Finite Group Databasebeta

← C5×#68117 #14288572

Subgroups — C5×#68117 #14288572

Order 1920.

472 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 106 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×#5483C211yesmaximal, characteristic
2C5×#4949C211yesmaximal, characteristic
3C5×#4722C212yesmaximal, complemented
4C3×C5×#323C211yesmaximal, characteristic
5C5×#4467C212yesmaximal, complemented
6C5×#1614—31nomaximal, Hall
7C2×C3×C5×C22⋊C4C2211yescharacteristic
8C2×C5×#217—21no—
9C2×C5×#217—21no—
10C2×C5×C4⋊Dic3C2211yescharacteristic
11C5×#924—21no—
12C5×#924C2212yes—
13C2×C5×Dic3⋊C4—21no—
14C5×#878—22no—
15C5×#878—22no—
16C3×C5×C22⋊C8C2212yes—
17C5×C3⋊#95C2211yescharacteristic
18C5×#818—22no—
19C5×#816—22no—
20#68117C511yesmaximal, characteristic, direct product factor, Hall
21C5×#543—31no—
22C5×#484—32no—
23C5×#399—31no—
24C5×#328—32no—
25C5×#323D311yescharacteristic
26C24×C3×C5D411yescharacteristic
27C22×C3×C4×C5D411yescharacteristic
28C22×C3×C4×C5C2311yescharacteristic
29C22×C5×Dic3—21no—
30C22×C5×Dic3—21no—
31C22×C5×Dic3—21no—
32C22×C5×Dic3—21no—
33C2×C5×Dic6—21no—
34C2×C5×Dic6D412yes—
35C2×C3×C5×C8—22no—
36C3×C5×C22⋊C4—22no—
37C5×#217—22no—
38C5×#217—22no—
39C5×#217—42no—
40C5×C4⋊Dic3—21no—
41C5×C4⋊Dic3—22no—
42C5×C4⋊Dic3D412yes—
43C5×Dic3⋊C4—22no—
44C5×Dic3⋊C4—22no—
45C5×Dic3⋊C4—42no—
46#5483C2×C511yescharacteristic
47#4949C2×C511yescharacteristic
48#4722C2×C512yescomplemented
49C3×#323C2×C511yescharacteristic
50#4467C2×C512yescomplemented
51C5×C22⋊Q8—32no—
52C5×C22⋊Q8—62no—
53C5×C22⋊Q8—62no—
54C5×C22⋊Q8—61no—
55C2×C5×C4⋊C4—31no—
56C2×C5×C4⋊C4—61no—
57C2×C5×C22⋊C4—61no—
58C2×C5×C22⋊C4—61no—
59C2×C5×C22⋊C4C2×D311yescharacteristic
60C5×#107—62no—
61C5×#103—62no—
62C5×C22⋊C8C2×D312yes—
63C5×#95—31no—
64#1614—31noSylow
65C23×C3×C5—21no—
66C23×C3×C5—21no—
67C23×C3×C5—41no—
68C23×C3×C5—41no—
69C23×C3×C5Q1612yes—
70C23×C3×C5C2×D411yescharacteristic
71C2×C3×C4×C5—21no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5C2×D412yes—
75C2×C5×Dic3—22no—
76C2×C5×Dic3—22no—
77C2×C5×Dic3—22no—
78C2×C5×Dic3—41no—
79C2×C5×Dic3—41no—
80C2×C5×Dic3—41no—
81C2×C5×Dic3—41no—
82C2×C5×Dic3—41no—
83C2×C5×Dic3—81no—
84C2×C5×Dic3—81no—
85C5×Dic6—42no—
86C5×Dic6—81no—
87C3×C5×C8—42no—
88C2×C3×C22⋊C4C22×C511yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×C4⋊Dic3C22×C511yescharacteristic
92#924—21no—
93#924C22×C512yes—
94C2×Dic3⋊C4—21no—
95#878—22no—
96#878—22no—
97C3×C22⋊C8C22×C512yes—
98C3⋊#95C22×C511yescharacteristic
99#818—22no—
100#816—22no—
101C24×C5D1211yescharacteristic
102C2×C5×Q8—32no—
103C2×C5×Q8—61no—
104C22×C4×C5—61no—
105C22×C4×C5—61no—
106C22×C4×C5—61no—
107C22×C4×C5—61no—
108C22×C4×C5D1211yescharacteristic
109C22×C4×C5C22×D311yescharacteristic
110C2×C5×C8—22no—
111C5×C4⋊C4—32no—
112C5×C4⋊C4—62no—
113C5×C4⋊C4—61no—
114C5×C4⋊C4—62no—
115C5×C4⋊C4—62no—
116C5×C4⋊C4—122no—
117C5×C22⋊C4—22no—
118C5×C22⋊C4—62no—
119C5×C22⋊C4—62no—
120C5×C22⋊C4—122no—
121#543—31no—
122#484—32no—
123#399—31no—
124#328—32no—
125#323C5×D311yescharacteristic
126C22×C3×C5—21no—
127C22×C3×C5—21no—
128C22×C3×C5—22no—
129C22×C3×C5—42no—
130C22×C3×C5—41no—
131C22×C3×C5—41no—
132C22×C3×C5—81no—
133C22×C3×C5#12011yescharacteristic
134C22×C3×C5C2×Q1611yescharacteristic
135C22×C3×C5#13711yescharacteristic
136C3×C4×C5—22no—
137C3×C4×C5—41no—
138C5×Dic3—42no—
139C5×Dic3—42no—
140C5×Dic3—42no—
141C5×Dic3—81no—
142C24×C3C5×D411yescharacteristic
143C22×C3×C4C5×D411yescharacteristic
144C22×C3×C4C23×C511yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C2×Dic6—21no—
150C2×Dic6C5×D412yes—
151C2×C3×C8—22no—
152C3×C22⋊C4—22no—
153#217—22no—
154#217—22no—
155#217—42no—
156C4⋊Dic3—21no—
157C4⋊Dic3—22no—
158C4⋊Dic3C5×D412yes—
159Dic3⋊C4—22no—
160Dic3⋊C4—22no—
161Dic3⋊C4—42no—
162C23×C5—21no—
163C23×C5—21no—
164C23×C5—41no—
165C23×C5—41no—
166C23×C5Dic1212yes—
