Finite Group Databasebeta

← C5×#85392 #14511130

Subgroups — C5×#85392 #14511130

Order 1920.

574 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×C5×#983C212yesmaximal, complemented
2C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#5421C212yesmaximal, complemented
4C5×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C4×C5×#226C211yesmaximal, characteristic, complemented
6C5×#3278—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2212yes—
8C5×#983—24no—
9C2×C5×GL(2,𝔽3)C2212yes—
10C2×C5×#226C2211yescharacteristic
11C5×#940—41nomaximal
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13C5×#859C2211yescharacteristic
14#85392C511yesmaximal, characteristic, direct product factor, Hall
15C2×C5×#137—32no—
16C5×#550—31no—
17C5×D4×Q8D311yescharacteristic, complemented
18C5×#492—31no—
19C5×#468—32no—
20C5×#462—31no—
21C5×#460—31no—
22C5×#452—32no—
23C5×#448—32no—
24C4×C5×Q16—31no—
25C5×#436—31no—
26C2×C3×C5×D4—41no—
27C2×C5×#67—42no—
28C2×C5×#67—42no—
29C2×C5×D12—41no—
30C2×C5×SL(2,𝔽3)—22no—
31C2×C5×SL(2,𝔽3)C2311yescharacteristic
32C5×GL(2,𝔽3)—22no—
33C5×#226D412yescomplemented
34C4×C5×Dic3—41no—
35C2×#983C2×C512yescomplemented
36D4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
37#5421C2×C512yescomplemented
38C4⋊GL(2,𝔽3)C2×C511yescharacteristic, complemented
39C4×#226C2×C511yescharacteristic, complemented
40C2×C5×#41—32no—
41C22×C5×Q8C2×D312yes—
42C5×#137—64no—
43C5×#137—64no—
44C2×C5×Q16—31no—
45C2×C5×Q16—32no—
46C2×C5×SD16—32no—
47C2×C5×SD16—32no—
48C2×C5×SD16—32no—
49C2×C5×M16—32no—
50C5×C4⋊Q8—31no—
51C5×#127—31no—
52C5×#127—61no—
53C5×C22⋊Q8—32no—
54C5×#121—32no—
55C5×#121—62no—
56C4×C5×Q8—31no—
57C4×C5×Q8C2×D311yescharacteristic
58C4×C5×D4—31no—
59C4×C5×D4—61no—
60C5×#107—31no—
61C5×C4⋊C8—31no—
62C5×#103—31no—
63C5×#103—32no—
64C5×#103—31no—
65C5×#102—32no—
66C5×C22⋊C8—32no—
67C4×C5×C8—31no—
68#3278—31noSylow
69C23×C3×C5—42no—
70C22×C5×D3—42no—
71C3×C5×D4—82no—
72C2×C3×C4×C5—41no—
73C5×#67—84no—
74C5×#67—84no—
75C2×C5×Dic3—41no—
76C2×C5×Dic3—41no—
77C5×D12—82no—
78C5×SL(2,𝔽3)C2×D411yescharacteristic
79C22×SL(2,𝔽3)C22×C512yes—
80#983—24no—
81C2×GL(2,𝔽3)C22×C512yes—
82C2×#226C22×C511yescharacteristic
83#940—41no—
84C4×SL(2,𝔽3)C22×C511yescharacteristic
85#859C22×C511yescharacteristic
86C5×#41—64no—
87C5×#41—122no—
88C2×C5×Q8—22no—
89C2×C5×Q8—32no—
90C2×C5×Q8—31no—
91C2×C5×Q8—62no—
92C2×C5×Q8C22×D311yescharacteristic
93C2×C5×D4—32no—
94C2×C5×D4—61no—
95C2×C5×D4—62no—
96C2×C5×D4—62no—
97C2×C5×D4—61no—
98C2×C5×D4S411yescharacteristic
99C22×C4×C5—32no—
100C22×C4×C5—62no—
101C5×Q16—32no—
102C5×Q16—32no—
103C5×Q16—64no—
104C5×SD16—64no—
105C5×SD16—62no—
106C5×SD16—62no—
107C5×SD16—122no—
108C5×M16—64no—
109C2×C5×C8—31no—
110C2×C5×C8—32no—
111C2×C5×C8—31no—
112C5×C4⋊C4—31no—
113C5×C4⋊C4—31no—
114C5×C4⋊C4—31no—
115C5×C4⋊C4—61no—
116C5×C4⋊C4—61no—
117C5×C22⋊C4—32no—
118C5×C22⋊C4—62no—
119C42×C5—31no—
120C42×C5—61no—
121C2×#137—32no—
122#550—31no—
123D4×Q8C5×D311yescharacteristic, complemented
124#492—31no—
125#468—32no—
126#462—31no—
127#460—31no—
128#452—32no—
129#448—32no—
130C4×Q16—31no—
131#436—31no—
132C22×C3×C5—41no—
133C22×C3×C5—82no—
134C22×C3×C5—82no—
135C22×C3×C5—82no—
136C2×C5×D3—82no—
137C2×C5×D3—82no—
138C2×C5×D3—82no—
139C3×C4×C5—42no—
140C5×Dic3—42no—
141C5×Dic3—42no—
142C2×C3×D4—41no—
143C2×#67—42no—
144C2×#67—42no—
145C2×D12—41no—
146C2×SL(2,𝔽3)—22no—
147C2×SL(2,𝔽3)C23×C511yescharacteristic
148GL(2,𝔽3)—22no—
149#226C5×D412yescomplemented
150C4×Dic3—41no—
151C23×C5—62no—
152C23×C5C2×S412yes—
153C5×Q8—31no—
154C5×Q8—32no—
155C5×Q8—62no—
156C5×Q8—62no—
157C5×Q8—61no—
158C5×Q8D3×D411yescharacteristic
159C5×D4—22no—
160C5×D4—62no—
161C5×D4—62no—
162C5×D4—124no—
163C5×D4—122no—
164C5×D4—241no—
