Finite Group Databasebeta

← #11754467

Subgroups — #11754467

Order 960.

476 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#1220C211yesmaximal, characteristic, complemented
2C5×#930C211yesmaximal, characteristic, complemented
3C3×#4072C211yesmaximal, characteristic, complemented
4#91557C212yesmaximal, complemented
5#91555C212yesmaximal, complemented
6#91466C211yesmaximal, characteristic, complemented
7#91437C211yesmaximal, characteristic, complemented
8#91435C211yesmaximal, characteristic, complemented
9#91405C212yesmaximal, complemented
10#91384C211yesmaximal, characteristic, complemented
11#91372C212yesmaximal, complemented
12#65948—31nomaximal, Hall
13C22×C3×C4×C5C2211yescharacteristic
14C2×#1220C2211yescharacteristic
15C2×C4×D15C2211yescharacteristic
16C2×C5×#67C2212yes—
17C2×C5×D12C2211yescharacteristic
18C2×#1203C2212yes—
19C2×#1200C2212yes—
20C2×D3×Dic5C2212yes—
21#6719C2211yescharacteristic
22#6714C2211yescharacteristic
23#6712C2211yescharacteristic
24C4×Dic15C2211yescharacteristic
25C5×(C2×D3)⋊C4C2212yes—
26C5×C4⋊Dic3C2211yescharacteristic
27C3×C22⋊Dic5C2212yes—
28C3×C4⋊Dic5C2211yescharacteristic
29C3×C4⋊Dic5C2211yescharacteristic
30C3×C4×Dic5C2212yes—
31Dic3⋊Dic5C2212yes—
32#6669C2212yes—
33#6668C2212yes—
34(C2×D3)⋊Dic5C2212yes—
35(C2×D3)⋊Dic5C2212yes—
36#5530—51nomaximal, Hall
37C5×#121—31no—
38#4084—32no—
39#4084—31no—
40#4081—32no—
41D4×Dic5—31no—
42C4×C22⋊D5—31no—
43#4072D311yescharacteristic, complemented
44D5×C4⋊C4—31no—
45#4036—31no—
46#4027—32no—
47#4025—32no—
48C23×C3×C5C2311yescharacteristic
49C22×D15C2311yescharacteristic
50C22×C5×D3C2312yes—
51C2×C3×C4×C5—22no—
52C2×C3×C4×C5C2311yescharacteristic
53C2×C3×C4×C5C2311yescharacteristic
54#1220—22no—
55C2×Dic15C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57C2×Dic15C2311yescharacteristic
58C4×D15—22no—
59C5×#67—42no—
60C2×C5×Dic3C2312yes—
61C5×D12—22no—
62C2×C3×Dic5C2312yes—
63C2×C3×Dic5C2312yes—
64#1203—24no—
65#1200—42no—
66D3×Dic5—42no—
67C3×#117—51no—
68#930D511yescharacteristic, complemented
69C4×#67—51no—
70#895—51no—
71#893—51no—
72D3×C4⋊C4—51no—
73#890—51no—
74#884—52no—
75#883—52no—
76#876—52no—
77C4×D12—52no—
78C2×C5×D4—32no—
79C2×C5×D4—31no—
80C22×C4×C5C2×D311yescharacteristic
81C2×C22⋊D5—31no—
82C2×C22⋊D5—32no—
83C22×Dic5—32no—
84C2×C4×D5—31no—
85C2×C4×D5—32no—
86C5×C4⋊C4—31no—
87C5×C22⋊C4—32no—
88C22⋊Dic5—31no—
89C22⋊Dic5—32no—
90C22⋊Dic5—32no—
91C22⋊Dic5C2×D312yes—
92#685—31no—
93C4⋊Dic5—32no—
94C4⋊Dic5C2×D311yescharacteristic
95C4⋊Dic5C2×D311yescharacteristic
96#683—31no—
97#683—32no—
98#683—32no—
99C4×Dic5—31no—
100C4×Dic5C2×D312yes—
101#553—151noSylow
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5C2411yescharacteristic
106C2×D15—21no—
107C2×D15—21no—
108C2×D15#4112yes—
109C2×C5×D3—22no—
110C2×C5×D3—22no—
111C2×C5×D3—22no—
112C3×C4×C5—21no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—21no—
116Dic15—21no—
117C3×Dic5—22no—
118C3×Dic5—22no—
119C5×Dic3—22no—
120C22×C3×C4C2×D511yescharacteristic
121C2×#67—51no—
122C2×#67C2×D512yes—
123C2×D12—52no—
124C2×D12C2×D511yescharacteristic
125C2×C4×D3—51no—
126C2×C4×D3—52no—
127C2×C4×D3—52no—
128C3×C4⋊C4—51no—
129C3×C4⋊C4—51no—
130C3×C22⋊C4—52no—
131C3×C42—52no—
132#217—51no—
133(C2×D3)⋊C4—52no—
134(C2×D3)⋊C4—52no—
135(C2×D3)⋊C4—51no—
136(C2×D3)⋊C4C2×D512yes—
137C4⋊Dic3—52no—
138C4⋊Dic3C2×D511yescharacteristic
139Dic3⋊C4—51no—
140Dic3⋊C4—52no—
141Dic3⋊C4—52no—
142C4×Dic3—51no—
143C23×C5—32no—
144C23×C5C22×D311yescharacteristic
145C22×D5—31no—
146C5×D4—62no—
147C5×D4—122no—
148C2×C4×C5—22no—
149C2×C4×C5—32no—
150C2×C4×C5C22×D311yescharacteristic
151C2×C4×C5C22×D311yescharacteristic
152C22⋊D5—64no—
