Finite Group Databasebeta

← #13087701

Subgroups — #13087701

Order 1440.

338 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 75 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×C3×Dic30C211yesmaximal, characteristic, complemented
2C2×#67156C211yesmaximal, characteristic
3C5×#3752C211yesmaximal, characteristic, complemented
4C3×#6714C211yesmaximal, characteristic, complemented
5#10623830C211yesmaximal, characteristic, complemented
6#10623829C211yesmaximal, characteristic, complemented
7#10623828C211yesmaximal, characteristic, complemented
8#91462—31nomaximal
9#91326—31nomaximal
10C22×C5×C3⋊D3C2211yescharacteristic
11C22×C3×D15C2211yescharacteristic
12C2×C32×C4×C5C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×Dic15C2211yescharacteristic
16C2×C3×Dic15C2211yescharacteristic
17C3×Dic30—22no—
18#67156—22no—
19#63907—51nomaximal, Hall
20C2×Dic30D311yescharacteristic, complemented
21C2×C3×Dic10—31no—
22C2×#1203—31no—
23C2×#1201—31no—
24#6714D311yescharacteristic, complemented
25C5×(C2×D3)⋊C4—31no—
26C5×(C2×D3)⋊C4—31no—
27C5×(C2×D3)⋊C4—61no—
28C3×#685—31no—
29#6667—31no—
30(C2×D3)⋊Dic5—31no—
31(C2×D3)⋊Dic5—31no—
32#6665—31no—
33Dic3×Dic5—31no—
34Dic3×Dic5—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C5×C3⋊D3—21no—
37C2×C5×C3⋊D3—21no—
38C2×C5×C3⋊D3—21no—
39C2×C3×D15—21no—
40C2×C3×D15—21no—
41C2×C3×D15—21no—
42C32×C4×C5—21no—
43C5×C3⋊Dic3—21no—
44C3×Dic15—21no—
45C3×Dic15—21no—
46C3×Dic15D412yes—
47#4031—91noHall
48C2×C3×Dic6—51no—
49C2×#626—51no—
50#3752D511yescharacteristic, complemented
51C3×(C2×D3)⋊C4—51no—
52#3722—51no—
53(C2×D3)⋊Dic3—51no—
54Dic32—51no—
55C22×D15C2×D311yescharacteristic
56C22×C5×D3—31no—
57C22×C5×D3—31no—
58C22×C5×D3—61no—
59C22×C3×D5—31no—
60C2×C3×C4×C5—21no—
61C2×C3×C4×C5C2×D311yescharacteristic
62C2×C3×C4×C5C2×D311yescharacteristic
63C2×Dic15C2×D311yescharacteristic
64C2×Dic15C2×D311yescharacteristic
65C2×Dic15C2×D311yescharacteristic
66Dic30—22no—
67C2×C5×Dic3—31no—
68C2×C5×Dic3—31no—
69C2×C5×Dic3—61no—
70C2×C3×Dic5—31no—
71C2×C3×Dic5—31no—
72C2×C3×Dic5—31no—
73C3×Dic10—62no—
74#1203—62no—
75#1201—62no—
76#947—151no—
77#885—151no—
78C2×C32×C5C2×D411yescharacteristic
79C2×C32×C5#4111yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C5×C3⋊D3—41no—
82C3×D15—41no—
83C2×C22⋊D5—91no—
84C2×Dic10—31no—
85C5×C22⋊C4—91no—
86C22⋊Dic5—91no—
87#685—31no—
88#685—91no—
89C4×Dic5—91no—
90C22×C3⋊D3C2×D511yescharacteristic
91C22×C3×D3—51no—
92C2×C32×C4C2×D511yescharacteristic
93C2×C3⋊Dic3C2×D511yescharacteristic
94C2×C3×Dic3—51no—
95C2×C3×Dic3—51no—
96C2×C3×Dic3—51no—
97C3×Dic6—102no—
98#626—102no—
99C22×C3×C5—21no—
100C22×C3×C5C22×D311yescharacteristic
101C22×C3×C5C22×D311yescharacteristic
102C2×D15—21no—
103C2×D15—21no—
104C2×D15—21no—
105C2×C5×D3—62no—
106C2×C5×D3—61no—
107C2×C5×D3—61no—
108C2×C5×D3—61no—
109C2×C5×D3—61no—
110C2×C5×D3—61no—
111C2×C5×D3—61no—
112C2×C5×D3—121no—
113C2×C5×D3—121no—
114C2×C3×D5—61no—
115C2×C3×D5—61no—
116C2×C3×D5—61no—
117C3×C4×C5—21no—
118C3×C4×C5—21no—
119C3×C4×C5—41no—
120Dic15—21no—
121Dic15—21no—
122Dic15#6712yes—
123C3×Dic5—32no—
124C3×Dic5—61no—
125C3×Dic5—61no—
126C5×Dic3—61no—
127C5×Dic3—61no—
128C5×Dic3—121no—
129C2×C3×Q8—151no—
130C2×#67—151no—
131C2×D12—151no—
132C2×Dic6—51no—
133C3×C22⋊C4—151no—
134#217—151no—
135(C2×D3)⋊C4—31no—
136(C2×D3)⋊C4—31no—
137(C2×D3)⋊C4—51no—
138(C2×D3)⋊C4—61no—
139(C2×D3)⋊C4—151no—
140(C2×D3)⋊C4—151no—
141(C2×D3)⋊C4—151no—
142C4×Dic3—151no—
143C4×Dic3—151no—
144C32×C5#12411yescharacteristic, complemented, Hall
145C23×C5—91no—
146C22×D5—31no—
147C2×C4×C5—91no—
148C2×C4×C5D3211yescharacteristic
149C22⋊D5—182no—
150C2×Dic5—31no—
151C2×Dic5—31no—
152C2×Dic5—31no—
153Dic10—62no—
154C22×C32C22×D511yescharacteristic
155C2×C3⋊D3—21no—
156C2×C3⋊D3—21no—
157C2×C3⋊D3—21no—
158C2×C3×D3—101no—
159C2×C3×D3—101no—
160C2×C3×D3—101no—
