Finite Group Databasebeta

← #11753391

Subgroups — #11753391

Order 960.

432 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#91784C211yesmaximal, characteristic
2C5×#888C211yesmaximal, characteristic, complemented
3C3×#4033C211yesmaximal, characteristic, complemented
4#91433C212yesmaximal, complemented
5#91424C212yesmaximal, complemented
6#91418C211yesmaximal, characteristic, complemented
7#91390C211yesmaximal, characteristic, complemented
8#91387C212yesmaximal, complemented
9#91386C211yesmaximal, characteristic, complemented
10#91375C212yesmaximal, complemented
11#91374C211yesmaximal, characteristic, complemented
12#66067—31nomaximal, Hall
13C2×D60C2211yescharacteristic
14C2×D60C2212yes—
15C2×C4×D15C2211yescharacteristic
16C2×C5×Dic6C2212yes—
17C2×C3×Dic10C2211yescharacteristic
18C2×#1200C2212yes—
19C3×C5×C4⋊C4C2211yescharacteristic
20#6714C2212yes—
21C5×C4⋊Dic3C2211yescharacteristic
22C5×Dic3⋊C4C2212yes—
23C4×C5×Dic3C2211yescharacteristic
24C3×#683C2212yes—
25C3×C4×Dic5C2211yescharacteristic
26C3×C4×Dic5C2212yes—
27Dic3⋊Dic5C2212yes—
28#6669C2211yescharacteristic
29#6669C2211yescharacteristic
30#6667C2212yes—
31#6667C2212yes—
32#6667C2211yescharacteristic
33#6667C2212yes—
34#6667C2212yes—
35#6667C2211yescharacteristic
36#5582—51nomaximal, Hall
37C5×C4⋊Q8—31no—
38#4093—32no—
39#4092—32no—
40#4044—32no—
41#4043—31no—
42#4041—31no—
43#4040—32no—
44#4039—31no—
45#4033D311yescharacteristic, complemented
46#4021—31no—
47#4018—31no—
48C22×D15C2311yescharacteristic
49C22×D15C2312yes—
50C2×C3×C4×C5C2312yes—
51C2×C3×C4×C5C2311yescharacteristic
52C2×Dic15C2311yescharacteristic
53D60—22no—
54D60—42no—
55C4×D15—22no—
56C2×C5×Dic3C2311yescharacteristic
57C2×C5×Dic3C2311yescharacteristic
58C2×C5×Dic3C2312yes—
59C5×Dic6—42no—
60C2×C3×Dic5C2311yescharacteristic
61C2×C3×Dic5C2312yes—
62C2×C3×Dic5C2311yescharacteristic
63C3×Dic10—22no—
64#1200—42no—
65C3×C4×Q8—51no—
66#947—51no—
67#946—51no—
68#896—52no—
69#895—51no—
70#894—52no—
71#888D511yescharacteristic, complemented
72#876—52no—
73#875—52no—
74C4×D12—51no—
75#872—51no—
76C2×C5×Q8—32no—
77C2×D20—31no—
78C2×D20—32no—
79C2×C4×D5—31no—
80C2×C4×D5—32no—
81C2×Dic10C2×D311yescharacteristic
82C5×C4⋊C4—32no—
83C5×C4⋊C4—31no—
84C5×C4⋊C4C2×D311yescharacteristic
85C42×C5—31no—
86#685—32no—
87#685—31no—
88#685—31no—
89#685—32no—
90#685—32no—
91#685—32no—
92#685—32no—
93C4⋊Dic5—32no—
94#683—31no—
95#683—31no—
96#683C2×D312yes—
97C4×Dic5C2×D311yescharacteristic
98C4×Dic5C2×D312yes—
99#542—151noSylow
100C22×C3×C5C2411yescharacteristic
101C2×D15—21no—
102C2×D15—22no—
103C2×D15—22no—
104C2×D15—21no—
105C2×D15—21no—
106C2×D15—22no—
107C3×C4×C5—22no—
108C3×C4×C5#4112yes—
109Dic15—21no—
110C3×Dic5—22no—
111C3×Dic5—21no—
112C3×Dic5—21no—
113C5×Dic3—22no—
114C5×Dic3—21no—
115C5×Dic3—21no—
116C2×C3×Q8—51no—
117C2×D12—51no—
118C2×D12—52no—
119C2×C4×D3—51no—
120C2×C4×D3—52no—
121C2×Dic6C2×D512yes—
122C3×C4⋊C4—52no—
123C3×C4⋊C4C2×D511yescharacteristic
124C3×C42—51no—
125C3×C42—52no—
126(C2×D3)⋊C4—52no—
127(C2×D3)⋊C4—51no—
128(C2×D3)⋊C4—51no—
129(C2×D3)⋊C4—52no—
130(C2×D3)⋊C4—52no—
131(C2×D3)⋊C4—52no—
132(C2×D3)⋊C4—52no—
133C4⋊Dic3C2×D511yescharacteristic
134Dic3⋊C4—51no—
135Dic3⋊C4—51no—
136Dic3⋊C4—52no—
137Dic3⋊C4C2×D512yes—
138C4×Dic3C2×D511yescharacteristic
139C22×D5—31no—
140C22×D5—32no—
141C5×Q8—122no—
142C2×C4×C5—31no—
143C2×C4×C5—31no—
144C2×C4×C5—32no—
145C2×C4×C5C22×D312yes—
146C2×C4×C5C22×D311yescharacteristic
147C2×Dic5—31no—
148C2×Dic5C22×D311yescharacteristic
