Finite Group Databasebeta

← C5×#85331 #14511069

Subgroups — C5×#85331 #14511069

Order 1920.

556 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 80 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
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C5×#5395C212yesmaximal, complemented
4C5×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C5×C4⋊#226C211yesmaximal, characteristic, complemented
6C5×#3037—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C5×#983—24no—
9C2×C5×GL(2,𝔽3)C2212yes—
10C2×C5×#226C2212yes—
11C5×#930—41nomaximal
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14#85331C511yesmaximal, characteristic, direct product factor, Hall
15C2×C5×#137—32no—
16C5×#532—31no—
17C2×C4×C5×Q8D311yescharacteristic, complemented
18C5×#464—32no—
19C5×#464—32no—
20C5×#462—31no—
21C5×#462—31no—
22C5×#460—31no—
23C5×#460—31no—
24C5×#423—31no—
25C5×#417—32no—
26C22×C3×C4×C5—41no—
27C2×C5×#67—42no—
28C2×C5×D12—41no—
29C2×C5×SL(2,𝔽3)D412yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31C5×GL(2,𝔽3)—22no—
32C5×#226—22no—
33C5×(C2×D3)⋊C4—42no—
34C5×C4⋊Dic3—41no—
35C2×#983C2×C512yescomplemented
36C2×C4×SL(2,𝔽3)C2×C511yescharacteristic, complemented
37#5395C2×C512yescomplemented
38C4⋊GL(2,𝔽3)C2×C511yescharacteristic, complemented
39C4⋊#226C2×C511yescharacteristic, complemented
40C2×C5×#41—32no—
41C22×C5×Q8C2×D311yescharacteristic
42C5×#137—64no—
43C5×#137—64no—
44C2×C5×Q16—32no—
45C2×C5×Q16—32no—
46C2×C5×SD16—32no—
47C2×C5×SD16—32no—
48C2×C5×M16—32no—
49C5×C4⋊Q8—31no—
50C5×#127—31no—
51C5×#124—32no—
52C5×#121—61no—
53C5×#121—61no—
54C4×C5×Q8—31no—
55C4×C5×Q8—31no—
56C4×C5×Q8C2×D311yescharacteristic
57C4×C5×Q8C2×D311yescharacteristic
58C4×C5×D4—61no—
59C4×C5×D4—61no—
60C2×C5×C4⋊C4—31no—
61C2×C42×C5—31no—
62C5×C4⋊C8—31no—
63C5×C4⋊C8—31no—
64C5×C4⋊C8—32no—
65C5×#103—32no—
66C5×#103—32no—
67C5×#102—32no—
68C5×#102—32no—
69#3037—31noSylow
70C23×C3×C5—41no—
71C22×C5×D3—42no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—82no—
75C5×#67—84no—
76C2×C5×Dic3—42no—
77C5×D12—82no—
78C5×SL(2,𝔽3)C2×D411yescharacteristic
79C22×SL(2,𝔽3)C22×C511yescharacteristic
80#983—24no—
81C2×GL(2,𝔽3)C22×C512yes—
82C2×#226C22×C512yes—
83#930—41no—
84C4×SL(2,𝔽3)C22×C511yescharacteristic
85C4×SL(2,𝔽3)C22×C511yescharacteristic
86C5×#41—64no—
87C5×#41—64no—
88C2×C5×Q8—31no—
89C2×C5×Q8—32no—
90C2×C5×Q8—32no—
91C2×C5×Q8D1212yescomplemented
92C2×C5×Q8C22×D311yescharacteristic
93C2×C5×D4—32no—
94C2×C5×D4—61no—
95C2×C5×D4—62no—
96C2×C5×D4—61no—
97C22×C4×C5—31no—
98C22×C4×C5—31no—
99C22×C4×C5—62no—
100C22×C4×C5S411yescharacteristic
101C5×Q16—64no—
102C5×Q16—62no—
103C5×Q16—62no—
104C5×SD16—64no—
105C5×SD16—62no—
106C5×SD16—62no—
107C5×M16—64no—
108C2×C5×C8—32no—
109C2×C5×C8—32no—
110C5×C4⋊C4—31no—
111C5×C4⋊C4—31no—
112C5×C4⋊C4—61no—
113C5×C4⋊C4—61no—
114C5×C4⋊C4—61no—
115C5×C22⋊C4—62no—
116C5×C22⋊C4—62no—
117C42×C5—31no—
118C42×C5—31no—
119C42×C5—31no—
120C42×C5—31no—
121C2×#137—32no—
122#532—31no—
123C2×C4×Q8C5×D311yescharacteristic, complemented
124#464—32no—
125#464—32no—
126#462—31no—
127#462—31no—
128#460—31no—
129#460—31no—
130#423—31no—
131#417—32no—
132C22×C3×C5—42no—
133C22×C3×C5—41no—
134C22×C3×C5—81no—
135C22×C3×C5—81no—
136C2×C5×D3—82no—
137C2×C5×D3—82no—
138C2×C5×D3—82no—
139C3×C4×C5—42no—
140C3×C4×C5—81no—
141C5×Dic3—82no—
142C22×C3×C4—41no—
143C2×#67—42no—
144C2×D12—41no—
145C2×SL(2,𝔽3)C5×D412yescomplemented
146C2×SL(2,𝔽3)C23×C511yescharacteristic
147GL(2,𝔽3)—22no—
148#226—22no—
149(C2×D3)⋊C4—42no—
150C4⋊Dic3—41no—
151C23×C5—62no—
152C23×C5C2×S411yescharacteristic
153C5×Q8—32no—
154C5×Q8—31no—
155C5×Q8—61no—
156C5×Q8—62no—
157C5×Q8—62no—
158C5×Q8C2×D1211yescharacteristic
159C5×D4—62no—
160C5×D4—62no—
161C5×D4—122no—
162C5×D4—124no—
163C5×D4—122no—
164C2×C4×C5—22no—
165C2×C4×C5—32no—
166C2×C4×C5—32no—
167C2×C4×C5—31no—
168C2×C4×C5—31no—
