Finite Group Databasebeta

← #14276856

Subgroups — #14276856

Order 1920.

456 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C4×#6763C211yesmaximal, characteristic
2C2×#91232C211yesmaximal, characteristic
3C3×D5×C22⋊C8C211yesmaximal, characteristic
4#10598559—31nomaximal, Hall
5C22×C3×C4×D5C2211yescharacteristic
6C2×#6763C412yes—
7C22×C3⋊#170C412yescomplemented
8C2×C4×ΓL(1,𝔽16)—21no—
9C2×C3×C8×D5—21no—
10#91235—21no—
11#91232—21no—
12#91232—21no—
13C3×C5×C22⋊C8C411yescharacteristic, complemented
14C3×C22⋊#168C411yescharacteristic, complemented
15C3×#3952—21no—
16#67678—51nomaximal, Hall
17C4×C22⋊AGL(1,𝔽5)—31no—
18C2×#4003—31no—
19D5×C22⋊C8D311yescharacteristic
20C23×C3×D5C2×C411yescharacteristic
21C22×ΓL(1,𝔽16)—22no—
22C22×C3×C4×C5C2×C411yescharacteristic
23C22×C3×Dic5C2×C411yescharacteristic
24C2×C3×C4×D5—42no—
25C2×C3×C4×D5D411yescharacteristic
26C2×C3×C4×D5D411yescharacteristic
27C2×C3×C4×D5D411yescharacteristic
28C2×C3×C4×D5Q811yescharacteristic
29#6763—22no—
30#6763C814yescomplemented
31C2×C3⋊#170—22no—
32C2×C3⋊#170—42no—
33C4×ΓL(1,𝔽16)—42no—
34C2×C3×C5×C8—21no—
35C2×C3×#168—21no—
36C3×C8×D5—42no—
37C2×C3×C22⋊C8—51no—
38C4×#217—51no—
39C2×#830—51no—
40C22×C4×D5C2×D311yescharacteristic
41C2×C22⋊AGL(1,𝔽5)—32no—
42C22×#170—32no—
43C2×C4×AGL(1,𝔽5)—61no—
44C2×C8×D5—21no—
45(C2×AGL(1,𝔽5))⋊C4—61no—
46#4003—61no—
47#4003—61no—
48C5×C22⋊C8Dic311yescharacteristic, complemented
49C22⋊#168Dic311yescharacteristic, complemented
50#3952—21no—
51#1282—151noSylow
52C23×C3×C5C4211yescharacteristic
53C22×C3×D5—41no—
54C22×C3×D5—41no—
55C22×C3×D5C22⋊C411yescharacteristic
56C22×C3×D5C4⋊C411yescharacteristic
57C22×C3×D5C2×C812yes—
58C22×C3×D5M1612yes—
59C2×ΓL(1,𝔽16)—24no—
60C2×ΓL(1,𝔽16)—42no—
61C2×ΓL(1,𝔽16)—42no—
62C2×C3×C4×C5—41no—
63C2×C3×C4×C5C22⋊C411yescharacteristic
64C2×C3×C4×C5C4⋊C411yescharacteristic
65C2×C3×Dic5—22no—
66C2×C3×Dic5C22⋊C411yescharacteristic
67C2×C3×Dic5C4⋊C411yescharacteristic
68C3×C4×D5—22no—
69C3×C4×D5—22no—
70C3×C4×D5—41no—
71C3×C4×D5—41no—
72C3⋊#170—42no—
73C3×C5×C8—41no—
74C3×#168—41no—
75C23×C3×C4—51no—
76C22×C3×C8—101no—
77C2×#217—52no—
78C2×C4×Dic3—101no—
79C22×C3⋊C8—52no—
80C3×C22⋊C8—51no—
81C3×C22⋊C8—101no—
82C3×C22⋊C8AGL(1,𝔽5)11yescharacteristic, complemented
83C3⋊#95—101no—
84#830—101no—
85#830—101no—
86C23×D5C4×D311yescharacteristic
87C22×AGL(1,𝔽5)—62no—
88C22×C4×C5C2×Dic311yescharacteristic
89C22×Dic5C4×D311yescharacteristic
90C2×C4×D5—42no—
91C2×C4×D5Dic611yescharacteristic
92C2×C4×D5D1211yescharacteristic
93C2×C4×D5#6711yescharacteristic
94C2×C4×D5#6711yescharacteristic
95C22⋊AGL(1,𝔽5)—34no—
96C22⋊AGL(1,𝔽5)—62no—
97C2×#170—62no—
98C2×#170—122no—
99C4×AGL(1,𝔽5)—122no—
100C2×C5×C8—21no—
