Finite Group Databasebeta

← #11749385

Subgroups — #11749385

Order 960.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C5×#942C211yesmaximal, characteristic, complemented
2C2×C3×#688C211yesmaximal, characteristic, complemented
3#91298C211yesmaximal, characteristic
4#91286C211yesmaximal, characteristic, complemented
5#91120C211yesmaximal, characteristic, complemented
6#90991C211yesmaximal, characteristic, complemented
7#90958C211yesmaximal, characteristic, complemented
8#65492—31nomaximal, Hall
9C2×C3×C5×Q8C2211yescharacteristic
10C5×#235C2211yescharacteristic
11C2×C3×D20C2211yescharacteristic
12#6710C2211yescharacteristic
13#6710C2211yescharacteristic
14C5×#216—21no—
15C5×#215—21no—
16C5×#208C2211yescharacteristic
17C3×#688—21no—
18C3×#688—21no—
19C2×C3×#168C2211yescharacteristic
20#6655—21no—
21#6652—21no—
22#6650—21no—
23D3×#168—21no—
24#5177—51nomaximal, Hall
25C5×#137—31no—
26#4095—31no—
27#4094—31no—
28C2×#688D311yescharacteristic, complemented
29#3968—31no—
30#3955—31no—
31#3954—31no—
32C22×C3×D5—21no—
33C3×C5×Q8—21no—
34C3×C5×Q8—21no—
35C2×C3×C4×C5—21no—
36C2×C3×C4×C5C2311yescharacteristic
37C5×#67—21no—
38C5×D12D411yescharacteristic
39C4×C5×D3—21no—
40C5×Dic6D411yescharacteristic
41C3×D20—21no—
42C3×D20—21no—
43C3⋊#168—21no—
44C3⋊#168—21no—
45C3×#168D411yescharacteristic
46C3×#168D411yescharacteristic
47C5×C3⋊C8—21no—
48C2×C3×SD16—51no—
49#942D511yescharacteristic, complemented
50#932—51no—
51#901—51no—
52#836—51no—
53#833—51no—
54#819—51no—
55C5×#41—31no—
56C2×C5×Q8C2×D311yescharacteristic
57C2×D20C2×D311yescharacteristic
58C5×Q16—61no—
59C5×SD16—61no—
60C5×M16—31no—
61#688—21no—
62#688—21no—
63#688—61no—
64#686—61no—
65#681—31no—
66#681—31no—
67#681—61no—
68C2×#168—61no—
69C2×#168C2×D311yescharacteristic
70#471—151noSylow
71C22×C3×C5#4111yescharacteristic
72C2×C5×D3—21no—
73C2×C3×D5—41no—
74C2×C3×D5—81no—
75C3×C4×C5—41no—
76C3×C4×C5C2×D411yescharacteristic
77C3×C4×C5C2×D411yescharacteristic
78C5×Dic3—21no—
79C2×C3×Q8C2×D511yescharacteristic
80C2×C3×D4—51no—
81#235C2×D511yescharacteristic
82C3×SD16—101no—
83C3×SD16—101no—
84C2×C3×C8—51no—
85#216—21no—
86#215—21no—
87#214—101no—
88#213—101no—
89#208—51no—
90#208—51no—
91#208C2×D511yescharacteristic
92ΓL(1,𝔽25)—101no—
93D3×C8—101no—
94C22×D5—21no—
95C5×Q8—21no—
96C5×Q8—21no—
97C5×Q8—31no—
98C5×D4—31no—
99C5×D4—61no—
100C2×C4×C5—21no—
101C2×C4×C5—61no—
102C2×C4×C5C22×D311yescharacteristic
103D20—21no—
104D20—21no—
105C5×C8—61no—
106#168—61no—
107#168—61no—
108#168#6711yescharacteristic
109#168#6711yescharacteristic
110#137—31no—
111AGL(1,ℤ/8)—151no—
112C2×SD16—51no—
113#131—151no—
114#108—151no—
115#101—151no—
116#100—151no—
117C2×C3×C5—21no—
118C2×C3×C5#12111yescharacteristic
119C3×D5—81no—
120C5×D3—41no—
121C23×C3—101no—
122C3×Q8—21no—
123C3×Q8—21no—
124C3×D4—101no—
125C3×D4—101no—
126C2×C3×C4—21no—
127C2×C3×C4C22×D511yescharacteristic
128#67—21no—
129D12C22⋊D511yescharacteristic
130C4×D3—21no—
131Dic6C22⋊D511yescharacteristic
132C3×C8—51no—
133C3×C8—51no—
134C3⋊C8—21no—
135C3⋊C8—101no—
136C3⋊C8—101no—
137C22×C5—61no—
138C22×C5#23711yescharacteristic
139C2×D5—41no—
140C2×D5—81no—
141C4×C5—41no—
142C4×C5—61no—
143C4×C5D3×D411yescharacteristic
144C4×C5C2×#6711yescharacteristic
145#41—31no—
146C2×Q8D3×D511yescharacteristic
147C2×D4—51no—
148Q16—61no—
149SD16—61no—
150SD16—101no—
151SD16—101no—
152SD16—301no—
153D8—301no—
154M16—31no—
155M16—151no—
156M16—151no—
157M16—301no—
158C2×C8—51no—
159C2×C8—301no—
160C3×C5#47111yescharacteristic, complemented, Hall
161C22×C3—201no—
162C22×C3—401no—
163C22×C3#71111yescharacteristic
164C2×D3—21no—
165C3×C4—41no—
166C3×C4D4×D511yescharacteristic
167C3×C4C2×C22⋊D511yescharacteristic
168Dic3—21no—
169C2×C5—21no—
170C2×C5—121no—
171C2×C5#93811yescharacteristic
172D5—81no—
173C23—101no—
174Q8—21no—
175Q8—21no—
176Q8—31no—
177D4—31no—
178D4—61no—
179D4—101no—
180D4—101no—
181C2×C4—21no—
182C2×C4—61no—
183C2×C4C2×D3×D511yescharacteristic
184C8—51no—
185C8—51no—
186C8—61no—
187C8—301no—
188C8—301no—
189C2×C3—21no—
190C2×C3—401no—
191C2×C3#408511yescharacteristic
192D3—41no—
193C5#517711yescharacteristic, complemented, Sylow
194C22—61no—
195C22—201no—
196C22—401no—
197C22#678011yescharacteristic
198C4—41no—
199C4—61no—
200C4D5×#6711yescharacteristic
201C4D3×C22⋊D511yescharacteristic
202C3#6549211yescharacteristic, complemented, Sylow
203C2—21no—
204C2—121no—
205C2—401no—
206C2#9156511yescharacteristic
subgroupG/H
centerC2#91565
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C2×D3×D5
FittingC2×C3×C5×Q8C22
socleC2×C3×C5#121
prime radical#91298C2
hypercenterC2×Q8D3×D5
hypocenterC3×C5#471
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C5×#942C2semidirect
2. C2×C3×#688C2semidirect
4. #91286C2semidirect
5. #91120C2semidirect
6. #90991C2semidirect
7. #90958C2semidirect
28. C2×#688D3semidirect
49. #942D5semidirect
160. C3×C5#471semidirect
193. C5#5177semidirect
202. C3#65492semidirect

Maximal subgroups

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

subgroupindex
C5×#9422
C2×C3×#6882
#912982
#912862
#911202
#909912
#909582
#654923
#51775