Finite Group Databasebeta

← C5×#71946 #14316736

Subgroups — C5×#71946 #14316736

Order 1920.

492 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C3×C5×#559C211yesmaximal, characteristic, complemented
2C5×#5077C211yesmaximal, characteristic
3C5×#5053C212yesmaximal, complemented
4C5×#4545C211yesmaximal, characteristic, complemented
5C5×#4539C212yesmaximal, complemented
6C5×#2198—31nomaximal, Hall
7C5×#1004C2211yescharacteristic
8C3×C5×#127C2211yescharacteristic
9C3×C5×#124—22no—
10C3×C5×#123—21no—
11C3×C5×#123C2212yes—
12C3×C4×C5×Q8—21no—
13C3×C4×C5×D4—22no—
14C5×#932—21no—
15C5×#929—22no—
16C5×#834C2212yes—
17C5×#833C2211yescharacteristic
18C5×#806—22no—
19C5×#805—21no—
20#71946C511yesmaximal, characteristic, direct product factor, Hall
21C5×#559D311yescharacteristic, complemented
22C5×#458—32no—
23C5×#453—31no—
24C5×#353—31no—
25C5×#352—32no—
26C2×C3×C5×Q8—21no—
27C2×C3×C5×D4—22no—
28C2×C3×C5×D4D411yescharacteristic
29C2×C3×C5×D4C2311yescharacteristic
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5D412yes—
32C2×C5×#67—22no—
33C5×#237—41no—
34C5×D3×D4—41no—
35C5×#235—21no—
36C3×C5×C4⋊C4—21no—
37C3×C5×C4⋊C4—22no—
38C3×C5×C22⋊C4—22no—
39C3×C5×C22⋊C4—21no—
40C3×C5×C22⋊C4—22no—
41C3×C5×C22⋊C4D412yes—
42C3×C42×C5—22no—
43C3×C42×C5D411yescharacteristic
44C5×#217—22no—
45C5×#214—41no—
46C5×#213—41no—
47C5×(C2×D3)⋊C4—42no—
48C5×Dic3⋊C4—42no—
49C5×#208—21no—
50C3×#559C2×C511yescharacteristic, complemented
51#5077C2×C511yescharacteristic
52#5053C2×C512yescomplemented
53#4545C2×C511yescharacteristic, complemented
54#4539C2×C512yescomplemented
55C5×H(5,𝔽2)—31no—
56C5×AGL(1,ℤ/8)—61no—
57C5×#127C2×D311yescharacteristic
58C5×#124—22no—
59C5×#123—21no—
60C5×#123—62no—
61C5×#123C2×D312yes—
62C4×C5×Q8—21no—
63C4×C5×D4—22no—
64C5×#104—61no—
65C5×#100—31no—
66C5×#99—32no—
67C5×#99—62no—
68#2198—31noSylow
69C23×C3×C5—21no—
70C23×C3×C5C2×D412yes—
71C22×C5×D3—22no—
72C3×C5×Q8—42no—
73C3×C5×D4—41no—
74C3×C5×D4—44no—
75C3×C5×D4—41no—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—42no—
80C2×C3×C4×C5—81no—
81C2×C3×C4×C5C2×D411yescharacteristic
82C5×#67—42no—
83C5×#67—41no—
84C2×C5×Dic3—22no—
85C2×C5×Dic3—42no—
86C5×D12—21no—
87C4×C5×D3—41no—
88C5×Dic6—21no—
89C5×C3⋊C8—41no—
90#1004C22×C511yescharacteristic
91C3×#127C22×C511yescharacteristic
92C3×#124—22no—
93C3×#123—21no—
94C3×#123C22×C512yes—
95C3×C4×Q8—21no—
96C3×C4×D4—22no—
97#932—21no—
98#929—22no—
99#834C22×C512yes—
100#833C22×C511yescharacteristic
101#806—22no—
102#805—21no—
103C5×#41—61no—
104C5×#41—121no—
105C2×C5×Q8—21no—
106C2×C5×D4—22no—
107C2×C5×D4—62no—
108C2×C5×D4—121no—
109C2×C5×D4#6711yescharacteristic
110C2×C5×D4C22×D311yescharacteristic
111C22×C4×C5—21no—
112C22×C4×C5#6712yes—
113C5×SD16—121no—
114C5×D8—121no—
115C5×M16—61no—
116C5×C4⋊C4—21no—
117C5×C4⋊C4—22no—
118C5×C4⋊C4—122no—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—21no—
121C5×C22⋊C4—22no—
122C5×C22⋊C4—62no—
123C5×C22⋊C4—122no—
124C5×C22⋊C4#6712yes—
125C42×C5—22no—
126C42×C5#6711yescharacteristic
127#559C5×D311yescharacteristic, complemented
128#458—32no—
129#453—31no—
130#353—31no—
131#352—32no—
132C22×C3×C5—22no—
133C22×C3×C5—42no—
134C22×C3×C5—42no—
135C22×C3×C5—41no—
136C22×C3×C5#12011yescharacteristic
137C2×C5×D3—41no—
138C2×C5×D3—82no—
139C3×C4×C5—22no—
140C3×C4×C5—21no—
141C3×C4×C5—42no—
142C3×C4×C5—42no—
143C5×Dic3—41no—
144C5×Dic3—82no—
145C2×C3×Q8—21no—
146C2×C3×D4—22no—
147C2×C3×D4C5×D411yescharacteristic
148C2×C3×D4C23×C511yescharacteristic
149C22×C3×C4—21no—
150C22×C3×C4C5×D412yes—
151C2×#67—22no—
152#237—41no—
153D3×D4—41no—
154#235—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—22no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4C5×D412yes—
161C3×C42—22no—
162C3×C42C5×D411yescharacteristic
163#217—22no—
164#214—41no—
