Finite Group Databasebeta

← #11753378

Subgroups — #11753378

Order 960.

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

normal subgroups only

This group has 384 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#91779C211yesmaximal, characteristic, complemented
2C5×#887C211yesmaximal, characteristic, complemented
3C3×#4043C211yesmaximal, characteristic
4#91385C211yesmaximal, characteristic, complemented
5#91381C212yesmaximal, complemented
6#91363C212yesmaximal
7#91362C212yesmaximal
8#91359C211yesmaximal, characteristic
9#91357C211yesmaximal, characteristic
10#91339C212yesmaximal, complemented
11Dic3×Dic10C211yesmaximal, characteristic, complemented
12#65917—31nomaximal, Hall
13C2×C5×Dic6C2211yescharacteristic
14C2×C3×C4×D5C2211yescharacteristic
15C2×C3×Dic10C2211yescharacteristic
16C3×C5×C4⋊C4C2211yescharacteristic
17C4⋊Dic15C2211yescharacteristic
18C4⋊Dic15C2212yes—
19#6712C2212yes—
20C4×Dic15C2211yescharacteristic
21C5×Dic3⋊C4C2212yes—
22C4×C5×Dic3C2211yescharacteristic
23C4×C5×Dic3C2212yes—
24C3×#685C2212yes—
25C3×C4⋊Dic5C2212yes—
26#6669C2212yes—
27#6669C2212yes—
28#6669C2212yes—
29#6668C2212yes—
30#6668C2211yescharacteristic
31#6668C2211yescharacteristic
32#6665C2211yescharacteristic
33#6665C2211yescharacteristic
34#6665C2212yes—
35Dic3×Dic5C2212yes—
36#5625—51nomaximal, Hall
37C4×C5×Q8—31no—
38#4092—31no—
39#4090—31no—
40#4044—32no—
41#4043D311yescharacteristic
42#4036—31no—
43#4035—32no—
44#4022—32no—
45#4018—31no—
46#4016—32no—
47C4×Dic10—31no—
48C22×C3×D5C2311yescharacteristic
49C2×C3×C4×C5C2312yes—
50C2×C3×C4×C5C2311yescharacteristic
51C2×Dic15C2312yes—
52C2×Dic15C2311yescharacteristic
53C2×Dic15C2311yescharacteristic
54C2×C5×Dic3C2311yescharacteristic
55C2×C5×Dic3C2311yescharacteristic
56C2×C5×Dic3C2312yes—
57C5×Dic6—22no—
58C2×C3×Dic5C2312yes—
59C2×C3×Dic5C2311yescharacteristic
60C3×C4×D5—22no—
61C3×Dic10—22no—
62C3×C22⋊Q8—51no—
63Q8×Dic3—51no—
64#944—51no—
65#926—51no—
66#924—51no—
67#890—51no—
68#890—52no—
69#889—52no—
70#887D511yescharacteristic, complemented
71#879—52no—
72#879—52no—
73C2×C5×Q8—31no—
74C2×C4×D5C2×D311yescharacteristic
75C2×Dic10C2×D311yescharacteristic
76C5×C4⋊C4—32no—
77C5×C4⋊C4C2×D311yescharacteristic
78C42×C5—31no—
79C42×C5—32no—
80#685—31no—
81#685—31no—
82#685—32no—
83#685C2×D312yes—
84C4⋊Dic5—31no—
85C4⋊Dic5—32no—
86C4⋊Dic5C2×D312yes—
87#683—32no—
88#683—32no—
89#683—31no—
90#683—31no—
91#683—32no—
92#683—32no—
93#683—32no—
94C4×Dic5—31no—
95C4×Dic5—32no—
96#541—151noSylow
97C22×C3×C5C2411yescharacteristic
98C2×C3×D5—21no—
99C2×C3×D5—21no—
100C2×C3×D5—21no—
101C3×C4×C5—22no—
102C3×C4×C5#4112yes—
103Dic15—22no—
104Dic15—21no—
105Dic15—21no—
106C3×Dic5—22no—
107C3×Dic5—21no—
108C5×Dic3—22no—
109C5×Dic3—21no—
110C5×Dic3—21no—
111C2×C3×Q8—51no—
112C22×C3×C4—51no—
113C2×Dic6C2×D511yescharacteristic
114C3×C4⋊C4—52no—
115C3×C4⋊C4C2×D511yescharacteristic
116C3×C22⋊C4—52no—
117#217—52no—
118#217—51no—
119#217—51no—
120C4⋊Dic3—52no—
121C4⋊Dic3—51no—
122C4⋊Dic3—52no—
123C4⋊Dic3—51no—
124C4⋊Dic3—51no—
125Dic3⋊C4—52no—
126Dic3⋊C4—52no—
127Dic3⋊C4—52no—
128Dic3⋊C4—52no—
129Dic3⋊C4C2×D512yes—
130C4×Dic3—51no—
131C4×Dic3—52no—
132C4×Dic3C2×D512yes—
133C4×Dic3C2×D511yescharacteristic
134C22×D5C22×D311yescharacteristic
135C5×Q8—62no—