153C22⋊D5—62no—
154C2×Dic5—31no—
155C2×Dic5—31no—
156C2×Dic5—31no—
157C2×Dic5—122no—
158C2×Dic5C22×D312yes—
159C2×Dic5C22×D312yes—
160C4×D5—62no—
161C4×D5—122no—
162#126—152no—
163#125—151no—
164#123—152no—
165#121—31no—
166#121—152no—
167#121—151no—
168C4×D4—152no—
169C4×D4—151no—
170C4×D4—151no—
171#117—51no—
172C2×C4⋊C4—151no—
173C2×C3×C5—41no—
174C2×C3×C5C2×#4111yescharacteristic
175C2×C3×C5C2×#4111yescharacteristic
176C2×C3×C5H(5,𝔽2)11yescharacteristic
177D15—22no—
178C5×D3—42no—
179C23×C3C22×D511yescharacteristic
180C22×D3—51no—
181C22×D3C22×D512yes—
182C2×C3×C4—22no—
183C2×C3×C4—52no—
184C2×C3×C4—52no—
185C2×C3×C4C22×D511yescharacteristic
186C2×C3×C4C22×D511yescharacteristic
187#67—42no—
188#67—102no—
189C2×Dic3—51no—
190C2×Dic3—51no—
191C2×Dic3—51no—
192C2×Dic3C22×D512yes—
193D12—22no—
194D12—104no—
195C4×D3—102no—
196C4×D3—202no—
197C4×D3—202no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—62no—
202C22×C5—62no—
203C22×C5—62no—
204C22×C5C23×D311yescharacteristic
205C2×D5—32no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—21no—
209C4×C5—62no—
210C4×C5#23512yes—
211Dic5—22no—
212Dic5—22no—
213Dic5—61no—
214Dic5—61no—
215Dic5—61no—
216C2×D4—32no—
217C2×D4—31no—
218C2×D4—152no—
219C2×D4—151no—
220C22×C4—152no—
221C22×C4—152no—
222C22×C4—151no—
223C22×C4D3×D511yescharacteristic
224C4⋊C4—31no—
225C4⋊C4—51no—
226C4⋊C4—51no—
227C4⋊C4—152no—
228C4⋊C4—152no—
229C4⋊C4—152no—
230C4⋊C4—151no—
231C22⋊C4—32no—
232C22⋊C4—52no—
233C22⋊C4—152no—
234C22⋊C4—152no—
235C22⋊C4—151no—
236C22⋊C4—151no—
237C42—52no—
238C42—151no—
239C3×C5#55311yescharacteristic, complemented, Hall
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3C23×D511yescharacteristic
244C2×D3—22no—
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—52no—
248C2×D3—101no—
249C2×D3—101no—
250C3×C4—21no—
251C3×C4—102no—
252C3×C4—102no—
253C3×C4#71112yes—
254Dic3—22no—
255Dic3—101no—
256Dic3—101no—
257Dic3—101no—
258C2×C5—41no—
259C2×C5—122no—
260C2×C5C2×#23511yescharacteristic
261C2×C5D3×#4111yescharacteristic
262C2×C5#100911yescharacteristic
263D5—62no—
264C23—32no—
265C23—151no—
266C23C2×D3×D511yescharacteristic
267D4—62no—
268D4—122no—
269D4—304no—
270D4—302no—
271C2×C4—22no—
272C2×C4—32no—
273C2×C4—52no—
274C2×C4—52no—
275C2×C4—151no—
276C2×C4—151no—
277C2×C4—151no—
278C2×C4—302no—
279C2×C4—602no—
280C2×C4—602no—
281C2×C4C2×D3×D511yescharacteristic
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—41no—
284C2×C3C2×#71111yescharacteristic
285C2×C3#414411yescharacteristic
286C2×C3D5×#4111yescharacteristic
287D3—42no—
288D3—102no—
289C5#553011yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22—21no—
293C22—62no—
294C22—62no—
295C22—62no—
296C22—152no—
297C22—301no—
298C22—301no—
299C22C22×D3×D511yescharacteristic
300C4—21no—
301C4—62no—
302C4—102no—
303C4—102no—
304C4—301no—
305C4—301no—
306C4—301no—
307C4#678212yes—
308C3#6594811yescharacteristic, complemented, Sylow
309C2—41no—
310C2—122no—
311C2—302no—
312C2#9201611yescharacteristic
313C2#9201811yescharacteristic
314C2C2×#678211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C4×#1220C2semidirect
2. C5×#930C2semidirect
3. C3×#4072C2semidirect
4. #91557C2semidirect
5. #91555C2semidirect
6. #91466C2semidirect
7. #91437C2semidirect
8. #91435C2semidirect
9. #91405C2semidirect
10. #91384C2semidirect
11. #91372C2semidirect
43. #4072D3semidirect
68. #930D5semidirect
239. C3×C5#553semidirect
289. C5#5530semidirect
308. C3#65948semidirect

Maximal subgroups

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

subgroupindex
C4×#12202
C5×#9302
C3×#40722
#915572
#915552
#914662
#914372
#914352
#914052
#913842
#913722
#659483
#55305