Finite Group Databasebeta

← C5×#68135 #14288392

Subgroups — C5×#68135 #14288392

Order 1920.

508 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 106 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×#5510C211yesmaximal, characteristic
2C5×#4981C211yesmaximal, characteristic, complemented
3C5×#4736C212yesmaximal, complemented
4C3×C5×#324C211yesmaximal, characteristic, complemented
5C5×#4470C212yesmaximal, complemented
6C5×#1610—31nomaximal, Hall
7C2×C5×#235—21no—
8C2×C3×C5×C4⋊C4C2211yescharacteristic
9C5×#930—21no—
10C5×#930C2212yes—
11C2×C5×C4⋊Dic3C2211yescharacteristic
12C2×C5×Dic3⋊C4—21no—
13C5×#897—22no—
14C5×#895—22no—
15C3×C5×C22⋊C8C2212yes—
16C5×C3⋊#95—21no—
17C5×C3⋊#95C2211yescharacteristic
18C5×#821—22no—
19C5×#818—22no—
20#68135C511yesmaximal, characteristic, direct product factor, Hall
21C5×#537—31no—
22C5×#480—32no—
23C5×#398—31no—
24C5×#327—32no—
25C5×#324D311yescharacteristic, complemented
26C22×C3×C4×C5D411yescharacteristic
27C22×C3×C4×C5D411yescharacteristic
28C22×C3×C4×C5C2311yescharacteristic
29C2×C5×#67—22no—
30C22×C5×Dic3—21no—
31C22×C5×Dic3—21no—
32C5×#235—42no—
33C5×#235—42no—
34C2×C5×D12—21no—
35C2×C5×D12—21no—
36C2×C5×D12D412yes—
37C2×C4×C5×D3—22no—
38C2×C5×Dic6—21no—
39C2×C3×C5×C8—22no—
40C3×C5×C4⋊C4—22no—
41C5×(C2×D3)⋊C4—22no—
42C5×(C2×D3)⋊C4—22no—
43C5×C4⋊Dic3—21no—
44C5×C4⋊Dic3—22no—
45C5×C4⋊Dic3D412yes—
46C5×Dic3⋊C4—42no—
47#5510C2×C511yescharacteristic
48#4981C2×C511yescharacteristic, complemented
49#4736C2×C512yescomplemented
50C3×#324C2×C511yescharacteristic, complemented
51#4470C2×C512yescomplemented
52C2×C5×#41—61no—
53C5×C22⋊Q8—62no—
54C5×#121—32no—
55C5×#121—62no—
56C5×#121—61no—
57C2×C5×C4⋊C4—31no—
58C2×C5×C4⋊C4—61no—
59C2×C5×C4⋊C4C2×D311yescharacteristic
60C5×#107—62no—
61C5×#102—62no—
62C5×C22⋊C8C2×D312yes—
63C5×#95—31no—
64C5×#95—61no—
65#1610—31noSylow
66C23×C3×C5C2×D411yescharacteristic
67C22×C5×D3—22no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5—21no—
70C2×C3×C4×C5—21no—
71C2×C3×C4×C5—41no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5D812yes—
74C2×C3×C4×C5C2×D412yes—
75C5×#67—42no—
76C5×#67—42no—
77C2×C5×Dic3—22no—
78C2×C5×Dic3—22no—
79C2×C5×Dic3—41no—
80C2×C5×Dic3—41no—
81C2×C5×Dic3—41no—
82C2×C5×Dic3—41no—
83C2×C5×Dic3—41no—
84C5×D12—42no—
85C5×D12—42no—
86C5×D12—81no—
87C4×C5×D3—44no—
88C5×Dic6—42no—
89C3×C5×C8—42no—
90C2×#235—21no—
91C2×C3×C4⋊C4C22×C511yescharacteristic
92#930—21no—
93#930C22×C512yes—
94C2×C4⋊Dic3C22×C511yescharacteristic
95C2×Dic3⋊C4—21no—
96#897—22no—
97#895—22no—
98C3×C22⋊C8C22×C512yes—
99C3⋊#95—21no—
100C3⋊#95C22×C511yescharacteristic
101#821—22no—
102#818—22no—
103C5×#41—122no—
104C5×#41—122no—
105C2×C5×Q8—61no—
106C2×C5×D4—32no—
107C2×C5×D4—61no—
108C2×C5×D4—61no—
109C2×C5×D4—62no—
110C22×C4×C5—62no—
111C22×C4×C5—61no—
112C22×C4×C5—61no—
113C22×C4×C5D1211yescharacteristic
114C22×C4×C5D1211yescharacteristic
115C22×C4×C5C22×D311yescharacteristic
116C2×C5×C8—22no—
117C5×C4⋊C4—22no—
118C5×C4⋊C4—32no—
119C5×C4⋊C4—61no—
120C5×C4⋊C4—62no—
121C5×C4⋊C4—122no—
122C5×C22⋊C4—62no—
123C5×C22⋊C4—62no—
124#537—31no—
125#480—32no—
126#398—31no—
127#327—32no—
128#324C5×D311yescharacteristic, complemented
129C22×C3×C5—21no—
130C22×C3×C5—21no—
131C22×C3×C5#12011yescharacteristic
132C22×C3×C5C2×D811yescharacteristic
133C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
134C2×C5×D3—42no—
135C2×C5×D3—42no—
136C2×C5×D3—42no—
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—41no—
140C5×Dic3—42no—
141C5×Dic3—42no—
142C5×Dic3—81no—
143C22×C3×C4C5×D411yescharacteristic
144C22×C3×C4C5×D411yescharacteristic
145C22×C3×C4C23×C511yescharacteristic
146C2×#67—22no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149#235—42no—
150#235—42no—
151C2×D12—21no—
152C2×D12—21no—
153C2×D12C5×D412yes—
154C2×C4×D3—22no—
155C2×Dic6—21no—
156C2×C3×C8—22no—
157C3×C4⋊C4—22no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—22no—
160C4⋊Dic3—21no—
161C4⋊Dic3—22no—
162C4⋊Dic3C5×D412yes—
163Dic3⋊C4—42no—
164C23×C5—62no—
165C23×C5C2×D1211yescharacteristic
166C5×Q8—122no—
