Finite Group Databasebeta

← C5×#70986 #14316733

Subgroups — C5×#70986 #14316733

Order 1920.

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

normal subgroups only

This group has 492 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×#5713C211yesmaximal, characteristic, complemented
2C3×C5×#453C211yesmaximal, characteristic
3C5×#5243C212yesmaximal, complemented
4C5×#4545C211yesmaximal, characteristic, complemented
5C5×#4476C212yesmaximal, complemented
6C5×#2198—31nomaximal, Hall
7C3×C5×H(5,𝔽2)C2211yescharacteristic
8C3×C5×AGL(1,ℤ/8)—21no—
9C3×C5×#127C2211yescharacteristic
10C5×#936—22no—
11C5×#935—21no—
12C5×#935—22no—
13C5×#935C2212yes—
14C5×D4×Dic3—22no—
15C4×C5×Dic6—21no—
16C3×C5×#104—21no—
17C3×C5×#100C2211yescharacteristic
18C5×#834—22no—
19C5×#834C2212yes—
20#70986C511yesmaximal, characteristic, direct product factor, Hall
21C5×#559—31no—
22C5×#458—32no—
23C5×#453D311yescharacteristic
24C5×#353—31no—
25C5×#352—32no—
26C3×C5×#41—21no—
27C3×C5×#41—41no—
28C2×C3×C5×D4—22no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—41no—
31C2×C3×C5×D4D411yescharacteristic
32C2×C3×C5×D4C2311yescharacteristic
33C22×C5×Dic3—21no—
34C22×C5×Dic3D412yes—
35C2×C5×Dic6—21no—
36C3×C5×SD16—41no—
37C3×C5×D8—41no—
38C3×C5×M16—21no—
39C3×C42×C5D411yescharacteristic
40C5×#217—22no—
41C5×#217—21no—
42C5×#217—22no—
43C5×#217—22no—
44C5×#217—42no—
45C5×#217D412yes—
46C5×C4⋊Dic3—21no—
47C5×Dic3⋊C4—22no—
48C5×Dic3⋊C4—42no—
49C4×C5×Dic3—22no—
50#5713C2×C511yescharacteristic, complemented
51C3×#453C2×C511yescharacteristic
52#5243C2×C512yescomplemented
53#4545C2×C511yescharacteristic, complemented
54#4476C2×C512yescomplemented
55C5×H(5,𝔽2)C2×D311yescharacteristic
56C5×AGL(1,ℤ/8)—21no—
57C5×#127C2×D311yescharacteristic
58C5×#124—62no—
59C5×#123—32no—
60C5×#123—61no—
61C5×#123—62no—
62C4×C5×Q8—61no—
63C4×C5×D4—62no—
64C5×#104—21no—
65C5×#100C2×D311yescharacteristic
66C5×#99—32no—
67C5×#99—62no—
68#2198—31noSylow
69C23×C3×C5—22no—
70C23×C3×C5—21no—
71C23×C3×C5C2×D412yes—
72C3×C5×Q8—21no—
73C3×C5×D4—21no—
74C3×C5×D4—41no—
75C3×C5×D4—44no—
76C3×C5×D4—41no—
77C3×C5×D4—42no—
78C3×C5×D4—41no—
79C2×C3×C4×C5—21no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5C2×D411yescharacteristic
83C2×C5×Dic3—22no—
84C2×C5×Dic3—22no—
85C2×C5×Dic3—42no—
86C2×C5×Dic3—42no—
87C2×C5×Dic3—81no—
88C5×Dic6—42no—
89C3×C5×C8—41no—
90C3×H(5,𝔽2)C22×C511yescharacteristic
91C3×AGL(1,ℤ/8)—21no—
92C3×#127C22×C511yescharacteristic
93#936—22no—
94#935—21no—
95#935—22no—
96#935C22×C512yes—
97D4×Dic3—22no—
98C4×Dic6—21no—
99C3×#104—21no—
100C3×#100C22×C511yescharacteristic
101#834—22no—
102#834C22×C512yes—
103C5×#41—21no—
104C5×#41—41no—
105C2×C5×Q8—61no—
106C2×C5×D4—22no—
107C2×C5×D4—22no—
108C2×C5×D4—41no—
109C2×C5×D4#6711yescharacteristic
110C2×C5×D4C22×D311yescharacteristic
111C22×C4×C5—32no—
112C22×C4×C5—61no—
113C5×SD16—41no—
114C5×D8—41no—
115C5×M16—21no—
116C5×C4⋊C4—61no—
117C5×C4⋊C4—62no—
118C5×C4⋊C4—122no—
119C5×C22⋊C4—32no—
120C5×C22⋊C4—62no—
121C5×C22⋊C4—61no—
122C5×C22⋊C4—62no—
123C5×C22⋊C4—62no—
124C5×C22⋊C4—122no—
125C42×C5—62no—
126C42×C5#6711yescharacteristic
127#559—31no—
128#458—32no—
129#453C5×D311yescharacteristic
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—41no—
137C22×C3×C5—82no—
138C22×C3×C5#12011yescharacteristic
139C3×C4×C5—22no—
140C3×C4×C5—21no—
141C3×C4×C5—41no—
142C5×Dic3—42no—
143C5×Dic3—42no—
144C5×Dic3—82no—
145C3×#41—21no—
146C3×#41—41no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—41no—
150C2×C3×D4C5×D411yescharacteristic
151C2×C3×D4C23×C511yescharacteristic
152C22×Dic3—21no—
153C22×Dic3C5×D412yes—
154C2×Dic6—21no—
155C3×SD16—41no—
156C3×D8—41no—
157C3×M16—21no—
