Finite Group Databasebeta

← #12000636

Subgroups — #12000636

Order 1280.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2822C211yesmaximal, characteristic
2C2×#65354C212yesmaximal
3C2×#64763C212yesmaximal
4#10600345C211yesmaximal, characteristic
5#10598121C211yesmaximal, characteristic
6C2×C5×#125C2212yes—
7C2×#4015C2211yescharacteristic
8C5×#400—22no—
9C5×#398—22no—
10C5×#398—22no—
11C5×#382C2211yescharacteristic
12#65354—24no—
13#65354—24no—
14C2×#3942—22no—
15C2×#3940—22no—
16C2×#3939—22no—
17C2×#3935C2211yescharacteristic
18C2×#3935C2212yes—
19#64763—24no—
20#10087—51nomaximal, Sylow
21C22×Dic10—21no—
22C5×#125—22no—
23C5×#125—22no—
24C5×#125—22no—
25C5×#125—22no—
26C2×C5×C4⋊C4—22no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4D412yes—
30C2×C5×C4⋊C4D412yes—
31C2×C5×C4⋊C4D412yes—
32C2×C42×C5C2311yescharacteristic
33C2×C4⋊Dic5—22no—
34C22×#168—22no—
35C22×#168—21no—
36#4015—22no—
37#4015—21no—
38#4015D412yes—
39C5×#95—24no—
40C5×#95—22no—
41C5×#95—22no—
42#3942—44no—
43#3940—28no—
44#3939—28no—
45#3935—22no—
46#3935—22no—
47#3935D414yes—
48C2×#501—52no—
49#2822D511yescharacteristic
50#1960—51no—
51C2×#330—52no—
52#1295—51no—
53C22×C4×C5—22no—
54C22×C4×C5—22no—
55C22×C4×C5—22no—
56C22×C4×C5C2×D412yes—
57C22×C4×C5C2×D411yescharacteristic
58C22×Dic5—41no—
59C2×Dic10—22no—
60C2×Dic10—42no—
61C2×Dic10—42no—
62C2×Dic10—41no—
63C5×C4⋊C4—22no—
64C5×C4⋊C4—22no—
65C5×C4⋊C4—22no—
66C5×C4⋊C4—22no—
67C5×C4⋊C4—42no—
68C5×C4⋊C4—42no—
69C5×C4⋊C4—42no—
70C5×C4⋊C4—42no—
71C5×C4⋊C4—42no—
72C5×C4⋊C4—82no—
73C42×C5C2×D411yescharacteristic
74C42×C5C2×D412yes—
75C42×C5#4111yescharacteristic
76C4⋊Dic5—24no—
77C4⋊Dic5—42no—
78C2×#168—28no—
79C2×#168—24no—
80C2×#168—22no—
81C2×#168—41no—
82C2×#168—41no—
83C2×C4⋊Q8—51no—
84C2×#125C2×D512yes—
85#501—104no—
86#501—104no—
87C2×#107—102no—
88C2×#106—102no—
89C2×#103—102no—
90C2×#97—52no—
91C2×#97—51no—
92#400—22no—
93#398—22no—
94#398—22no—
95#382C2×D511yescharacteristic
96#330—104no—
97C23×C5#12011yescharacteristic
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—24no—
101C2×C4×C5—42no—
102C2×C4×C5—42no—
103C2×C4×C5—42no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—44no—
107C2×C4×C5—44no—
108C2×C4×C5#12012yes—
109C2×C4×C5#12111yescharacteristic
110C2×C4×C5#12112yes—
111C2×C4×C5#12711yescharacteristic
112C2×Dic5—42no—
113C2×Dic5—81no—
114C2×Dic5—81no—
115Dic10—44no—
116Dic10—82no—
117#168—28no—
118#168—42no—
119C22×Q8—101no—
120C22×C8—102no—
121C22×C8—101no—
122C4⋊Q8—52no—
123C4⋊Q8—102no—
124C4⋊Q8—101no—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—102no—
133C2×C4⋊C4C22⋊D512yes—
134C2×C4⋊C4C22⋊D512yes—
135C2×C4⋊C4C22⋊D512yes—
136C2×C42C22×D511yescharacteristic
137#107—108no—
138#106—108no—
139#103—204no—
140#97—54no—
141#97—102no—
142#97—102no—
143#95—24no—
144#95—22no—
145#95—22no—
146C22×C5#39211yescharacteristic
147C22×C5#45311yescharacteristic
148C22×C5#45411yescharacteristic
149C22×C5#45614yes—
150C4×C5—22no—
151C4×C5—22no—
152C4×C5—21no—
153C4×C5—21no—
154C4×C5—82no—
155C4×C5—82no—
156C4×C5—82no—
157Dic5—82no—
158C2×Q8—102no—
159C2×Q8—202no—
160C2×Q8—202no—
161C2×Q8—201no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—201no—
166C22×C4C2×C22⋊D512yes—
167C22×C4C2×C22⋊D511yescharacteristic
168C2×C8—108no—
169C2×C8—104no—
170C2×C8—102no—
171C2×C8—201no—
172C2×C8—201no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—104no—
184C4⋊C4—202no—
185C42D4×D511yescharacteristic
186C42D4×D512yes—
187C42#71111yescharacteristic
188C2×C5#166012yes—
189C2×C5#166312yes—
190C2×C5#200512yes—
191C2×C5#200611yescharacteristic
192C23#409911yescharacteristic
193Q8—204no—
194Q8—402no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—202no—
206C2×C4—401no—
207C2×C4—401no—
208C2×C4#408312yes—
209C2×C4#408411yescharacteristic
210C2×C4#408512yes—
211C2×C4#408611yescharacteristic
212C8—108no—
213C8—202no—
214C5#1008711yescharacteristic, complemented, Sylow
215C22#6551211yescharacteristic
216C22#6553011yescharacteristic
217C22#6553411yescharacteristic
218C22#6553514yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—82no—
226C4—402no—
227C2#1060266112yes—
228C2#1060267211yescharacteristic
229C2#1060274912yes—
230C2#1060275412yes—
subgroupG/H
centerC23#4099
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#2822C2
socle, prime radicalC23×C5#120
hypercenter#2822D5
hypocenterC5#10087
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
214. C5#10087semidirect

Maximal subgroups

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

subgroupindex
C5×#28222
C2×#653542
C2×#647632
#106003452
#105981212
#100875