149C2×Dic5C22×D311yescharacteristic
150C2×Dic5C22×D312yes—
151D20—62no—
152D20—122no—
153C4×D5—62no—
154C4×D5—122no—
155Dic10—22no—
156C4⋊Q8—31no—
157#126—152no—
158#124—151no—
159#124—152no—
160#123—152no—
161C22⋊Q8—151no—
162C22⋊Q8—152no—
163#121—151no—
164C4×Q8—51no—
165C4×D4—151no—
166#117—151no—
167C2×C3×C5C2×#4111yescharacteristic
168C2×C3×C5H(5,𝔽2)11yescharacteristic
169C2×C3×C52−1+411yescharacteristic
170D15—41no—
171D15—42no—
172C22×D3—52no—
173C22×D3—51no—
174C3×Q8—102no—
175C2×C3×C4—51no—
176C2×C3×C4—52no—
177C2×C3×C4—51no—
178C2×C3×C4C22×D512yes—
179C2×C3×C4C22×D511yescharacteristic
180C2×Dic3—51no—
181C2×Dic3C22×D511yescharacteristic
182C2×Dic3C22×D511yescharacteristic
183C2×Dic3C22×D512yes—
184D12—102no—
185D12—202no—
186C4×D3—102no—
187C4×D3—202no—
188Dic6—42no—
189C22×C5C23×D311yescharacteristic
190C2×D5—61no—
191C2×D5—62no—
192C2×D5—61no—
193C2×D5—62no—
194C2×D5—62no—
195C2×D5—61no—
196C4×C5—22no—
197C4×C5—61no—
198C4×C5—62no—
199C4×C5—61no—
200C4×C5#23512yes—
201Dic5—22no—
202Dic5—21no—
203Dic5—21no—
204Dic5—61no—
205C2×Q8—32no—
206C2×Q8—51no—
207C2×D4—151no—
208C2×D4—152no—
209C22×C4—152no—
210C22×C4—151no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—52no—
214C4⋊C4—151no—
215C4⋊C4—151no—
216C4⋊C4—152no—
217C4⋊C4D3×D511yescharacteristic
218C22⋊C4—151no—
219C22⋊C4—151no—
220C22⋊C4—152no—
221C22⋊C4—152no—
222C22⋊C4—152no—
223C22⋊C4—152no—
224C22⋊C4—152no—
225C42—31no—
226C42—51no—
227C42—52no—
228C3×C5#54211yescharacteristic, complemented, Hall
229C22×C3C23×D511yescharacteristic
230C2×D3—102no—
231C2×D3—101no—
232C2×D3—101no—
233C2×D3—102no—
234C2×D3—102no—
235C2×D3—101no—
236C3×C4—22no—
237C3×C4—101no—
238C3×C4—102no—
239C3×C4—101no—
240C3×C4#71312yes—
241Dic3—21no—
242Dic3—22no—
243Dic3—21no—
244Dic3—101no—
245C2×C5C2×#23511yescharacteristic
246C2×C5#100711yescharacteristic
247C2×C5#100911yescharacteristic
248D5—121no—
249D5—122no—
250C23—151no—
251C23—152no—
252Q8—102no—
253Q8—122no—
254D4—302no—
255D4—602no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—32no—
259C2×C4—51no—
260C2×C4—52no—
261C2×C4—51no—
262C2×C4—151no—
263C2×C4—302no—
264C2×C4—602no—
265C2×C4C2×D3×D512yes—
266C2×C4C2×D3×D511yescharacteristic
267C2×C3#414411yescharacteristic
268C2×C3C2×#71311yescharacteristic
269C2×C3#414711yescharacteristic
270D3—202no—
271D3—201no—
272C5#558211yescharacteristic, complemented, Sylow
273C22—301no—
274C22—302no—
275C22—302no—
276C22—301no—
277C22—301no—
278C22—302no—
279C22C22×D3×D511yescharacteristic
280C4—22no—
281C4—62no—
282C4—61no—
283C4—61no—
284C4—102no—
285C4—101no—
286C4—101no—
287C4—301no—
288C4#677312yes—
289C3#6606711yescharacteristic, complemented, Sylow
290C2—601no—
291C2—602no—
292C2C2×#677311yescharacteristic
293C2#9202711yescharacteristic
294C2#9202911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
prime radical#91784C2
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#542
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#888C2semidirect
3. C3×#4033C2semidirect
4. #91433C2semidirect
5. #91424C2semidirect
6. #91418C2semidirect
7. #91390C2semidirect
8. #91387C2semidirect
9. #91386C2semidirect
10. #91375C2semidirect
11. #91374C2semidirect
45. #4033D3semidirect
71. #888D5semidirect
228. C3×C5#542semidirect
272. C5#5582semidirect
289. C3#66067semidirect

Maximal subgroups

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

subgroupindex
#917842
C5×#8882
C3×#40332
#914332
#914242
#914182
#913902
#913872
#913862
#913752
#913742
#660673
#55825