167C5×D4—122no—
168C5×D4—122no—
169C5×D4—122no—
170C5×D4—122no—
171C5×D4—241no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5—62no—
178C2×C4×C5—62no—
179C2×C4×C5—124no—
180C2×C4×C5—121no—
181C2×C4×C5—121no—
182C2×C4×C5—121no—
183C2×C4×C5—121no—
184C2×C4×C5—121no—
185C2×C4×C5D2412yes—
186C2×C4×C5D3×D412yes—
187C5×C8—42no—
188C2×#41—61no—
189C22⋊Q8—62no—
190#121—32no—
191#121—62no—
192#121—61no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4C2×C5×D311yescharacteristic
196#107—62no—
197#102—62no—
198C22⋊C8C2×C5×D312yes—
199#95—31no—
200#95—61no—
201C2×C3×C5—21no—
202C2×C3×C5—21no—
203C2×C3×C5#44711yescharacteristic
204C2×C3×C5#45611yescharacteristic
205C2×C3×C5#45811yescharacteristic
206C5×D3—82no—
207C23×C3C2×C5×D411yescharacteristic
208C22×D3—22no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4C5×D812yes—
215C2×C3×C4C2×C5×D412yes—
216#67—42no—
217#67—42no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225D12—42no—
226D12—42no—
227D12—81no—
228C4×D3—44no—
229Dic6—42no—
230C3×C8—42no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—122no—
234C22×C5—122no—
235C22×C5—122no—
236C22×C5#88211yescharacteristic
237C22×C5C2×D2411yescharacteristic
238C22×C5#90811yescharacteristic
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—41no—
242C4×C5—122no—
243C4×C5—122no—
244C4×C5—241no—
245#41—122no—
246#41—122no—
247C2×Q8—61no—
248C2×D4—32no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—62no—
252C22×C4—62no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4C5×D1211yescharacteristic
256C22×C4C5×D1211yescharacteristic
257C22×C4C22×C5×D311yescharacteristic
258C2×C8—22no—
259C4⋊C4—22no—
260C4⋊C4—32no—
261C4⋊C4—61no—
262C4⋊C4—62no—
263C4⋊C4—122no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C3×C5#161011yescharacteristic, complemented, Hall
267C22×C3—21no—
268C22×C3—21no—
269C22×C3C5×#12011yescharacteristic
270C22×C3C2×C5×D811yescharacteristic
271C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
272C2×D3—42no—
273C2×D3—42no—
274C2×D3—42no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—41no—
278Dic3—42no—
279Dic3—42no—
280Dic3—81no—
281C2×C5—21no—
282C2×C5—21no—
283C2×C5—242no—
284C2×C5#473211yescharacteristic
285C2×C5#474211yescharacteristic
286C2×C5#475211yescharacteristic
287C23—62no—
288C23C2×C5×D1211yescharacteristic
289Q8—122no—
290D4—122no—
291D4—122no—
292D4—122no—
293D4—122no—
294D4—241no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—124no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4C5×D2412yes—
309C2×C4C5×D3×D412yes—
310C8—42no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3C5×#44711yescharacteristic
314C2×C3C5×#45611yescharacteristic
315C2×C3C5×#45811yescharacteristic
316D3—82no—
317C5#6813511yescharacteristic, direct product factor, Sylow
318C22—21no—
319C22—21no—
320C22—122no—
321C22—122no—
322C22—122no—
323C22C5×#88211yescharacteristic
324C22C2×C5×D2411yescharacteristic
325C22C5×#90811yescharacteristic
326C4—22no—
327C4—22no—
328C4—41no—
329C4—122no—
330C4—122no—
331C4—241no—
332C3C5×#161011yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—242no—
336C2C5×#473211yescharacteristic
337C2C5×#474211yescharacteristic
338C2C5×#475211yescharacteristic
subgroupG/H
centerC22×C5#882
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#324C2
socleC22×C3×C5#120
prime radicalC5×#5510C2
hypercenterC5×#324D3
hypocenterC3C5×#1610
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#324

Complements

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

NH ≅ G/Ntype
2. C5×#4981C2semidirect
3. C5×#4736C2semidirect
4. C3×C5×#324C2semidirect
5. C5×#4470C2semidirect
20. #68135C5direct (N × H)
25. C5×#324D3semidirect
48. #4981C2×C5semidirect
49. #4736C2×C5semidirect
50. C3×#324C2×C5semidirect
51. #4470C2×C5semidirect
128. #324C5×D3semidirect
266. C3×C5#1610semidirect
317. C5#68135direct (N × H)
332. C3C5×#1610semidirect

Maximal subgroups

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

subgroupindex
C5×#55102
C5×#49812
C5×#47362
C3×C5×#3242
C5×#44702
C5×#16103
#681355