Finite Group Databasebeta

← C2×#13102940 #72221343

Subgroups — C2×#13102940 #72221343

Order 3200.

3376 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 3376 conjugacy classes of subgroups — too many to draw. Only its 395 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×#11717517C211yesmaximal, characteristic, complemented
2C2×C52×#127C211yesmaximal, characteristic, complemented
3C2×#11717252C212yesmaximal, complemented
4C2×#11716700C211yesmaximal, characteristic, complemented
5#13102940C218yesmaximal, direct product factor
6C2×#11716672C212yesmaximal, complemented
7C22×C52×D4—22no—
8C22×C52×D4C2211yescharacteristic
9C22×C52×D4C2211yescharacteristic, complemented
10C22×C22⋊(C5⋊D5)—22no—
11#11717517—22no—
12#11717517—24no—
13#11717517C2214yescomplemented
14C2×#87902—22no—
15C2×D4×C5⋊D5—22no—
16C2×#87900C2212yes—
17C52×#127—22no—
18C52×#127C2214yescomplemented
19C2×C42×C52C2211yescharacteristic, complemented
20C2×#87810—22no—
21#11717252—24no—
22#11717252C2218yescomplemented
23C2×#87809—22no—
24C2×#87804C2212yescomplemented
25#11716700C2214yescomplemented
26#11716672C2218yescomplemented
27C2×#65366—56nomaximal
28C24×C52—21no—
29C24×C52D412yes—
30C23×C5⋊D5—22no—
31C2×C52×D4—28no—
32C2×C52×D4—22no—
33C2×C52×D4—22no—
34C2×C52×D4—22no—
35C2×C52×D4—22no—
36C2×C52×D4—21no—
37C2×C52×D4—48no—
38C2×C52×D4D412yes—
39C2×C52×D4C2312yes—
40C2×C52×D4C2312yescomplemented
41C22×C4×C52—21no—
42C22×C4×C52C2311yescharacteristic
43C2×C22⋊(C5⋊D5)—24no—
44C2×C22⋊(C5⋊D5)—28no—
45C2×C22⋊(C5⋊D5)—22no—
46C2×C22⋊(C5⋊D5)—28no—
47C22×#1039—22no—
48#87902—28no—
49#87902—28no—
50D4×C5⋊D5—28no—
51D4×C5⋊D5—28no—
52#87900—22no—
53#87900—22no—
54#87900D414yescomplemented
55#87900C2314yes—
56C2×C4⋊(C5⋊D5)D412yescomplemented
57C2×C4×C5⋊D5—22no—
58C2×#6045D412yescomplemented
59C42×C52—21no—
60C42×C52C2312yescomplemented
61#87810—28no—
62#87809—28no—
63#87804—22no—
64#87804C2314yescomplemented
65C2×#6029—22no—
66C2×#4144—56no—
67C2×C5×#127D516yescomplemented
68C2×#4078—512no—
69C2×#3965—56no—
70#65366—548no—
71C2×#3937—512no—
72C23×C52—22no—
73C23×C52—22no—
74C23×C52—22no—
75C23×C52—44no—
76C23×C52—44no—
77C23×C52—41no—
78C23×C52—41no—
79C23×C52C2×D411yescharacteristic
80C23×C52C2×D414yes—
81C22×C5⋊D5—22no—
82C22×C5⋊D5—28no—
83C22×C5⋊D5—84no—
84C52×D4—24no—
85C52×D4—24no—
86C52×D4—24no—
87C52×D4—416no—
88C52×D4—42no—
89C2×C4×C52—24no—
90C2×C4×C52—22no—
91C2×C4×C52C2×D411yescharacteristic
92C2×C4×C52C2×D412yes—
93C2×C4×C52C2×D412yes—
94C2×C4×C52C2411yescharacteristic
95C22⋊(C5⋊D5)—216no—
96C22⋊(C5⋊D5)—24no—
97C22⋊(C5⋊D5)—24no—
98C2×#1039—28no—
99C2×#1039—22no—
100C4⋊(C5⋊D5)—22no—
101C4⋊(C5⋊D5)C2×D414yescomplemented
102C4×C5⋊D5—24no—
103C4×C5⋊D5—24no—
104#6045—22no—
105#6045C2×D414yescomplemented
106#6029—24no—
107C22×C5×D4—212no—
108C22×C5×D4C2×D516yes—
109C22×C5×D4C2×D516yescomplemented
110C22×C22⋊D5—1012no—
111#4144—524no—
112#4144—1024no—
113#4144—1012no—
114C2×#711—1012no—
