Finite Group Databasebeta

← #14307160

Subgroups — #14307160

Order 1920.

520 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 122 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
1#11752498C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
3C3×C4×#700C211yesmaximal, characteristic
4#11746524C212yesmaximal
5#11746524C212yesmaximal
6#10604592—31nomaximal, Hall
7C2×C4⋊ΓL(1,𝔽16)—21no—
8C2×C4×ΓL(1,𝔽16)C2212yes—
9C2×C3×#700C2212yes—
10#91759C411yescharacteristic, complemented
11C5×#870C411yescharacteristic, complemented
12C3×C42×D5C2211yescharacteristic
13#91468C411yescharacteristic, complemented
14#91468C411yescharacteristic, complemented
15#91358C2211yescharacteristic
16#91358C2211yescharacteristic
17#91349—21no—
18#91349—21no—
19#91323—21no—
20#91323—21no—
21#91235—21no—
22C3×#4002—21no—
23C3×C4×#170—21no—
24#69868—51nomaximal, Hall
25#66022—31no—
26C4×C4⋊AGL(1,𝔽5)—31no—
27C4×#700D311yescharacteristic
28#64875—32no—
29#64875—32no—
30C22×ΓL(1,𝔽16)—22no—
31C2×C3×C4×D5D411yescharacteristic
32C2×C3×C4×D5D411yescharacteristic
33C2×C3×C4×D5C2311yescharacteristic
34C2×#1204—21no—
35C2×#1204—21no—
36C4⋊ΓL(1,𝔽16)—24no—
37C4×ΓL(1,𝔽16)—24no—
38C2×C3×#170—22no—
39C3×#700—22no—
40C3×#700Q814yescomplemented
41C3×C42×C5C2×C411yescharacteristic
42C4⋊Dic15C2×C411yescharacteristic
43C4⋊Dic15C2×C411yescharacteristic
44#6712—21no—
45#6712—21no—
46C5×C4⋊Dic3C2×C411yescharacteristic
47C5×C4⋊Dic3C2×C411yescharacteristic
48C5×Dic3⋊C4—21no—
49C5×Dic3⋊C4—21no—
50C3×C4×Dic5C2×C411yescharacteristic
51C3×C4×Dic5D411yescharacteristic
52C3×C4×Dic5D411yescharacteristic
53#6668—21no—
54#6668—21no—
55#6665—21no—
56#6665—21no—
57Dic3×Dic5—21no—
58Dic3×Dic5—21no—
59#5470—51no—
60C3×C4×M16—51no—
61C4×C4⋊Dic3—51no—
62#4555—52no—
63#4555—52no—
64C2×C4⋊AGL(1,𝔽5)—61no—
65C2×C4×AGL(1,𝔽5)—32no—
66C2×#700C2×D312yes—
67C5×#125—31no—
68#4042—61no—
69#4042—61no—
70#4038—31no—
71#4038—31no—
72#4034—31no—
73#4034—31no—
74#4033—61no—
75#4033—61no—
76C42×D5C2×D311yescharacteristic
77#4016—31no—
78(C2×AGL(1,𝔽5))⋊C4—61no—
79#4002—21no—
80C4×#170—21no—
81#1986—151noSylow
82C22×C3×D5C2×D411yescharacteristic
83C2×ΓL(1,𝔽16)—24no—
84C2×ΓL(1,𝔽16)—42no—
85C2×ΓL(1,𝔽16)—42no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22×C411yescharacteristic
89C2×Dic15—21no—
90C2×Dic15—21no—
91C2×C5×Dic3—21no—
92C2×C5×Dic3—21no—
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C22⋊C411yescharacteristic
95C2×C3×Dic5C2×D411yescharacteristic
96C3×C4×D5—22no—
97C3×C4×D5—41no—
98C3×C4×D5C2×Q812yes—
99C3×C4×D5#4112yes—
100#1204—42no—
101#1204—42no—
102C3×#170—24no—
103C2×C3×M16—52no—
104C2×C3×C42—51no—
105#926—51no—
106#926—51no—
107C2×C4⋊Dic3—101no—
108#924—101no—
109#924—101no—
110C2×C4×Dic3—52no—
111#870—51no—
112#870AGL(1,𝔽5)11yescharacteristic, complemented
113#869—51no—
114#869—51no—
115C4×Dic6—101no—
116C4×Dic6—101no—
117C3×#97—101no—
118C3×C4×C8—101no—
119C3⋊#95—101no—
120C22×AGL(1,𝔽5)—62no—
121C2×C4×D5D1211yescharacteristic
122C2×C4×D5D1211yescharacteristic