167C23×C5C2×D1211yescharacteristic
168C5×Q8—122no—
169C5×Q8—241no—
170C2×C4×C5—21no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—62no—
174C2×C4×C5—62no—
175C2×C4×C5—62no—
176C2×C4×C5—121no—
177C2×C4×C5—121no—
178C2×C4×C5—121no—
179C2×C4×C5—121no—
180C2×C4×C5—121no—
181C2×C4×C5—241no—
182C2×C4×C5—241no—
183C2×C4×C5D3×D412yes—
184C5×C8—42no—
185C22⋊Q8—32no—
186C22⋊Q8—62no—
187C22⋊Q8—62no—
188C22⋊Q8—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4C2×C5×D311yescharacteristic
194#107—62no—
195#103—62no—
196C22⋊C8C2×C5×D312yes—
197#95—31no—
198C2×C3×C5—21no—
199C2×C3×C5—21no—
200C2×C3×C5—42no—
201C2×C3×C5#45111yescharacteristic
202C2×C3×C5#45311yescharacteristic
203C2×C3×C5#45811yescharacteristic
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3C5×Q1612yes—
209C23×C3C2×C5×D411yescharacteristic
210C2×C3×C4—21no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4C2×C5×D412yes—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—22no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224Dic6—42no—
225Dic6—81no—
226C3×C8—42no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—22no—
230C22×C5—42no—
231C22×C5—41no—
232C22×C5—41no—
233C22×C5—81no—
234C22×C5#88211yescharacteristic
235C22×C5C2×Dic1211yescharacteristic
236C22×C5#90911yescharacteristic
237C4×C5—22no—
238C4×C5—41no—
239C4×C5—122no—
240C4×C5—122no—
241C4×C5—122no—
242C4×C5—241no—
243C24C5×D1211yescharacteristic
244C2×Q8—32no—
245C2×Q8—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C5×D1211yescharacteristic
251C22×C4C22×C5×D311yescharacteristic
252C2×C8—22no—
253C4⋊C4—32no—
254C4⋊C4—62no—
255C4⋊C4—61no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C4⋊C4—122no—
259C22⋊C4—22no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22⋊C4—122no—
263C3×C5#161411yescharacteristic, complemented, Hall
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—42no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—81no—
271C22×C3C5×#12011yescharacteristic
272C22×C3C2×C5×Q1611yescharacteristic
273C22×C3C5×#13711yescharacteristic
274C3×C4—22no—
275C3×C4—41no—
276Dic3—42no—
277Dic3—42no—
278Dic3—42no—
279Dic3—81no—
280C2×C5—21no—
281C2×C5—21no—
282C2×C5—42no—
283C2×C5#473911yescharacteristic
284C2×C5#474211yescharacteristic
285C2×C5#474711yescharacteristic
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23C5×Dic1212yes—
291C23C2×C5×D1211yescharacteristic
292Q8—122no—
293Q8—241no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4C5×D3×D412yes—
308C8—42no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—42no—
312C2×C3C5×#45111yescharacteristic
313C2×C3C5×#45311yescharacteristic
314C2×C3C5×#45811yescharacteristic
315C5#6811711yescharacteristic, direct product factor, Sylow
316C22—21no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C5×#88211yescharacteristic
324C22C2×C5×Dic1211yescharacteristic
325C22C5×#90911yescharacteristic
326C4—22no—
327C4—41no—
328C4—122no—
329C4—122no—
330C4—122no—
331C4—241no—
332C3C5×#161411yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—42no—
336C2C5×#473911yescharacteristic
337C2C5×#474211yescharacteristic
338C2C5×#474711yescharacteristic
subgroupG/H
centerC22×C5#882
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#323C2
socleC22×C3×C5#120
prime radicalC24×C3×C5D4
hypercenterC5×#323D3
hypocenterC3C5×#1614
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#323

Complements

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

NH ≅ G/Ntype
3. C5×#4722C2semidirect
5. C5×#4467C2semidirect
20. #68117C5direct (N × H)
48. #4722C2×C5semidirect
50. #4467C2×C5semidirect
263. C3×C5#1614semidirect
315. C5#68117direct (N × H)
332. C3C5×#1614semidirect

Maximal subgroups

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

subgroupindex
C5×#54832
C5×#49492
C5×#47222
C3×C5×#3232
C5×#44672
C5×#16143
#681175