Finite Group Databasebeta

← #13021823

Subgroups — #13021823

Order 1280.

1162 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 212 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×#65850C211yesmaximal, characteristic
2C22×#4005C212yesmaximal
3C2×#64867C214yesmaximal
4C23×Dic10C2211yescharacteristic
5C22×#704C2214yes—
6C2×C5×#117C411yescharacteristic
7C2×#4072C411yescharacteristic
8#65850—28no—
9C2×#4033—22no—
10C2×#4023C412yes—
11C2×#4023C2212yes—
12C2×C4×Dic10—22no—
13C2×#4005—24no—
14C2×#4005—22no—
15C2×#4005—22no—
16C2×#4005C418yescomplemented
17#64867—28no—
18#10770—51nomaximal, Sylow
19C23×C4×C5C2×C411yescharacteristic
20C23×Dic5C2×C411yescharacteristic
21C23×Dic5C2311yescharacteristic
22C22×Dic10—24no—
23C22×Dic10—24no—
24C22×Dic10C2×C414yes—
25C22×Dic10D412yes—
26C22×Dic10D412yes—
27C22×Dic10D412yes—
28C22×Dic10Q812yes—
29C2×#704—24no—
30C2×#704—24no—
31C2×#704—24no—
32C2×#704—24no—
33C2×#704—24no—
34C2×#704C2×C418yes—
35C22×#170—24no—
36C5×#117—24no—
37C2×C5×C4⋊C4—21no—
38C2×C5×C22⋊C4C2×C412yes—
39C2×C42×C5—21no—
40C2×C22⋊Dic5C2×C412yes—
41#4072—24no—
42C2×C4⋊Dic5—21no—
43C2×#683—22no—
44C2×#683—22no—
45C2×C4×Dic5—21no—
46C2×C4×Dic5—22no—
47C2×C4×Dic5—22no—
48#4033—216no—
49#4023—24no—
50#4023—24no—
51#4023—24no—
52#4023D418yes—
53C4×Dic10—216no—
54#4005—28no—
55#4005—28no—
56C2×#519—51no—
57C22×#101—52no—
58C2×#344—54no—
59C24×C5C22×C411yescharacteristic
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5C4212yes—
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5C4⋊C412yes—
67C22×Dic5—22no—
68C22×Dic5—22no—
69C22×Dic5—22no—
70C22×Dic5C4212yes—
71C22×Dic5C22⋊C412yes—
72C22×Dic5C4⋊C412yes—
73C22×Dic5C22×C412yes—
74C22×Dic5C2×D411yescharacteristic
75C22×Dic5C2×D412yes—
76C22×Dic5C2×D414yes—
77C22×Dic5C2×Q811yescharacteristic
78C22×Dic5#4114yes—
79C2×Dic10—28no—
80C2×Dic10—28no—
81C2×Dic10—24no—
82C2×Dic10—24no—
83C2×Dic10—28no—
84C2×Dic10—28no—
85C2×Dic10—24no—
86C2×Dic10—24no—
87C2×Dic10C22⋊C414yes—
88C2×Dic10C22⋊C414yes—
89C2×Dic10C4⋊C414yes—
90C2×Dic10C4⋊C414yes—
91#704—28no—
92#704—28no—
93#704—44no—
94#704—44no—
95C2×#170—28no—
96C2×#170—44no—
97C2×#170—44no—
98C5×C4⋊C4—24no—
99C5×C22⋊C4—22no—
100C5×C22⋊C4C22⋊C414yes—
101C42×C5—24no—
102C22⋊Dic5—22no—
103C22⋊Dic5C22⋊C414yes—
104C4⋊Dic5—24no—
105#683—28no—
106#683—28no—
107C4×Dic5—28no—
108C4×Dic5—24no—
109C4×Dic5—28no—
110C23×Q8—51no—
111C22×M16—54no—
112#519—108no—
113C2×C4×Q8—102no—
114C2×C4×Q8—102no—
115C2×#117—51no—
116C2×#117—52no—
117C2×#117—52no—
118C2×#117AGL(1,𝔽5)11yescharacteristic
119C2×#101—58no—
120C2×#101—104no—
121C2×#101—102no—
122C2×#101—102no—
123#344—108no—
124C23×C5—42no—
125C23×C5C2×C4211yescharacteristic
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5C2×C22⋊C412yes—
128C23×C5C2×C4⋊C411yescharacteristic
129C23×C5#11712yes—
130C2×C4×C5—24no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—24no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—24no—
137C2×C4×C5—44no—
138C2×C4×C5#9512yes—
139C2×C4×C5#9512yes—
140C2×Dic5—24no—
141C2×Dic5—22no—
142C2×Dic5—24no—
143C2×Dic5—22no—
144C2×Dic5—24no—
145C2×Dic5—44no—
146C2×Dic5#9514yes—
147C2×Dic5C2×C22⋊C412yes—
148C2×Dic5C2×C4⋊C412yes—
149C2×Dic5C4×D414yes—
150C2×Dic5C4×D414yes—
151C2×Dic5#12014yes—
152C2×Dic5#12114yes—
153C2×Dic5C22⋊Q814yes—
154C2×Dic5#12314yes—
155Dic10—28no—
156Dic10—28no—
157Dic10—28no—
158Dic10—28no—
159#170—48no—
160C22×Q8—54no—
161C22×Q8—52no—
162C22×Q8—52no—
163C22×Q8—52no—
164C22×Q8—52no—
165C22×Q8—104no—
166C22×Q8—104no—
167C23×C4—51no—
168C23×C4—51no—
