Finite Group Databasebeta

← #14296308

Subgroups — #14296308

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91221C212yesmaximal
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
3D5⋊(C4⋊(C3⋊C8))C211yesmaximal, characteristic
4C3×C4×C8×D5C211yesmaximal, characteristic
5#11746763C212yesmaximal
6#10603842—31nomaximal, Hall
7C2×C4⋊ΓL(1,𝔽16)C412yes—
8C2×C4×ΓL(1,𝔽16)C2212yes—
9C2×D5⋊(C3⋊C8)C2212yes—
10C2×C3×C8×D5C2211yescharacteristic
11C2×C3×C8×D5C2211yescharacteristic
12C3×C42×D5C2211yescharacteristic
13#91235C412yes—
14#91233C412yescomplemented
15C4⋊(C3⋊#170)C412yescomplemented
16#91221—24no—
17C3×C4×C5×C8C411yescharacteristic, complemented
18C3×C8×Dic5C411yescharacteristic, complemented
19C3×C8×Dic5C411yescharacteristic, complemented
20C3×C4×#168C411yescharacteristic, complemented
21#69757—51nomaximal, Hall
22C2×#3992—32no—
23C4×C4⋊AGL(1,𝔽5)—31no—
24D5⋊(C4⋊C8)—31no—
25C4×C8×D5D311yescharacteristic
26(C2×AGL(1,𝔽5))⋊C8—32no—
27C22×ΓL(1,𝔽16)C2×C414yes—
28C2×C3×C4×D5C2×C411yescharacteristic
29C2×C3×C4×D5C2×C411yescharacteristic
30C2×C3×C4×D5C2311yescharacteristic
31C2×C3⋊#170C2×C414yes—
32C4⋊ΓL(1,𝔽16)—24no—
33C4×ΓL(1,𝔽16)—22no—
34C4×ΓL(1,𝔽16)—22no—
35D5⋊(C3⋊C8)—22no—
36D5⋊(C3⋊C8)—22no—
37C2×C3×C5×C8C2×C411yescharacteristic
38C2×C3×C5×C8C2×C411yescharacteristic
39C3×C42×C5C2×C411yescharacteristic
40C3×C4×Dic5C2×C411yescharacteristic
41C3×C4×Dic5C2×C411yescharacteristic
42C3×C4×Dic5C2×C411yescharacteristic
43C2×C3×#168C2×C411yescharacteristic
44C2×C3×#168C2×C411yescharacteristic
45C3×C8×D5—21no—
46C3×C8×D5—21no—
47C3×C8×D5D412yes—
48C3×C8×D5Q812yes—
49C2×C3×C4×C8—51no—
50C2×#815—52no—
51C4×C4⋊Dic3—51no—
52C2×C4⋊(C3⋊C8)—51no—
53#4550—52no—
54C2×C4⋊AGL(1,𝔽5)—32no—
55C2×C4×AGL(1,𝔽5)—32no—
56C2×D5⋊C8—32no—
57C2×C8×D5C2×D311yescharacteristic
58C2×C8×D5C2×D311yescharacteristic
59C42×D5C2×D311yescharacteristic
60(C2×AGL(1,𝔽5))⋊C4—32no—
61#4004—32no—
62C4⋊#170—32no—
63#3992—64no—
64C4×C5×C8Dic311yescharacteristic, complemented
65C8×Dic5Dic311yescharacteristic, complemented
66C8×Dic5Dic311yescharacteristic, complemented
67C4×#168Dic311yescharacteristic, complemented
68#1770—151noSylow
69C22×C3×D5C22×C411yescharacteristic
70C2×ΓL(1,𝔽16)—24no—
71C2×ΓL(1,𝔽16)—24no—
72C2×ΓL(1,𝔽16)—24no—
73C2×C3×C4×C5C4211yescharacteristic
74C2×C3×C4×C5C4211yescharacteristic
75C2×C3×C4×C5C22×C411yescharacteristic
76C2×C3×Dic5C4211yescharacteristic
77C2×C3×Dic5C4211yescharacteristic
78C2×C3×Dic5C22×C411yescharacteristic