115C2×D4×D5—1012no—
116C2×#709—512no—
117C5×#127—212no—
118C5×#127C2×D5124yescomplemented
119C2×C42×C5C2×D516yescomplemented
120C2×#687—1012no—
121#4078—548no—
122#4078—1024no—
123C2×#686—1012no—
124C2×#681—512no—
125#3965—524no—
126#3937—548no—
127C2×#453—251noSylow
128C22×C52—22no—
129C22×C52—24no—
130C22×C52—44no—
131C22×C52—44no—
132C22×C52—44no—
133C22×C52—42no—
134C22×C52—82no—
135C22×C52—82no—
136C22×C52—82no—
137C22×C52#12011yescharacteristic
138C22×C52#12011yescharacteristic
139C22×C52C22×D411yescharacteristic
140C2×C5⋊D5—24no—
141C2×C5⋊D5—44no—
142C2×C5⋊D5—88no—
143C4×C52—24no—
144C4×C52#12012yes—
145C4×C52C22×D412yes—
146#1039—24no—
147C24×C5—26no—
148C24×C5C22⋊D5112yes—
149C23×D5—1012no—
150C2×C5×D4—248no—
151C2×C5×D4—212no—
152C2×C5×D4—212no—
153C2×C5×D4—212no—
154C2×C5×D4—212no—
155C2×C5×D4—26no—
156C2×C5×D4—448no—
157C2×C5×D4C22⋊D5112yes—
158C2×C5×D4C22×D5112yes—
159C2×C5×D4C22×D5112yescomplemented
160C22×C4×C5—26no—
161C22×C4×C5C22×D516yes—
162C2×C22⋊D5—1024no—
163C2×C22⋊D5—1048no—
164C2×C22⋊D5—1012no—
165C2×C22⋊D5—1048no—
166C22×Dic5—1012no—
167#711—1048no—
168#711—1048no—
169D4×D5—1048no—
170D4×D5—1048no—
171#709—524no—
172#709—524no—
173#709—1012no—
174#709—1012no—
175C2×D20—512no—
176C2×C4×D5—1012no—
177C2×Dic10—512no—
178C42×C5—26no—
179C42×C5C22×D5112yescomplemented
180#687—1048no—
181#686—1048no—
182#681—524no—
183#681—1012no—
184C2×#168—1012no—
185C2×H(5,𝔽2)—251no—
186C2×AGL(1,ℤ/8)—252no—
187C2×#127C5⋊D511yescharacteristic, complemented
188#453—258no—
189C2×#104—252no—
190C2×#100—251no—
191C2×C52—21no—
192C2×C52—21no—
193C2×C52—44no—
194C2×C52—82no—
195C2×C52#45312yescomplemented
196C2×C52C2×#12011yescharacteristic
197C5⋊D5—44no—
198C23×C5—212no—
199C23×C5—212no—
200C23×C5—212no—
201C23×C5—424no—
202C23×C5—424no—
203C23×C5—46no—
204C23×C5—46no—
205C23×C5C2×C22⋊D516yes—
206C23×C5C2×C22⋊D5124yes—
207C22×D5—1012no—
208C22×D5—1048no—
209C22×D5—4024no—
210C5×D4—224no—
211C5×D4—224no—
212C5×D4—224no—
213C5×D4—496no—
214C5×D4—412no—
215C2×C4×C5—224no—
216C2×C4×C5—212no—
217C2×C4×C5D4×D5112yes—
218C2×C4×C5D4×D5112yes—
219C2×C4×C5C2×C22⋊D516yes—
220C2×C4×C5C23×D516yes—
221C22⋊D5—1096no—
222C22⋊D5—1024no—
223C22⋊D5—1024no—
224C2×Dic5—1048no—
225C2×Dic5—1012no—
226D20—524no—
227D20—1012no—
228C4×D5—1024no—
229C4×D5—1024no—
230Dic10—524no—
231Dic10—1012no—
232#168—1024no—
233H(5,𝔽2)—254no—
234H(5,𝔽2)—504no—
235H(5,𝔽2)—502no—
236C2×#41—252no—
237C2×#41—502no—
238C22×D4—22no—
239C22×D4—502no—
240C22×D4—502no—
241C22×D4C2×C5⋊D511yescharacteristic
242C22×D4C2×C5⋊D511yescharacteristic, complemented
243AGL(1,ℤ/8)—258no—
244AGL(1,ℤ/8)—504no—
245C2×SD16—502no—
246C2×D8—502no—
247C2×M16—252no—
248#127—22no—
249#127C2×C5⋊D514yescomplemented
250C2×C42C2×C5⋊D511yescharacteristic, complemented
251#104—258no—
252#100—254no—
253C52C2×#45311yescharacteristic, complemented, Sylow
254C22×C5—212no—
255C22×C5—224no—
256C22×C5—424no—
257C22×C5—424no—
258C22×C5—424no—
