Finite Group Databasebeta

← C5×#70989 #14316932

Subgroups — C5×#70989 #14316932

Order 1920.

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

normal subgroups only

This group has 432 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×#5672C211yesmaximal, characteristic, complemented
2C3×C5×#454C211yesmaximal, characteristic
3C5×#5246C212yesmaximal, complemented
4C5×#4542C211yesmaximal, characteristic, complemented
5C5×#4478C212yesmaximal, complemented
6C5×#2203—31nomaximal, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×AGL(1,ℤ/8)—21no—
9C3×C5×#124C2211yescharacteristic
10C5×#951—22no—
11C5×#944C2212yes—
12C5×#935—21no—
13C5×#879—22no—
14C5×#878—22no—
15C5×#870—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic
18C5×#837—22no—
19C5×#836C2212yes—
20#70989C511yesmaximal, characteristic, direct product factor, Hall
21C5×#563—31no—
22C5×#456—32no—
23C5×#454D311yescharacteristic
24C5×#356—32no—
25C5×#355—31no—
26C3×C5×#41—22no—
27C3×C5×#41—21no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8—41no—
30C2×C3×C5×Q8C2311yescharacteristic
31C2×C3×C5×D4D411yescharacteristic
32C22×C5×Dic3—21no—
33C2×C5×Dic6D412yes—
34C3×C5×SD16—41no—
35C3×C5×D8—41no—
36C3×C5×M16—21no—
37C3×C5×C22⋊C4—22no—
38C3×C42×C5D411yescharacteristic
39C5×#217—22no—
40C5×#217—21no—
41C5×#216—42no—
42C5×#214—42no—
43C5×C4⋊Dic3—21no—
44C5×C4⋊Dic3—21no—
45C5×Dic3⋊C4—22no—
46C5×Dic3⋊C4—22no—
47C5×Dic3⋊C4—22no—
48C4×C5×Dic3D412yes—
49C5×#208—22no—
50#5672C2×C511yescharacteristic, complemented
51C3×#454C2×C511yescharacteristic
52#5246C2×C512yescomplemented
53#4542C2×C511yescharacteristic, complemented
54#4478C2×C512yescomplemented
55C5×2−1+4C2×D311yescharacteristic
56C5×#137—62no—
57C5×AGL(1,ℤ/8)—21no—
58C5×C4⋊Q8—32no—
59C5×#126—62no—
60C5×#125—61no—
61C5×#124C2×D311yescharacteristic
62C5×#123—61no—
63C5×C22⋊Q8—62no—
64C5×#104—21no—
65C5×#104—62no—
66C5×#101—32no—
67C5×#101C2×D311yescharacteristic
68#2203—31noSylow
69C23×C3×C5—21no—
70C3×C5×Q8—22no—
71C3×C5×Q8—21no—
72C3×C5×Q8—41no—
73C3×C5×D4—22no—
74C3×C5×D4—21no—
75C3×C5×D4—41no—
76C3×C5×D4—41no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—42no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5C2×D412yes—
81C2×C3×C4×C5C2×D411yescharacteristic
82C2×C5×Dic3—22no—
83C2×C5×Dic3—22no—
84C2×C5×Dic3—81no—
85C5×Dic6—42no—
86C3×C5×C8—41no—
87C5×C3⋊C8—42no—
88C3×2−1+4C22×C511yescharacteristic
89C3×AGL(1,ℤ/8)—21no—
90C3×#124C22×C511yescharacteristic
91#951—22no—
92#944C22×C512yes—
93#935—21no—
94#879—22no—
95#878—22no—
96#870—21no—
97C3×#104—21no—
98C3×#101C22×C511yescharacteristic
99#837—22no—
100#836C22×C512yes—
101C5×#41—22no—
102C5×#41—21no—
103C5×#41—41no—
104C2×C5×Q8—32no—
105C2×C5×Q8—41no—
106C2×C5×Q8C22×D311yescharacteristic
107C2×C5×D4#6711yescharacteristic
108C22×C4×C5—61no—
109C5×Q16—122no—
110C5×SD16—41no—
111C5×SD16—122no—
112C5×D8—41no—
113C5×M16—21no—
114C5×M16—62no—
115C5×C4⋊C4—61no—
116C5×C4⋊C4—62no—
117C5×C4⋊C4—61no—
118C5×C4⋊C4—62no—
119C5×C4⋊C4—62no—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—62no—
122C5×C22⋊C4—61no—
123C42×C5—32no—
124C42×C5#6711yescharacteristic
125#563—31no—
126#456—32no—
127#454C5×D311yescharacteristic
128#356—32no—
129#355—31no—
130C22×C3×C5—42no—
131C22×C3×C5—41no—
132C22×C3×C5—41no—
133C22×C3×C5#12011yescharacteristic
134C3×C4×C5—22no—
135C3×C4×C5—21no—
136C3×C4×C5—41no—
137C3×C4×C5—41no—
138C5×Dic3—42no—
139C5×Dic3—42no—
140C3×#41—22no—
141C3×#41—21no—
142C3×#41—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8C23×C511yescharacteristic
145C2×C3×D4C5×D411yescharacteristic
146C22×Dic3—21no—
147C2×Dic6C5×D412yes—
148C3×SD16—41no—
149C3×D8—41no—
150C3×M16—21no—
151C3×C22⋊C4—22no—
