Finite Group Databasebeta

← #13085920

Subgroups — #13085920

Order 1440.

430 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C2×#67152C211yesmaximal, characteristic
2C2×#67137C211yesmaximal, characteristic, complemented
3C2×C3×D3×Dic5C211yesmaximal, characteristic, complemented
4#10623831C211yesmaximal, characteristic, complemented
5#10623813C211yesmaximal, characteristic, complemented
6C5×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
7C3×Dic3×Dic5C211yesmaximal, characteristic, complemented
8#91552—31nomaximal
9#91405—31nomaximal
10C22×C3⋊D15C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C5×C3⋊Dic3C2211yescharacteristic
13C2×C3×Dic15C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67152C414yescomplemented
18#67137—22no—
19C3×D3×Dic5—22no—
20C4×#626—51nomaximal, Hall
21C2×#1220—31no—
22C22×C3×Dic5—31no—
23C2×#1203—31no—
24C2×#1200—31no—
25C2×#1200—31no—
26C2×#1200—61no—
27C2×D3×Dic5D311yescharacteristic, complemented
28#6714—31no—
29C5×#217—31no—
30C5×(C2×D3)⋊C4—31no—
31C3×C4×Dic5—31no—
32#6669—31no—
33#6668—31no—
34#6667—31no—
35Dic3×Dic5D311yescharacteristic, complemented
36C22×C32×C5C2311yescharacteristic
37C2×C3⋊D15—21no—
38C2×C3⋊D15—21no—
39C2×C3⋊D15C2×C412yescomplemented
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C2×C3×C5×D3C2×C412yescomplemented
43C5×C3⋊Dic3—21no—
44C3×Dic15—21no—
45C3×Dic15C2×C412yescomplemented
46C3×C5×Dic3—21no—
47C32×Dic5D412yescomplemented
48#4027—91noHall
49C2×C4×C3⋊D3—51no—
50C2×C3×C4×D3—51no—
51C2×#626—51no—
52C3×C4×Dic3—51no—
53Dic3⋊Dic3—51no—
54#3722—51no—
55(C2×D3)⋊Dic3D511yescharacteristic, complemented
56C23×C3×C5—31no—
57C22×D15—31no—
58C22×D15—31no—
59C22×D15—61no—
60C22×C5×D3C2×D311yescharacteristic
61C2×C3×C4×C5—31no—
62#1220—34no—
63C2×Dic15C2×D311yescharacteristic
64C2×Dic15C2×D311yescharacteristic
65C2×C5×Dic3—31no—
66C2×C5×Dic3—31no—
67C2×C5×Dic3—61no—
68C2×C5×Dic3C2×D311yescharacteristic
69C2×C3×Dic5—21no—
70C2×C3×Dic5—31no—
71C2×C3×Dic5—31no—
72C2×C3×Dic5—62no—
73C2×C3×Dic5C2×D311yescharacteristic
74C2×C3×Dic5C2×D311yescharacteristic
75#1203—34no—
76#1200—64no—
77#1200—62no—
78#1200—62no—
79D3×Dic5—22no—
80C4×#67—151no—
81C4×D12—151no—
82C2×C32×C5C22×C411yescharacteristic
83C2×C32×C5C2×D411yescharacteristic
84C2×C32×C5#4111yescharacteristic
85C3⋊D15—22no—
86C3×C5×D3—22no—
87C2×C22⋊D5—91no—
88C22×Dic5—31no—
89C2×C4×D5—91no—
90C5×C22⋊C4—91no—
91#685—91no—
92#683—91no—
93C4×Dic5—31no—
94C22×C3⋊D3—51no—
95C22×C3×D3C2×D511yescharacteristic
96C2×C32×C4—51no—
97C2×C3⋊Dic3C2×D511yescharacteristic
98C4×C3⋊D3—102no—
99C2×C3×Dic3—51no—
100C2×C3×Dic3—51no—
101C2×C3×Dic3C2×D511yescharacteristic
102C3×C4×D3—102no—
103#626—54no—
104C22×C3×C5—21no—
105C22×C3×C5—32no—
106C22×C3×C5—61no—
107C22×C3×C5—61no—
