Finite Group Databasebeta

← #14285355

Subgroups — #14285355

Order 1920.

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

normal subgroups only

This group has 456 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
1D3×#4090C211yesmaximal, characteristic
2#11751804C212yesmaximal
3#11746802C211yesmaximal, characteristic, complemented
4C3×#64931C211yesmaximal, characteristic, complemented
5#11746511C212yesmaximal, complemented
6#10604972—31nomaximal, Hall
7C2×C5×D3×Q8C411yescharacteristic
8C2×D3×Dic10C411yescharacteristic
9#91833C411yescharacteristic
10C3×#4090C2211yescharacteristic
11D3×#683—22no—
12C4×D3×Dic5C2211yescharacteristic
13#91385C411yescharacteristic
14#91376C2211yescharacteristic
15#91329—22no—
16#91233C2212yes—
17C3×#4004C2212yes—
18(C2×D3)⋊#170—22no—
19#91169—22no—
20#68487—51nomaximal, Hall
21C2×#4090—31no—
22#65757—32no—
23#64931—31no—
24#64931—32no—
25#64931D311yescharacteristic, complemented
26C2×C3×C5×Q8C2×C411yescharacteristic
27C2×Dic30C2×C411yescharacteristic
28C5×D3×Q8—42no—
29C2×C4×C5×D3—21no—
30C2×C4×C5×D3C2×C411yescharacteristic
31C2×C5×Dic6—21no—
32C2×C5×Dic6C2×C411yescharacteristic
33C2×C3×Dic10C2×C411yescharacteristic
34C2×#1204—21no—
35C2×D3×Dic5—21no—
36C2×D3×Dic5D412yes—
37D3×Dic10—42no—
38C2×C3⋊#170—22no—
39C2×C3×#170—22no—
40#6712—22no—
41C4×Dic15C2×C411yescharacteristic
42C3×#683—22no—
43C3×C4×Dic5C2311yescharacteristic
44#6669—22no—
45#6668—22no—
46Dic3×Dic5D412yes—
47D3×C4⋊Q8—51no—
48#4835—52no—
49C3×#333—51no—
50#4546—51no—
51#4481—52no—
52C22×C5×Q8—31no—
53C22×Dic10—31no—
54#4090—31no—
55#4090—31no—
56#4090—31no—
57#4090—62no—
58#4090C2×D311yescharacteristic
59C2×#683—62no—
60C2×C4×Dic5—31no—
61C22⋊#170—62no—
62#4004—32no—
63#4004C2×D312yes—
64#4002—62no—
65#1558—151noSylow
66C22×C5×D3C22⋊C411yescharacteristic
67C3×C5×Q8—41no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5C22×C411yescharacteristic
70C2×Dic15—21no—
71C2×Dic15C22⋊C412yes—
72Dic30—41no—
73C2×C5×Dic3—21no—
74C2×C5×Dic3C22⋊C411yescharacteristic
75C4×C5×D3—22no—
76C4×C5×D3—42no—
77C5×Dic6—42no—
78C5×Dic6—41no—
79C2×C3×Dic5—21no—
80C2×C3×Dic5C2×D412yes—
81C3×Dic10—41no—
82#1204—42no—
83D3×Dic5—22no—
84D3×Dic5—22no—
85D3×Dic5—42no—
86C3⋊#170—42no—
87C3×#170—42no—
88C2×D3×Q8—51no—
89C2×D3×Q8AGL(1,𝔽5)11yescharacteristic
90C3×C4⋊Q8—51no—
91#944—51no—
92#944—51no—
93D3×C4⋊C4—102no—
94#888—102no—
95C42×D3—51no—
96#869—51no—
97C3×C4⋊C8—52no—
98(C2×D3)⋊C8—102no—
99#815—102no—
100C4⋊(C3⋊C8)—52no—
101C2×C5×Q8—31no—
102C2×C5×Q8—61no—
103C2×C5×Q8—122no—
104C2×C5×Q8C4×D311yescharacteristic
105C22×C4×C5—31no—
106C22×C4×C5—61no—
107C22×Dic5—32no—
108C22×Dic5—61no—
109C2×Dic10—31no—
110C2×Dic10—61no—
111C2×Dic10—122no—
112C2×Dic10C4×D311yescharacteristic
113C2×#170—22no—
114C2×#170—62no—
115#683—22no—
116#683—62no—
117#683—62no—
118#683—62no—
119C4×Dic5—31no—
120C4×Dic5—32no—
121C4×Dic5C22×D311yescharacteristic
122C2×C4⋊Q8—151no—
123#432—152no—
124#333—51no—
125#333—152no—
126#333—151no—
127C22×C3×C5C2×C22⋊C411yescharacteristic
128C2×C5×D3—22no—
129C2×C5×D3#10012yes—
130C3×C4×C5—21no—
131C3×C4×C5—41no—
132Dic15—22no—
133Dic15—41no—
134C3×Dic5—41no—
135C3×Dic5#13712yes—
136C3×Dic5#13712yes—
137C5×Dic3—21no—
138C5×Dic3—41no—
139C2×C3×Q8—51no—
140C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
141D3×Q8—42no—
142D3×Q8—202no—
143C2×C4×D3—21no—
144C2×C4×D3—52no—
145C2×C4×D3—101no—
146C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
147C2×Dic6—21no—
148C2×Dic6—51no—
149C2×Dic6—101no—
150C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
151C2×C3×C8—102no—