79C3×C4×D5—22no—
80C3×C4×D5M1614yes—
81C3×C4×D5C2×D411yescharacteristic
82C3×C4×D5C2×Q811yescharacteristic
83C3×C4×D5#4111yescharacteristic
84C3×C4×D5#4111yescharacteristic
85C3⋊#170—24no—
86C3×C5×C8—21no—
87C3×C5×C8C4⋊C412yescomplemented
88C3×#168—21no—
89C3×#168C4⋊C412yescomplemented
90C22×C3×C8—51no—
91C22×C3×C8—51no—
92C2×C3×C42—51no—
93C2×C4⋊Dic3—52no—
94C2×C4×Dic3—52no—
95C22×C3⋊C8—52no—
96C3×C4×C8—51no—
97C3×C4×C8—51no—
98C3×C4×C8—51no—
99C3×C4×C8AGL(1,𝔽5)11yescharacteristic, complemented
100C3⋊#95—52no—
101#815—104no—
102C4⋊(C3⋊C8)—52no—
103C4⋊(C3⋊C8)—52no—
104C22×AGL(1,𝔽5)—34no—
105C2×C4×D5C4×D311yescharacteristic
106C2×C4×D5C4×D311yescharacteristic
107C2×C4×D5C22×D311yescharacteristic
108C2×#170—34no—
109C4⋊AGL(1,𝔽5)—64no—
110C4×AGL(1,𝔽5)—62no—
111C4×AGL(1,𝔽5)—62no—
112D5⋊C8—62no—
113D5⋊C8—62no—
114C2×C5×C8C2×Dic311yescharacteristic
115C2×C5×C8C2×Dic311yescharacteristic
116C42×C5C2×Dic311yescharacteristic
117C4×Dic5C4×D311yescharacteristic
118C4×Dic5C4×D311yescharacteristic
119C4×Dic5C2×Dic311yescharacteristic
120C2×#168C2×Dic311yescharacteristic
121C2×#168C2×Dic311yescharacteristic
122C8×D5—21no—
123C8×D5—21no—
124C8×D5Dic612yes—
125C8×D5D1212yes—
126C2×C4⋊C8—151no—
127C2×#97—152no—
128C2×C4×C8—51no—
129C4×C4⋊C4—151no—
130#336—152no—
131C22×C3×C5C2×C4211yescharacteristic
132C2×C3×D5C4×D411yescharacteristic
133C2×C3×D5C4×Q811yescharacteristic
134C2×C3×D5C2×M1612yes—
135C2×C3×D5#13112yes—
136ΓL(1,𝔽16)—44no—
137C3×C4×C5—21no—
138C3×C4×C5#9712yescomplemented
139C3×C4×C5C2×C4⋊C411yescharacteristic
140C3×C4×C5#11711yescharacteristic
141C3×Dic5—21no—
142C3×Dic5#9712yescomplemented
143C3×Dic5C4×D411yescharacteristic
144C3×Dic5C4×Q811yescharacteristic
145C22×C3×C4—51no—
146C22×C3×C4—51no—
147C22×C3×C4—51no—
148C22×Dic3—54no—
149C2×C3×C8—51no—
150C2×C3×C8—52no—
151C2×C3×C8—52no—
152C2×C3×C8—51no—
153C2×C3×C8—101no—
154C2×C3×C8—101no—
155C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
156C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
157C3×C42—51no—
158C3×C42—51no—
159C3×C42—51no—
160C3×C42C2×AGL(1,𝔽5)11yescharacteristic
161C4⋊Dic3—104no—
162C4×Dic3—102no—
163C4×Dic3—102no—
164C2×C3⋊C8—54no—
165C2×C3⋊C8—102no—
166C2×C3⋊C8—102no—
167C22×D5C2×C4×D311yescharacteristic
168C2×AGL(1,𝔽5)—64no—
169C2×AGL(1,𝔽5)—64no—
170C2×AGL(1,𝔽5)—64no—
171C2×C4×C5C4×Dic311yescharacteristic
172C2×C4×C5C4×Dic311yescharacteristic
173C2×C4×C5C22×Dic311yescharacteristic
