Finite Group Databasebeta

← C5×D4×#226 #14511128

Subgroups — C5×D4×#226 #14511128

Order 1920.

550 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C5×#226C212yesmaximal, complemented
2C5×D4×SL(2,𝔽3)C211yesmaximal, characteristic
3C5×#5419C212yesmaximal, complemented
4C5×C4⋊#226C211yesmaximal, characteristic, complemented
5C4×C5×#226C211yesmaximal, characteristic, complemented
6C5×D4×Q16—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2212yes—
8C2×C5×#226—24no—
9C2×C5×#226C2212yes—
10C2×C5×#226C2211yescharacteristic
11C5×D4×Dic3—41nomaximal
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13C5×#859C2211yescharacteristic
14D4×#226C511yesmaximal, characteristic, direct product factor, Hall
15C22×C5×Q16—32no—
16C5×D4×Q8—31no—
17C5×D4×Q8D311yescharacteristic
18C5×#494—31no—
19C5×#467—32no—
20C5×#462—31no—
21C5×#462—31no—
22C5×#451—32no—
23C5×#451—32no—
24C4×C5×Q16—31no—
25C5×C8×D4—31no—
26C2×C3×C5×D4—41no—
27C22×C5×Dic3—42no—
28C2×C5×SL(2,𝔽3)—22no—
29C2×C5×SL(2,𝔽3)C2311yescharacteristic
30C5×#226—22no—
31C5×#226D412yesdirect product factor
32C5×#217—42no—
33C5×C4⋊Dic3—41no—
34C4×C5×Dic3—41no—
35C22×#226C2×C512yescomplemented
36D4×SL(2,𝔽3)C2×C511yescharacteristic
37#5419C2×C512yescomplemented
38C4⋊#226C2×C511yescharacteristic, complemented
39C4×#226C2×C511yescharacteristic, complemented
40C22×C5×Q8—32no—
41C22×C5×Q8C2×D312yes—
42C2×C5×Q16—31no—
43C2×C5×Q16—32no—
44C2×C5×Q16—32no—
45C2×C5×Q16—32no—
46C2×C5×Q16—32no—
47C2×C5×Q16—64no—
48C2×C5×Q16—64no—
49C22×C5×C8—32no—
50C5×C4⋊Q8—31no—
51C5×C4⋊Q8—31no—
52C5×C4⋊Q8—61no—
53C5×C22⋊Q8—32no—
54C5×C22⋊Q8—32no—
55C5×C22⋊Q8—62no—
56C4×C5×Q8—31no—
57C4×C5×Q8C2×D311yescharacteristic
58C4×C5×D4—31no—
59C4×C5×D4—61no—
60C5×#107—31no—
61C5×C4⋊C8—31no—
62C5×#103—31no—
63C5×#103—32no—
64C5×#103—32no—
65C5×#103—31no—
66C5×C22⋊C8—32no—
67C4×C5×C8—31no—
68D4×Q16—31noSylow
69C23×C3×C5—42no—
70C3×C5×D4—44no—
71C2×C3×C4×C5—41no—
72C2×C5×Dic3—41no—
73C2×C5×Dic3—42no—
74C2×C5×Dic3—41no—
75C2×C5×Dic3—84no—
76C5×SL(2,𝔽3)C2×D411yescharacteristic
77C22×SL(2,𝔽3)C22×C512yes—
78C2×#226—24no—
79C2×#226C22×C512yes—
80C2×#226C22×C511yescharacteristic
81D4×Dic3—41no—
82C4×SL(2,𝔽3)C22×C511yescharacteristic
83#859C22×C511yescharacteristic
84C2×C5×Q8—22no—
85C2×C5×Q8—32no—
86C2×C5×Q8—31no—
87C2×C5×Q8—32no—
88C2×C5×Q8—62no—
89C2×C5×Q8—62no—
90C2×C5×Q8—64no—
91C2×C5×Q8—122no—
92C2×C5×Q8C22×D311yescharacteristic
93C2×C5×D4S411yescharacteristic
94C22×C4×C5—32no—