152C3×C42C5×D411yescharacteristic
153#217—22no—
154#217—21no—
155#216—42no—
156#214—42no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159Dic3⋊C4—22no—
160Dic3⋊C4—22no—
161Dic3⋊C4—22no—
162C4×Dic3C5×D412yes—
163#208—22no—
164C23×C5—21no—
165C5×Q8—21no—
166C5×Q8—22no—
167C5×Q8—41no—
168C5×Q8—122no—
169C5×D4—21no—
170C5×D4—22no—
171C5×D4—41no—
172C5×D4—41no—
173C2×C4×C5—21no—
174C2×C4×C5—41no—
175C2×C4×C5—42no—
176C2×C4×C5—62no—
177C2×C4×C5—62no—
178C2×C4×C5—241no—
179C2×C4×C5D3×D412yes—
180C2×C4×C5C2×#6711yescharacteristic
181C5×C8—41no—
182C5×C8—122no—
1832−1+4C2×C5×D311yescharacteristic
184#137—62no—
185AGL(1,ℤ/8)—21no—
186C4⋊Q8—32no—
187#126—62no—
188#125—61no—
189#124C2×C5×D311yescharacteristic
190#123—61no—
191C22⋊Q8—62no—
192#104—21no—
193#104—62no—
194#101—32no—
195#101C2×C5×D311yescharacteristic
196C2×C3×C5—21no—
197C2×C3×C5—81no—
198C2×C3×C5—81no—
199C2×C3×C5B(4,𝔽2)11yescharacteristic
200C23×C3—21no—
201C3×Q8—22no—
202C3×Q8—21no—
203C3×Q8—41no—
204C3×D4—22no—
205C3×D4—21no—
206C3×D4—41no—
207C3×D4—41no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4C2×C5×D412yes—
212C2×C3×C4C2×C5×D411yescharacteristic
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—81no—
216Dic6—42no—
217C3×C8—41no—
218C3⋊C8—42no—
219C22×C5—42no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5#93711yescharacteristic
223C4×C5—21no—
224C4×C5—22no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—122no—
228C4×C5—122no—
229#41—22no—
230#41—21no—
231#41—41no—
232C2×Q8—32no—
233C2×Q8—41no—
234C2×Q8C22×C5×D311yescharacteristic
235C2×D4C5×#6711yescharacteristic
236C22×C4—61no—
237Q16—122no—
238SD16—41no—
239SD16—122no—
240D8—41no—
241M16—21no—
242M16—62no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C4⋊C4—61no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C22⋊C4—22no—
249C22⋊C4—62no—
250C22⋊C4—61no—
251C42—32no—
252C42C5×#6711yescharacteristic
253C3×C5#220311yescharacteristic, complemented, Hall
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3C5×#12011yescharacteristic
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262Dic3—42no—
263Dic3—42no—
264C2×C5—21no—
265C2×C5—81no—
266C2×C5—81no—
267C2×C5#503211yescharacteristic
268C23—21no—
269Q8—21no—
270Q8—22no—
271Q8—41no—
272Q8—122no—
273D4—21no—
274D4—22no—
275D4—41no—
276D4—41no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—241no—
283C2×C4C5×D3×D412yes—
284C2×C4C2×C5×#6711yescharacteristic
285C8—41no—
286C8—122no—
287C2×C3—21no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3C5×B(4,𝔽2)11yescharacteristic
291C5#7098911yescharacteristic, direct product factor, Sylow
292C22—42no—
293C22—41no—
294C22—41no—
295C22C5×#93711yescharacteristic
296C4—21no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—122no—
301C4—122no—
302C3C5×#220311yescharacteristic, complemented, Sylow
303C2—21no—
304C2—81no—
305C2—81no—
306C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×Q8C23×C5
FrattiniC2×Q8C22×C5×D3
Fitting, prime radicalC3×C5×#454C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×#454D3
hypocenterC3C5×#2203
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×Q8 ◁ C5×#454

Complements

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

NH ≅ G/Ntype
1. C5×#5672C2semidirect
3. C5×#5246C2semidirect
4. C5×#4542C2semidirect
5. C5×#4478C2semidirect
20. #70989C5direct (N × H)
50. #5672C2×C5semidirect
52. #5246C2×C5semidirect
53. #4542C2×C5semidirect
54. #4478C2×C5semidirect
253. C3×C5#2203semidirect
291. C5#70989direct (N × H)
302. C3C5×#2203semidirect

Maximal subgroups

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

subgroupindex
C5×#56722
C3×C5×#4542
C5×#52462
C5×#45422
C5×#44782
C5×#22033
#709895