136C2×C4×C5—31no—
137C2×C4×C5—31no—
138C2×C4×C5—32no—
139C2×C4×C5C22×D312yes—
140C2×C4×C5C22×D311yescharacteristic
141C2×Dic5—31no—
142C2×Dic5—31no—
143C2×Dic5—32no—
144C2×Dic5C22×D312yes—
145C2×Dic5C22×D311yescharacteristic
146C4×D5—22no—
147Dic10—22no—
148C4⋊Q8—151no—
149#126—152no—
150#126—152no—
151#125—152no—
152#125—152no—
153#125—151no—
154C22⋊Q8—51no—
155C22⋊Q8—151no—
156C4×Q8—31no—
157C4×Q8—151no—
158#117—151no—
159C2×C3×C5C2×#4111yescharacteristic
160C2×C3×C52−1+411yescharacteristic
161C2×C3×C52−1+411yescharacteristic
162C3×D5—41no—
163C23×C3—51no—
164C3×Q8—102no—
165C2×C3×C4—52no—
166C2×C3×C4—51no—
167C2×C3×C4—102no—
168C2×C3×C4C22×D512yes—
169C2×C3×C4C22×D511yescharacteristic
170C2×Dic3—51no—
171C2×Dic3—52no—
172C2×Dic3—51no—
173C2×Dic3C22×D511yescharacteristic
174C2×Dic3C22×D511yescharacteristic
175C2×Dic3C22×D512yes—
176Dic6—22no—
177C22×C5C23×D311yescharacteristic
178C2×D5—21no—
179C2×D5—21no—
180C2×D5—21no—
181C4×C5—22no—
182C4×C5—62no—
183C4×C5—61no—
184C4×C5—61no—
185C4×C5#23712yes—
186Dic5—22no—
187Dic5—21no—
188Dic5—62no—
189Dic5—61no—
190Dic5—61no—
191C2×Q8—31no—
192C2×Q8—51no—
193C22×C4—51no—
194C4⋊C4—32no—
195C4⋊C4—52no—
196C4⋊C4—151no—
197C4⋊C4—151no—
198C4⋊C4—152no—
199C4⋊C4—152no—
200C4⋊C4—152no—
201C4⋊C4—152no—
202C4⋊C4—151no—
203C4⋊C4—152no—
204C4⋊C4—152no—
205C4⋊C4D3×D511yescharacteristic
206C22⋊C4—52no—
207C22⋊C4—151no—
208C22⋊C4—151no—
209C22⋊C4—152no—
210C42—31no—
211C42—32no—
212C42—152no—
213C42—151no—
214C3×C5#54111yescharacteristic, complemented, Hall
215C22×C3—101no—
216C22×C3—101no—
217C22×C3—101no—
218C22×C3C23×D511yescharacteristic
219C3×C4—22no—
220C3×C4—101no—
221C3×C4—102no—
222C3×C4#70912yes—
223Dic3—22no—
224Dic3—21no—
225Dic3—21no—
226Dic3—102no—
227Dic3—101no—
228Dic3—101no—
229C2×C5C2×#23711yescharacteristic
230C2×C5#100711yescharacteristic
231C2×C5#101011yescharacteristic
232D5—41no—
233C23—51no—
234Q8—62no—
235Q8—102no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—52no—
240C2×C4—51no—
241C2×C4—102no—
242C2×C4—151no—
243C2×C4—152no—
244C2×C4—151no—
245C2×C4C2×D3×D512yes—
246C2×C4C2×D3×D511yescharacteristic
247C2×C3—201no—
248C2×C3C2×#70911yescharacteristic
249C2×C3#414711yescharacteristic
250C2×C3#415011yescharacteristic
251C5#562511yescharacteristic, complemented, Sylow
252C22—101no—
253C22—101no—
254C22—101no—
255C22C22×D3×D511yescharacteristic
256C4—22no—
257C4—62no—
258C4—61no—
259C4—61no—
260C4—101no—
261C4—102no—
262C4—301no—
263C4—302no—
264C4—301no—
265C4#676512yes—
266C3#6591711yescharacteristic, complemented, Sylow
267C2—201no—
268C2C2×#676511yescharacteristic
269C2#9202011yescharacteristic
270C2#9202911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
prime radicalC22×C3×D5C23
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#541
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. #91779C2semidirect
2. C5×#887C2semidirect
4. #91385C2semidirect
5. #91381C2semidirect
10. #91339C2semidirect
11. Dic3×Dic10C2semidirect
70. #887D5semidirect
214. C3×C5#541semidirect
251. C5#5625semidirect
266. C3#65917semidirect

Maximal subgroups

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

subgroupindex
#917792
C5×#8872
C3×#40432
#913852
#913812
#913632
#913622
#913592
#913572
#913392
Dic3×Dic102
#659173
#56255