108C22×C3×C5C22×D311yescharacteristic
109C22×C3×C5C22×D311yescharacteristic
110C2×D15—32no—
111C2×D15—32no—
112C2×D15—62no—
113C2×D15—62no—
114C2×D15—62no—
115C2×D15—61no—
116C2×D15—61no—
117C2×D15—61no—
118C2×D15—61no—
119C2×C5×D3—21no—
120C2×C5×D3—21no—
121C2×C5×D3C4×D312yescomplemented
122C3×C4×C5—61no—
123Dic15—21no—
124Dic15C4×D312yescomplemented
125C3×Dic5—22no—
126C3×Dic5—32no—
127C3×Dic5—61no—
128C3×Dic5D1212yescomplemented
129C3×Dic5#6712yescomplemented
130C5×Dic3—21no—
131C5×Dic3—62no—
132C5×Dic3—61no—
133C5×Dic3—61no—
134C22×C3×C4—151no—
135C2×#67—151no—
136C2×D12—151no—
137C2×C4×D3—51no—
138C2×C4×D3—151no—
139C2×C4×D3—151no—
140C2×C4×D3—301no—
141C3×C42—151no—
142#217—31no—
143(C2×D3)⋊C4—31no—
144(C2×D3)⋊C4—151no—
145(C2×D3)⋊C4—151no—
146C4⋊Dic3—151no—
147Dic3⋊C4—151no—
148C4×Dic3—51no—
149C32×C5C4×D411yescharacteristic, complemented, Hall
150C23×C5—31no—
151C22×D5—91no—
152C2×C4×C5—31no—
153C2×C4×C5—91no—
154C22⋊D5—94no—
155C2×Dic5—31no—
156C2×Dic5—31no—
157C2×Dic5—62no—
158C2×Dic5D3211yescharacteristic
159C4×D5—182no—
160C22×C32C22×D511yescharacteristic
161C2×C3⋊D3—52no—
162C2×C3⋊D3—101no—
163C2×C3⋊D3—101no—
164C2×C3×D3—21no—
165C2×C3×D3—21no—
166C2×C3×D3C4×D512yescomplemented
167C32×C4—52no—
168C3⋊Dic3—21no—
169C3×Dic3—21no—
170C3×Dic3—52no—
171C3×Dic3—101no—
172C4×D4—451noSylow
173C2×C3×C5—21no—
174C2×C3×C5—21no—
175C2×C3×C5—21no—
176C2×C3×C5—62no—
177C2×C3×C5C2×C4×D311yescharacteristic
178C2×C3×C5C2×C4×D311yescharacteristic
179C2×C3×C5C2×D1211yescharacteristic
180C2×C3×C5#23511yescharacteristic
181C2×C3×C5#23511yescharacteristic
182C2×C3×C5C2×#6711yescharacteristic
183D15—64no—
184D15—62no—
185D15—62no—
186C5×D3—22no—
187C23×C3—31no—
188C22×D3—151no—
189C22×D3—151no—
190C22×D3—301no—
191C22×D3D3×D511yescharacteristic
192C2×C3×C4—31no—
193C2×C3×C4—51no—
194C2×C3×C4—51no—
195C2×C3×C4—101no—
196C2×C3×C4—151no—
197C2×C3×C4—151no—
198C2×C3×C4—302no—
199#67—154no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—51no—
203C2×Dic3—51no—
204C2×Dic3—61no—
205C2×Dic3D3×D511yescharacteristic
206D12—154no—
207C4×D3—102no—
208C4×D3—302no—
209C4×D3—302no—
210C4×D3—304no—
211C22×C5—32no—
212C22×C5—61no—
213C22×C5—61no—
214C22×C5C2×D3211yescharacteristic
215C2×D5—92no—
216C2×D5—181no—
217C2×D5—181no—
218C4×C5—61no—
219C4×C5—181no—
220Dic5—32no—
221Dic5—61no—
222Dic5#62612yescomplemented
223C2×C32C2×C4×D511yescharacteristic
224C2×C32D4×D511yescharacteristic
225C2×C32#71111yescharacteristic
226C3⋊D3—102no—
227C3×D3—22no—
228C2×D4—451no—
229C22×C4—151no—
230C22×C4—451no—
231C4⋊C4—451no—
232C22⋊C4—91no—
233C22⋊C4—451no—
234C42—151no—
235C3×C5—21no—
236C3×C5C4×D1211yescharacteristic, complemented
237C3×C5C4×#6711yescharacteristic, complemented
238C22×C3—21no—
