Finite Group Databasebeta

← #11753631

Subgroups — #11753631

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⋊C4×D15C211yesmaximal, characteristic, complemented
2C5×#890C211yesmaximal, characteristic, complemented
3C3×#4041C211yesmaximal, characteristic, complemented
4#91465C212yesmaximal, complemented
5#91465C211yesmaximal, characteristic, complemented
6C4×#1202C211yesmaximal, characteristic, complemented
7Dic3×D20C211yesmaximal, characteristic, complemented
8#91407C212yesmaximal, complemented
9#91396C212yesmaximal, complemented
10#91361C212yesmaximal, complemented
11#91358C211yesmaximal, characteristic, complemented
12#66029—31nomaximal, Hall
13C2×C4×D15C2211yescharacteristic
14C2×C4×D15C2212yes—
15C2×C3×D20C2211yescharacteristic
16C2×C3×D20C2212yes—
17C2×C3×C4×D5C2211yescharacteristic
18C2×#1202C2212yes—
19C2×#1202C2211yescharacteristic
20C2×D5×Dic3C2212yes—
21C3×C5×C4⋊C4C2211yescharacteristic
22C4⋊Dic15C2211yescharacteristic
23#6712C2212yes—
24C5×C4⋊Dic3C2211yescharacteristic
25C5×C4⋊Dic3C2212yes—
26C5×Dic3⋊C4C2212yes—
27C4×C5×Dic3C2211yescharacteristic
28C3×#685C2212yes—
29Dic3⋊Dic5C2211yescharacteristic
30#6667C2211yescharacteristic
31#6665C2211yescharacteristic
32#6665C2212yes—
33#6665C2212yes—
34#6665C2212yes—
35Dic3×Dic5C2212yes—
36#5614—51nomaximal, Hall
37C5×#125—31no—
38#4044—32no—
39#4041—32no—
40#4041—31no—
41#4041D311yescharacteristic, complemented
42#4040—32no—
43#4039—32no—
44#4038—31no—
45D5×C4⋊C4—31no—
46C4×D20—31no—
47C4×D20—31no—
48C22×D15C2311yescharacteristic
49C22×C3×D5C2311yescharacteristic
50C22×C3×D5C2312yes—
51C2×C3×C4×C5C2312yes—
52C2×C3×C4×C5C2311yescharacteristic
53C2×Dic15C2312yes—
54C2×Dic15C2311yescharacteristic
55C4×D15—22no—
56C4×D15—42no—
57C2×C5×Dic3C2312yes—
58C2×C5×Dic3C2311yescharacteristic
59C2×C5×Dic3C2311yescharacteristic
60C2×C3×Dic5C2311yescharacteristic
61C3×D20—22no—
62C3×D20—42no—
63C3×C4×D5—22no—
64#1202—24no—
65#1202—22no—
66D5×Dic3—42no—
67C3×#121—51no—
68#938—52no—
69#938—51no—
70#935—52no—
71D4×Dic3—51no—
72C4×#67—51no—
73#926—51no—
74D3×C4⋊C4—51no—
75#890D511yescharacteristic, complemented
76#67⋊C4—52no—
77#879—52no—
78C2×D20—31no—
79C2×D20—32no—
80C2×D20C2×D311yescharacteristic
81C2×D20C2×D312yes—
82C2×C4×D5—31no—
83C2×C4×D5—32no—
84C2×C4×D5—32no—
85C2×C4×D5C2×D311yescharacteristic
86C5×C4⋊C4—32no—
87C5×C4⋊C4—31no—
88C5×C4⋊C4—32no—
89C5×C4⋊C4C2×D311yescharacteristic
90C42×C5—31no—
91#685—31no—
92#685—32no—
93#685—32no—
94#685—32no—
95#685—31no—
96#685C2×D312yes—
97C4⋊Dic5—31no—
98C4⋊Dic5—31no—
99#683—32no—
100C4×Dic5—32no—
101#553—151noSylow
102C22×C3×C5C2411yescharacteristic
103C2×D15—21no—
104C2×D15—21no—
105C2×D15#4112yes—
106C2×C3×D5—21no—
107C2×C3×D5—22no—
108C2×C3×D5—21no—
109C2×C3×D5—22no—
110C2×C3×D5—22no—
111C2×C3×D5—21no—
112C3×C4×C5—22no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—22no—
116C3×Dic5—21no—
117C5×Dic3—22no—
118C5×Dic3—21no—
119C5×Dic3—21no—
120C2×C3×D4—51no—
121C2×C3×D4—52no—
122C22×C3×C4—51no—
123C2×#67—52no—
124C2×#67—51no—
125C22×Dic3—52no—
126C2×C4×D3—51no—
127C2×C4×D3—52no—
128C3×C4⋊C4C2×D511yescharacteristic
129C3×C22⋊C4—52no—
130#217—52no—
131#217—52no—
132#217—52no—
133#217—51no—
134(C2×D3)⋊C4—51no—
135C4⋊Dic3—51no—
136C4⋊Dic3C2×D512yes—
137C4⋊Dic3C2×D511yescharacteristic
138Dic3⋊C4—52no—
139Dic3⋊C4—51no—
140Dic3⋊C4C2×D512yes—
141C4×Dic3—52no—
142C4×Dic3C2×D511yescharacteristic
143C22×D5—31no—
144C22×D5C22×D311yescharacteristic
145C22×D5C22×D312yes—
146C2×C4×C5—32no—
147C2×C4×C5—31no—
148C2×C4×C5—31no—
149C2×C4×C5C22×D312yes—
150C2×C4×C5C22×D311yescharacteristic
151C2×Dic5—32no—
152C2×Dic5—31no—
