Finite Group Databasebeta

← #13023250

Subgroups — #13023250

Order 1280.

398 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 98 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#10610357C211yesmaximal, characteristic
2C2×C22⋊#674C212yesmaximal, complemented
3#10598782C212yesmaximal, complemented
4#10598781C212yesmaximal, complemented
5#66125C411yescharacteristic
6C23×#168C2211yescharacteristic
7C2×#4053C411yescharacteristic
8C22×#674—22no—
9C5×#407C411yescharacteristic
10#65418C2211yescharacteristic
11#65411C411yescharacteristic
12#65408C2211yescharacteristic
13#65019—22no—
14#65014—22no—
15C22⋊#674—22no—
16C22⋊#674C2214yes—
17#11468—51nomaximal, Sylow
18C23×C4×C5C2×C411yescharacteristic
19C2×#709C2×C411yescharacteristic
20C2×C5×M16C2×C411yescharacteristic
21C2×C5×M16C2×C411yescharacteristic
22#4099—21no—
23#4076C812yes—
24#4075—21no—
25C4×C22⋊D5—22no—
26#4072C2×C411yescharacteristic
27#4070C812yes—
28C22×#168—42no—
29C22×#168C2×C411yescharacteristic
30C22×#168D412yes—
31C22×#168D412yes—
32C22×#168C2311yescharacteristic
33#4053—42no—
34C2×#676—21no—
35C2×C8×D5—21no—
36C2×#674—24no—
37C2×#674—44no—
38C5×C22⋊C8—22no—
39#3952—22no—
40#3947—21no—
41#3946—22no—
42C8×Dic5—21no—
43#2929—51no—
44C2×C22⋊C16—52no—
45#1323—52no—
46#1315—52no—
47C24×C5M1611yescharacteristic
48C22×C4×C5—41no—
49C22×C4×C5—41no—
50C22×C4×C5C22⋊C412yes—
51C22×C4×C5C2×C812yes—
52C22×C4×C5M1611yescharacteristic
53C22×C4×C5C22×C411yescharacteristic
54C2×C22⋊D5—22no—
55C2×C22⋊D5—21no—
56#709—41no—
57#709—41no—
58C2×D20C2×C811yescharacteristic
59C2×C4×D5—21no—
60C2×Dic10C2×C811yescharacteristic
61C5×M16—41no—
62C5×M16—41no—
63C2×C5×C8—21no—
64C2×C5×C8—21no—
65C22⋊Dic5—22no—
66C22⋊Dic5—21no—
67#685—22no—
68C4⋊Dic5C2×C812yes—
69#683—22no—
70C4×Dic5—21no—
71C2×#168—24no—
72C2×#168—22no—
73C2×#168—42no—
74C2×#168—41no—
75C2×#168C22⋊C411yescharacteristic
76C2×#168C22⋊C411yescharacteristic
77C2×#168C2×D412yes—
78#676—42no—
79C8×D5—42no—
80#674—44no—
81C2×#131—51no—
82C23×C8—51no—
83#525—51no—
84C22×C16—102no—
85#435—102no—
86C8×D4—102no—
87#424—51no—
88#408—51no—
89#407AGL(1,𝔽5)11yescharacteristic
90#405—51no—
91C22⋊C16—54no—
92C22⋊C16—102no—
93C23×C5—24no—
94C23×C5—41no—
95C23×C5C22⋊C812yes—
96C23×C5C2×M1611yescharacteristic
97C22×D5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5—22no—
102C2×C4×C5—42no—
103C2×C4×C5—41no—
104C2×C4×C5C22⋊C812yes—
105C2×C4×C5C2×C22⋊C411yescharacteristic
106C2×C4×C5C22×C811yescharacteristic
107C22⋊D5—42no—
108C22⋊D5—44no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111D20—41no—
112C4×D5—41no—
113C4×D5—41no—
114Dic10—41no—
115C5×C8—41no—
116C5×C8—41no—
117#168—22no—
118#168—21no—
119#168#9912yes—
120C2×#41—51no—
121C23×C4C2×AGL(1,𝔽5)11yescharacteristic
122#131—202no—
