Finite Group Databasebeta

← #14308920

Subgroups — #14308920

Order 1920.

564 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 120 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×#5500C211yesmaximal, characteristic, complemented
2C3×#65277C211yesmaximal, characteristic
3#11748366C212yesmaximal
4#11746351C211yesmaximal, characteristic, complemented
5#11745977C212yesmaximal, complemented
6#10602696—31nomaximal, Hall
7C22×C3×D20C2211yescharacteristic
8C2×C3×C5×C4⋊C4C2211yescharacteristic
9C2×#6710C2212yes—
10C5×#926C2212yes—
11C5×#924—22no—
12C5×#890—22no—
13C5×#888—22no—
14C2×C3×#685—22no—
15C2×C3×C4×Dic5C2211yescharacteristic
16#90964—22no—
17#90964—22no—
18#69643—51nomaximal, Hall
19C5×#544—31no—
20#65290—32no—
21#65277D311yescharacteristic
22#64752—32no—
23#64750—31no—
24C23×C3×D5—21no—
25C22×C3×C4×C5D412yes—
26C22×C3×C4×C5C2311yescharacteristic
27C2×C5×Dic6—22no—
28C22×C3×Dic5—21no—
29C2×C3×D20—22no—
30C2×C3×D20—22no—
31C2×C3×D20—21no—
32C2×C3×D20Q812yescomplemented
33C3×C5×C4⋊C4—22no—
34#6710—22no—
35#6710—22no—
36C2×C3⋊#168—22no—
37C5×#217—22no—
38C5×C4⋊Dic3—22no—
39C5×C4⋊Dic3D412yes—
40C5×C4⋊Dic3D412yes—
41C5×Dic3⋊C4—24no—
42C4×C5×Dic3—22no—
43C3×#685—44no—
44C3×C4×Dic5—22no—
45#5500D511yescharacteristic, complemented
46C3×#390—51no—
47#5215—52no—
48#4530—51no—
49#4523—52no—
50C22×D20C2×D311yescharacteristic
51C5×C4⋊Q8—62no—
52C5×#125—62no—
53C5×C22⋊Q8—62no—
54C5×#117—32no—
55C2×C5×C4⋊C4C2×D311yescharacteristic
56C2×#685—22no—
57C2×C4×Dic5C2×D311yescharacteristic
58C2×#681—32no—
59#3941—62no—
60#3941—62no—
61#2027—151noSylow
62C23×C3×C5C2×D411yescharacteristic
63C22×C3×D5—22no—
64C22×C3×D5—42no—
65C22×C3×D5—42no—
66C22×C3×D5—41no—
67C22×C3×D5—41no—
68C2×C3×C4×C5—22no—
69C2×C3×C4×C5—42no—
70C2×C3×C4×C5C2×D412yes—
71C2×C3×C4×C5C2×Q811yescharacteristic
72C2×C3×C4×C5#4112yes—
73C2×C3×C4×C5#4111yescharacteristic
74C2×C5×Dic3—22no—
75C2×C5×Dic3—22no—
76C5×Dic6—82no—
77C2×C3×Dic5—22no—
78C2×C3×Dic5—41no—
79C2×C3×Dic5—41no—
80C3×D20—24no—
81C3×D20—42no—
82C3⋊#168—42no—
83C22×C3×D4—51no—
84C2×C3×C4⋊C4C2×D511yescharacteristic
85C2×C3×C22⋊C4—102no—
86C2×C3×C42—51no—
87#926C2×D512yes—
88#924—22no—
89C2×#208—52no—
90#890—22no—
91#888—22no—
92#832—102no—
93#832—102no—
94C23×D5—21no—
95C2×C5×Q8—62no—
96C22×C4×C5#6712yes—
97C22×C4×C5C22×D311yescharacteristic
98C22×Dic5—21no—
99C2×D20—22no—
100C2×D20—22no—
101C2×D20—21no—
102C2×D20Dic612yescomplemented
103C5×C4⋊C4—22no—
104C5×C4⋊C4—32no—
105C5×C4⋊C4—32no—
106C5×C4⋊C4—62no—
107C5×C4⋊C4—64no—
108C5×C22⋊C4—62no—
109C42×C5—62no—
110#685—44no—
111C4×Dic5—22no—
112#681—62no—
113#681—62no—
114C2×#168—62no—
115#544—31no—
116#418—152no—
117#390—51no—
118#343—151no—
119#339—152no—
120C22×C3×C5—21no—
121C22×C3×C5—21no—
122C22×C3×C5#12011yescharacteristic
123C22×C3×C5C22⋊Q811yescharacteristic
124C22×C3×C5#12411yescharacteristic
125C2×C3×D5—44no—
126C2×C3×D5—42no—
127C2×C3×D5—82no—
128C2×C3×D5—81no—
129C2×C3×D5—81no—
130C3×C4×C5—42no—
131C3×C4×C5C22⋊Q812yes—
132C3×C4×C5#12412yes—
133C3×Dic5—42no—
134C5×Dic3—42no—
135C5×Dic3—42no—
136C24×C3—101no—
137C2×C3×D4—52no—
138C2×C3×D4—102no—
139C2×C3×D4—101no—
140C2×C3×D4—102no—
141C22×C3×C4—101no—
142C22×C3×C4C22⋊D512yes—
143C22×C3×C4C22×D511yescharacteristic
144C2×Dic6—22no—
145C3×C4⋊C4—22no—
146C3×C22⋊C4—204no—
147C3×C42—102no—
148#217—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3C22⋊D512yes—
151C4⋊Dic3C22⋊D512yes—
152Dic3⋊C4—24no—
153C4×Dic3—22no—
154#208—102no—
155#208—102no—
156C2×C3⋊C8—102no—
157C23×C5C2×#6711yescharacteristic
158C22×D5—22no—
159C22×D5—42no—
160C22×D5—42no—
161C22×D5—41no—
162C22×D5—41no—
163C5×Q8—242no—
164C2×C4×C5—22no—
165C2×C4×C5—42no—
166C2×C4×C5—62no—
167C2×C4×C5—62no—
168C2×C4×C5C2×Dic611yescharacteristic
169C2×C4×C5#23511yescharacteristic