152C3×C4⋊C4—102no—
153C3×C42—51no—
154C4⋊Dic3—102no—
155Dic3⋊C4—102no—
156Dic3⋊C4—102no—
157C4×Dic3—51no—
158C4×Dic3—52no—
159C2×C3⋊C8—102no—
160C23×C5—31no—
161C5×Q8—41no—
162C5×Q8—121no—
163C5×Q8—122no—
164C2×C4×C5—21no—
165C2×C4×C5—31no—
166C2×C4×C5—62no—
167C2×C4×C5—61no—
168C2×C4×C5—122no—
169C2×C4×C5C2×C4×D311yescharacteristic
170C2×Dic5—21no—
171C2×Dic5—32no—
172C2×Dic5—62no—
173C2×Dic5—62no—
174C2×Dic5—61no—
175C2×Dic5—122no—
176C2×Dic5D3×D412yes—
177Dic10—41no—
178Dic10—122no—
179Dic10—121no—
180#170—42no—
181#170—122no—
182C22×Q8—31no—
183C22×Q8—151no—
184C4⋊Q8—51no—
185C4⋊Q8—151no—
186C4⋊Q8—151no—
187C4⋊Q8—151no—
188C4⋊Q8—302no—
189C2×C4⋊C4—302no—
190C2×C42—151no—
191C4⋊C8—52no—
192C4⋊C8—152no—
193C22⋊C8—302no—
194#97—302no—
195C2×C3×C5C2×#10011yescharacteristic
196C2×C3×C5#41912yes—
197C5×D3—22no—
198C22×D3C22⋊AGL(1,𝔽5)11yescharacteristic
199C3×Q8—41no—
200C3×Q8—201no—
201C2×C3×C4—21no—
202C2×C3×C4—52no—
203C2×C3×C4—101no—
204C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—52no—
207C2×Dic3—101no—
208C2×Dic3C22⋊AGL(1,𝔽5)11yescharacteristic
209C4×D3—22no—
210C4×D3—42no—
211C4×D3—102no—
212C4×D3—102no—
213C4×D3—202no—
214Dic6—42no—
215Dic6—41no—
216Dic6—202no—
217Dic6—201no—
218C3×C8—202no—
219C3⋊C8—202no—
220C22×C5—32no—
221C22×C5—62no—
222C22×C5D3×C22⋊C411yescharacteristic
223C4×C5—21no—
224C4×C5—41no—
225C4×C5—61no—
226C4×C5—121no—
227Dic5—41no—
228Dic5—62no—
229Dic5—121no—
230Dic5#91512yes—
231Dic5#91812yes—
232C2×Q8—31no—
233C2×Q8—51no—
234C2×Q8—61no—
235C2×Q8—122no—
236C2×Q8—151no—
237C2×Q8—301no—
238C2×Q8—602no—
239C2×Q8D3×AGL(1,𝔽5)11yescharacteristic
240C22×C4—31no—
241C22×C4—61no—
242C22×C4—152no—
243C22×C4—301no—
244C2×C8—102no—
245C2×C8—302no—
246C4⋊C4—102no—
247C4⋊C4—302no—
248C4⋊C4—302no—
249C4⋊C4—302no—
250C42—51no—
251C42—152no—
252C42—151no—
253C3×C5#155811yescharacteristic, complemented, Hall
254C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
255C2×D3—22no—
256C2×D3#401312yes—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—52no—
260C3×C4—52no—
261C3×C4—201no—
262Dic3—21no—
263Dic3—41no—
264Dic3—102no—
265Dic3—201no—
266C2×C5—62no—
267C2×C5D3×#10011yescharacteristic
268C2×C5#480212yes—
269C23—31no—
270Q8—41no—
271Q8—121no—
272Q8—122no—
273Q8—201no—
274Q8—601no—
275Q8—602no—
276C2×C4—21no—
277C2×C4—31no—
278C2×C4—52no—
279C2×C4—62no—
280C2×C4—61no—
281C2×C4—101no—
282C2×C4—122no—
283C2×C4—152no—
284C2×C4—302no—
285C2×C4—302no—
286C2×C4—301no—
287C2×C4—602no—
288C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
289C8—202no—
290C8—602no—
291C2×C3#6578212yes—
292C2×C3C2×#401311yescharacteristic
293D3—22no—
294C5#6848711yescharacteristic, complemented, Sylow
295C22—32no—
296C22—62no—
297C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
298C4—21no—
299C4—41no—
300C4—52no—
301C4—52no—
302C4—61no—
303C4—121no—
304C4—201no—
305C4—302no—
306C4—601no—
307C3#1060497211yescharacteristic, complemented, Sylow
308C2—62no—
309C2#1175196212yes—
310C2D3×#401311yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC2×C3×C5×Q8C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C5×D3C22⋊C4
cosocleC3×C4×Dic5C23
hypercenterC2×Q8D3×AGL(1,𝔽5)
hypocenterC3×C5#1558
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11746802C2semidirect
4. C3×#64931C2semidirect
5. #11746511C2semidirect
25. #64931D3semidirect
253. C3×C5#1558semidirect
294. C5#68487semidirect
307. C3#10604972semidirect

Maximal subgroups

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

subgroupindex
D3×#40902
#117518042
#117468022
C3×#649312
#117465112
#106049723
#684875