101C2×#168—21no—
102C8×D5—42no—
103C2×C22⋊C8—51no—
104C2×C22⋊C8—151no—
105C4×C22⋊C4—151no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5#9511yescharacteristic
109C22×C3×C5C4×C811yescharacteristic
110C22×C3×C5#9711yescharacteristic
111C2×C3×D5—21no—
112C2×C3×D5—21no—
113C2×C3×D5—21no—
114C2×C3×D5—21no—
115C2×C3×D5—42no—
116C2×C3×D5—42no—
117C2×C3×D5C22⋊C812yes—
118C2×C3×D5C4⋊C812yes—
119ΓL(1,𝔽16)—44no—
120C3×C4×C5—21no—
121C3×C4×C5—21no—
122C3×Dic5—21no—
123C3×Dic5#9912yes—
124C24×C3—51no—
125C22×C3×C4—51no—
126C22×C3×C4—51no—
127C22×C3×C4—51no—
128C22×C3×C4—51no—
129C22×C3×C4—51no—
130C22×C3×C4—202no—
131C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
132C22×Dic3—102no—
133C2×C3×C8—21no—
134C2×C3×C8—101no—
135C2×C3×C8—202no—
136#217—54no—
137#217—102no—
138C4×Dic3—202no—
139C2×C3⋊C8—102no—
140C2×C3⋊C8—202no—
141C23×C5C4×Dic311yescharacteristic
142C22×D5—41no—
143C22×D5—41no—
144C22×D5D3×C812yes—
145C22×D5ΓL(1,𝔽25)12yes—
146C22×D5Dic3⋊C411yescharacteristic
147C22×D5(C2×D3)⋊C411yescharacteristic
148C2×AGL(1,𝔽5)—64no—
149C2×AGL(1,𝔽5)—122no—
150C2×AGL(1,𝔽5)—122no—
151C2×C4×C5—41no—
152C2×C4×C5C4⋊Dic311yescharacteristic
153C2×C4×C5#21711yescharacteristic
154C2×Dic5—22no—
155C2×Dic5Dic3⋊C411yescharacteristic
156C2×Dic5(C2×D3)⋊C411yescharacteristic
157C4×D5—22no—
158C4×D5—22no—
159C4×D5—41no—
160C4×D5—41no—
161#170—122no—
162C5×C8—41no—
163#168—41no—
164C23×C4—51no—
165C22×C8—101no—
166C22×C8—152no—
167C2×C22⋊C4—152no—
168C2×C42—301no—
169C22⋊C8—51no—
170C22⋊C8—101no—
171C22⋊C8—301no—
172C22⋊C8—301no—
173C22⋊C8ΓL(1,𝔽16)11yescharacteristic, complemented
174#95—301no—
175C2×C3×C5—21no—
176C2×C3×C5—21no—
177C2×C3×C5#33611yescharacteristic
178C2×C3×C5#34211yescharacteristic
179C2×C3×C5#34411yescharacteristic
180C3×D5—22no—
181C3×D5—22no—
182C23×C3—51no—
183C23×C3—51no—
184C23×C3—52no—
185C23×C3—52no—
186C23×C3—201no—
187C23×C3—201no—
188C23×C3C4×AGL(1,𝔽5)11yescharacteristic
189C2×C3×C4—41no—
190C2×C3×C4—51no—
191C2×C3×C4—51no—
192C2×C3×C4—102no—
193C2×C3×C4—102no—
194C2×C3×C4—102no—
195C2×C3×C4—201no—
196C2×C3×C4—201no—
197C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic
198C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
199C2×Dic3—104no—
200C2×Dic3—202no—
201C2×Dic3—202no—
202C3×C8—41no—
203C3×C8—201no—
204C3⋊C8—202no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5C8×Dic311yescharacteristic
208C22×C5#81511yescharacteristic
209C22×C5C3⋊#9511yescharacteristic
210C2×D5—21no—
211C2×D5—21no—
212C2×D5—21no—
213C2×D5—21no—
214C2×D5—42no—
215C2×D5—42no—
216C2×D5Dic3⋊C812yes—
217C2×D5(C2×D3)⋊C812yes—
218AGL(1,𝔽5)—124no—
219C4×C5—21no—
220C4×C5—21no—
221Dic5—21no—
222Dic5#80612yes—
