Finite Group Databasebeta

← C5×#70994 #14316971

Subgroups — C5×#70994 #14316971

Order 1920.

472 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 82 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×#5734C211yesmaximal, characteristic, complemented
2C3×C5×#456C211yesmaximal, characteristic, complemented
3C5×#5245C212yesmaximal, complemented
4C5×#4548C211yesmaximal, characteristic, complemented
5C5×#4477C212yesmaximal, complemented
6C5×#2204—31nomaximal, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×C4⋊Q8C2211yescharacteristic
10C5×#949—22no—
11C5×#947—21no—
12C5×#947C2212yes—
13C5×#895—22no—
14C5×#892—22no—
15C4×C5×D12—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic
18C5×#837—22no—
19C5×#836C2212yes—
20#70994C511yesmaximal, characteristic, direct product factor, Hall
21C5×#555—31no—
22C5×#456D311yescharacteristic, complemented
23C5×#454—32no—
24C5×#356—31no—
25C5×#355—32no—
26C3×C5×#41—22no—
27C3×C5×#41—21no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8—41no—
30C2×C3×C5×Q8D411yescharacteristic
31C2×C3×C5×Q8C2311yescharacteristic
32C2×C5×D12—21no—
33C2×C5×D12—21no—
34C2×C5×D12D412yes—
35C2×C4×C5×D3—22no—
36C3×C5×Q16—41no—
37C3×C5×SD16—41no—
38C3×C5×M16—21no—
39C3×C5×C4⋊C4—22no—
40C3×C42×C5D411yescharacteristic
41C5×#215—42no—
42C5×#213—42no—
43C5×(C2×D3)⋊C4—22no—
44C5×(C2×D3)⋊C4—22no—
45C5×(C2×D3)⋊C4—22no—
46C5×C4⋊Dic3—21no—
47C4×C5×Dic3—21no—
48C4×C5×Dic3D412yes—
49C5×#208—22no—
50#5734C2×C511yescharacteristic, complemented
51C3×#456C2×C511yescharacteristic, complemented
52#5245C2×C512yescomplemented
53#4548C2×C511yescharacteristic, complemented
54#4477C2×C512yescomplemented
55C5×2−1+4C2×D311yescharacteristic
56C5×#137—21no—
57C5×AGL(1,ℤ/8)—62no—
58C5×C4⋊Q8C2×D311yescharacteristic
59C5×#124—32no—
60C5×#124—61no—
61C5×#121—62no—
62C4×C5×D4—61no—
63C5×#117—62no—
64C5×#104—21no—
65C5×#104—62no—
66C5×#101—32no—
67C5×#101C2×D311yescharacteristic
68#2204—31noSylow
69C22×C5×D3—22no—
70C3×C5×Q8—22no—
71C3×C5×Q8—21no—
72C3×C5×Q8—41no—
73C3×C5×Q8—41no—
74C3×C5×D4—22no—
75C3×C5×D4—21no—
76C3×C5×D4—41no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—42no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5C2×D412yes—
82C2×C3×C4×C5C2×D411yescharacteristic
83C2×C5×Dic3—22no—
84C5×D12—42no—
85C5×D12—42no—
86C5×D12—81no—
87C4×C5×D3—44no—
88C3×C5×C8—41no—
89C5×C3⋊C8—42no—
90C3×2−1+4C22×C511yescharacteristic
91C3×#137—21no—
92C3×C4⋊Q8C22×C511yescharacteristic
93#949—22no—
94#947—21no—
95#947C22×C512yes—
96#895—22no—
97#892—22no—
98C4×D12—21no—
99C3×#104—21no—
100C3×#101C22×C511yescharacteristic
101#837—22no—
102#836C22×C512yes—
103C5×#41—22no—
104C5×#41—21no—
105C5×#41—41no—
106C2×C5×Q8—41no—
107C2×C5×Q8#6711yescharacteristic
108C2×C5×Q8C22×D311yescharacteristic
109C2×C5×D4—32no—
110C2×C5×D4—61no—
111C2×C5×D4—61no—
112C22×C4×C5—62no—
113C5×Q16—41no—
114C5×SD16—41no—
115C5×SD16—122no—
116C5×D8—122no—
117C5×M16—21no—
118C5×M16—62no—
119C5×C4⋊C4—22no—
120C5×C4⋊C4—61no—
121C5×C22⋊C4—62no—
122C5×C22⋊C4—62no—
123C5×C22⋊C4—62no—
124C42×C5—32no—
125C42×C5—61no—
126C42×C5#6711yescharacteristic
127#555—31no—
128#456C5×D311yescharacteristic, complemented
129#454—32no—
130#356—31no—
131#355—32no—
132C22×C3×C5—41no—
133C22×C3×C5#12011yescharacteristic
134C2×C5×D3—42no—
135C2×C5×D3—42no—
136C2×C5×D3—42no—
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—21no—
140C3×C4×C5—41no—
141C3×C4×C5—41no—
142C5×Dic3—42no—
143C3×#41—22no—
144C3×#41—21no—
145C3×#41—41no—
146C2×C3×Q8—41no—
147C2×C3×Q8C5×D411yescharacteristic
148C2×C3×Q8C23×C511yescharacteristic
149C2×D12—21no—
150C2×D12—21no—
151C2×D12C5×D412yes—
152C2×C4×D3—22no—
153C3×Q16—41no—
154C3×SD16—41no—
155C3×M16—21no—
156C3×C4⋊C4—22no—
157C3×C42C5×D411yescharacteristic
