Finite Group Databasebeta

← C5×#70995 #14317007

Subgroups — C5×#70995 #14317007

Order 1920.

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

normal subgroups only

This group has 448 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×#5722C211yesmaximal, characteristic, complemented
2C3×C5×#456C211yesmaximal, characteristic
3C5×#5246C212yesmaximal
4C5×#4548C211yesmaximal, characteristic, complemented
5C5×#4478C212yesmaximal, complemented
6C5×#2205—31nomaximal, Hall
7C3×C5×2−1+4C2211yescharacteristic
8C3×C5×#137—21no—
9C3×C5×C4⋊Q8C2211yescharacteristic
10C5×#951—22no—
11C5×#944—21no—
12C5×#944C2212yes—
13C5×#888—22no—
14C5×#887—22no—
15C4×C5×Dic6—21no—
16C3×C5×#104—21no—
17C3×C5×#101C2211yescharacteristic
18C5×#837—22no—
19C5×#836C2212yes—
20#70995C511yesmaximal, characteristic, direct product factor, Hall
21C5×Q82—31no—
22C5×#456—32no—
23C5×#456D311yescharacteristic
24C5×#356—32no—
25C5×#356—31no—
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×Dic6—21no—
33C2×C5×Dic6—21no—
34C2×C5×Dic6D412yes—
35C3×C5×Q16—41no—
36C3×C5×SD16—41no—
37C3×C5×M16—21no—
38C3×C5×C4⋊C4—22no—
39C3×C42×C5D411yescharacteristic
40C5×#216—42no—
41C5×#214—42no—
42C5×C4⋊Dic3—21no—
43C5×Dic3⋊C4—22no—
44C5×Dic3⋊C4—22no—
45C5×Dic3⋊C4—22no—
46C4×C5×Dic3—22no—
47C4×C5×Dic3—21no—
48C4×C5×Dic3D412yes—
49C5×#208—22no—
50#5722C2×C511yescharacteristic, complemented
51C3×#456C2×C511yescharacteristic
52#5246C2×C512yes—
53#4548C2×C511yescharacteristic, complemented
54#4478C2×C512yescomplemented
55C5×2−1+4C2×D311yescharacteristic
56C5×#137—21no—
57C5×#137—62no—
58C5×C4⋊Q8—32no—
59C5×C4⋊Q8—62no—
60C5×C4⋊Q8—61no—
61C5×C4⋊Q8C2×D311yescharacteristic
62C4×C5×Q8—62no—
63C4×C5×Q8—61no—
64C5×#104—21no—
65C5×#104—62no—
66C5×#101—32no—
67C5×#101C2×D311yescharacteristic
68#2205—31noSylow
69C3×C5×Q8—22no—
70C3×C5×Q8—21no—
71C3×C5×Q8—41no—
72C3×C5×Q8—41no—
73C3×C5×D4—22no—
74C3×C5×D4—21no—
75C3×C5×D4—41no—
76C2×C3×C4×C5—21no—
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—
84C5×Dic6—24no—
85C5×Dic6—42no—
86C5×Dic6—81no—
87C3×C5×C8—41no—
88C5×C3⋊C8—42no—
89C3×2−1+4C22×C511yescharacteristic
90C3×#137—21no—
91C3×C4⋊Q8C22×C511yescharacteristic
92#951—22no—
93#944—21no—
94#944C22×C512yes—
95#888—22no—
96#887—22no—
97C4×Dic6—21no—
98C3×#104—21no—
99C3×#101C22×C511yescharacteristic
100#837—22no—
101#836C22×C512yes—
102C5×#41—22no—
103C5×#41—21no—
104C5×#41—41no—
105C2×C5×Q8—32no—
106C2×C5×Q8—41no—
107C2×C5×Q8—61no—
108C2×C5×Q8—61no—
109C2×C5×Q8#6711yescharacteristic
110C2×C5×Q8C22×D311yescharacteristic
111C5×Q16—41no—
112C5×Q16—122no—
113C5×SD16—41no—
114C5×SD16—122no—
115C5×M16—21no—
116C5×M16—62no—
117C5×C4⋊C4—22no—
118C5×C4⋊C4—62no—
119C5×C4⋊C4—61no—
120C5×C4⋊C4—62no—
121C5×C4⋊C4—62no—
122C42×C5—32no—
123C42×C5—62no—
124C42×C5—61no—
125C42×C5#6711yescharacteristic
126Q82—31no—
127#456—32no—
128#456C5×D311yescharacteristic
129#356—31no—
130#356—32no—
131C22×C3×C5—41no—
132C22×C3×C5#12011yescharacteristic
133C3×C4×C5—22no—
134C3×C4×C5—22no—
135C3×C4×C5—21no—
136C3×C4×C5—41no—
137C3×C4×C5—41no—
138C5×Dic3—24no—
139C5×Dic3—42no—
140C3×#41—22no—
141C3×#41—21no—
142C3×#41—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8C5×D411yescharacteristic
145C2×C3×Q8C23×C511yescharacteristic
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×Dic6C5×D412yes—
149C3×Q16—41no—
150C3×SD16—41no—
151C3×M16—21no—