165C5×D4—242no—
166C2×C4×C5—31no—
167C2×C4×C5—31no—
168C2×C4×C5—32no—
169C2×C4×C5—62no—
170C2×C4×C5—62no—
171C2×C4×C5—61no—
172C2×C4×C5—61no—
173C2×C4×C5—124no—
174C2×C4×C5C2×S411yescharacteristic
175C5×C8—32no—
176C5×C8—62no—
177C5×C8—61no—
178C2×#41—32no—
179C22×Q8C2×C5×D312yes—
180#137—64no—
181#137—64no—
182C2×Q16—31no—
183C2×Q16—32no—
184C2×SD16—32no—
185C2×SD16—32no—
186C2×SD16—32no—
187C2×M16—32no—
188C4⋊Q8—31no—
189#127—31no—
190#127—61no—
191C22⋊Q8—32no—
192#121—32no—
193#121—62no—
194C4×Q8—31no—
195C4×Q8C2×C5×D311yescharacteristic
196C4×D4—31no—
197C4×D4—61no—
198#107—31no—
199C4⋊C8—31no—
200#103—31no—
201#103—32no—
202#103—31no—
203#102—32no—
204C22⋊C8—32no—
205C4×C8—31no—
206C2×C3×C5—41no—
207C2×C3×C5—41no—
208C2×C3×C5—41no—
209C2×C3×C5—162no—
210C5×D3—162no—
211C23×C3—42no—
212C22×D3—42no—
213C3×D4—82no—
214C2×C3×C4—41no—
215#67—84no—
216#67—84no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219D12—82no—
220SL(2,𝔽3)C2×C5×D411yescharacteristic
221C22×C5—22no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—122no—
225C22×C5—122no—
226C22×C5—122no—
227C22×C5C22×S411yescharacteristic
228C4×C5—31no—
229C4×C5—31no—
230C4×C5—62no—
231C4×C5—62no—
232C4×C5—61no—
233C4×C5—121no—
234C4×C5#98312yescomplemented
235#41—64no—
236#41—122no—
237C2×Q8—22no—
238C2×Q8—32no—
239C2×Q8—31no—
240C2×Q8—62no—
241C2×Q8C22×C5×D311yescharacteristic
242C2×D4—32no—
243C2×D4—61no—
244C2×D4—62no—
245C2×D4—62no—
246C2×D4—61no—
247C2×D4C5×S411yescharacteristic
248C22×C4—32no—
249C22×C4—62no—
250Q16—32no—
251Q16—32no—
252Q16—64no—
253SD16—64no—
254SD16—62no—
255SD16—62no—
256SD16—122no—
257M16—64no—
258C2×C8—31no—
259C2×C8—31no—
260C2×C8—32no—
261C4⋊C4—31no—
262C4⋊C4—31no—
263C4⋊C4—31no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C22⋊C4—32no—
267C22⋊C4—62no—
268C42—31no—
269C42—61no—
270C3×C5—41noHall
271C22×C3—41no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—82no—
275C2×D3—82no—
276C2×D3—82no—
277C2×D3—82no—
278C3×C4—42no—
279Dic3—42no—
280Dic3—42no—
281C2×C5—42no—
282C2×C5—242no—
283C2×C5D4×S411yescharacteristic
284C2×C5C2×#98311yescharacteristic
285C2×C5#592611yescharacteristic
286C23—62no—
287C23C2×C5×S412yes—
288Q8—31no—
289Q8—32no—
290Q8—62no—
291Q8—62no—
292Q8—61no—
293Q8C5×D3×D411yescharacteristic
294D4—22no—
295D4—62no—
296D4—62no—
297D4—124no—
298D4—122no—
299D4—241no—
300D4—242no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—124no—
309C2×C4C2×C5×S411yescharacteristic
310C8—32no—
311C8—62no—
312C8—61no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—162no—
317D3—162no—
318C5#8539211yescharacteristic, direct product factor, Sylow
319C22—22no—
320C22—22no—
321C22—22no—
322C22—122no—
323C22—122no—
324C22—122no—
325C22C22×C5×S411yescharacteristic
326C4—31no—
327C4—31no—
328C4—62no—
329C4—62no—
330C4—61no—
331C4—121no—
332C4C5×#98312yescomplemented
333C3—41noSylow
334C2—42no—
335C2—242no—
336C2C5×D4×S411yescharacteristic
337C2C2×C5×#98311yescharacteristic
338C2C5×#592611yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×D4×Q8D3
hypercenterC2×C5×D4S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#983C2semidirect
2. C5×D4×SL(2,𝔽3)C2semidirect
3. C5×#5421C2semidirect
4. C5×C4⋊GL(2,𝔽3)C2semidirect
5. C4×C5×#226C2semidirect
14. #85392C5direct (N × H)
17. C5×D4×Q8D3semidirect
33. C5×#226D4semidirect
35. C2×#983C2×C5semidirect
36. D4×SL(2,𝔽3)C2×C5semidirect
37. #5421C2×C5semidirect
38. C4⋊GL(2,𝔽3)C2×C5semidirect
39. C4×#226C2×C5semidirect
123. D4×Q8C5×D3semidirect
149. #226C5×D4semidirect
234. C4×C5#983semidirect
318. C5#85392direct (N × H)
332. C4C5×#983semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9832
C5×D4×SL(2,𝔽3)2
C5×#54212
C5×C4⋊GL(2,𝔽3)2
C4×C5×#2262
C5×#32783
C5×#9404
#853925