95C22×C4×C5—62no—
96C5×Q16—32no—
97C5×Q16—32no—
98C5×Q16—64no—
99C5×Q16—62no—
100C5×Q16—64no—
101C5×Q16—62no—
102C5×Q16—122no—
103C2×C5×C8—31no—
104C2×C5×C8—32no—
105C2×C5×C8—31no—
106C2×C5×C8—64no—
107C5×C4⋊C4—31no—
108C5×C4⋊C4—31no—
109C5×C4⋊C4—31no—
110C5×C4⋊C4—61no—
111C5×C4⋊C4—61no—
112C5×C4⋊C4—62no—
113C5×C4⋊C4—61no—
114C5×C4⋊C4—61no—
115C5×C22⋊C4—32no—
116C5×C22⋊C4—62no—
117C42×C5—31no—
118C42×C5—61no—
119C22×Q16—32no—
120D4×Q8—31no—
121D4×Q8C5×D311yescharacteristic
122#494—31no—
123#467—32no—
124#462—31no—
125#462—31no—
126#451—32no—
127#451—32no—
128C4×Q16—31no—
129C8×D4—31no—
130C22×C3×C5—44no—
131C22×C3×C5—41no—
132C22×C3×C5—82no—
133C22×C3×C5—82no—
134C3×C4×C5—42no—
135C5×Dic3—42no—
136C5×Dic3—82no—
137C5×Dic3—81no—
138C2×C3×D4—41no—
139C22×Dic3—42no—
140C2×SL(2,𝔽3)—22no—
141C2×SL(2,𝔽3)C23×C511yescharacteristic
142#226—22no—
143#226C5×D412yesdirect product factor
144#217—42no—
145C4⋊Dic3—41no—
146C4×Dic3—41no—
147C23×C5C2×S412yes—
148C5×Q8—31no—
149C5×Q8—32no—
150C5×Q8—62no—
151C5×Q8—62no—
152C5×Q8—61no—
153C5×Q8—62no—
154C5×Q8—62no—
155C5×Q8—124no—
156C5×Q8D3×D411yescharacteristic
157C5×D4#22614yesdirect product factor
158C2×C4×C5—31no—
159C2×C4×C5—31no—
160C2×C4×C5—32no—
161C2×C4×C5—62no—
162C2×C4×C5—62no—
163C2×C4×C5—61no—
164C2×C4×C5—61no—
165C2×C4×C5—62no—
166C2×C4×C5—124no—
167C2×C4×C5C2×S411yescharacteristic
168C5×C8—32no—
169C5×C8—62no—
170C5×C8—61no—
171C22×Q8—32no—
172C22×Q8C2×C5×D312yes—
173C2×Q16—31no—
174C2×Q16—32no—
175C2×Q16—32no—
176C2×Q16—32no—
177C2×Q16—32no—
178C2×Q16—64no—
179C2×Q16—64no—
180C22×C8—32no—
181C4⋊Q8—31no—
182C4⋊Q8—31no—
183C4⋊Q8—61no—
184C22⋊Q8—32no—
185C22⋊Q8—32no—
186C22⋊Q8—62no—
187C4×Q8—31no—
188C4×Q8C2×C5×D311yescharacteristic
189C4×D4—31no—
190C4×D4—61no—
191#107—31no—
192C4⋊C8—31no—
193#103—31no—
194#103—32no—
195#103—32no—
196#103—31no—
197C22⋊C8—32no—
198C4×C8—31no—
199C2×C3×C5—41no—
200C2×C3×C5—41no—
201C2×C3×C5—41no—
202C2×C3×C5—84no—
203C23×C3—42no—
204C3×D4—44no—
205C2×C3×C4—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—84no—
210SL(2,𝔽3)C2×C5×D411yescharacteristic
211C22×C5—22no—
212C22×C5—22no—
213C22×C5C2×#22614yescomplemented
214C22×C5C22×S411yescharacteristic
215C4×C5—31no—
216C4×C5—31no—
217C4×C5—62no—
218C4×C5—62no—
219C4×C5—61no—
220C4×C5—122no—
221C4×C5—121no—
222C4×C5C2×#22612yescomplemented
223C2×Q8—22no—
224C2×Q8—32no—
225C2×Q8—31no—
226C2×Q8—32no—
227C2×Q8—62no—
228C2×Q8—62no—
229C2×Q8—64no—
230C2×Q8—122no—