153C2×Dic5C22×D311yescharacteristic
154D20—22no—
155D20—42no—
156D20—62no—
157D20—64no—
158C4×D5—22no—
159C4×D5—62no—
160C4×D5—122no—
161C4×D5—122no—
162#126—152no—
163#125—31no—
164#123—152no—
165#121—51no—
166#121—152no—
167#121—151no—
168C4×D4—151no—
169C4×D4—151no—
170C4×D4—152no—
171#117—151no—
172C2×C4⋊C4—151no—
173C2×C3×C5C2×#4111yescharacteristic
174C2×C3×C5C2×#4111yescharacteristic
175C2×C3×C5H(5,𝔽2)11yescharacteristic
176D15—22no—
177C3×D5—41no—
178C3×D5—42no—
179C23×C3—51no—
180C23×C3—52no—
181C22×D3—51no—
182C3×D4—102no—
183C3×D4—202no—
184C2×C3×C4—51no—
185C2×C3×C4—102no—
186C2×C3×C4C22×D512yes—
187C2×C3×C4C22×D511yescharacteristic
188#67—104no—
189#67—102no—
190C2×Dic3—52no—
191C2×Dic3—51no—
192C2×Dic3—202no—
193C2×Dic3C22×D512yes—
194C2×Dic3C22×D511yescharacteristic
195C2×Dic3C22×D511yescharacteristic
196C4×D3—102no—
197C4×D3—202no—
198C22×C5C23×D311yescharacteristic
199C2×D5—21no—
200C2×D5—22no—
201C2×D5—22no—
202C2×D5—21no—
203C2×D5—21no—
204C2×D5—22no—
205C2×D5—32no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—22no—
209C4×C5—61no—
210C4×C5—62no—
211C4×C5—61no—
212C4×C5#23712yes—
213Dic5—21no—
214Dic5—61no—
215Dic5—62no—
216C2×D4—51no—
217C2×D4—52no—
218C2×D4—151no—
219C2×D4—152no—
220C22×C4—51no—
221C22×C4—152no—
222C22×C4—151no—
223C22×C4—152no—
224C4⋊C4—31no—
225C4⋊C4—32no—
226C4⋊C4—32no—
227C4⋊C4—151no—
228C4⋊C4—151no—
229C4⋊C4—152no—
230C4⋊C4D3×D511yescharacteristic
231C22⋊C4—52no—
232C22⋊C4—151no—
233C22⋊C4—152no—
234C22⋊C4—152no—
235C22⋊C4—151no—
236C22⋊C4—152no—
237C42—31no—
238C42—152no—
239C3×C5#55311yescharacteristic, complemented, Hall
240C22×C3—101no—
241C22×C3—102no—
242C22×C3—101no—
243C22×C3—102no—
244C22×C3—102no—
245C22×C3—101no—
246C22×C3C23×D511yescharacteristic
247C2×D3—52no—
248C2×D3—101no—
249C2×D3—101no—
250C3×C4—22no—
251C3×C4—101no—
252C3×C4#71312yes—
253Dic3—22no—
254Dic3—21no—
255Dic3—21no—
256Dic3—101no—
257Dic3—102no—
258C2×C5C2×#23711yescharacteristic
259C2×C5#100411yescharacteristic
260C2×C5D3×#4111yescharacteristic
261D5—41no—
262D5—42no—
263D5—62no—
264C23—52no—
265C23—51no—
266C23—151no—
267D4—102no—
268D4—202no—
269D4—302no—
270D4—304no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—32no—
274C2×C4—51no—
275C2×C4—102no—
276C2×C4—152no—
277C2×C4—151no—
278C2×C4—302no—
279C2×C4—602no—
280C2×C4—602no—
281C2×C4C2×D3×D512yes—
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—201no—
284C2×C3—202no—
285C2×C3C2×#71311yescharacteristic
286C2×C3D5×#4111yescharacteristic
287C2×C3#414911yescharacteristic
288D3—102no—
289C5#561411yescharacteristic, complemented, Sylow
290C22—101no—
291C22—102no—
292C22—102no—
293C22—101no—
294C22—102no—
295C22—101no—
296C22—152no—
297C22—301no—
298C22—301no—
299C22C22×D3×D511yescharacteristic
300C4—22no—
301C4—61no—
302C4—61no—
303C4—62no—
304C4—101no—
305C4—301no—
306C4—302no—
307C4#676612yes—
308C3#6602911yescharacteristic, complemented, Sylow
309C2—201no—
310C2—202no—
311C2—302no—
312C2C2×#676611yescharacteristic
313C2#9202611yescharacteristic
314C2#9203411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×D15C2semidirect
2. C5×#890C2semidirect
3. C3×#4041C2semidirect
4. #91465C2semidirect
5. #91465C2semidirect
6. C4×#1202C2semidirect
7. Dic3×D20C2semidirect
8. #91407C2semidirect
9. #91396C2semidirect
10. #91361C2semidirect
11. #91358C2semidirect
41. #4041D3semidirect
75. #890D5semidirect
239. C3×C5#553semidirect
289. C5#5614semidirect
308. C3#66029semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D152
C5×#8902
C3×#40412
#914652
C4×#12022
Dic3×D202
#914072
#913962
#913612
#913582
#660293
#56145