174C2×Dic5C4×Dic311yescharacteristic
175C2×Dic5C4×Dic311yescharacteristic
176C2×Dic5C2×C4×D311yescharacteristic
177C4×D5—22no—
178C4×D5ΓL(1,𝔽25)14yes—
179C4×D5C2×Dic611yescharacteristic
180C4×D5C2×D1211yescharacteristic
181C4×D5#23511yescharacteristic
182C4×D5#23511yescharacteristic
183#170—64no—
184C5×C8—21no—
185C5×C8C4⋊Dic312yescomplemented
186#168—21no—
187#168C4⋊Dic312yescomplemented
188C22×C8—51no—
189C22×C8—51no—
190C22×C8—152no—
191C2×C4⋊C4—152no—
192C2×C42—51no—
193C2×C42—152no—
194C4⋊C8—152no—
195C4⋊C8—152no—
196#97—304no—
197C4×C8—51no—
198C4×C8—51no—
199C4×C8—51no—
200C4×C8ΓL(1,𝔽16)11yescharacteristic, complemented
201#95—152no—
202C2×C3×C5C4×C4⋊C411yescharacteristic
203C2×C3×C5C2×#9711yescharacteristic
204C2×C3×C5#40511yescharacteristic
205C3×D5#43612yes—
206C3×D5#44612yes—
207C23×C3—51no—
208C2×C3×C4—51no—
209C2×C3×C4—51no—
210C2×C3×C4—51no—
211C2×C3×C4—51no—
212C2×C3×C4—51no—
213C2×C3×C4—51no—
214C2×C3×C4—54no—
215C2×C3×C4—51no—
216C2×C3×C4—102no—
217C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
218C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
219C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
220C2×Dic3—104no—
221C2×Dic3—104no—
222C2×Dic3—104no—
223C3×C8—21no—
224C3×C8—52no—
225C3×C8—101no—
226C3×C8C4⋊AGL(1,𝔽5)12yescomplemented
227C3⋊C8—104no—
228C22×C5C2×C4×Dic311yescharacteristic
229C2×D5C4×Dic611yescharacteristic
230C2×D5C4×D1211yescharacteristic
231C2×D5C2×ΓL(1,𝔽25)12yes—
232C2×D5#90112yes—
233AGL(1,𝔽5)—124no—
234C4×C5—21no—
235C4×C5#81512yescomplemented
236C4×C5C2×C4⋊Dic311yescharacteristic
237C4×C5#92611yescharacteristic
238Dic5—21no—
239Dic5#81512yescomplemented
240Dic5C4×Dic611yescharacteristic
241Dic5C4×D1211yescharacteristic
242C22×C4—51no—
243C22×C4—51no—
244C22×C4—51no—
245C22×C4—154no—
246C2×C8—51no—
247C2×C8—52no—
248C2×C8—52no—
249C2×C8—51no—
250C2×C8—101no—
251C2×C8—101no—
252C2×C8—154no—
253C2×C8—302no—
254C2×C8—302no—
255C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
256C2×C8C2×ΓL(1,𝔽16)11yescharacteristic
257C4⋊C4—304no—
258C42—51no—
259C42—51no—
260C42—51no—
261C42—302no—
262C42—302no—
263C42C2×ΓL(1,𝔽16)11yescharacteristic
264C3×C5#177011yescharacteristic, complemented, Hall
265C22×C3—51no—
266C22×C3—51no—
267C22×C3—52no—
268C22×C3—52no—
269C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
270C3×C4—21no—
271C3×C4—51no—
272C3×C4—51no—
273C3×C4—52no—
274C3×C4—101no—
275C3×C4#399212yescomplemented