158#215—42no—
159#213—42no—
160(C2×D3)⋊C4—22no—
161(C2×D3)⋊C4—22no—
162(C2×D3)⋊C4—22no—
163C4⋊Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3C5×D412yes—
166#208—22no—
167C23×C5—62no—
168C5×Q8—22no—
169C5×Q8—21no—
170C5×Q8—41no—
171C5×Q8—41no—
172C5×D4—22no—
173C5×D4—21no—
174C5×D4—41no—
175C5×D4—122no—
176C5×D4—122no—
177C5×D4—241no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—41no—
181C2×C4×C5—42no—
182C2×C4×C5—62no—
183C2×C4×C5—124no—
184C2×C4×C5D3×D412yes—
185C2×C4×C5C2×#6711yescharacteristic
186C5×C8—41no—
187C5×C8—122no—
1882−1+4C2×C5×D311yescharacteristic
189#137—21no—
190AGL(1,ℤ/8)—62no—
191C4⋊Q8C2×C5×D311yescharacteristic
192#124—32no—
193#124—61no—
194#121—62no—
195C4×D4—61no—
196#117—62no—
197#104—21no—
198#104—62no—
199#101—32no—
200#101C2×C5×D311yescharacteristic
201C2×C3×C5—21no—
202C2×C3×C5—81no—
203C2×C3×C5B(4,𝔽2)11yescharacteristic
204C5×D3—82no—
205C22×D3—22no—
206C3×Q8—22no—
207C3×Q8—21no—
208C3×Q8—41no—
209C3×Q8—41no—
210C3×D4—22no—
211C3×D4—21no—
212C3×D4—41no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4C2×C5×D412yes—
218C2×C3×C4C2×C5×D411yescharacteristic
219C2×Dic3—22no—
220D12—42no—
221D12—42no—
222D12—81no—
223C4×D3—44no—
224C3×C8—41no—
225C3⋊C8—42no—
226C22×C5—41no—
227C22×C5—122no—
228C22×C5—122no—
229C22×C5—122no—
230C22×C5#93711yescharacteristic
231C4×C5—22no—
232C4×C5—21no—
233C4×C5—22no—
234C4×C5—41no—
235C4×C5—41no—
236C4×C5—122no—
237#41—22no—
238#41—21no—
239#41—41no—
240C2×Q8—41no—
241C2×Q8C5×#6711yescharacteristic
242C2×Q8C22×C5×D311yescharacteristic
243C2×D4—32no—
244C2×D4—61no—
245C2×D4—61no—
246C22×C4—62no—
247Q16—41no—
248SD16—41no—
249SD16—122no—
250D8—122no—
251M16—21no—
252M16—62no—
253C4⋊C4—22no—
254C4⋊C4—61no—
255C22⋊C4—62no—
256C22⋊C4—62no—
257C22⋊C4—62no—
258C42—32no—
259C42—61no—
260C42C5×#6711yescharacteristic
261C3×C5#220411yescharacteristic, complemented, Hall
262C22×C3—41no—
263C22×C3C5×#12011yescharacteristic
264C2×D3—42no—
265C2×D3—42no—
266C2×D3—42no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—41no—
272Dic3—42no—
273C2×C5—21no—
274C2×C5—81no—
275C2×C5—242no—
276C2×C5#503211yescharacteristic
277C23—62no—
278Q8—22no—
279Q8—21no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—21no—
284D4—41no—
285D4—122no—
286D4—122no—
287D4—241no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—62no—
293C2×C4—124no—
294C2×C4C5×D3×D412yes—
295C2×C4C2×C5×#6711yescharacteristic
296C8—41no—
297C8—122no—
298C2×C3—21no—
299C2×C3—81no—
300C2×C3C5×B(4,𝔽2)11yescharacteristic
301D3—82no—
302C5#7099411yescharacteristic, direct product factor, Sylow
303C22—41no—
304C22—122no—
305C22—122no—
306C22—122no—
307C22C5×#93711yescharacteristic
308C4—22no—
309C4—21no—
310C4—22no—
311C4—41no—
312C4—41no—
313C4—122no—
314C3C5×#220411yescharacteristic, complemented, Sylow
315C2—21no—
316C2—81no—
317C2—242no—
318C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×Q8C23×C5
FrattiniC2×Q8C22×C5×D3
FittingC3×C5×#456C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×#456D3
hypocenterC3C5×#2204
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×Q8 ◁ C5×#456

Complements

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

NH ≅ G/Ntype
1. C5×#5734C2semidirect
2. C3×C5×#456C2semidirect
3. C5×#5245C2semidirect
4. C5×#4548C2semidirect
5. C5×#4477C2semidirect
20. #70994C5direct (N × H)
22. C5×#456D3semidirect
50. #5734C2×C5semidirect
51. C3×#456C2×C5semidirect
52. #5245C2×C5semidirect
53. #4548C2×C5semidirect
54. #4477C2×C5semidirect
128. #456C5×D3semidirect
261. C3×C5#2204semidirect
302. C5#70994direct (N × H)
314. C3C5×#2204semidirect

Maximal subgroups

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

subgroupindex
C5×#57342
C3×C5×#4562
C5×#52452
C5×#45482
C5×#44772
C5×#22043
#709945