Finite Group Databasebeta

← #13023946

Subgroups — #13023946

Order 1280.

354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×#65162C211yesmaximal, characteristic
2#10604710C211yesmaximal, characteristic
3#10604698C211yesmaximal, characteristic
4#10598608C214yesmaximal, complemented
5C22×Dic20C411yescharacteristic
6C2×#4033—21no—
7#65759—21no—
8C2×C5×#107C411yescharacteristic
9C2×C8×Dic5C2211yescharacteristic
10C2×#3942—21no—
11#65162—21no—
12#65162—21no—
13#65162C412yes—
14#65162C2212yes—
15C22⋊#674—21no—
16#64888—21no—
17#64886C2212yes—
18#64884—21no—
19#64881C2212yes—
20#11668—51nomaximal, Sylow
21C22×Dic10—21no—
22C22×C5×C8C2×C411yescharacteristic
23C2×C5×C4⋊C4—21no—
24C2×#683—41no—
25C2×C4×Dic5—41no—
26C2×C4×Dic5D411yescharacteristic
27C22×#168D411yescharacteristic
28C2×Dic20—21no—
29C2×Dic20—24no—
30C2×Dic20C2×C412yes—
31#4033—22no—
32#4033—22no—
33#4033—41no—
34#4033—41no—
35C22⋊#170—41no—
36#4002—41no—
37C4⋊#170—22no—
38C2×#674—22no—
39C5×#107—21no—
40C5×#107C2×C412yes—
41C8×Dic5C2×C411yescharacteristic
42C8×Dic5D411yescharacteristic
43C8×Dic5D411yescharacteristic
44C8×Dic5C2311yescharacteristic
45#3942—22no—
46#3942—22no—
47C2×C4×Q16—51no—
48#2163—51no—
49#1568—51no—
50#1334—54no—
51C22×C4×C5—41no—
52C22×C4×C5C22⋊C411yescharacteristic
53C22×Dic5—41no—
54C22×Dic5D811yescharacteristic
55C22×Dic5SD1611yescharacteristic
56C2×Dic10—24no—
57C2×Dic10—22no—
58C2×Dic10—42no—
59C2×Dic10—41no—
60C2×#170—42no—
61C2×C5×C8—22no—
62C2×C5×C8C22⋊C411yescharacteristic
63C2×C5×C8C22×C411yescharacteristic
64C5×C4⋊C4—22no—
65C5×C4⋊C4—41no—
66#683—42no—
67#683—42no—
68C4×Dic5—42no—
69C4×Dic5—42no—
70C4×Dic5C22⋊C411yescharacteristic
71C4×Dic5D811yescharacteristic
72C4×Dic5SD1611yescharacteristic
73C4×Dic5C2×D411yescharacteristic
74C2×#168—41no—
75C2×#168C22⋊C411yescharacteristic
76C2×#168C2×D411yescharacteristic
77Dic20—24no—
78Dic20—22no—
79Dic20M1614yes—
80#674—42no—
81C22×Q16—51no—
82C2×C4×Q8—101no—
83C4×Q16—52no—
84C4×Q16—52no—
85C4×Q16—101no—
86C4×Q16—101no—
87#432—101no—
88C2×#107AGL(1,𝔽5)11yescharacteristic
89C2×#103—101no—
90C2×C4×C8—51no—
91C4⋊C16—101no—
92C22⋊C16—101no—
93#346—52no—
94#335—101no—
95#334—52no—
96C23×C5#10211yescharacteristic
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—42no—
100C2×C4×C5—82no—
101C2×C4×C5#10211yescharacteristic
102C2×C4×C5C2×C22⋊C411yescharacteristic
103C2×Dic5—22no—
104C2×Dic5—41no—
105C2×Dic5—44no—
106C2×Dic5—42no—
107C2×Dic5#10211yescharacteristic
108C2×Dic5#10211yescharacteristic
109C2×Dic5C2×D811yescharacteristic
110C2×Dic5C2×SD1611yescharacteristic
111Dic10—24no—
112Dic10—44no—
113Dic10—42no—
114#170—82no—
115C5×C8—21no—
116C5×C8C2×M1612yes—
117#168—21no—
118#168—21no—
