Finite Group Databasebeta

← #14411532

Subgroups — #14411532

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×ΓL(1,𝔽16)C211yesmaximal, characteristic
2D5⋊#948C211yesmaximal, characteristic
3C3×D5×#131C211yesmaximal, characteristic
4#11752387C213yesmaximal, complemented
5M16×ΓL(1,𝔽16)C213yesmaximal, complemented
6#11752377C213yesmaximal, complemented
7C2×#91221C213yesmaximal, complemented
8#10617415—31nomaximal, Hall
9C3×D5×#41C2211yescharacteristic
10Q8×ΓL(1,𝔽16)C412yes—
11#91992C412yescomplemented
12D4×ΓL(1,𝔽16)C416yes—
13#91990C416yescomplemented
14#91989C2213yes—
15C2×C4×ΓL(1,𝔽16)C2213yes—
16D5⋊#208C2213yes—
17C2×D5⋊(C3⋊C8)C2213yes—
18C3×C5×#131C411yescharacteristic, complemented
19C3×#4094C411yescharacteristic, complemented
20C3×#4053C413yescomplemented
21C3×D5×M16C2213yes—
22C3×#4047C413yescomplemented
23C2×C3×C8×D5C2213yes—
24#91221C2211yescharacteristic
25#91221C2213yes—
26#91221C2213yes—
27#91221C2213yes—
28C8×ΓL(1,𝔽16)C2216yes—
29#83782—51nomaximal, Hall
30#41×AGL(1,𝔽5)—31no—
31D5⋊#131—31no—
32D5×#131D311yescharacteristic
33#65728—33no—
34M16×AGL(1,𝔽5)—33no—
35#65718—33no—
36C2×#3992—33no—
37C22×ΓL(1,𝔽16)C2×C416yes—
38C3×C5×#41C2×C411yescharacteristic
39C3×#713C2×C411yescharacteristic
40C3×Q8×D5C2×C411yescharacteristic
41C3×#711C2×C413yes—
42C3×D4×D5C2×C413yes—
43C3×#709C2×C413yes—
44C2×C3×C4×D5C2313yes—
45#6763C2×C416yes—
46#6762C2×C416yescomplemented
47C2×C3⋊#170C2×C416yescomplemented
48C4⋊ΓL(1,𝔽16)C2×C416yes—
49C4×ΓL(1,𝔽16)C2×C416yes—
50C4×ΓL(1,𝔽16)C2311yescharacteristic
51C4×ΓL(1,𝔽16)C2313yes—
52#6758C2×C416yescomplemented
53D5⋊(C3⋊C8)C2×C416yescomplemented
54D5⋊(C3⋊C8)C2311yescharacteristic
55D5⋊(C3⋊C8)C2313yes—
56C3×C5×M16C2×C413yescomplemented
57C2×C3×C5×C8C2×C413yescomplemented
58C3×#681C2×C413yescomplemented
59C2×C3×#168C2×C413yescomplemented
60C3×#676C2×C416yescomplemented
61C3×C8×D5C2×C413yescomplemented
62C3×C8×D5C2×C413yescomplemented
63C3×C8×D5C2313yes—
64C3×C8×D5C2311yescharacteristic
65C2×C3×#131—51no—
66Dic3×#41—51no—
67C2×#948—51no—
68#5122—53no—
69Dic3×M16—53no—
70#5101—53no—
71C2×#815—53no—
72D5×#41C2×D311yescharacteristic
73Q8×AGL(1,𝔽5)—32no—
74#4133—32no—
75D4×AGL(1,𝔽5)—36no—
76#4131—36no—
77#4130—33no—
78C2×C4×AGL(1,𝔽5)—33no—
79D5⋊M16—33no—
80C2×D5⋊C8—33no—
81C5×#131Dic311yescharacteristic, complemented
82#4094Dic311yescharacteristic, complemented
83#4053Dic313yescomplemented
84D5×M16C2×D313yes—
85#4047Dic313yescomplemented
86C2×C8×D5C2×D313yes—
87#3992—31no—
88#3992—33no—
89#3992—33no—
90#3992—33no—
91C8×AGL(1,𝔽5)—36no—
92#2875—151noSylow
93C22×C3×D5C22×C413yes—
94C2×ΓL(1,𝔽16)—46no—
95C2×ΓL(1,𝔽16)C22×C412yes—
96C2×ΓL(1,𝔽16)C22×C416yes—
97C3×C5×Q8C4211yescharacteristic
98C3×C5×D4C4213yes—
99C2×C3×C4×C5C22×C413yes—
100C3×C22⋊D5C4216yes—
101C2×C3×Dic5C22×C413yes—
102C3×D20C4213yes—
103C3×C4×D5C22×C413yes—
104C3×C4×D5C22×C413yes—
105C3×C4×D5C22×C413yes—
106C3×C4×D5C2411yescharacteristic
