Finite Group Databasebeta

← #11747312

Subgroups — #11747312

Order 960.

260 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C3×#4041C211yesmaximal, characteristic, complemented
2#91449C211yesmaximal, characteristic, complemented
3C2×#6652C211yesmaximal, characteristic, complemented
4C2×#6649C211yesmaximal, characteristic, complemented
5#91092C211yesmaximal, characteristic, complemented
6C5×#809C211yesmaximal, characteristic, complemented
7#90965C211yesmaximal, characteristic, complemented
8#65071—31nomaximal, Hall
9C2×C5×D12C2211yescharacteristic
10C2×C3×D20—21no—
11C2×C3×D20C2211yescharacteristic
12C2×C3×C4×D5C2211yescharacteristic
13C2×#1201—21no—
14C3×C5×C4⋊C4C2211yescharacteristic
15C4⋊Dic15C2211yescharacteristic
16C2×C3⋊#168C2211yescharacteristic
17C2×C5×C3⋊C8C2211yescharacteristic
18C3×#685—21no—
19(C2×D3)⋊Dic5—21no—
20#6652—22no—
21#6649—22no—
22#5039—51nomaximal, Hall
23#4084—31no—
24C2×#686—31no—
25C2×#676—31no—
26#4041D311yescharacteristic, complemented
27C5×#102—31no—
28#3953—31no—
29#3940—31no—
30C22×C5×D3—21no—
31C22×C3×D5—21no—
32C22×C3×D5D411yescharacteristic
33C2×C3×C4×C5—21no—
34C2×C3×C4×C5C2311yescharacteristic
35C2×Dic15—21no—
36C5×D12—21no—
37C5×D12—21no—
38C2×C3×Dic5D411yescharacteristic
39C3×D20—21no—
40C3×D20—21no—
41C3×D20—42no—
42C3×C4×D5—21no—
43C3×C4×D5—21no—
44#1201—42no—
45C3⋊#168D412yescomplemented
46C5×C3⋊C8—21no—
47C3×#121—51no—
48C2×#213—51no—
49#930—51no—
50C2×#208—51no—
51#832—51no—
52#809D511yescharacteristic, complemented
53#808—51no—
54C2×C5×D4—31no—
55C2×C22⋊D5—61no—
56C2×D20—21no—
57C2×D20C2×D311yescharacteristic
58C2×C4×D5C2×D311yescharacteristic
59C2×C5×C8—31no—
60C5×C4⋊C4C2×D311yescharacteristic
61C22⋊Dic5—61no—
62#686—62no—
63#685—21no—
64C4⋊Dic5—31no—
65C2×#168—31no—
66#676—62no—
67#469—151noSylow
68C22×C3×C5C2×D411yescharacteristic
69C2×C5×D3—41no—
70C2×C5×D3—41no—
71C2×C5×D3—41no—
72C2×C3×D5—21no—
73C2×C3×D5—21no—
74C2×C3×D5—21no—
75C2×C3×D5—41no—
76C2×C3×D5—41no—
77C2×C3×D5—41no—
78C3×C4×C5—41no—
79C3×C4×C5C2×D411yescharacteristic
80C3×C4×C5#4111yescharacteristic
81Dic15—41no—
82C3×Dic5—21no—
83C2×C3×D4—51no—
84C2×C3×D4—101no—
85C22×C3×C4—51no—
86C2×#67—101no—
87C2×D12C2×D511yescharacteristic
88C3×C4⋊C4C2×D511yescharacteristic
89C3×C22⋊C4—101no—
90#213—102no—
91(C2×D3)⋊C4—101no—
92C4⋊Dic3—51no—
93#208—102no—
94C2×C3⋊C8—51no—
95C2×C3⋊C8C2×D511yescharacteristic
96C23×C5—61no—
97C22×D5—21no—
98C22×D5#6711yescharacteristic
99C5×D4—61no—
100C5×D4—61no—
101C2×C4×C5—21no—
102C2×C4×C5C22×D311yescharacteristic
103C22⋊D5—122no—
104C2×Dic5—61no—
105C2×Dic5#6711yescharacteristic
106D20—21no—
107D20—21no—
108D20—42no—
109C4×D5—21no—
110C4×D5—21no—
111C5×C8—61no—
112#168—32no—
113C2×D8—151no—
114C2×M16—151no—
115#121—51no—
116#121—151no—
117#106—151no—
118#102—31no—
119#102—151no—
120C2×C3×C5#12111yescharacteristic
121C2×C3×C5AGL(1,ℤ/8)11yescharacteristic
122C2×C3×C5AGL(1,ℤ/8)11yescharacteristic
123C3×D5—41no—
