Finite Group Databasebeta

← #14307156

Subgroups — #14307156

Order 1920.

520 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 126 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#11752506C211yesmaximal, characteristic
2C4×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
3C3×C4×#700C211yesmaximal, characteristic
4#11746524C214yesmaximal
5#10604680—31nomaximal, Hall
6C2×C4⋊ΓL(1,𝔽16)—21no—
7C2×C4×ΓL(1,𝔽16)C2212yes—
8C2×C3×#700C2212yes—
9#91758C411yescharacteristic, complemented
10C5×#869C411yescharacteristic, complemented
11C3×C42×D5C2211yescharacteristic
12#91388C412yescomplemented
13#91360—21no—
14#91359—21no—
15#91358C2212yes—
16#91333—21no—
17Dic5×Dic6—21no—
18#91235—21no—
19C3×#4002—21no—
20C3×C4×#170—21no—
21#69868—51nomaximal, Hall
22#66062—31no—
23C4×C4⋊AGL(1,𝔽5)—31no—
24C4×#700D311yescharacteristic
25#64875—34no—
26C22×ΓL(1,𝔽16)—22no—
27C2×Dic30—21no—
28C2×C5×Dic6—21no—
29C2×C3×C4×D5D411yescharacteristic
30C2×C3×C4×D5D411yescharacteristic
31C2×C3×C4×D5C2311yescharacteristic
32C4⋊ΓL(1,𝔽16)—24no—
33C4×ΓL(1,𝔽16)—24no—
34C2×C3×#170—22no—
35C3×#700—22no—
36C3×#700Q814yescomplemented
37C3×C42×C5C2×C411yescharacteristic
38C4⋊Dic15—21no—
39C4⋊Dic15C2×C412yes—
40C5×C4⋊Dic3—21no—
41C5×C4⋊Dic3C2×C412yes—
42C3×C4×Dic5C2×C411yescharacteristic
43C3×C4×Dic5D411yescharacteristic
44C3×C4×Dic5D411yescharacteristic
45Dic3⋊Dic5—22no—
46#6669—22no—
47#6665—22no—
48Dic3×Dic5—22no—
49#5470—51no—
50C3×C4×M16—51no—
51C4×C4⋊Dic3—51no—
52#4555—54no—
53C2×C4⋊AGL(1,𝔽5)—61no—
54C2×C4×AGL(1,𝔽5)—32no—
55C2×#700C2×D312yes—
56C5×C4⋊Q8—31no—
57#4092—61no—
58Q8×Dic5—61no—
59#4043—61no—
60#4038—32no—
61#4036—32no—
62#4033—61no—
63C42×D5C2×D311yescharacteristic
64#4015—31no—
65(C2×AGL(1,𝔽5))⋊C4—61no—
66#4002—21no—
67C4×#170—21no—
68#1986—151noSylow
69C22×C3×D5C2×D411yescharacteristic
70C2×ΓL(1,𝔽16)—24no—
71C2×ΓL(1,𝔽16)—42no—
72C2×ΓL(1,𝔽16)—42no—
73C2×C3×C4×C5C22⋊C411yescharacteristic
74C2×C3×C4×C5C22⋊C411yescharacteristic
75C2×C3×C4×C5C22×C411yescharacteristic
76C2×Dic15—22no—
77Dic30—42no—
78C2×C5×Dic3—22no—
79C5×Dic6—42no—
80C2×C3×Dic5C22⋊C411yescharacteristic
81C2×C3×Dic5C22⋊C411yescharacteristic
82C2×C3×Dic5C2×D411yescharacteristic
83C3×C4×D5—22no—
84C3×C4×D5—41no—
85C3×C4×D5C2×Q812yes—
86C3×C4×D5#4112yes—
87C3×#170—24no—
88C2×C3×M16—52no—
89C2×C3×C42—51no—
90#926—52no—
91C2×C4⋊Dic3—101no—
92#924—101no—
93#924—101no—
94C2×C4×Dic3—52no—
95#870—52no—
96#869—51no—
97#869AGL(1,𝔽5)11yescharacteristic, complemented
98C4×Dic6—101no—
99C4×Dic6—101no—
100C3×#97—101no—
101C3×C4×C8—101no—
102C3⋊#95—101no—
103C22×AGL(1,𝔽5)—62no—
104C2×C5×Q8—61no—
105C2×C4×D5D1211yescharacteristic
