidirect
7. SL(2,𝔽3)×Dic10C2semidirect
8. #10617776C3semidirect
22. #65904C2×C3semidirect
25. #65903C2×C3semidirect
26. #65903C2×C3semidirect
28. Q8×Dic10C2×C3semidirect
35. C2×C5×SL(2,𝔽3)Q8semidirect
120. C2×C5×Q8C3×Q8semidirect
197. C2×SL(2,𝔽3)Dic10semidirect
322. C2×Q8C3×Dic10semidirect
414. C5#3502⋊C3semidirect

Maximal subgroups

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

subgroupindex
C5×#519⋊C32
#66146⋊C32
C2×Dic5×SL(2,𝔽3)2
#65904⋊C32
#65903⋊C32
SL(2,𝔽3)×Dic102
#106177763
C3×#40244
#3502⋊C35