Finite Group Databasebeta

← #14513754

Subgroups — #14513754

Order 1920.

402 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 65 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#11752139C211yesmaximal, characteristic
3C2×#91184C211yesmaximal, characteristic
4#11752110C211yesmaximal, characteristic, complemented
5#11752108C211yesmaximal, characteristic, complemented
6#11752108C211yesmaximal, characteristic, complemented
7#11752105C211yesmaximal, characteristic, complemented
8#10614281—31nomaximal, Hall
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10#91767—41nomaximal
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13#91184C2211yescharacteristic
14#91184C2211yescharacteristic
15#91184C2211yescharacteristic
16#91184C2211yescharacteristic
17#85351—51nomaximal, Hall
18C5×#519D311yescharacteristic
19#65853—31no—
20#65514—31no—
21C2×#3967—31no—
22#65311—31no—
23#65311—31no—
24#65311—31no—
25#65311—31no—
26#65310—31no—
27#65310—31no—
28#65309—31no—
29#65309—31no—
30#65288—31no—
31#65287—31no—
32#65284—31no—
33C22×Dic15—41no—
34C2×Dic30—41no—
35C2×C5×SL(2,𝔽3)Q812yescomplemented
36C2×C5×SL(2,𝔽3)C2311yescharacteristic
37C3×C5×C22⋊C4—41no—
38#6719—41no—
39C4⋊Dic15—41no—
40#6712—41no—
41#6712—41no—
42#519⋊C3D511yescharacteristic
43#5420—51no—
44C2×#859—51no—
45#5391—51no—
46#5389—51no—
47#5389—51no—
48#5386—51no—
49C22×C5×Q8C2×D311yescharacteristic
50C4×C5×Q8—31no—
51C4×C5×Q8—31no—
52C4×C5×Q8C2×D311yescharacteristic
53C4×C5×Q8C2×D311yescharacteristic
54C5×#117—31no—
55C5×#117—31no—
56#4072—31no—
57C2×C4⋊Dic5—31no—
58C2×#681—31no—
59C22×#168—31no—
60#4036—61no—
61#4034—61no—
62#4024—61no—
63#4024—61no—
64#4016—31no—
65#4016—31no—
66#4015—31no—
67#4015—31no—
68#3967—31no—
69#3967—31no—
70#3967—31no—
71#3967—31no—
72#3967—31no—
73#3967—31no—
74#3967—31no—
75#3967—31no—
76#3940—31no—
77#3940—31no—
78#3940—31no—
79#3940—31no—
80#3939—31no—
81#3939—31no—
82#3939—31no—
83#3939—31no—
84C4⋊#168—31no—
85C4⋊#168—31no—
86C4⋊#168—31no—
87C4⋊#168—31no—
88#3082—151noSylow
89C23×C3×C5—41no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5—41no—
92C2×Dic15—41no—
93C2×Dic15—41no—
94C2×Dic15—41no—
95C2×Dic15—41no—
96C2×Dic15—82no—
97Dic30—82no—
98C5×SL(2,𝔽3)C2×Q811yescharacteristic
99C22×SL(2,𝔽3)C2×D511yescharacteristic
100#878—201no—
101C4×SL(2,𝔽3)C2×D511yescharacteristic
102C4×SL(2,𝔽3)C2×D511yescharacteristic
103#859—51no—
104#859—51no—
105#859—51no—
106#859—51no—
107C2×C5×Q8—31no—
108C2×C5×Q8—32no—
109C2×C5×Q8Dic612yescomplemented
110C2×C5×Q8C22×D311yescharacteristic
111C22×C4×C5—31no—
112C22×Dic5—61no—
113C2×Dic10—61no—
114C2×Dic10—61no—
115C5×C4⋊C4—31no—
116C5×C4⋊C4—31no—
117C5×C4⋊C4—31no—
118C5×C4⋊C4—31no—
119C5×C22⋊C4—31no—
120C5×C22⋊C4S411yescharacteristic
121C42×C5—31no—
122C42×C5—31no—
123C42×C5—61no—
124C22⋊Dic5—61no—
125C4⋊Dic5—31no—
126C4⋊Dic5—31no—
127C4⋊Dic5—31no—
128C4⋊Dic5—31no—
129C4⋊Dic5—61no—
130C4⋊Dic5—61no—
131C4⋊Dic5—61no—
132#683—61no—
133#683—61no—
134#683—61no—
135#683—61no—
136C4×Dic5—61no—
137#681—62no—
138C2×#168—31no—
139C2×#168—31no—
140C2×#168—31no—
141C2×#168—31no—
142C2×#168—62no—
143#544—151no—
144#519D1511yescharacteristic
145#479—151no—
146#479—151no—
147#479—151no—
148#479—151no—
149#477—151no—
150#477—151no—
151#475—151no—
152#475—151no—
153#428—151no—
154#427—151no—
155#424—151no—
156#419—151no—
157C2×#103—151no—
158C22×C3×C5—42no—
159C22×C3×C5—41no—
160C22×C3×C5—81no—
161C22×C3×C5—81no—
162C3×C4×C5—81no—
163C3×C4×C5—81no—
164Dic15—42no—
165Dic15—81no—
166Dic15—81no—
167Dic15—81no—
168C22×Dic3—201no—
169C2×Dic6—201no—
170C2×SL(2,𝔽3)Dic1012yescomplemented
171C2×SL(2,𝔽3)C22×D511yescharacteristic
172C3×C22⋊C4—41no—
173#217—201no—
174C4⋊Dic3—201no—
175Dic3⋊C4—201no—
176Dic3⋊C4—201no—
177C23×C5C2×S411yescharacteristic
178C5×Q8—32no—
179C5×Q8—31no—
180C5×Q8—61no—
181C5×Q8C2×Dic611yescharacteristic
182C2×C4×C5—32no—
183C2×C4×C5—32no—
184C2×C4×C5—31no—