107C3×Dic10C4213yes—
108C3⋊#170C22×C412yes—
109C3⋊#170C22×C416yes—
110C3×C5×C8C22×C411yescharacteristic
111C3×C5×C8C22×C413yes—
112C3×#168C22×C411yescharacteristic
113C3×#168C22×C413yes—
114C2×C3×#41—51no—
115C3×#131—51no—
116C3×#131—53no—
117C3×#131—53no—
118C3×#131AGL(1,𝔽5)11yescharacteristic, complemented
119C2×C3×M16—53no—
120C22×C3×C8—53no—
121#948—56no—
122#948—52no—
123Q8×Dic3—52no—
124D4×Dic3—56no—
125#926—53no—
126C2×C4×Dic3—53no—
127C2×#208—53no—
128C22×C3⋊C8—53no—
129#815—53no—
130#815—51no—
131#815—53no—
132#815—53no—
133C8×Dic3—56no—
134C22×AGL(1,𝔽5)—36no—
135C5×#41C2×Dic311yescharacteristic
136#713C4×D311yescharacteristic
137Q8×D5C4×D311yescharacteristic
138#711C4×D313yes—
139D4×D5C4×D313yes—
140#709C2×Dic313yes—
141C2×C4×D5C22×D313yes—
142C22⋊AGL(1,𝔽5)—36no—
143#704—36no—
144C2×#170—36no—
145C4⋊AGL(1,𝔽5)—36no—
146C4×AGL(1,𝔽5)—36no—
147C4×AGL(1,𝔽5)—31no—
148C4×AGL(1,𝔽5)—33no—
149#700—36no—
150D5⋊C8—36no—
151D5⋊C8—31no—
152D5⋊C8—33no—
153C5×M16C2×Dic313yescomplemented
154C2×C5×C8C2×Dic313yescomplemented
155#681C2×Dic313yescomplemented
156C2×#168C2×Dic313yescomplemented
157#676C2×Dic316yescomplemented
158C8×D5C2×Dic313yescomplemented
159C8×D5C2×Dic313yescomplemented
160C8×D5C22×D313yes—
161C8×D5C22×D311yescharacteristic
162C2×#131—51no—
163C2×#131—151no—
164C4×#41—151no—
165#405—153no—
166#405—153no—
167C4×M16—153no—
168C2×#97—153no—
169C22×C3×C5C2×C4213yes—
170C2×C3×D5—26no—
171C2×C3×D5C2×C4213yes—
172C2×C3×D5C23×C411yescharacteristic
173ΓL(1,𝔽16)—22no—
174ΓL(1,𝔽16)—26no—
175C3×C4×C5C2×C4213yes—
176C3×C4×C5C23×C411yescharacteristic
177C3×Dic5C2×C4213yes—
178C3×Dic5C23×C411yescharacteristic
179C3×#41—51no—
180C3×#41—53no—
181C3×#41—53no—
182C3×#41C2×AGL(1,𝔽5)11yescharacteristic
183C2×C3×Q8—51no—
184C2×C3×D4—53no—
185C22×C3×C4—53no—
186C22×Dic3—56no—
187C3×M16—53no—
188C3×M16—56no—
189C3×M16C2×AGL(1,𝔽5)13yescomplemented
190C2×C3×C8—53no—
191C2×C3×C8—53no—
192C2×C3×C8—53no—
193C2×C3×C8—53no—
194C2×C3×C8—51no—
195C2×C3×C8C2×AGL(1,𝔽5)13yescomplemented
196#217—56no—
197C4⋊Dic3—56no—
198C4×Dic3—53no—
199C4×Dic3—51no—
200C4×Dic3—56no—
201#208—56no—
202#208—56no—
203C2×C3⋊C8—53no—
204C2×C3⋊C8—51no—
205C2×C3⋊C8—56no—
206C2×C3⋊C8—56no—
207C22×D5C2×C4×D313yes—
208C2×AGL(1,𝔽5)—32no—
209C2×AGL(1,𝔽5)—36no—
210C2×AGL(1,𝔽5)—126no—
211C5×Q8C4×Dic311yescharacteristic
212C5×D4C4×Dic313yes—
213C2×C4×C5C22×Dic313yes—
214C22⋊D5C4×Dic316yes—
215C2×Dic5C2×C4×D313yes—
216D20C4×Dic313yes—
217C4×D5C2×C4×D313yes—
218C4×D5C2×C4×D313yes—
219C4×D5C22×Dic313yes—
220C4×D5C23×D311yescharacteristic
221Dic10C4×Dic313yes—
222#170—32no—
223#170—36no—
224C5×C8C22×Dic311yescharacteristic
225C5×C8C22×Dic313yes—
226#168C22×Dic311yescharacteristic
227#168C22×Dic313yes—
228C2×#41—51no—
229#131—51no—
230#131—53no—
231#131—53no—
232#131—156no—
233#131—152no—
234#131ΓL(1,𝔽16)11yescharacteristic, complemented
235C2×M16—53no—
236C2×M16—153no—
237C22×C8—53no—