170C2×C4×C5D3×D412yes—
171C2×C4×C5#23712yes—
172C2×Dic5—22no—
173C2×Dic5—41no—
174C2×Dic5—41no—
175D20—24no—
176D20—42no—
177#168—122no—
178C22×D4—51no—
179C2×M16—152no—
180C4⋊Q8—62no—
181#125—62no—
182C22⋊Q8—62no—
183#117—32no—
184C2×C4⋊C4D3×D511yescharacteristic
185C2×C22⋊C4—102no—
186C2×C42—51no—
187#102—302no—
188#102—302no—
189C2×C3×C5—21no—
190C2×C3×C5—21no—
191C2×C3×C5#39311yescharacteristic
192C2×C3×C5#45311yescharacteristic
193C2×C3×C5#45411yescharacteristic
194C3×D5—82no—
195C23×C3—102no—
196C23×C3—202no—
197C23×C3—202no—
198C23×C3—201no—
199C23×C3—201no—
200C23×C3C2×C22⋊D511yescharacteristic
201C3×D4—104no—
202C3×D4—202no—
203C2×C3×C4—22no—
204C2×C3×C4—42no—
205C2×C3×C4—102no—
206C2×C3×C4—201no—
207C2×C3×C4—201no—
208C2×C3×C4Q8×D511yescharacteristic
209C2×C3×C4#71312yes—
210C2×C3×C4#71311yescharacteristic
211C2×C3×C4C2×C22⋊D512yes—
212C2×Dic3—22no—
213C2×Dic3—22no—
214Dic6—82no—
215C3⋊C8—202no—
216C22×C5—21no—
217C22×C5—21no—
218C22×C5#92411yescharacteristic
219C22×C5#93611yescharacteristic
220C22×C5#93711yescharacteristic
221C2×D5—44no—
222C2×D5—42no—
223C2×D5—82no—
224C2×D5—81no—
225C2×D5—81no—
226C4×C5—42no—
227C4×C5—122no—
228C4×C5—122no—
229C4×C5#87812yes—
230C4×C5#88512yes—
231Dic5—42no—
232C24—101no—
233C2×Q8—62no—
234C2×D4—52no—
235C2×D4—102no—
236C2×D4—101no—
237C2×D4—102no—
238C22×C4—101no—
239C22×C4#120112yes—
240C22×C4C2×D3×D511yescharacteristic
241M16—302no—
242M16—302no—
243C2×C8—302no—
244C4⋊C4—22no—
245C4⋊C4—32no—
246C4⋊C4—32no—
247C4⋊C4—62no—
248C4⋊C4—64no—
249C22⋊C4—62no—
250C22⋊C4—204no—
251C42—62no—
252C42—102no—
253C3×C5#202711yescharacteristic, complemented, Hall
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—204no—
257C22×C3—202no—
258C22×C3—402no—
259C22×C3—401no—
260C22×C3—401no—
261C22×C3#409211yescharacteristic
262C22×C3#409311yescharacteristic
263C22×C3#409911yescharacteristic
264C3×C4—42no—
265C3×C4—202no—
266C3×C4#409212yes—
267C3×C4#409312yes—
268Dic3—42no—
269Dic3—42no—
270C2×C5—21no—
271C2×C5—21no—
272C2×C5#494711yescharacteristic
273C2×C5#506211yescharacteristic
274C2×C5#507711yescharacteristic
275D5—82no—
276C23—102no—
277C23—202no—
278C23—202no—
279C23—201no—
280C23—201no—
281C23C2×#120111yescharacteristic
282Q8—242no—
283D4—104no—
284D4—202no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—102no—
290C2×C4—201no—
291C2×C4—201no—
292C2×C4D5×Dic611yescharacteristic
293C2×C4#676612yes—
294C2×C4#677311yescharacteristic
295C2×C4D3×C22⋊D512yes—
296C8—602no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—402no—
300C2×C3#6552011yescharacteristic
301C2×C3#6553011yescharacteristic
302C2×C3#6553411yescharacteristic
303C5#6964311yescharacteristic, complemented, Sylow
304C22—21no—
305C22—21no—
306C22—204no—
307C22—202no—
308C22—402no—
309C22—401no—
310C22—401no—
311C22#9135911yescharacteristic
312C22#9136811yescharacteristic
313C22#9156811yescharacteristic
314C4—42no—
315C4—122no—
316C4—122no—
317C4—202no—
318C4#9142112yes—
319C4#9145512yes—
320C3#1060269611yescharacteristic, complemented, Sylow
321C2—21no—
322C2—21no—
323C2—402no—
324C2#1174877311yescharacteristic
325C2#1174914011yescharacteristic
326C2#1174938611yescharacteristic
subgroupG/H
centerC22#91568
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5#120
prime radicalC22×C3×D20C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2027
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C5×#5500C2semidirect
4. #11746351C2semidirect
5. #11745977C2semidirect
32. C2×C3×D20Q8semidirect
45. #5500D5semidirect
102. C2×D20Dic6semidirect
253. C3×C5#2027semidirect
303. C5#69643semidirect
320. C3#10602696semidirect

Maximal subgroups

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

subgroupindex
C5×#55002
C3×#652772
#117483662
#117463512
#117459772
#106026963
#696435