123C2×C4×D5C22×D311yescharacteristic
124C2×Dic10—61no—
125C2×Dic10—61no—
126C2×#170—22no—
127C4⋊AGL(1,𝔽5)—64no—
128C4×AGL(1,𝔽5)—64no—
129#700—22no—
130#700Dic614yescomplemented
131C5×C4⋊C4—31no—
132C5×C4⋊C4—31no—
133C5×C4⋊C4—61no—
134C5×C4⋊C4—61no—
135C42×C5C4×D311yescharacteristic
136#685—61no—
137#685—61no—
138C4⋊Dic5—31no—
139C4⋊Dic5—31no—
140#683—61no—
141#683—61no—
142#683—61no—
143#683—61no—
144C4×Dic5—61no—
145C4×Dic5—61no—
146C4×Dic5C4×D311yescharacteristic
147C4×Dic5#6711yescharacteristic
148C4×Dic5#6711yescharacteristic
149#533—151no—
150C4×M16—51no—
151C4×C4⋊C4—151no—
152#339—152no—
153#339—152no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C2×C3×D5—21no—
156C2×C3×D5—21no—
157C2×C3×D5#12411yescharacteristic
158C2×C3×D5C4⋊Q811yescharacteristic
159ΓL(1,𝔽16)—44no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162C3×C4×C5C4×Q812yes—
163Dic15—41no—
164Dic15—41no—
165C3×Dic5—21no—
166C3×Dic5#10412yes—
167C3×Dic5C22⋊Q812yes—
168C5×Dic3—41no—
169C5×Dic3—41no—
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—51no—
173C22×Dic3—102no—
174C2×Dic6—101no—
175C2×Dic6—101no—
176C3×M16—54no—
177C3×M16—102no—
178C2×C3×C8—102no—
179C3×C42—51no—
180C3×C42—51no—
181C3×C42—51no—
182C3×C42C2×AGL(1,𝔽5)11yescharacteristic
183#217—101no—
184#217—101no—
185C4⋊Dic3—51no—
186C4⋊Dic3—51no—
187C4⋊Dic3—101no—
188C4⋊Dic3—101no—
189C4⋊Dic3—104no—
190C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
191C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
192Dic3⋊C4—21no—
193Dic3⋊C4—21no—
194Dic3⋊C4—101no—
195Dic3⋊C4—101no—
196C4×Dic3—101no—
197C4×Dic3—101no—
198C4×Dic3—104no—
199C22×D5C2×D1211yescharacteristic
200C2×AGL(1,𝔽5)—64no—
201C2×AGL(1,𝔽5)—122no—
202C2×AGL(1,𝔽5)—122no—
203C2×C4×C5—61no—
204C2×C4×C5—61no—
205C2×C4×C5(C2×D3)⋊C411yescharacteristic
206C2×C4×C5(C2×D3)⋊C411yescharacteristic
207C2×C4×C5C2×C4×D311yescharacteristic
208C2×Dic5—61no—
209C2×Dic5—61no—
210C2×Dic5(C2×D3)⋊C411yescharacteristic
211C2×Dic5(C2×D3)⋊C411yescharacteristic
212C2×Dic5C2×#6711yescharacteristic
213C4×D5—22no—
214C4×D5—41no—
215C4×D5C2×Dic612yes—
216C4×D5#23512yes—
217Dic10—122no—
218Dic10—122no—
219#170—24no—
220C2×M16—52no—
221C4⋊Q8—151no—
222C4⋊Q8—151no—
223#125—31no—
224#125—151no—
225C22⋊Q8—301no—
226C22⋊Q8—301no—
227C4×Q8—301no—
228C4×Q8—301no—
229#117—151no—
230#117—151no—
231C2×C4⋊C4—301no—
232C2×C42—51no—
233C2×C42—152no—
234#97—101no—
235C4×C8—101no—
236#95—301no—
237C2×C3×C5#39111yescharacteristic
238C2×C3×C5C2×#10411yescharacteristic
239C2×C3×C5#42111yescharacteristic
240C3×D5—21no—
241C3×D5—21no—
242C23×C3—51no—
243C2×C3×C4—51no—
244C2×C3×C4—51no—
245C2×C3×C4—52no—
246C2×C3×C4—52no—
247C2×C3×C4—51no—
248C2×C3×C4—102no—
249C2×C3×C4—201no—
250C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
251C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
252C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—104no—
256C2×Dic3—101no—
257C2×Dic3—101no—
258C2×Dic3—202no—
259C2×Dic3—202no—