165#213—41no—
166(C2×D3)⋊C4—42no—
167Dic3⋊C4—42no—
168#208—21no—
169C23×C5—21no—
170C23×C5—62no—
171C23×C5C2×#6712yes—
172C5×Q8—42no—
173C5×Q8—61no—
174C5×D4—41no—
175C5×D4—41no—
176C5×D4—44no—
177C5×D4—61no—
178C5×D4—122no—
179C5×D4—121no—
180C2×C4×C5—21no—
181C2×C4×C5—22no—
182C2×C4×C5—22no—
183C2×C4×C5—42no—
184C2×C4×C5—62no—
185C2×C4×C5—81no—
186C2×C4×C5—121no—
187C2×C4×C5—122no—
188C2×C4×C5C2×#6711yescharacteristic
189C5×C8—121no—
190H(5,𝔽2)—31no—
191AGL(1,ℤ/8)—61no—
192#127C2×C5×D311yescharacteristic
193#124—22no—
194#123—21no—
195#123—62no—
196#123C2×C5×D312yes—
197C4×Q8—21no—
198C4×D4—22no—
199#104—61no—
200#100—31no—
201#99—32no—
202#99—62no—
203C2×C3×C5—21no—
204C2×C3×C5—42no—
205C2×C3×C5—81no—
206C2×C3×C5B(4,𝔽2)11yescharacteristic
207C5×D3—81no—
208C23×C3—21no—
209C23×C3C2×C5×D412yes—
210C22×D3—22no—
211C3×Q8—42no—
212C3×D4—41no—
213C3×D4—41no—
214C3×D4—44no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4C2×C5×D411yescharacteristic
221#67—42no—
222#67—41no—
223C2×Dic3—22no—
224C2×Dic3—42no—
225D12—21no—
226C4×D3—41no—
227Dic6—21no—
228C3⋊C8—41no—
229C22×C5—22no—
230C22×C5—42no—
231C22×C5—42no—
232C22×C5—41no—
233C22×C5—121no—
234C22×C5—242no—
235C22×C5#95311yescharacteristic
236C4×C5—22no—
237C4×C5—21no—
238C4×C5—42no—
239C4×C5—42no—
240C4×C5—121no—
241C4×C5—242no—
242#41—61no—
243#41—121no—
244C2×Q8—21no—
245C2×D4—22no—
246C2×D4—62no—
247C2×D4—121no—
248C2×D4C5×#6711yescharacteristic
249C2×D4C22×C5×D311yescharacteristic
250C22×C4—21no—
251C22×C4C5×#6712yes—
252SD16—121no—
253D8—121no—
254M16—61no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—122no—
258C22⋊C4—22no—
259C22⋊C4—21no—
260C22⋊C4—22no—
261C22⋊C4—62no—
262C22⋊C4—122no—
263C22⋊C4C5×#6712yes—
264C42—22no—
265C42C5×#6711yescharacteristic
266C3×C5#219811yescharacteristic, complemented, Hall
267C22×C3—22no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—41no—
271C22×C3C5×#12011yescharacteristic
272C2×D3—41no—
273C2×D3—82no—
274C3×C4—22no—
275C3×C4—21no—
276C3×C4—42no—
277C3×C4—42no—
278Dic3—41no—
279Dic3—82no—
280C2×C5—21no—
281C2×C5—42no—
282C2×C5—81no—
283C2×C5—241no—
284C2×C5#524411yescharacteristic
285C23—21no—
286C23—62no—
287C23C2×C5×#6712yes—
288Q8—42no—
289Q8—61no—
290D4—41no—
291D4—41no—
292D4—44no—
293D4—61no—
294D4—122no—
295D4—121no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—42no—
300C2×C4—62no—
301C2×C4—81no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4C2×C5×#6711yescharacteristic
305C8—121no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—81no—
309C2×C3C5×B(4,𝔽2)11yescharacteristic
310D3—81no—
311C5#7194611yescharacteristic, direct product factor, Sylow
312C22—22no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—121no—
317C22—242no—
318C22C5×#95311yescharacteristic
319C4—22no—
320C4—21no—
321C4—42no—
322C4—42no—
323C4—121no—
324C4—242no—
325C3C5×#219811yescharacteristic, complemented, Sylow
326C2—21no—
327C2—42no—
328C2—81no—
329C2—241no—
330C2C5×#524411yescharacteristic
subgroupG/H
centerC2×C5#5244
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#559C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC5×#5077C2
hypercenterC5×#559D3
hypocenterC3C5×#2198
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×#559

Complements

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

NH ≅ G/Ntype
1. C3×C5×#559C2semidirect
3. C5×#5053C2semidirect
4. C5×#4545C2semidirect
5. C5×#4539C2semidirect
20. #71946C5direct (N × H)
21. C5×#559D3semidirect
50. C3×#559C2×C5semidirect
52. #5053C2×C5semidirect
53. #4545C2×C5semidirect
54. #4539C2×C5semidirect
127. #559C5×D3semidirect
266. C3×C5#2198semidirect
311. C5#71946direct (N × H)
325. C3C5×#2198semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5592
C5×#50772
C5×#50532
C5×#45452
C5×#45392
C5×#21983
#719465