158C3×C42C5×D411yescharacteristic
159#217—22no—
160#217—21no—
161#217—22no—
162#217—22no—
163#217—42no—
164#217C5×D412yes—
165C4⋊Dic3—21no—
166Dic3⋊C4—22no—
167Dic3⋊C4—42no—
168C4×Dic3—22no—
169C23×C5—22no—
170C23×C5—21no—
171C23×C5D3×D412yes—
172C5×Q8—21no—
173C5×Q8—122no—
174C5×D4—21no—
175C5×D4—41no—
176C5×D4—41no—
177C5×D4—44no—
178C5×D4—42no—
179C5×D4—41no—
180C2×C4×C5—21no—
181C2×C4×C5—22no—
182C2×C4×C5—41no—
183C2×C4×C5—62no—
184C2×C4×C5—62no—
185C2×C4×C5—122no—
186C2×C4×C5—122no—
187C2×C4×C5—241no—
188C2×C4×C5C2×#6711yescharacteristic
189C5×C8—41no—
190H(5,𝔽2)C2×C5×D311yescharacteristic
191AGL(1,ℤ/8)—21no—
192#127C2×C5×D311yescharacteristic
193#124—62no—
194#123—32no—
195#123—61no—
196#123—62no—
197C4×Q8—61no—
198C4×D4—62no—
199#104—21no—
200#100C2×C5×D311yescharacteristic
201#99—32no—
202#99—62no—
203C2×C3×C5—21no—
204C2×C3×C5—42no—
205C2×C3×C5—81no—
206C2×C3×C5—81no—
207C2×C3×C5B(4,𝔽2)11yescharacteristic
208C23×C3—22no—
209C23×C3—21no—
210C23×C3C2×C5×D412yes—
211C3×Q8—21no—
212C3×D4—21no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—44no—
216C3×D4—42no—
217C3×D4—41no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4C2×C5×D411yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—81no—
227Dic6—42no—
228C3×C8—41no—
229C22×C5—22no—
230C22×C5—42no—
231C22×C5—42no—
232C22×C5—41no—
233C22×C5—41no—
234C22×C5—82no—
235C22×C5#93711yescharacteristic
236C4×C5—22no—
237C4×C5—21no—
238C4×C5—41no—
239C4×C5—122no—
240C4×C5—122no—
241C4×C5—242no—
242#41—21no—
243#41—41no—
244C2×Q8—61no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4C5×#6711yescharacteristic
249C2×D4C22×C5×D311yescharacteristic
250C22×C4—32no—
251C22×C4—61no—
252SD16—41no—
253D8—41no—
254M16—21no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—122no—
258C22⋊C4—32no—
259C22⋊C4—62no—
260C22⋊C4—61no—
261C22⋊C4—62no—
262C22⋊C4—62no—
263C22⋊C4—122no—
264C42—62no—
265C42C5×#6711yescharacteristic
266C3×C5#219811yescharacteristic, complemented, Hall
267C22×C3—22no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—82no—
273C22×C3C5×#12011yescharacteristic
274C3×C4—22no—
275C3×C4—21no—
276C3×C4—41no—
277Dic3—42no—
278Dic3—42no—
279Dic3—82no—
280C2×C5—21no—
281C2×C5—42no—
282C2×C5—81no—
283C2×C5—81no—
284C2×C5#503211yescharacteristic
285C23—22no—
286C23—21no—
287C23C5×D3×D412yes—
288Q8—21no—
289Q8—122no—
290D4—21no—
291D4—41no—
292D4—41no—
293D4—44no—
294D4—42no—
295D4—41no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—241no—
304C2×C4C2×C5×#6711yescharacteristic
305C8—41no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3C5×B(4,𝔽2)11yescharacteristic
311C5#7098611yescharacteristic, direct product factor, Sylow
312C22—22no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—82no—
318C22C5×#93711yescharacteristic
319C4—22no—
320C4—21no—
321C4—41no—
322C4—122no—
323C4—122no—
324C4—242no—
325C3C5×#219811yescharacteristic, complemented, Sylow
326C2—21no—
327C2—42no—
328C2—81no—
329C2—81no—
330C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
Fitting, prime radicalC3×C5×#453C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×#453D3
hypocenterC3C5×#2198
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×#453

Complements

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

NH ≅ G/Ntype
1. C5×#5713C2semidirect
3. C5×#5243C2semidirect
4. C5×#4545C2semidirect
5. C5×#4476C2semidirect
20. #70986C5direct (N × H)
50. #5713C2×C5semidirect
52. #5243C2×C5semidirect
53. #4545C2×C5semidirect
54. #4476C2×C5semidirect
266. C3×C5#2198semidirect
311. C5#70986direct (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
C5×#57132
C3×C5×#4532
C5×#52432
C5×#45452
C5×#44762
C5×#21983
#709865