185C2×C4×C5—31no—
186C2×C4×C5—31no—
187C2×C4×C5—31no—
188C2×C4×C5C2×S411yescharacteristic
189C2×C4×C5C2×S411yescharacteristic
190C2×Dic5—61no—
191C2×Dic5—61no—
192C2×Dic5—61no—
193C2×Dic5—61no—
194C2×Dic5—241no—
195Dic10—241no—
196Dic10—241no—
197#168—61no—
198#168—61no—
199#168—61no—
200#168—61no—
201C22×Q8C2×D1511yescharacteristic
202C2×M16—151no—
203C22×C8—151no—
204C4⋊Q8—151no—
205C4⋊Q8—151no—
206C4⋊Q8—301no—
207#125—151no—
208#125—151no—
209#125—301no—
210C22⋊Q8—301no—
211C22⋊Q8—301no—
212C4×Q8—31no—
213C4×Q8—31no—
214C4×Q8C2×D1511yescharacteristic
215C4×Q8C2×D1511yescharacteristic
216#117—31no—
217#117—31no—
218#117—151no—
219C2×C4⋊C4—151no—
220#107—151no—
221#107—151no—
222#107—151no—
223#107—151no—
224#106—151no—
225#106—151no—
226#106—151no—
227#106—151no—
228C4⋊C8—151no—
229C4⋊C8—151no—
230C4⋊C8—151no—
231C4⋊C8—151no—
232#103—151no—
233#103—151no—
234#103—151no—
235#103—151no—
236#103—151no—
237#103—151no—
238#103—151no—
239#103—151no—
240C2×C3×C5—41no—
241C2×C3×C5—41no—
242C2×C3×C5—41no—
243C2×C3×C5—82no—
244C23×C3—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×Dic3—201no—
248C2×Dic3—201no—
249C2×Dic3—201no—
250C2×Dic3—201no—
251C2×Dic3—402no—
252Dic6—402no—
253SL(2,𝔽3)C2×Dic1011yescharacteristic
254C22×C5—21no—
255C22×C5—21no—
256C22×C5A4⋊Q812yes—
257C22×C5C22×S411yescharacteristic
258C4×C5—21no—
259C4×C5—21no—
260C4×C5—31no—
261C4×C5—31no—
262C4×C5—32no—
263C4×C5—61no—
264C4×C5—61no—
265Dic5—121no—
266Dic5—121no—
267Dic5—121no—
268Dic5—121no—
269C2×Q8—31no—
270C2×Q8—32no—
271C2×Q8—301no—
272C2×Q8—301no—
273C2×Q8Dic3012yescomplemented
274C2×Q8C22×D1511yescharacteristic
275C22×C4—31no—
276C22×C4—301no—
277M16—302no—
278C2×C8—151no—
279C2×C8—151no—
280C2×C8—151no—
281C2×C8—151no—
282C2×C8—302no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—151no—
288C4⋊C4—151no—
289C4⋊C4—151no—
290C4⋊C4—151no—
291C4⋊C4—301no—
292C4⋊C4—301no—
293C4⋊C4—301no—
294C4⋊C4—301no—
295C4⋊C4—301no—
296C4⋊C4—301no—
297C4⋊C4—301no—
298C22⋊C4—31no—
299C22⋊C4—301no—
300C22⋊C4C5⋊S411yescharacteristic
301C42—31no—
302C42—31no—
303C42—61no—
304C42—301no—
305C3×C5—41noHall
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—81no—
309C22×C3—81no—
310C3×C4—81no—
311C3×C4—81no—
312Dic3—202no—
313Dic3—401no—
314Dic3—401no—
315Dic3—401no—
316C2×C5—22no—
317C2×C5C2×A4⋊Q811yescharacteristic
318C2×C5#592611yescharacteristic
319C2×C5#592711yescharacteristic
320C23C2×C5⋊S411yescharacteristic
321Q8—32no—
322Q8—31no—
323Q8—61no—
324Q8—1201no—
325Q8—1201no—
326Q8C2×Dic3011yescharacteristic
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—301no—
334C2×C4—301no—
335C2×C4—301no—
336C2×C4—301no—
337C2×C4—1201no—
338C2×C4C2×C5⋊S411yescharacteristic
339C2×C4C2×C5⋊S411yescharacteristic
340C8—301no—
341C8—301no—
342C8—301no—
343C8—301no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—82no—
348C5#8535111yescharacteristic, complemented, Sylow
349C22—21no—
350C22—21no—
351C22#9194812yes—
352C22C22×C5⋊S411yescharacteristic
353C4—21no—
354C4—21no—
355C4—31no—
356C4—31no—
357C4—32no—
358C4—61no—
359C4—61no—
360C4—601no—
361C4—601no—
362C4—601no—
363C4—601no—
364C3—41noSylow
365C2—22no—
366C2C2×#9194811yescharacteristic
367C2#1175691011yescharacteristic
368C2#1175691111yescharacteristic
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC5×#519D3
socleC22×C5C22×S4
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC22⋊C4C5⋊S4
hypocenterC5×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #11752110C2semidirect
5. #11752108C2semidirect
6. #11752108C2semidirect
7. #11752105C2semidirect
35. C2×C5×SL(2,𝔽3)Q8semidirect
109. C2×C5×Q8Dic6semidirect
170. C2×SL(2,𝔽3)Dic10semidirect
273. C2×Q8Dic30semidirect
348. C5#85351semidirect

Maximal subgroups

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

subgroupindex
C5×#519⋊C32
#117521392
C2×#911842
#117521102
#117521082
#117521052
#106142813
#917674
#853515