223C24—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—202no—
230C22×C4—302no—
231C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
232C2×C8—21no—
233C2×C8—101no—
234C2×C8—202no—
235C2×C8—302no—
236C2×C8—602no—
237C22⋊C4—154no—
238C22⋊C4—302no—
239C42—602no—
240C3×C5#128211yescharacteristic, complemented, Hall
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—52no—
244C22×C3—52no—
245C22×C3—101no—
246C22×C3—101no—
247C22×C3—101no—
248C22×C3—101no—
249C22×C3—202no—
250C22×C3—202no—
251C22×C3C8×AGL(1,𝔽5)11yescharacteristic
252C22×C3#399211yescharacteristic
253C22×C3(C2×AGL(1,𝔽5))⋊C411yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—52no—
257C3×C4—101no—
258Dic3—204no—
259C2×C5—21no—
260C2×C5—21no—
261C2×C5#452711yescharacteristic
262C2×C5#455011yescharacteristic
263C2×C5#455911yescharacteristic
264D5—22no—
265D5—22no—
266C23—51no—
267C23—51no—
268C23—52no—
269C23—52no—
270C23—201no—
271C23—201no—
272C23C4×ΓL(1,𝔽16)11yescharacteristic
273C2×C4—41no—
274C2×C4—51no—
275C2×C4—51no—
276C2×C4—102no—
277C2×C4—102no—
278C2×C4—102no—
279C2×C4—201no—
280C2×C4—201no—
281C2×C4—304no—
282C2×C4—602no—
283C2×C4—602no—
284C2×C4C4⋊ΓL(1,𝔽16)11yescharacteristic
285C2×C4#676311yescharacteristic
286C8—41no—
287C8—201no—
288C8—602no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—102no—
292C2×C3—102no—
293C2×C3#6486611yescharacteristic
294C2×C3(C2×AGL(1,𝔽5))⋊C811yescharacteristic
295C2×C3#6490211yescharacteristic
296C5#6767811yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—52no—
300C22—52no—
301C22—101no—
302C22—101no—
303C22—101no—
304C22—101no—
305C22—202no—
306C22—202no—
307C22C8×ΓL(1,𝔽16)11yescharacteristic
308C22#9122111yescharacteristic
309C22#9123511yescharacteristic
310C4—21no—
311C4—21no—
312C4—52no—
313C4—101no—
314C4—604no—
315C3#1059855911yescharacteristic, complemented, Sylow
316C2—21no—
317C2—21no—
318C2—102no—
319C2—102no—
320C2#1174673711yescharacteristic
321C2#1174676311yescharacteristic
322C2#1174677311yescharacteristic
subgroupG/H
centerC22#91235
commutatorC22×C3×C5C4×C8
FrattiniC22×C4C2×ΓL(1,𝔽16)
FittingC3×C5×C22⋊C8C4
socleC22×C3×C5#95
prime radicalC23×C3×D5C2×C4
cosocleC22×C3×C4×D5C22
hypercenterC22⋊C8ΓL(1,𝔽16)
hypocenterC3×C5#1282
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
7. C22×C3⋊#170C4semidirect
13. C3×C5×C22⋊C8C4semidirect
14. C3×C22⋊#168C4semidirect
30. #6763C8semidirect
48. C5×C22⋊C8Dic3semidirect
49. C22⋊#168Dic3semidirect
82. C3×C22⋊C8AGL(1,𝔽5)semidirect
173. C22⋊C8ΓL(1,𝔽16)semidirect
240. C3×C5#1282semidirect
296. C5#67678semidirect
315. C3#10598559semidirect

Maximal subgroups

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

subgroupindex
C4×#67632
C2×#912322
C3×D5×C22⋊C82
#105985593
#676785