169C2×C4×C5—31no—
170C2×C4×C5—31no—
171C2×C4×C5—62no—
172C2×C4×C5—124no—
173C2×C4×C5—121no—
174C2×C4×C5C2×S411yescharacteristic
175C2×C4×C5C2×S411yescharacteristic
176C5×C8—62no—
177C5×C8—62no—
178C2×#41—32no—
179C22×Q8C2×C5×D311yescharacteristic
180#137—64no—
181#137—64no—
182C2×Q16—32no—
183C2×Q16—32no—
184C2×SD16—32no—
185C2×SD16—32no—
186C2×M16—32no—
187C4⋊Q8—31no—
188#127—31no—
189#124—32no—
190#121—61no—
191#121—61no—
192C4×Q8—31no—
193C4×Q8—31no—
194C4×Q8C2×C5×D311yescharacteristic
195C4×Q8C2×C5×D311yescharacteristic
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—31no—
199C2×C42—31no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202C4⋊C8—32no—
203#103—32no—
204#103—32no—
205#102—32no—
206#102—32no—
207C2×C3×C5—41no—
208C2×C3×C5—41no—
209C2×C3×C5—41no—
210C2×C3×C5—82no—
211C5×D3—162no—
212C23×C3—41no—
213C22×D3—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—82no—
217#67—84no—
218C2×Dic3—42no—
219D12—82no—
220SL(2,𝔽3)C2×C5×D411yescharacteristic
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—122no—
224C22×C5—122no—
225C22×C5—122no—
226C22×C5C4⋊S412yes—
227C22×C5C22×S411yescharacteristic
228C4×C5—21no—
229C4×C5—31no—
230C4×C5—32no—
231C4×C5—31no—
232C4×C5—61no—
233C4×C5—61no—
234C4×C5—122no—
235C4×C5#98312yescomplemented
236#41—64no—
237#41—64no—
238C2×Q8—31no—
239C2×Q8—32no—
240C2×Q8—32no—
241C2×Q8C5×D1212yescomplemented
242C2×Q8C22×C5×D311yescharacteristic
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—62no—
246C2×D4—61no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—62no—
250C22×C4C5×S411yescharacteristic
251Q16—64no—
252Q16—62no—
253Q16—62no—
254SD16—64no—
255SD16—62no—
256SD16—62no—
257M16—64no—
258C2×C8—32no—
259C2×C8—32no—
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C42—31no—
268C42—31no—
269C42—31no—
270C42—31no—
271C3×C5—41noHall
272C22×C3—42no—
273C22×C3—41no—
274C22×C3—81no—
275C22×C3—81no—
276C2×D3—82no—
277C2×D3—82no—
278C2×D3—82no—
279C3×C4—42no—
280C3×C4—81no—
281Dic3—82no—
282C2×C5—22no—
283C2×C5—242no—
284C2×C5C2×C4⋊S411yescharacteristic
285C2×C5C2×#98311yescharacteristic
286C2×C5#592311yescharacteristic
287C23—62no—
288C23C2×C5×S411yescharacteristic
289Q8—32no—
290Q8—31no—
291Q8—61no—
292Q8—62no—
293Q8—62no—
294Q8C2×C5×D1211yescharacteristic
295D4—62no—
296D4—62no—
297D4—122no—
298D4—124no—
299D4—122no—
300C2×C4—22no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—62no—
308C2×C4—124no—
309C2×C4—121no—
310C2×C4C2×C5×S411yescharacteristic
311C2×C4C2×C5×S411yescharacteristic
312C8—62no—
313C8—62no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—82no—
318D3—162no—
319C5#8533111yescharacteristic, direct product factor, Sylow
320C22—21no—
321C22—21no—
322C22—122no—
323C22—122no—
324C22—122no—
325C22C5×C4⋊S412yes—
326C22C22×C5×S411yescharacteristic
327C4—21no—
328C4—31no—
329C4—32no—
330C4—31no—
331C4—61no—
332C4—61no—
333C4—122no—
334C4C5×#98312yescomplemented
335C3—41noSylow
336C2—22no—
337C2—242no—
338C2C2×C5×C4⋊S411yescharacteristic
339C2C2×C5×#98311yescharacteristic
340C2C5×#592311yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
hypercenterC22×C4×C5S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×#983C2semidirect
2. C2×C4×C5×SL(2,𝔽3)C2semidirect
3. C5×#5395C2semidirect
4. C5×C4⋊GL(2,𝔽3)C2semidirect
5. C5×C4⋊#226C2semidirect
14. #85331C5direct (N × H)
17. C2×C4×C5×Q8D3semidirect
29. C2×C5×SL(2,𝔽3)D4semidirect
35. C2×#983C2×C5semidirect
36. C2×C4×SL(2,𝔽3)C2×C5semidirect
37. #5395C2×C5semidirect
38. C4⋊GL(2,𝔽3)C2×C5semidirect
39. C4⋊#226C2×C5semidirect
91. C2×C5×Q8D12semidirect
123. C2×C4×Q8C5×D3semidirect
145. C2×SL(2,𝔽3)C5×D4semidirect
235. C4×C5#983semidirect
241. C2×Q8C5×D12semidirect
319. C5#85331direct (N × H)
334. 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
C2×C4×C5×SL(2,𝔽3)2
C5×#53952
C5×C4⋊GL(2,𝔽3)2
C5×C4⋊#2262
C5×#30373
C5×#9304
#853315