231C2×Q8C22×C5×D311yescharacteristic
232C2×D4C5×S411yescharacteristic
233C22×C4—32no—
234C22×C4—62no—
235Q16—32no—
236Q16—32no—
237Q16—64no—
238Q16—62no—
239Q16—64no—
240Q16—62no—
241Q16—122no—
242C2×C8—31no—
243C2×C8—31no—
244C2×C8—32no—
245C2×C8—64no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—62no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C22⋊C4—32no—
255C22⋊C4—62no—
256C42—31no—
257C42—61no—
258C3×C5—41noHall
259C22×C3—44no—
260C22×C3—41no—
261C22×C3—82no—
262C22×C3—82no—
263C3×C4—42no—
264Dic3—42no—
265Dic3—81no—
266Dic3—82no—
267C2×C5—24no—
268C2×C5D4×S411yescharacteristic
269C2×C5C22×#22611yescharacteristic
270C2×C5#592711yescharacteristic
271C23C2×C5×S412yes—
272Q8—31no—
273Q8—32no—
274Q8—62no—
275Q8—62no—
276Q8—61no—
277Q8—62no—
278Q8—62no—
279Q8—124no—
280Q8C5×D3×D411yescharacteristic
281D4C5×#22614yesdirect product factor
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—32no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—62no—
290C2×C4—124no—
291C2×C4C2×C5×S411yescharacteristic
292C8—32no—
293C8—62no—
294C8—61no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—84no—
299C5D4×#22611yescharacteristic, direct product factor, Sylow
300C22—22no—
301C22—22no—
302C22C2×C5×#22614yescomplemented
303C22C22×C5×S411yescharacteristic
304C4—31no—
305C4—31no—
306C4—62no—
307C4—62no—
308C4—61no—
309C4—122no—
310C4—121no—
311C4C2×C5×#22612yescomplemented
312C3—41noSylow
313C2—24no—
314C2C5×D4×S411yescharacteristic
315C2C22×C5×#22611yescharacteristic
316C2C5×#592711yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×D4×Q8D3
prime radicalC5×D4×SL(2,𝔽3)C2
hypercenterC2×C5×D4S4
hypocenterSL(2,𝔽3)C2×C5×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C2×C5×D4

Complements

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

NH ≅ G/Ntype
1. C22×C5×#226C2semidirect
3. C5×#5419C2semidirect
4. C5×C4⋊#226C2semidirect
5. C4×C5×#226C2semidirect
14. D4×#226C5direct (N × H)
31. C5×#226D4direct (N × H)
35. C22×#226C2×C5semidirect
37. #5419C2×C5semidirect
38. C4⋊#226C2×C5semidirect
39. C4×#226C2×C5semidirect
143. #226C5×D4direct (N × H)
157. C5×D4#226direct (N × H)
213. C22×C5C2×#226semidirect
222. C4×C5C2×#226semidirect
281. D4C5×#226direct (N × H)
299. C5D4×#226direct (N × H)
302. C22C2×C5×#226semidirect
311. C4C2×C5×#226semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#2262
C5×D4×SL(2,𝔽3)2
C5×#54192
C5×C4⋊#2262
C4×C5×#2262
C5×D4×Q163
C5×D4×Dic34
D4×#2265