123C2×M16—101no—
124C2×M16C2×AGL(1,𝔽5)11yescharacteristic
125C2×M16C2×AGL(1,𝔽5)11yescharacteristic
126C22×C8—52no—
127C22×C8—51no—
128C22×C8—52no—
129C22×C8—51no—
130C22×C8—101no—
131C22×C8—202no—
132#123—101no—
133C22⋊Q8—52no—
134#121—52no—
135#120—101no—
136C4×D4—102no—
137#117—51no—
138C2×C16—104no—
139C2×C16—204no—
140C4⋊C8—102no—
141C22⋊C8—22no—
142C22⋊C8—102no—
143#97—101no—
144C4×C8—101no—
145C22×C5—22no—
146C22×C5—22no—
147C22×C5—22no—
148C22×C5—21no—
149C22×C5—44no—
150C22×C5#35014yes—
151C22×C5C2×C22⋊C811yescharacteristic
152C2×D5—42no—
153C2×D5—41no—
154C4×C5—21no—
155C4×C5—22no—
156C4×C5C2×#9911yescharacteristic
157C4×C5#41311yescharacteristic
158Dic5—42no—
159Dic5—41no—
160C24#70011yescharacteristic
161#41—201no—
162#41—201no—
163C2×Q8—51no—
164C2×D4—51no—
165C2×D4—101no—
166C2×D4—102no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—101no—
170C22×C4D5⋊C812yes—
171C22×C4#70011yescharacteristic
172C22×C4C22⋊AGL(1,𝔽5)12yes—
173C22×C4C22×AGL(1,𝔽5)11yescharacteristic
174M16—41no—
175M16—41no—
176M16—202no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—51no—
180C2×C8—51no—
181C2×C8—52no—
182C2×C8—104no—
183C2×C8—102no—
184C2×C8—202no—
185C2×C8—201no—
186C2×C8—202no—
187C4⋊C4—52no—
188C4⋊C4—102no—
189C22⋊C4—101no—
190C22⋊C4—102no—
191C22⋊C4—102no—
192C42—101no—
193C16—204no—
194C2×C5—24no—
195C2×C5—41no—
196C2×C5#145911yescharacteristic
197C2×C5C2×#35012yes—
198D5—81no—
199C23—24no—
200C23—41no—
201C23—101no—
202C23#400312yes—
203C23C2×#70011yescharacteristic
204Q8—201no—
205D4—201no—
206D4—202no—
207D4—204no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—102no—
215C2×C4—101no—
216C2×C4—201no—
217C2×C4—201no—
218C2×C4#400312yes—
219C2×C4C2×D5⋊C811yescharacteristic
220C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
221C8—41no—
222C8—41no—
223C8—52no—
224C8—102no—
225C8—101no—
226C5#1146811yescharacteristic, complemented, Sylow
227C22—22no—
228C22—22no—
229C22—21no—
230C22—22no—
231C22—44no—
232C22—201no—
233C22—202no—
234C22#6489814yes—
235C22C2×#400311yescharacteristic
236C4—21no—
237C4—22no—
238C4—201no—
239C4—202no—
240C4D5⋊#9911yescharacteristic
241C4#6577711yescharacteristic
242C2—24no—
243C2—41no—
244C2—401no—
245C2#1060420111yescharacteristic
246C2C2×#6489812yes—
subgroupG/H
centerC2×C4C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#407C4
socleC22×C5C2×C22⋊C8
prime radical#66125C4
cosocleC22×#168C23
hypercenter#407AGL(1,𝔽5)
hypocenterC5#11468
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#674C2semidirect
3. #10598782C2semidirect
4. #10598781C2semidirect
226. C5#11468semidirect

Maximal subgroups

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

subgroupindex
#106103572
C2×C22⋊#6742
#105987822
#105987812
#114685