238C22×C8—153no—
239C4×Q8—152no—
240C4×D4—156no—
241#117—153no—
242C2×C42—153no—
243#97—151no—
244#97—153no—
245#97—153no—
246#97—153no—
247C4×C8—156no—
248C2×C3×C5—23no—
249C2×C3×C5C22×C4211yescharacteristic
250C3×D5—23no—
251C3×D5#56812yes—
252C23×C3—53no—
253C3×Q8—53no—
254C3×Q8C4×AGL(1,𝔽5)11yescharacteristic
255C3×D4—56no—
256C3×D4—53no—
257C3×D4C4×AGL(1,𝔽5)13yes—
258C2×C3×C4—53no—
259C2×C3×C4—53no—
260C2×C3×C4—53no—
261C2×C3×C4—53no—
262C2×C3×C4—51no—
263C2×C3×C4C22×AGL(1,𝔽5)13yes—
264C2×Dic3—52no—
265C2×Dic3—56no—
266C2×Dic3—206no—
267C3×C8—51no—
268C3×C8—53no—
269C3×C8C22×AGL(1,𝔽5)11yescharacteristic
270C3×C8C22×AGL(1,𝔽5)13yes—
271C3⋊C8—52no—
272C3⋊C8—56no—
273C22×C5C2×C4×Dic313yes—
274C2×D5—26no—
275C2×D5C2×C4×Dic313yes—
276C2×D5C22×C4×D311yescharacteristic
277AGL(1,𝔽5)—66no—
278AGL(1,𝔽5)—62no—
279C4×C5C2×C4×Dic313yes—
280C4×C5C23×Dic311yescharacteristic
281Dic5C2×C4×Dic313yes—
282Dic5C22×C4×D311yescharacteristic
283#41—51no—
284#41—53no—
285#41—53no—
286#41C2×ΓL(1,𝔽16)11yescharacteristic
287C2×Q8—51no—
288C2×D4—53no—
289C22×C4—53no—
290C22×C4—156no—
291M16—53no—
292M16—56no—
293M16—156no—
294M16—156no—
295M16C2×ΓL(1,𝔽16)13yescomplemented
296C2×C8—53no—
297C2×C8—53no—
298C2×C8—51no—
299C2×C8—53no—
300C2×C8—53no—
301C2×C8—153no—
302C2×C8—151no—
303C2×C8—156no—
304C2×C8—156no—
305C2×C8C2×ΓL(1,𝔽16)13yescomplemented
306C4⋊C4—156no—
307C22⋊C4—156no—
308C42—153no—
309C42—151no—
310C42—156no—
311C3×C5#287511yescharacteristic, complemented, Hall
312C22×C3—51no—
313C22×C3—53no—
314C22×C3—106no—
315C22×C3C2×C4×AGL(1,𝔽5)13yes—
316C3×C4—53no—
317C3×C4—51no—
318C3×C4C2×C4×AGL(1,𝔽5)13yes—
319C3×C4C23×AGL(1,𝔽5)11yescharacteristic
320Dic3—106no—
321Dic3—102no—
322C2×C5—23no—
323C2×C5C22×C4×Dic311yescharacteristic
324D5—23no—
325D5#575012yes—
326C23—53no—
327Q8—53no—
328Q8C4×ΓL(1,𝔽16)11yescharacteristic
329D4—56no—
330D4—53no—
331D4C4×ΓL(1,𝔽16)13yes—
332C2×C4—53no—
333C2×C4—53no—
334C2×C4—53no—
335C2×C4—51no—
336C2×C4—53no—
337C2×C4—152no—
338C2×C4—156no—
339C2×C4—606no—
340C2×C4C22×ΓL(1,𝔽16)13yes—
341C8—51no—
342C8—53no—
343C8—152no—
344C8—156no—
345C8C22×ΓL(1,𝔽16)11yescharacteristic
346C8C22×ΓL(1,𝔽16)13yes—
347C2×C3—23no—
348C2×C3—52no—
349C2×C3—103no—
350C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
351C5#8378211yescharacteristic, complemented, Sylow
352C22—51no—
353C22—53no—
354C22—106no—
355C22C2×C4×ΓL(1,𝔽16)13yes—
356C4—51no—
357C4—53no—
358C4—302no—
359C4—306no—
360C4C2×C4×ΓL(1,𝔽16)13yes—
361C4C23×ΓL(1,𝔽16)11yescharacteristic
362C3#1061741511yescharacteristic, complemented, Sylow
363C2—23no—
364C2—52no—
365C2—103no—
366C2C22×C4×ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
center, FrattiniC4C23×ΓL(1,𝔽16)
commutator, socleC2×C3×C5C22×C42
FittingC3×C5×#131C4
prime radicalC3×D5×#41C22
cosocleC3×C4×D5C24
hypercenter#131ΓL(1,𝔽16)
hypocenterC3×C5#2875
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ #131