169C23×C4C2×AGL(1,𝔽5)11yescharacteristic
170C2×M16—58no—
171C2×M16—104no—
172C2×M16—104no—
173C2×M16—104no—
174C2×M16—104no—
175C2×M16—104no—
176C22×C8—104no—
177C4×Q8—1016no—
178C4×Q8—1016no—
179#117—24no—
180#117—58no—
181#117—104no—
182#117—104no—
183#117—104no—
184#117—104no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—102no—
187C2×C4⋊C4—102no—
188C2×C4⋊C4—101no—
189C2×C22⋊C4—52no—
190C2×C22⋊C4C2×AGL(1,𝔽5)12yes—
191C2×C42—21no—
192C2×C42—102no—
193C2×C42—102no—
194C2×C42—101no—
195#101—108no—
196#101—108no—
197C22×C5—22no—
198C22×C5—24no—
199C22×C5—41no—
200C22×C5—42no—
201C22×C5—41no—
202C22×C5C2×#9511yescharacteristic
203C22×C5C4×C22⋊C412yes—
204C22×C5#37911yescharacteristic
205C22×C5#38012yes—
206C22×C5#38111yescharacteristic
207C4×C5—22no—
208C4×C5—22no—
209C4×C5—44no—
210Dic5—24no—
211Dic5—44no—
212Dic5#38514yes—
213Dic5#38614yes—
214C24C22×AGL(1,𝔽5)11yescharacteristic
215C2×Q8—54no—
216C2×Q8—54no—
217C2×Q8—54no—
218C2×Q8—54no—
219C2×Q8—104no—
220C2×Q8—104no—
221C2×Q8—108no—
222C2×Q8—108no—
223C2×Q8—104no—
224C2×Q8—104no—
225C2×Q8—108no—
226C2×Q8—108no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—52no—
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—54no—
234C22×C4—52no—
235C22×C4—54no—
236C22×C4—52no—
237C22×C4—52no—
238C22×C4—52no—
239C22×C4—102no—
240C22×C4—102no—
241C22×C4—102no—
242C22×C4C4×AGL(1,𝔽5)12yes—
243C22×C4C4⋊AGL(1,𝔽5)12yes—
244C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
245C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
246M16—108no—
247M16—108no—
248M16—204no—
249M16—204no—
250C2×C8—108no—
251C2×C8—204no—
252C2×C8—204no—
253C4⋊C4—24no—
254C4⋊C4—108no—
255C4⋊C4—108no—
256C4⋊C4—104no—
257C22⋊C4—22no—
258C22⋊C4—54no—
259C22⋊C4—102no—
260C22⋊C4C22⋊AGL(1,𝔽5)14yes—
261C42—24no—
262C42—108no—
263C42—104no—
264C42—108no—
265C2×C5—21no—
266C2×C5—21no—
267C2×C5—22no—
268C2×C5—22no—
269C2×C5C22⋊#9511yescharacteristic
270C2×C5#183012yes—
271C23—42no—
272C23C2×C4×AGL(1,𝔽5)11yescharacteristic
273C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
274C23#413012yes—
275C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
276C23C2×C22⋊AGL(1,𝔽5)12yes—
277Q8—108no—
278Q8—108no—
279Q8—108no—
280Q8—108no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—54no—
290C2×C4—52no—
291C2×C4—52no—
292C2×C4—54no—
293C2×C4—54no—
294C2×C4—54no—
295C2×C4—54no—
296C2×C4—54no—
297C2×C4—54no—
298C2×C4—102no—
299C2×C4—104no—
300C2×C4—102no—
301C2×C4—104no—
302C2×C4—104no—
303C2×C4—204no—
304C2×C4(C2×AGL(1,𝔽5))⋊C412yes—
305C2×C4(C2×AGL(1,𝔽5))⋊C412yes—
306C8—208no—
307C5#1077011yescharacteristic, complemented, Sylow
308C22—22no—
309C22—24no—
310C22—41no—
311C22—41no—
312C22—42no—
313C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
314C22C4×C22⋊AGL(1,𝔽5)12yes—
315C22#6576411yescharacteristic
316C22#6576512yes—
317C22#6580011yescharacteristic
318C4—22no—
319C4—22no—
320C4—44no—
321C4—54no—
322C4—54no—
323C4—104no—
324C4—204no—
325C2—21no—
326C2—21no—
327C2—22no—
328C2—22no—
329C2#1060438712yes—
330C2#1060490911yescharacteristic
subgroupG/H
centerC22#65764
commutatorC23×C5C2×C42
FrattiniC24C22×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC22×C5#381
prime radicalC24×C5C22×C4
cosocleC23×Dic5C23
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5#10770
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
16. C2×#4005C4semidirect
307. C5#10770semidirect

Maximal subgroups

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

subgroupindex
C2×#658502
C22×#40052
C2×#648672
#107705