106C2×C4×D5D1211yescharacteristic
107C2×C4×D5C22×D311yescharacteristic
108C2×Dic10—61no—
109C2×#170—22no—
110C4⋊AGL(1,𝔽5)—64no—
111C4×AGL(1,𝔽5)—64no—
112#700—22no—
113#700Dic614yescomplemented
114C5×C4⋊C4—32no—
115C5×C4⋊C4—61no—
116C42×C5C4×D311yescharacteristic
117#685—62no—
118C4⋊Dic5—32no—
119C4⋊Dic5—61no—
120C4⋊Dic5—62no—
121#683—62no—
122C4×Dic5—62no—
123C4×Dic5C4×D311yescharacteristic
124C4×Dic5#6711yescharacteristic
125C4×Dic5#6711yescharacteristic
126#533—151no—
127C4×M16—51no—
128C4×C4⋊C4—151no—
129#339—154no—
130C22×C3×C5C2×C22⋊C411yescharacteristic
131C2×C3×D5—21no—
132C2×C3×D5—21no—
133C2×C3×D5#12411yescharacteristic
134C2×C3×D5C4⋊Q811yescharacteristic
135ΓL(1,𝔽16)—44no—
136C3×C4×C5—21no—
137C3×C4×C5#10412yes—
138C3×C4×C5C4×Q812yes—
139Dic15—42no—
140C3×Dic5—21no—
141C3×Dic5—21no—
142C3×Dic5C22⋊Q812yes—
143C5×Dic3—42no—
144C22×C3×C4—51no—
145C22×C3×C4—51no—
146C22×C3×C4—51no—
147C22×Dic3—102no—
148C2×Dic6—21no—
149C2×Dic6—101no—
150C3×M16—54no—
151C3×M16—102no—
152C2×C3×C8—102no—
153C3×C42—51no—
154C3×C42—51no—
155C3×C42—51no—
156C3×C42C2×AGL(1,𝔽5)11yescharacteristic
157#217—102no—
158C4⋊Dic3—21no—
159C4⋊Dic3—52no—
160C4⋊Dic3—104no—
161C4⋊Dic3—101no—
162C4⋊Dic3C2×AGL(1,𝔽5)12yes—
163Dic3⋊C4—102no—
164Dic3⋊C4—102no—
165C4×Dic3—102no—
166C4×Dic3—104no—
167C22×D5C2×D1211yescharacteristic
168C2×AGL(1,𝔽5)—64no—
169C2×AGL(1,𝔽5)—122no—
170C2×AGL(1,𝔽5)—122no—
171C5×Q8—122no—
172C2×C4×C5—62no—
173C2×C4×C5(C2×D3)⋊C411yescharacteristic
174C2×C4×C5(C2×D3)⋊C411yescharacteristic
175C2×C4×C5C2×C4×D311yescharacteristic
176C2×Dic5—62no—
177C2×Dic5(C2×D3)⋊C411yescharacteristic
178C2×Dic5(C2×D3)⋊C411yescharacteristic
179C2×Dic5C2×#6711yescharacteristic
180C4×D5—22no—
181C4×D5—41no—
182C4×D5C2×Dic612yes—
183C4×D5#23512yes—
184Dic10—122no—
185#170—24no—
186C2×M16—52no—
187C4⋊Q8—31no—
188C4⋊Q8—151no—
189#125—152no—
190C22⋊Q8—301no—
191C22⋊Q8—301no—
192C4×Q8—301no—
193C4×Q8—301no—
194#117—152no—
195C2×C4⋊C4—301no—
196C2×C42—51no—
197C2×C42—152no—
198#97—101no—
199C4×C8—101no—
200#95—301no—
201C2×C3×C5#39111yescharacteristic
202C2×C3×C5C2×#10411yescharacteristic
203C2×C3×C5#42111yescharacteristic
204C3×D5—21no—
205C3×D5—21no—
206C23×C3—51no—
207C2×C3×C4—51no—
208C2×C3×C4—52no—
209C2×C3×C4—52no—
210C2×C3×C4—51no—
211C2×C3×C4—51no—
212C2×C3×C4—102no—
213C2×C3×C4—201no—
214C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
215C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
216C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
217C2×Dic3—22no—
218C2×Dic3—104no—
219C2×Dic3—102no—
220C2×Dic3—202no—
221C2×Dic3—202no—
222Dic6—42no—
223Dic6—202no—
