Finite Group Databasebeta

← #14313765

Subgroups — #14313765

Order 1920.

460 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5724C211yesmaximal, characteristic
2#11749372C212yesmaximal
3#11746369C211yesmaximal, characteristic, complemented
4C3×#64768C211yesmaximal, characteristic, complemented
5#11746016C212yesmaximal, complemented
6#10602666—31nomaximal, Hall
7C2×C5×#239C2211yescharacteristic
8C2×C5×D3×Q8C411yescharacteristic
9C3×C5×C4⋊Q8C2211yescharacteristic
10C5×#947C411yescharacteristic
11C5×#944C2211yescharacteristic
12C5×#896—22no—
13C5×#894—22no—
14C5×#875C411yescharacteristic
15C5×#872C411yescharacteristic
16D3×#681—22no—
17#91120C2212yes—
18C3×#3968C2212yes—
19#90959—22no—
20#69173—51nomaximal, Hall
21C5×#536—31no—
22#65526—32no—
23#64768—32no—
24#64768—31no—
25#64768D311yescharacteristic, complemented
26C2×C3×C5×Q8C2×C411yescharacteristic
27C2×C3×C5×Q8C2311yescharacteristic
28C5×#239—42no—
29C5×D3×Q8—42no—
30C2×C5×D12C2×C411yescharacteristic
31C2×C5×D12D412yes—
32C2×C4×C5×D3—21no—
33C2×C4×C5×D3C2×C411yescharacteristic
34C2×C4×C5×D3D412yes—
35C2×C5×Dic6—21no—
36C2×C5×Dic6C2×C411yescharacteristic
37C3×C5×C4⋊C4—22no—
38C3×C42×C5C2×C411yescharacteristic
39#6710—22no—
40C5×(C2×D3)⋊C4—21no—
41C5×(C2×D3)⋊C4—22no—
42C5×(C2×D3)⋊C4—22no—
43C5×Dic3⋊C4—22no—
44C5×Dic3⋊C4—21no—
45C4×C5×Dic3C2×C411yescharacteristic
46C3×#681—22no—
47#6650—42no—
48D3×#168—42no—
49#5724D511yescharacteristic
50#4751—52no—
51C3×#356—51no—
52#4548—51no—
53#4478—52no—
54C2×C5×#41—31no—
55C22×C5×Q8—31no—
56C5×C4⋊Q8—31no—
57C5×C4⋊Q8C2×D311yescharacteristic
58C5×#124—31no—
59C5×#124—31no—
60C5×#123—62no—
61C5×C22⋊Q8—62no—
62C5×#117—31no—
63C2×#681—62no—
64#3968—32no—
65#3968C2×D312yes—
66#3965—62no—
67#2135—151noSylow
68C22×C5×D3C22⋊C412yes—
69C22×C5×D3C22⋊C411yescharacteristic
70C3×C5×Q8—41no—
71C3×C5×Q8—41no—
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5C22×C411yescharacteristic
75C2×C3×C4×C5C2×D412yes—
76C2×C5×Dic3—21no—
77C2×C5×Dic3—21no—
78C2×C5×Dic3C22⋊C411yescharacteristic
79C5×D12—41no—
80C5×D12—42no—
81C4×C5×D3—22no—
82C4×C5×D3—22no—
83C4×C5×D3—22no—
84C4×C5×D3—42no—
85C5×Dic6—42no—
86C5×Dic6—41no—
87C3⋊#168—42no—
88C3×#168—42no—
89C2×#239C2×D511yescharacteristic
90C2×D3×Q8Dic511yescharacteristic
91C3×C4⋊Q8C2×D511yescharacteristic
92#947Dic511yescharacteristic
93#944C2×D511yescharacteristic
94D3×M16—102no—
95#896—22no—
96#894—22no—
97#875Dic511yescharacteristic
98#872Dic511yescharacteristic
99C3×#101—52no—
100#836—52no—
101#823—102no—
102C5×#41—122no—
103C2×C5×Q8—31no—
104C2×C5×Q8—61no—
105C2×C5×Q8—122no—
106C2×C5×Q8C4×D311yescharacteristic
107C2×C5×Q8C22×D311yescharacteristic
108C2×C5×D4—31no—
109C2×C5×D4—32no—
110C22×C4×C5—31no—
111C22×C4×C5—32no—
112C22×C4×C5—61no—
113C5×C4⋊C4—22no—
114C5×C4⋊C4—61no—
115C5×C4⋊C4—62no—
116C5×C22⋊C4—61no—
117C5×C22⋊C4—62no—
118C5×C22⋊C4—62no—
119C42×C5—31no—
120C42×C5C4×D311yescharacteristic
121#681—22no—
122#681—62no—
123#681—122no—
124C2×#168—122no—
125#536—31no—
126#413—152no—
127#356—51no—
128#356—152no—
129#356—151no—
130C22×C3×C5C2×C22⋊C411yescharacteristic
131C2×C5×D3—21no—
132C2×C5×D3—22no—
133C2×C5×D3—42no—
134C2×C5×D3—41no—
135C3×C4×C5—22no—
136C3×C4×C5—21no—
137C3×C4×C5—41no—
138C3×C4×C5—41no—
139C5×Dic3—41no—
140C5×Dic3—41no—
141C5×Dic3#9912yes—
142C2×C3×Q8C2×Dic511yescharacteristic
143C2×C3×Q8C22×D511yescharacteristic
144#239—42no—
145D3×Q8—42no—
146C2×D12C2×Dic511yescharacteristic
147C2×D12C22⋊D512yes—
148C2×C4×D3—21no—
149C2×C4×D3C2×Dic511yescharacteristic
150C2×C4×D3C22⋊D512yes—
151C2×Dic6—21no—
152C2×Dic6C2×Dic511yescharacteristic
153C3×M16—102no—
154C3×C4⋊C4—22no—
155C3×C42C2×Dic511yescharacteristic
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—22no—
