Finite Group Databasebeta

← C5×#69169 #14317013

Subgroups — C5×#69169 #14317013

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
2C5×#5095C211yesmaximal, characteristic
3C5×#4752C212yesmaximal
4C3×C5×#356C211yesmaximal, characteristic, complemented
5C5×#4478C212yesmaximal, complemented
6C5×#2205—31nomaximal, Hall
7C5×#1007C2211yescharacteristic
8C3×C5×C4⋊Q8C2211yescharacteristic
9C5×#944—21no—
10C5×#944C2212yes—
11C5×#942—21no—
12C5×#909—22no—
13C5×#888—22no—
14C5×#887—22no—
15C4×C5×Dic6—21no—
16C3×C5×#101C2212yes—
17C5×#836C2211yescharacteristic
18C5×#825—22no—
19C5×#805—21no—
20#69169C511yesmaximal, characteristic, direct product factor, Hall
21C5×Q82—31no—
22C5×#456—32no—
23C5×#456—31no—
24C5×#356—32no—
25C5×#356D311yescharacteristic, complemented
26C2×C3×C5×Q8D411yescharacteristic
27C2×C3×C5×Q8C2311yescharacteristic
28C5×#239—41no—
29C5×D3×Q8—41no—
30C5×#235—22no—
31C5×#235—21no—
32C2×C5×Dic6—21no—
33C2×C5×Dic6—21no—
34C2×C5×Dic6D412yes—
35C3×C5×M16—22no—
36C3×C5×C4⋊C4—22no—
37C3×C42×C5D411yescharacteristic
38C5×#216—41no—
39C5×#215—41no—
40C5×C4⋊Dic3—21no—
41C5×Dic3⋊C4—22no—
42C5×Dic3⋊C4—22no—
43C5×Dic3⋊C4—22no—
44C4×C5×Dic3—22no—
45C4×C5×Dic3—21no—
46C4×C5×Dic3D412yes—
47C5×#208—21no—
48C5×Dic12—42no—
49C5×#204—42no—
50#5722C2×C511yescharacteristic, complemented
51#5095C2×C511yescharacteristic
52#4752C2×C512yes—
53C3×#356C2×C511yescharacteristic, complemented
54#4478C2×C512yescomplemented
55C5×2−1+4—31no—
56C5×#137—62no—
57C5×#137—61no—
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—62no—
65C5×#104—61no—
66C5×#101—31no—
67C5×#101C2×D312yes—
68#2205—31noSylow
69C3×C5×Q8—41no—
70C3×C5×Q8—41no—
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5C2×D411yescharacteristic
74C2×C3×C4×C5C2×D412yes—
75C5×#67—41no—
76C2×C5×Dic3—22no—
77C2×C5×Dic3—22no—
78C5×D12—22no—
79C5×D12—21no—
80C4×C5×D3—41no—
81C4×C5×D3—42no—
82C5×Dic6—24no—
83C5×Dic6—22no—
84C5×Dic6—21no—
85C5×Dic6—42no—
86C5×Dic6—81no—
87C3×C5×C8—42no—
88C5×C3⋊C8—41no—
89#1007C22×C511yescharacteristic
90C3×C4⋊Q8C22×C511yescharacteristic
91#944—21no—
92#944C22×C512yes—
93#942—21no—
94#909—22no—
95#888—22no—
96#887—22no—
97C4×Dic6—21no—
98C3×#101C22×C512yes—
99#836C22×C511yescharacteristic
100#825—22no—
101#805—21no—
102C5×#41—61no—
103C5×#41—62no—
104C5×#41—121no—
105C2×C5×Q8—32no—
106C2×C5×Q8—61no—
107C2×C5×Q8—61no—
108C2×C5×Q8—121no—
109C2×C5×Q8D1211yescharacteristic
110C2×C5×Q8C22×D311yescharacteristic
111C5×Q16—122no—
112C5×Q16—121no—
113C5×SD16—122no—
114C5×SD16—121no—
115C5×M16—22no—
116C5×M16—61no—
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×C5D1211yescharacteristic
126Q82—31no—
127#456—31no—
128#456—32no—
129#356—32no—
130#356C5×D311yescharacteristic, complemented
131C22×C3×C5#12011yescharacteristic
132C2×C5×D3—41no—
133C3×C4×C5—22no—
134C3×C4×C5—21no—
135C3×C4×C5—22no—
136C3×C4×C5—41no—
137C5×Dic3—24no—
138C5×Dic3—41no—
139C5×Dic3—42no—
140C2×C3×Q8C5×D411yescharacteristic
141C2×C3×Q8C23×C511yescharacteristic
142#239—41no—
143D3×Q8—41no—
144#235—21no—
145#235—22no—
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×Dic6C5×D412yes—
149C3×M16—22no—
150C3×C4⋊C4—22no—
151C3×C42C5×D411yescharacteristic
152#216—41no—
153#215—41no—
154C4⋊Dic3—21no—
155Dic3⋊C4—22no—
156Dic3⋊C4—22no—
157Dic3⋊C4—22no—
158C4×Dic3—22no—
159C4×Dic3—21no—
160C4×Dic3C5×D412yes—