260Dic6—202no—
261Dic6—202no—
262C3×C8—104no—
263C22×C5C2×(C2×D3)⋊C411yescharacteristic
264C2×D5—21no—
265C2×D5—21no—
266C2×D5#86911yescharacteristic
267C2×D5#87511yescharacteristic
268AGL(1,𝔽5)—124no—
269C4×C5—21no—
270C4×C5—21no—
271C4×C5—121no—
272C4×C5—121no—
273C4×C5C4×Dic612yes—
274Dic5—21no—
275Dic5—121no—
276Dic5—121no—
277Dic5#82512yes—
278Dic5#92412yes—
279C2×Q8—301no—
280C2×Q8—301no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—302no—
285M16—54no—
286M16—102no—
287C2×C8—102no—
288C4⋊C4—31no—
289C4⋊C4—31no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—151no—
293C4⋊C4—151no—
294C4⋊C4—301no—
295C4⋊C4—301no—
296C4⋊C4—304no—
297C4⋊C4—301no—
298C4⋊C4—301no—
299C22⋊C4—301no—
300C22⋊C4—301no—
301C42—51no—
302C42—51no—
303C42—51no—
304C42—301no—
305C42—301no—
306C42—304no—
307C42D3×AGL(1,𝔽5)11yescharacteristic
308C3×C5#198611yescharacteristic, complemented, Hall
309C22×C3—51no—
310C22×C3—51no—
311C22×C3—101no—
312C22×C3—101no—
313C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—52no—
317C3×C4—52no—
318C3×C4—101no—
319C3×C4Q8×AGL(1,𝔽5)12yes—
320Dic3—41no—
321Dic3—41no—
322Dic3—204no—
323Dic3—201no—
324Dic3—201no—
325C2×C5#492911yescharacteristic
326C2×C5C2×#82511yescharacteristic
327C2×C5#513911yescharacteristic
328D5—21no—
329D5—21no—
330C23—51no—
331Q8—602no—
332Q8—602no—
333C2×C4—51no—
334C2×C4—51no—
335C2×C4—52no—
336C2×C4—52no—
337C2×C4—51no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—102no—
341C2×C4—201no—
342C2×C4—304no—
343C2×C4—301no—
344C2×C4—301no—
345C2×C4—602no—
346C2×C4—602no—
347C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
348C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
349C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
350C8—104no—
351C2×C3—101no—
352C2×C3—101no—
353C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
354C2×C3D5⋊#10411yescharacteristic
355C2×C3#6579511yescharacteristic
356C5#6986811yescharacteristic, complemented, Sylow
357C22—51no—
358C22—51no—
359C22—101no—
360C22—101no—
361C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
362C4—21no—
363C4—21no—
364C4—52no—
365C4—52no—
366C4—101no—
367C4—121no—
368C4—121no—
369C4—604no—
370C4—601no—
371C4—601no—
372C4AGL(1,𝔽5)×Dic612yes—
373C3#1060459211yescharacteristic, complemented, Sylow
374C2—101no—
375C2—101no—
376C2D5⋊#82511yescharacteristic
377C2#1175171711yescharacteristic
378C2#1175174911yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C3×C4×D5C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#1986
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
10. #91759C4semidirect
11. C5×#870C4semidirect
13. #91468C4semidirect
14. #91468C4semidirect
40. C3×#700Q8semidirect
112. #870AGL(1,𝔽5)semidirect
130. #700Dic6semidirect
308. C3×C5#1986semidirect
356. C5#69868semidirect
373. C3#10604592semidirect

Maximal subgroups

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

subgroupindex
#117524982
C4×C4⋊ΓL(1,𝔽16)2
C3×C4×#7002
#117465242
#106045923
#698685