224C3×C8—104no—
225C22×C5C2×(C2×D3)⋊C411yescharacteristic
226C2×D5—21no—
227C2×D5—21no—
228C2×D5#86911yescharacteristic
229C2×D5#87511yescharacteristic
230AGL(1,𝔽5)—124no—
231C4×C5—21no—
232C4×C5—122no—
233C4×C5#82512yes—
234C4×C5C4×Dic612yes—
235Dic5—21no—
236Dic5—21no—
237Dic5—122no—
238Dic5#92412yes—
239C2×Q8—61no—
240C2×Q8—301no—
241C22×C4—51no—
242C22×C4—51no—
243C22×C4—51no—
244C22×C4—302no—
245M16—54no—
246M16—102no—
247C2×C8—102no—
248C4⋊C4—32no—
249C4⋊C4—61no—
250C4⋊C4—152no—
251C4⋊C4—302no—
252C4⋊C4—302no—
253C4⋊C4—304no—
254C4⋊C4—301no—
255C22⋊C4—302no—
256C42—51no—
257C42—51no—
258C42—51no—
259C42—302no—
260C42—304no—
261C42D3×AGL(1,𝔽5)11yescharacteristic
262C3×C5#198611yescharacteristic, complemented, Hall
263C22×C3—51no—
264C22×C3—51no—
265C22×C3—101no—
266C22×C3—101no—
267C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
268C3×C4—21no—
269C3×C4—52no—
270C3×C4—101no—
271C3×C4—101no—
272C3×C4#401012yes—
273C3×C4Q8×AGL(1,𝔽5)12yes—
274Dic3—42no—
275Dic3—204no—
276Dic3—202no—
277C2×C5#492911yescharacteristic
278C2×C5C2×#82511yescharacteristic
279C2×C5#513911yescharacteristic
280D5—21no—
281D5—21no—
282C23—51no—
283Q8—122no—
284Q8—602no—
285C2×C4—51no—
286C2×C4—52no—
287C2×C4—52no—
288C2×C4—51no—
289C2×C4—51no—
290C2×C4—62no—
291C2×C4—102no—
292C2×C4—201no—
293C2×C4—304no—
294C2×C4—302no—
295C2×C4—602no—
296C2×C4—602no—
297C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
298C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
299C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
300C8—104no—
301C2×C3—101no—
302C2×C3—101no—
303C2×C3#6576911yescharacteristic
304C2×C3C2×#401011yescharacteristic
305C2×C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
306C5#6986811yescharacteristic, complemented, Sylow
307C22—51no—
308C22—51no—
309C22—101no—
310C22—101no—
311C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
312C4—21no—
313C4—52no—
314C4—101no—
315C4—101no—
316C4—122no—
317C4—604no—
318C4—602no—
319C4#9115712yes—
320C4AGL(1,𝔽5)×Dic612yes—
321C3#1060468011yescharacteristic, complemented, Sylow
322C2—101no—
323C2—101no—
324C2#1175171811yescharacteristic
325C2C2×#9115711yescharacteristic
326C2#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
9. #91758C4semidirect
10. C5×#869C4semidirect
12. #91388C4semidirect
36. C3×#700Q8semidirect
97. #869AGL(1,𝔽5)semidirect
113. #700Dic6semidirect
262. C3×C5#1986semidirect
306. C5#69868semidirect
321. C3#10604680semidirect

Maximal subgroups

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

subgroupindex
#117525062
C4×C4⋊ΓL(1,𝔽16)2
C3×C4×#7002
#117465242
#106046803
#698685