Complements

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

NH ≅ G/Ntype
4. #11752387C2semidirect
5. M16×ΓL(1,𝔽16)C2semidirect
6. #11752377C2semidirect
7. C2×#91221C2semidirect
11. #91992C4semidirect
13. #91990C4semidirect
18. C3×C5×#131C4semidirect
19. C3×#4094C4semidirect
20. C3×#4053C4semidirect
22. C3×#4047C4semidirect
46. #6762C2×C4semidirect
47. C2×C3⋊#170C2×C4semidirect
52. #6758C2×C4semidirect
53. D5⋊(C3⋊C8)C2×C4semidirect
56. C3×C5×M16C2×C4semidirect
57. C2×C3×C5×C8C2×C4semidirect
58. C3×#681C2×C4semidirect
59. C2×C3×#168C2×C4semidirect
60. C3×#676C2×C4semidirect
61. C3×C8×D5C2×C4semidirect
62. C3×C8×D5C2×C4semidirect
81. C5×#131Dic3semidirect
82. #4094Dic3semidirect
83. #4053Dic3semidirect
85. #4047Dic3semidirect
118. C3×#131AGL(1,𝔽5)semidirect
153. C5×M16C2×Dic3semidirect
154. C2×C5×C8C2×Dic3semidirect
155. #681C2×Dic3semidirect
156. C2×#168C2×Dic3semidirect
157. #676C2×Dic3semidirect
158. C8×D5C2×Dic3semidirect
159. C8×D5C2×Dic3semidirect
189. C3×M16C2×AGL(1,𝔽5)semidirect
195. C2×C3×C8C2×AGL(1,𝔽5)semidirect
234. #131ΓL(1,𝔽16)semidirect
295. M16C2×ΓL(1,𝔽16)semidirect
305. C2×C8C2×ΓL(1,𝔽16)semidirect
311. C3×C5#2875semidirect
351. C5#83782semidirect
362. C3#10617415semidirect

Maximal subgroups

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

subgroupindex
#41×ΓL(1,𝔽16)2
D5⋊#9482
C3×D5×#1312
#117523872
M16×ΓL(1,𝔽16)2
#117523772
C2×#912212
#106174153
#837825