239C22×C3—32no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3C2×D3×D511yescharacteristic
243C22×C3C2×D3×D511yescharacteristic
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—152no—
247C2×D3—152no—
248C2×D3—302no—
249C2×D3—302no—
250C2×D3—302no—
251C2×D3—301no—
252C2×D3—301no—
253C2×D3—301no—
254C2×D3—301no—
255C2×D3#120012yescomplemented
256C3×C4—52no—
257C3×C4—52no—
258C3×C4—61no—
259C3×C4—102no—
260C3×C4—152no—
261C3×C4—301no—
262Dic3—21no—
263Dic3—52no—
264Dic3—62no—
265Dic3—61no—
266Dic3—61no—
267Dic3—101no—
268C2×C5—62no—
269C2×C5#379811yescharacteristic
270C2×C5C4×D3211yescharacteristic
271C2×C5C2×#62611yescharacteristic
272D5—182no—
273C32#402711yescharacteristic, complemented, Sylow
274C23—31no—
275C23—451no—
276D4—454no—
277C2×C4—31no—
278C2×C4—51no—
279C2×C4—91no—
280C2×C4—151no—
281C2×C4—151no—
282C2×C4—302no—
283C2×C4—902no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—62no—
288C2×C3C4×D3×D511yescharacteristic
289C2×C3D5×D1211yescharacteristic
290C2×C3#678211yescharacteristic
291C2×C3#678211yescharacteristic
292C2×C3C2×#120011yescharacteristic
293C2×C3D5×#6711yescharacteristic
294D3—22no—
295D3—304no—
296D3—302no—
297D3—302no—
298C5C4×#62611yescharacteristic, complemented, Sylow
299C22—32no—
300C22—61no—
301C22—61no—
302C22—452no—
303C22—901no—
304C22—901no—
305C22D32×D511yescharacteristic
306C4—52no—
307C4—61no—
308C4—152no—
309C4—181no—
310C4—301no—
311C3—21no—
312C3#9140511yescharacteristic, complemented
313C3#9155211yescharacteristic, complemented
314C2—62no—
315C2—902no—
316C2D3×#120011yescharacteristic
317C2D5×#62611yescharacteristic
318C2#1062404811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutatorC2×C32×C5C22×C4
Fitting, socle, cosocleC22×C32×C5C23
prime radicalC2×#67152C2
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#67137C2semidirect
3. C2×C3×D3×Dic5C2semidirect
4. #10623831C2semidirect
5. #10623813C2semidirect
6. C5×(C2×D3)⋊Dic3C2semidirect
7. C3×Dic3×Dic5C2semidirect
17. #67152C4semidirect
27. C2×D3×Dic5D3semidirect
35. Dic3×Dic5D3semidirect
39. C2×C3⋊D15C2×C4semidirect
42. C2×C3×C5×D3C2×C4semidirect
45. C3×Dic15C2×C4semidirect
47. C32×Dic5D4semidirect
55. (C2×D3)⋊Dic3D5semidirect
121. C2×C5×D3C4×D3semidirect
124. Dic15C4×D3semidirect
128. C3×Dic5D12semidirect
129. C3×Dic5#67semidirect
149. C32×C5C4×D4semidirect
166. C2×C3×D3C4×D5semidirect
222. Dic5#626semidirect
236. C3×C5C4×D12semidirect
237. C3×C5C4×#67semidirect
255. C2×D3#1200semidirect
273. C32#4027semidirect
298. C5C4×#626semidirect
312. C3#91405semidirect
313. C3#91552semidirect

Maximal subgroups

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

subgroupindex
C2×#671522
C2×#671372
C2×C3×D3×Dic52
#106238312
#106238132
C5×(C2×D3)⋊Dic32
C3×Dic3×Dic52
#915523
#914053
C4×#6265