161#208—21no—
162Dic12—42no—
163#204—42no—
164C5×Q8—41no—
165C5×Q8—41no—
166C5×Q8—64no—
167C5×Q8—62no—
168C5×Q8—61no—
169C5×Q8—122no—
170C5×Q8—241no—
171C5×D4—62no—
172C5×D4—61no—
173C5×D4—121no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—62no—
177C2×C4×C5—62no—
178C2×C4×C5—122no—
179C2×C4×C5—121no—
180C2×C4×C5C2×D1211yescharacteristic
181C2×C4×C5D3×D412yes—
182C5×C8—42no—
183C5×C8—121no—
1842−1+4—31no—
185#137—62no—
186#137—61no—
187C4⋊Q8—32no—
188C4⋊Q8—62no—
189C4⋊Q8—61no—
190C4⋊Q8C2×C5×D311yescharacteristic
191C4×Q8—62no—
192C4×Q8—61no—
193#104—62no—
194#104—61no—
195#101—31no—
196#101C2×C5×D312yes—
197C2×C3×C5—21no—
198C2×C3×C5B(4,𝔽2)11yescharacteristic
199C5×D3—81no—
200C3×Q8—41no—
201C3×Q8—41no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4C2×C5×D411yescharacteristic
205C2×C3×C4C2×C5×D412yes—
206#67—41no—
207C2×Dic3—22no—
208C2×Dic3—22no—
209D12—22no—
210D12—21no—
211C4×D3—41no—
212C4×D3—42no—
213Dic6—24no—
214Dic6—22no—
215Dic6—21no—
216Dic6—42no—
217Dic6—81no—
218C3×C8—42no—
219C3⋊C8—41no—
220C22×C5—121no—
221C22×C5#88211yescharacteristic
222C4×C5—22no—
223C4×C5—22no—
224C4×C5—21no—
225C4×C5—41no—
226C4×C5—64no—
227C4×C5—122no—
228C4×C5—121no—
229#41—61no—
230#41—62no—
231#41—121no—
232C2×Q8—32no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—121no—
236C2×Q8C5×D1211yescharacteristic
237C2×Q8C22×C5×D311yescharacteristic
238Q16—122no—
239Q16—121no—
240SD16—122no—
241SD16—121no—
242M16—22no—
243M16—61no—
244C4⋊C4—22no—
245C4⋊C4—62no—
246C4⋊C4—61no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C42—32no—
250C42—62no—
251C42—61no—
252C42C5×D1211yescharacteristic
253C3×C5#220511yescharacteristic, complemented, Hall
254C22×C3C5×#12011yescharacteristic
255C2×D3—41no—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—41no—
260Dic3—24no—
261Dic3—41no—
262Dic3—42no—
263C2×C5—21no—
264C2×C5—241no—
265C2×C5#474111yescharacteristic
266Q8—41no—
267Q8—41no—
268Q8—64no—
269Q8—62no—
270Q8—61no—
271Q8—122no—
272Q8—241no—
273D4—62no—
274D4—61no—
275D4—121no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—122no—
281C2×C4—121no—
282C2×C4C2×C5×D1211yescharacteristic
283C2×C4C5×D3×D412yes—
284C8—42no—
285C8—121no—
286C2×C3—21no—
287C2×C3C5×B(4,𝔽2)11yescharacteristic
288D3—81no—
289C5#6916911yescharacteristic, direct product factor, Sylow
290C22—121no—
291C22C5×#88211yescharacteristic
292C4—22no—
293C4—22no—
294C4—21no—
295C4—41no—
296C4—64no—
297C4—122no—
298C4—121no—
299C3C5×#220511yescharacteristic, complemented, Sylow
300C2—21no—
301C2—241no—
302C2C5×#474111yescharacteristic
subgroupG/H
centerC2×C5#4741
commutator, cosocleC2×C3×Q8C23×C5
FrattiniC2×Q8C22×C5×D3
FittingC3×C5×#356C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC5×#1007C22
hypercenterC5×#356D3
hypocenterC3C5×#2205
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×Q8 ◁ C5×#356

Complements

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

NH ≅ G/Ntype
1. C5×#5722C2semidirect
4. C3×C5×#356C2semidirect
5. C5×#4478C2semidirect
20. #69169C5direct (N × H)
25. C5×#356D3semidirect
50. #5722C2×C5semidirect
53. C3×#356C2×C5semidirect
54. #4478C2×C5semidirect
130. #356C5×D3semidirect
253. C3×C5#2205semidirect
289. C5#69169direct (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
C5×#50952
C5×#47522
C3×C5×#3562
C5×#44782
C5×#22053
#691695