124C3×D5—81no—
125C5×D3—81no—
126C23×C3—51no—
127C23×C3—101no—
128C22×D3—21no—
129C3×D4—101no—
130C3×D4—101no—
131C3×D4—202no—
132C2×C3×C4—21no—
133C2×C3×C4—51no—
134C2×C3×C4—101no—
135C2×C3×C4—101no—
136C2×C3×C4C22×D511yescharacteristic
137#67—202no—
138C2×Dic3—101no—
139D12—21no—
140D12—21no—
141C3⋊C8—21no—
142C3⋊C8—52no—
143C22×C5—121no—
144C22×C5—121no—
145C22×C5—121no—
146C22×C5C2×#6711yescharacteristic
147C2×D5—21no—
148C2×D5—21no—
149C2×D5—21no—
150C2×D5—41no—
151C2×D5—41no—
152C2×D5—41no—
153C4×C5—41no—
154C4×C5D3×D411yescharacteristic
155C4×C5#23711yescharacteristic
156Dic5—21no—
157Dic5—121no—
158C2×D4—31no—
159C2×D4—51no—
160C2×D4—101no—
161C2×D4—301no—
162C22×C4—51no—
163D8—302no—
164M16—302no—
165C2×C8—31no—
166C2×C8—151no—
167C4⋊C4—151no—
168C4⋊C4D3×D511yescharacteristic
169C22⋊C4—101no—
170C22⋊C4—301no—
171C3×C5#46911yescharacteristic, complemented, Hall
172C22×C3—101no—
173C22×C3—101no—
174C22×C3—101no—
175C22×C3—201no—
176C22×C3—201no—
177C22×C3—201no—
178C22×C3D4×D511yescharacteristic
179C2×D3—41no—
180C2×D3—41no—
181C2×D3—41no—
182C3×C4—41no—
183C3×C4—101no—
184C3×C4#70911yescharacteristic
185C3×C4D4×D511yescharacteristic
186Dic3—201no—
187C2×C5—241no—
188C2×C5#93211yescharacteristic
189C2×C5#93911yescharacteristic
190C2×C5#94911yescharacteristic
191D5—41no—
192D5—81no—
193C23—51no—
194C23—61no—
195C23—101no—
196D4—61no—
197D4—61no—
198D4—101no—
199D4—101no—
200D4—202no—
201D4—602no—
202C2×C4—21no—
203C2×C4—51no—
204C2×C4—101no—
205C2×C4—101no—
206C2×C4—301no—
207C2×C4C2×D3×D511yescharacteristic
208C8—61no—
209C8—152no—
210C2×C3—201no—
211C2×C3—401no—
212C2×C3#403011yescharacteristic
213C2×C3#405711yescharacteristic
214C2×C3#406011yescharacteristic
215D3—81no—
216C5#503911yescharacteristic, complemented, Sylow
217C22—101no—
218C22—101no—
219C22—101no—
220C22—121no—
221C22—121no—
222C22—121no—
223C22—201no—
224C22—201no—
225C22—201no—
226C22D5×#6711yescharacteristic
227C4—41no—
228C4—101no—
229C4—601no—
230C4#676511yescharacteristic
231C4#677711yescharacteristic
232C3#6507111yescharacteristic, complemented, Sylow
233C2—201no—
234C2—241no—
235C2—401no—
236C2#9140811yescharacteristic
237C2#9148911yescharacteristic
238C2#9150211yescharacteristic
subgroupG/H
centerC22D5×#67
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C2×D3×D5
FittingC3×C5×C4⋊C4C22
socleC22×C3×C5C2×D4
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#469
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#4041C2semidirect
2. #91449C2semidirect
3. C2×#6652C2semidirect
4. C2×#6649C2semidirect
5. #91092C2semidirect
6. C5×#809C2semidirect
7. #90965C2semidirect
26. #4041D3semidirect
45. C3⋊#168D4semidirect
52. #809D5semidirect
171. C3×C5#469semidirect
216. C5#5039semidirect
232. C3#65071semidirect

Maximal subgroups

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

subgroupindex
C3×#40412
#914492
C2×#66522
C2×#66492
#910922
C5×#8092
#909652
#650713
#50395