158(C2×D3)⋊C4—22no—
159Dic3⋊C4—22no—
160Dic3⋊C4—21no—
161C4×Dic3C2×Dic511yescharacteristic
162#208—102no—
163ΓL(1,𝔽25)—202no—
164D3×C8—202no—
165C23×C5—32no—
166C23×C5—31no—
167C5×Q8—41no—
168C5×Q8—41no—
169C5×Q8—121no—
170C5×Q8—122no—
171C5×D4—122no—
172C5×D4—121no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—31no—
176C2×C4×C5—62no—
177C2×C4×C5—62no—
178C2×C4×C5—62no—
179C2×C4×C5—61no—
180C2×C4×C5—61no—
181C2×C4×C5—122no—
182C2×C4×C5C2×C4×D311yescharacteristic
183C2×C4×C5D3×D412yes—
184#168—42no—
185#168—122no—
186C2×#41—31no—
187C22×Q8—31no—
188C2×M16—302no—
189C4⋊Q8—31no—
190C4⋊Q8D3×D511yescharacteristic
191#124—31no—
192#124—31no—
193#123—62no—
194C22⋊Q8—62no—
195#117—31no—
196#101—52no—
197#101—152no—
198#100—302no—
199C2×C3×C5—21no—
200C2×C3×C5C2×#9911yescharacteristic
201C5×D3—41no—
202C5×D3—42no—
203C22×D3C22⋊Dic512yes—
204C22×D3C22⋊Dic511yescharacteristic
205C3×Q8—41no—
206C3×Q8—41no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4C22×Dic511yescharacteristic
210C2×C3×C4C2×C22⋊D512yes—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3C22⋊Dic511yescharacteristic
214D12—41no—
215D12—42no—
216C4×D3—22no—
217C4×D3—22no—
218C4×D3—22no—
219C4×D3—42no—
220Dic6—42no—
221Dic6—41no—
222C3×C8—202no—
223C3⋊C8—202no—
224C22×C5—61no—
225C22×C5—62no—
226C22×C5—122no—
227C22×C5—121no—
228C22×C5D3×C22⋊C411yescharacteristic
229C4×C5—22no—
230C4×C5—21no—
231C4×C5—32no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5—121no—
235C4×C5—121no—
236#41—122no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—122no—
240C2×Q8D3×Dic511yescharacteristic
241C2×Q8C2×D3×D511yescharacteristic
242C2×D4—31no—
243C2×D4—32no—
244C22×C4—31no—
245C22×C4—32no—
246C22×C4—61no—
247M16—102no—
248M16—302no—
249M16—602no—
250C2×C8—602no—
251C4⋊C4—22no—
252C4⋊C4—61no—
253C4⋊C4—62no—
254C22⋊C4—61no—
255C22⋊C4—62no—
256C22⋊C4—62no—
257C42—31no—
258C42D3×Dic511yescharacteristic
259C3×C5#213511yescharacteristic, complemented, Hall
260C22×C3C2×C22⋊Dic511yescharacteristic
261C2×D3—21no—
262C2×D3—22no—
263C2×D3—42no—
264C2×D3—41no—
265C3×C4—22no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269Dic3—41no—
270Dic3—41no—
271Dic3#396612yes—
272C2×C5—21no—
273C2×C5—121no—
274C2×C5—122no—
275C2×C5D3×#9911yescharacteristic
276C23—32no—
277C23—31no—
278Q8—41no—
279Q8—41no—
280Q8—121no—
281Q8—122no—
282D4—122no—
283D4—121no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—31no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—122no—
293C2×C4C2×D3×Dic511yescharacteristic
294C2×C4D3×C22⋊D512yes—
295C8—202no—
296C8—602no—
297C2×C3—21no—
298C2×C3C2×#396611yescharacteristic
299D3—41no—
300D3—42no—
301C5#6917311yescharacteristic, complemented, Sylow
302C22—61no—
303C22—62no—
304C22—122no—
305C22—121no—
306C22D3×C22⋊Dic511yescharacteristic
307C4—22no—
308C4—21no—
309C4—32no—
310C4—41no—
311C4—41no—
312C4—121no—
313C4—121no—
314C3#1060266611yescharacteristic, complemented, Sylow
315C2—21no—
316C2—121no—
317C2—122no—
318C2D3×#396611yescharacteristic
subgroupG/H
centerC2D3×#3966
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×Q8C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC2×C3×C5C2×#99
prime radicalC2×C5×#239C22
cosocleC2×C3×C5×Q8C23
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#2135
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11746369C2semidirect
4. C3×#64768C2semidirect
5. #11746016C2semidirect
25. #64768D3semidirect
259. C3×C5#2135semidirect
301. C5#69173semidirect
314. C3#10602666semidirect

Maximal subgroups

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

subgroupindex
C5×#57242
#117493722
#117463692
C3×#647682
#117460162
#106026663
#691735