119C22×Q8—101no—
120C2×Q16—52no—
121C2×Q16—104no—
122C2×Q16—101no—
123C22×C8—51no—
124C22×C8C2×AGL(1,𝔽5)11yescharacteristic
125C4×Q8—102no—
126C4×Q8—102no—
127C4×Q8—201no—
128C4×Q8—201no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—201no—
131C2×C42—51no—
132C2×C42—201no—
133C2×C16—102no—
134#107—21no—
135#107C2×AGL(1,𝔽5)12yes—
136C4⋊C8—102no—
137#103—102no—
138#103—102no—
139C22⋊C8—201no—
140#97—201no—
141C4×C8—51no—
142C4×C8—51no—
143C4×C8—51no—
144C4×C8—51no—
145C22×C5—21no—
146C22×C5—21no—
147C22×C5#36212yes—
148C22×C5C2×#10211yescharacteristic
149C4×C5—21no—
150C4×C5—82no—
151C4×C5#40711yescharacteristic
152C4×C5#42111yescharacteristic
153Dic5—21no—
154Dic5—21no—
155Dic5—21no—
156Dic5—21no—
157Dic5—44no—
158C2×Q8—104no—
159C2×Q8—102no—
160C2×Q8—202no—
161C2×Q8—201no—
162C22×C4—41no—
163C22×C4—51no—
164C22×C4—51no—
165C22×C4—201no—
166C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
167Q16—54no—
168Q16—104no—
169Q16—102no—
170C2×C8—22no—
171C2×C8—51no—
172C2×C8—51no—
173C2×C8—201no—
174C2×C8—202no—
175C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
176C2×C8C22×AGL(1,𝔽5)11yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—41no—
179C4⋊C4—202no—
180C4⋊C4—202no—
181C42—51no—
182C42—51no—
183C42—51no—
184C42—51no—
185C42—202no—
186C42—202no—
187C16—202no—
188C2×C5—22no—
189C2×C5#148611yescharacteristic
190C2×C5#214511yescharacteristic
191C2×C5C2×#36211yescharacteristic
192C23#400711yescharacteristic
193Q8—104no—
194Q8—204no—
195Q8—202no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—51no—
200C2×C4—51no—
201C2×C4—51no—
202C2×C4—51no—
203C2×C4—82no—
204C2×C4—102no—
205C2×C4—201no—
206C2×C4—204no—
207C2×C4—202no—
208C2×C4#400711yescharacteristic
209C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
210C8—21no—
211C8—101no—
212C8—101no—
213C8—402no—
214C8D5⋊M1612yes—
215C5#1166811yescharacteristic, complemented, Sylow
216C22—21no—
217C22—21no—
218C22#6491012yes—
219C22C2×#400711yescharacteristic
220C4—21no—
221C4—82no—
222C4—101no—
223C4—101no—
224C4—101no—
225C4—101no—
226C4—204no—
227C4#6575511yescharacteristic
228C4#6578411yescharacteristic
229C2—22no—
230C2#1060427211yescharacteristic
231C2#1060478411yescharacteristic
232C2C2×#6491011yescharacteristic
subgroupG/H
centerC22C2×#4007
commutatorC2×C5×C8C22×C4
FrattiniC2×C8C22×AGL(1,𝔽5)
FittingC2×C5×#107C4
socleC22×C5C2×#102
prime radicalC23×C5#102
cosocleC8×Dic5C23
hypercenterC2×#107AGL(1,𝔽5)
hypocenterC5#11668
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
4. #10598608C2semidirect
215. C5#11668semidirect

Maximal subgroups

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

subgroupindex
C2×#651622
#106047102
#106046982
#105986082
#116685