276C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
277C3×C4#413011yescharacteristic
278Dic3—204no—
279C2×C5C4×C4⋊Dic311yescharacteristic
280C2×C5C2×#81511yescharacteristic
281C2×C5#510111yescharacteristic
282D5#467912yes—
283D5#468812yes—
284C23—51no—
285C2×C4—51no—
286C2×C4—51no—
287C2×C4—51no—
288C2×C4—51no—
289C2×C4—54no—
290C2×C4—51no—
291C2×C4—51no—
292C2×C4—51no—
293C2×C4—102no—
294C2×C4—304no—
295C2×C4—304no—
296C2×C4—304no—
297C2×C4C4×ΓL(1,𝔽16)11yescharacteristic
298C2×C4C4×ΓL(1,𝔽16)11yescharacteristic
299C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
300C8—21no—
301C8—52no—
302C8—101no—
303C8—304no—
304C8C4⋊ΓL(1,𝔽16)12yescomplemented
305C2×C3—52no—
306C2×C3—52no—
307C2×C3C4×C4⋊AGL(1,𝔽5)11yescharacteristic
308C2×C3C2×#399211yescharacteristic
309C2×C3#6571811yescharacteristic
310C5#6975711yescharacteristic, complemented, Sylow
311C22—51no—
312C22—51no—
313C22—52no—
314C22—52no—
315C22C2×C4×ΓL(1,𝔽16)11yescharacteristic
316C4—21no—
317C4—51no—
318C4—51no—
319C4—52no—
320C4—101no—
321C4—604no—
322C4#9122112yescomplemented
323C4C2×C4⋊ΓL(1,𝔽16)11yescharacteristic
324C4#9198911yescharacteristic
325C3#1060384211yescharacteristic, complemented, Sylow
326C2—52no—
327C2—52no—
328C2C4×C4⋊ΓL(1,𝔽16)11yescharacteristic
329C2C2×#9122111yescharacteristic
330C2#1175237711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C2×C42
FittingC3×C4×C5×C8C4
prime radicalC22×C3×D5C22×C4
cosocleC2×C3×C4×D5C23
hypercenterC4×C8ΓL(1,𝔽16)
hypocenterC3×C5#1770
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
14. #91233C4semidirect
15. C4⋊(C3⋊#170)C4semidirect
17. C3×C4×C5×C8C4semidirect
18. C3×C8×Dic5C4semidirect
19. C3×C8×Dic5C4semidirect
20. C3×C4×#168C4semidirect
64. C4×C5×C8Dic3semidirect
65. C8×Dic5Dic3semidirect
66. C8×Dic5Dic3semidirect
67. C4×#168Dic3semidirect
87. C3×C5×C8C4⋊C4semidirect
89. C3×#168C4⋊C4semidirect
99. C3×C4×C8AGL(1,𝔽5)semidirect
138. C3×C4×C5#97semidirect
142. C3×Dic5#97semidirect
185. C5×C8C4⋊Dic3semidirect
187. #168C4⋊Dic3semidirect
200. C4×C8ΓL(1,𝔽16)semidirect
226. C3×C8C4⋊AGL(1,𝔽5)semidirect
235. C4×C5#815semidirect
239. Dic5#815semidirect
264. C3×C5#1770semidirect
275. C3×C4#3992semidirect
304. C8C4⋊ΓL(1,𝔽16)semidirect
310. C5#69757semidirect
322. C4#91221semidirect
325. C3#10603842semidirect

Maximal subgroups

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

subgroupindex
C2×#912212
C4×C4⋊ΓL(1,𝔽16)2
D5⋊(C4⋊(C3⋊C8))2
C3×C4×C8×D52
#117467632
#106038423
#697575