Finite Group Databasebeta

← #12012647

Subgroups — #12012647

Order 1280.

310 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3202C211yesmaximal, characteristic
2#10601883C212yesmaximal, complemented
3#10600798C211yesmaximal, characteristic
4#10598271C212yesmaximal, complemented
5#10598181C211yesmaximal, characteristic
6C5×#536—21no—
7C5×#485—22no—
8C5×#470C2212yes—
9C5×#467—21no—
10C5×#451—22no—
11C5×#408C2211yescharacteristic
12#65525—21no—
13#65414C2211yescharacteristic
14#65391—21no—
15#64785—22no—
16#64777C2211yescharacteristic
17#64773C2212yes—
18#64707—22no—
19#12302—51nomaximal, Sylow
20C2×C5×#41D411yescharacteristic
21C22×C5×Q8—21no—
22C2×C5×Q16—21no—
23C2×C5×Q16D412yes—
24C2×C5×M16C2311yescharacteristic
25C22×C5×C8D411yescharacteristic
26C5×C4⋊Q8—41no—
27C5×#124—41no—
28C5×#123—41no—
29C5×#123—41no—
30C5×C22⋊Q8—22no—
31C5×C22⋊Q8—22no—
32C5×#117—21no—
33#4072—21no—
34C2×#681—21no—
35C5×#107—21no—
36C5×#106D412yes—
37C5×#103—22no—
38C5×#103—22no—
39C5×C22⋊C8—22no—
40#3969—42no—
41#3951—22no—
42#3946—42no—
43#3944—22no—
44#3202D511yescharacteristic
45#2239—52no—
46#2098—51no—
47#1340—52no—
48#1321—51no—
49C5×#41—41no—
50C5×#41—41no—
51C2×C5×Q8—22no—
52C2×C5×Q8—41no—
53C2×C5×Q8—81no—
54C2×C5×Q8—81no—
55C2×C5×Q8SD1611yescharacteristic
56C2×C5×D4—21no—
57C2×C5×D4SD1611yescharacteristic
58C22×C4×C5—21no—
59C22×C4×C5—41no—
60C22×C4×C5C2×D411yescharacteristic
61C5×Q16—42no—
62C5×Q16—81no—
63C5×M16—41no—
64C2×C5×C8—21no—
65C2×C5×C8—41no—
66C2×C5×C8C2×D412yes—
67C5×C4⋊C4—22no—
68C5×C4⋊C4—42no—
69C5×C4⋊C4—42no—
70C5×C22⋊C4—41no—
71C5×C22⋊C4—42no—
72C5×C22⋊C4—42no—
73C42×C5—41no—
74C22⋊Dic5—41no—
75C4⋊Dic5—41no—
76C4×Dic5—22no—
77#681—41no—
78#681—41no—
79C2×#168—41no—
80#672—42no—
81#536—21no—
82#485—22no—
83#470C2×D512yes—
84#467—21no—
85#451—22no—
86#430—51no—
87#424—101no—
88#421—101no—
89#408C2×D511yescharacteristic
90#365—102no—
91#362—102no—
92#349—52no—
93H(3,ℤ/4)—51no—
94C23×C5—21no—
95C23×C5AGL(1,ℤ/8)11yescharacteristic
96C5×Q8—41no—
97C5×Q8—42no—
98C5×Q8—42no—
99C5×Q8—82no—
100C5×D4—41no—
101C5×D4—81no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—21no—
105C2×C4×C5—41no—
106C2×C4×C5—41no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5—82no—
110C2×C4×C5#12011yescharacteristic
111C2×C4×C5C2×SD1611yescharacteristic
112C2×Dic5—42no—
113C5×C8—22no—
114C5×C8—41no—
115#168—81no—
116C2×#41C22⋊D511yescharacteristic
117C22×Q8—21no—
118C2×Q16—21no—
119C2×Q16C22⋊D512yes—
120C2×M16—101no—
121C2×M16C22×D511yescharacteristic
122C22×C8C22⋊D511yescharacteristic
123C4⋊Q8—41no—
124#124—41no—
125#123—41no—
126#123—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129#117—21no—
130#117—101no—
131M32—102no—
132#108—102no—
133#107—21no—
134#106C22⋊D512yes—
135C4⋊C8—202no—
136#104—202no—
137#103—22no—
138#103—22no—
139C22⋊C8—22no—
140C22×C5—21no—
141C22×C5—41no—
142C22×C5—42no—
143C22×C5—41no—
144C22×C5#45011yescharacteristic
145C4×C5—21no—
146C4×C5—41no—
147C4×C5—82no—
148C4×C5—82no—
149C4×C5#45311yescharacteristic
150C4×C5#45811yescharacteristic
151Dic5—82no—
152#41—41no—
153#41—41no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—81no—
157C2×Q8—81no—
158C2×Q8#68711yescharacteristic
159C2×D4—21no—
160C2×D4#68711yescharacteristic
161C22×C4—21no—
162C22×C4—41no—
163C22×C4C2×C22⋊D511yescharacteristic
164Q16—42no—
165Q16—81no—
166M16—41no—
167M16—201no—
168M16—201no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—201no—
172C2×C8C2×C22⋊D512yes—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—201no—
177C22⋊C4—41no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—201no—
181C42—41no—
182C42—102no—
183C16—202no—
184C2×C5—21no—
185C2×C5—41no—
186C2×C5—81no—
187C2×C5#161311yescharacteristic
188C23—21no—
189C23#407811yescharacteristic
190Q8—41no—
191Q8—42no—
192Q8—42no—
193Q8—82no—
194D4—41no—
195D4—81no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—82no—
204C2×C4—202no—
205C2×C4C2×#68711yescharacteristic
206C2×C4#409911yescharacteristic
207C8—22no—
208C8—41no—
209C8—401no—
210C5#1230211yescharacteristic, complemented, Sylow
211C22—21no—
212C22—41no—
213C22—42no—
214C22—41no—
215C22#6550711yescharacteristic
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—402no—
221C4#6553011yescharacteristic
222C4#6553211yescharacteristic
223C2—21no—
224C2—41no—
225C2—81no—
226C2#1060263711yescharacteristic
subgroupG/H
centerC2#10602637
commutator, cosocleC2×C5×M16C23
FrattiniC2×M16C22×D5
FittingC5×#3202C2
socleC2×C5#1613
prime radicalC2×C5×#41D4
hypercenter#3202D5
hypocenterC5#12302
LCSG ▷ C2×C5×M16 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3202

Complements

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

NH ≅ G/Ntype
2. #10601883C2semidirect
4. #10598271C2semidirect
210. C5#12302semidirect

Maximal subgroups

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

subgroupindex
C5×#32022
#106018832
#106007982
#105982712
#105981812
#123025