161C32×C4—21no—
162C3⋊Dic3—21no—
163C3×Dic3—52no—
164C3×Dic3—101no—
165C3×Dic3—101no—
166#124—451noSylow
167C2×C3×C5—21no—
168C2×C3×C5—21no—
169C2×C3×C5—21no—
170C2×C3×C5#23511yescharacteristic
171C2×C3×C5D3×D411yescharacteristic
172C2×C3×C5#23711yescharacteristic
173C2×C3×C5#23911yescharacteristic
174C2×C3×C5#23911yescharacteristic
175C2×C3×C5C2×#6711yescharacteristic
176D15—41no—
177C3×D5—121no—
178C5×D3—122no—
179C5×D3—121no—
180C5×D3—121no—
181C23×C3—151no—
182C22×D3—31no—
183C22×D3—31no—
184C22×D3—51no—
185C22×D3—61no—
186C3×Q8—302no—
187C2×C3×C4—21no—
188C2×C3×C4—151no—
189C2×C3×C4—151no—
190C2×C3×C4—151no—
191C2×C3×C4D3×D511yescharacteristic
192C2×C3×C4D3×D511yescharacteristic
193#67—302no—
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—51no—
197C2×Dic3—51no—
198C2×Dic3—51no—
199C2×Dic3—61no—
200D12—302no—
201Dic6—102no—
202C22×C5—181no—
203C22×C5—181no—
204C22×C5—181no—
205C22×C5C2×D3211yescharacteristic
206C2×D5—61no—
207C2×D5—61no—
208C2×D5—61no—
209C4×C5—21no—
210C4×C5—181no—
211Dic5—32no—
212Dic5—61no—
213Dic5—61no—
214C2×C32#70911yescharacteristic
215C2×C32D4×D511yescharacteristic
216C2×C32#71111yescharacteristic
217C3⋊D3—41no—
218C3×D3—201no—
219C2×Q8—151no—
220C2×D4—451no—
221C22⋊C4—91no—
222C22⋊C4—151no—
223C22⋊C4—451no—
224C22⋊C4—451no—
225C42—451no—
226C3×C5—21no—
227C3×C5#88511yescharacteristic, complemented
228C3×C5#94711yescharacteristic, complemented
229C22×C3—21no—
230C22×C3—301no—
231C22×C3—301no—
232C22×C3—301no—
233C22×C3C2×D3×D511yescharacteristic
234C22×C3C2×D3×D511yescharacteristic
235C2×D3—62no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—61no—
242C2×D3—101no—
243C2×D3—101no—
244C2×D3—101no—
245C2×D3—121no—
246C2×D3—121no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—41no—
250C3×C4—152no—
251C3×C4—301no—
252C3×C4—301no—
253Dic3—52no—
254Dic3—61no—
255Dic3—61no—
256Dic3—101no—
257Dic3—101no—
258Dic3—121no—
259C2×C5—361no—
260C2×C5#379611yescharacteristic
261C2×C5#379911yescharacteristic
262C2×C5D3×#6711yescharacteristic
263D5—121no—
264C32#403111yescharacteristic, complemented, Sylow
265C23—91no—
266C23—151no—
267Q8—302no—
268D4—902no—
269C2×C4—91no—
270C2×C4—151no—
271C2×C4—151no—
272C2×C4—151no—
273C2×C4(C3⋊D3)⋊D511yescharacteristic
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—601no—
278C2×C3#676511yescharacteristic
279C2×C3#676811yescharacteristic
280C2×C3#676911yescharacteristic
281C2×C3#677211yescharacteristic
282C2×C3D5×#6711yescharacteristic
283C2×C3#679011yescharacteristic
284D3—122no—
285D3—121no—
286D3—121no—
287D3—201no—
288C5#6390711yescharacteristic, complemented, Sylow
289C22—181no—
290C22—181no—
291C22—181no—
292C22—301no—
293C22—301no—
294C22—301no—
295C22C2×(C3⋊D3)⋊D511yescharacteristic
296C4—21no—
297C4—152no—
298C4—181no—
299C4—301no—
300C4—301no—
301C3—21no—
302C3#9132611yescharacteristic, complemented
303C3#9146211yescharacteristic, complemented
304C2—361no—
305C2—601no—
306C2#1062422011yescharacteristic
307C2#1062422411yescharacteristic
308C2#1062423511yescharacteristic
subgroupG/H
center, FrattiniC22C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC22×C32×C5C23
FittingC2×C32×C4×C5C22
prime radicalC2×#67156C2
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5#124
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×Dic30C2semidirect
3. C5×#3752C2semidirect
4. C3×#6714C2semidirect
5. #10623830C2semidirect
6. #10623829C2semidirect
7. #10623828C2semidirect
20. C2×Dic30D3semidirect
24. #6714D3semidirect
50. #3752D5semidirect
144. C32×C5#124semidirect
227. C3×C5#885semidirect
228. C3×C5#947semidirect
264. C32#4031semidirect
288. C5#63907semidirect
302. C3#91326semidirect
303. C3#91462semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Dic302
C2×#671562
C5×#37522
C3×#67142
#106238302
#106238292
#106238282
#914623
#913263
#639075