152C3×C4⋊C4—22no—
153C3×C42C5×D411yescharacteristic
154#216—42no—
155#214—42no—
156C4⋊Dic3—21no—
157Dic3⋊C4—22no—
158Dic3⋊C4—22no—
159Dic3⋊C4—22no—
160C4×Dic3—22no—
161C4×Dic3—21no—
162C4×Dic3C5×D412yes—
163#208—22no—
164C5×Q8—21no—
165C5×Q8—22no—
166C5×Q8—41no—
167C5×Q8—41no—
168C5×Q8—64no—
169C5×Q8—122no—
170C5×Q8—241no—
171C5×D4—21no—
172C5×D4—22no—
173C5×D4—41no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—41no—
177C2×C4×C5—42no—
178C2×C4×C5—62no—
179C2×C4×C5—62no—
180C2×C4×C5D3×D412yes—
181C2×C4×C5C2×#6711yescharacteristic
182C5×C8—41no—
183C5×C8—122no—
1842−1+4C2×C5×D311yescharacteristic
185#137—21no—
186#137—62no—
187C4⋊Q8—32no—
188C4⋊Q8—62no—
189C4⋊Q8—61no—
190C4⋊Q8C2×C5×D311yescharacteristic
191C4×Q8—62no—
192C4×Q8—61no—
193#104—21no—
194#104—62no—
195#101—32no—
196#101C2×C5×D311yescharacteristic
197C2×C3×C5—21no—
198C2×C3×C5—81no—
199C2×C3×C5B(4,𝔽2)11yescharacteristic
200C3×Q8—22no—
201C3×Q8—21no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×D4—22no—
205C3×D4—21no—
206C3×D4—41no—
207C2×C3×C4—21no—
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—
215Dic6—24no—
216Dic6—42no—
217Dic6—81no—
218C3×C8—41no—
219C3⋊C8—42no—
220C22×C5—41no—
221C22×C5#93711yescharacteristic
222C4×C5—22no—
223C4×C5—21no—
224C4×C5—22no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—64no—
228C4×C5—122no—
229#41—22no—
230#41—21no—
231#41—41no—
232C2×Q8—32no—
233C2×Q8—41no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8C5×#6711yescharacteristic
237C2×Q8C22×C5×D311yescharacteristic
238Q16—41no—
239Q16—122no—
240SD16—41no—
241SD16—122no—
242M16—21no—
243M16—62no—
244C4⋊C4—22no—
245C4⋊C4—62no—
246C4⋊C4—61no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C42—32no—
250C42—62no—
251C42—61no—
252C42C5×#6711yescharacteristic
253C3×C5#220511yescharacteristic, complemented, Hall
254C22×C3—41no—
255C22×C3C5×#12011yescharacteristic
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—21no—
259C3×C4—41no—
260C3×C4—41no—
261Dic3—24no—
262Dic3—42no—
263C2×C5—21no—
264C2×C5—81no—
265C2×C5#503211yescharacteristic
266Q8—21no—
267Q8—22no—
268Q8—41no—
269Q8—41no—
270Q8—64no—
271Q8—122no—
272Q8—241no—
273D4—21no—
274D4—22no—
275D4—41no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4C5×D3×D412yes—
283C2×C4C2×C5×#6711yescharacteristic
284C8—41no—
285C8—122no—
286C2×C3—21no—
287C2×C3—81no—
288C2×C3C5×B(4,𝔽2)11yescharacteristic
289C5#7099511yescharacteristic, direct product factor, Sylow
290C22—41no—
291C22C5×#93711yescharacteristic
292C4—22no—
293C4—21no—
294C4—22no—
295C4—41no—
296C4—41no—
297C4—64no—
298C4—122no—
299C3C5×#220511yescharacteristic, complemented, Sylow
300C2—21no—
301C2—81no—
302C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×Q8C23×C5
FrattiniC2×Q8C22×C5×D3
FittingC3×C5×#456C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC3×C5×2−1+4C22
hypercenterC5×#456D3
hypocenterC3C5×#2205
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×#5722C2semidirect
4. C5×#4548C2semidirect
5. C5×#4478C2semidirect
20. #70995C5direct (N × H)
50. #5722C2×C5semidirect
53. #4548C2×C5semidirect
54. #4478C2×C5semidirect
253. C3×C5#2205semidirect
289. C5#70995direct (N × H)
299. C3C5×#2205semidirect

Maximal subgroups

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

subgroupindex
C5×#57222
C3×C5×#4562
C5×#52462
C5×#45482
C5×#44782
C5×#22053
#709955