259C22×C5—412no—
260C22×C5—812no—
261C22×C5—812no—
262C22×C5—812no—
263C22×C5#408316yes—
264C22×C5#408316yes—
265C22×C5C22×C22⋊D516yes—
266C2×D5—1024no—
267C2×D5—2024no—
268C2×D5—4048no—
269C4×C5—224no—
270C4×C5#4083112yes—
271C4×C5C2×D4×D5112yes—
272Dic5—1024no—
273C24—21no—
274C24—502no—
275C24C22⋊(C5⋊D5)12yes—
276#41—254no—
277#41—254no—
278#41—508no—
279#41—508no—
280#41—502no—
281#41—502no—
282C2×Q8—252no—
283C2×D4—28no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—48no—
290C2×D4—252no—
291C2×D4—508no—
292C2×D4—508no—
293C2×D4—504no—
294C2×D4—508no—
295C2×D4—502no—
296C2×D4—508no—
297C2×D4C22⋊(C5⋊D5)12yes—
298C2×D4C22×C5⋊D512yes—
299C2×D4C22×C5⋊D512yescomplemented
300C22×C4—21no—
301C22×C4—502no—
302C22×C4—502no—
303C22×C4C22×C5⋊D511yescharacteristic
304SD16—508no—
305D8—508no—
306M16—254no—
307M16—502no—
308C2×C8—502no—
309C42—21no—
310C42C22×C5⋊D512yescomplemented
311C2×C5—26no—
312C2×C5—26no—
313C2×C5—424no—
314C2×C5—812no—
315C2×C5#65366112yescomplemented
316C2×C5C2×#408316yes—
317D5—2024no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—44no—
322C23—44no—
323C23—41no—
324C23—41no—
325C23—508no—
326C23—502no—
327C23—2004no—
328C23C2×C22⋊(C5⋊D5)11yescharacteristic
329C23C2×C22⋊(C5⋊D5)14yes—
330Q8—254no—
331Q8—502no—
332D4—24no—
333D4—24no—
334D4—24no—
335D4—416no—
336D4—42no—
337D4—254no—
338D4—5016no—
339D4—504no—
340D4—504no—
341D4—502no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—508no—
345C2×C4—502no—
346C2×C4—504no—
347C2×C4—504no—
348C2×C4D4×C5⋊D512yes—
349C2×C4D4×C5⋊D512yes—
350C2×C4C2×C22⋊(C5⋊D5)11yescharacteristic
351C2×C4C23×C5⋊D511yescharacteristic
352C8—504no—
353C5C2×#6536616yescomplemented
354C22—22no—
355C22—24no—
356C22—44no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—504no—
364C22—1004no—
365C22—2008no—
366C22#1171725711yescharacteristic
367C22#1171725711yescharacteristic
368C22C22×C22⋊(C5⋊D5)11yescharacteristic
369C4—24no—
370C4—504no—
371C4#1171725712yes—
372C4C2×D4×C5⋊D512yes—
373C2—21no—
374C2—21no—
375C2—44no—
376C2—82no—
377C2—1004no—
378C2#1310294012yesdirect product factor
379C2C2×#1171725711yescharacteristic
subgroupG/H
centerC22#11717257
commutator, cosocleC2×C4×C52C24
FrattiniC2×C4C23×C5⋊D5
FittingC2×C52×#127C2
socleC22×C52#120
hypercenterC2×#127C5⋊D5
hypocenterC52C2×#453
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#11717517C2semidirect
2. C2×C52×#127C2semidirect
3. C2×#11717252C2semidirect
4. C2×#11716700C2semidirect
5. #13102940C2direct (N × H)
6. C2×#11716672C2semidirect
9. C22×C52×D4C22semidirect
13. #11717517C22semidirect
18. C52×#127C22semidirect
19. C2×C42×C52C22semidirect
22. #11717252C22semidirect
24. C2×#87804C22semidirect
25. #11716700C22semidirect
26. #11716672C22semidirect
40. C2×C52×D4C23semidirect
54. #87900D4semidirect
56. C2×C4⋊(C5⋊D5)D4semidirect
58. C2×#6045D4semidirect
60. C42×C52C23semidirect
64. #87804C23semidirect
67. C2×C5×#127D5semidirect
101. C4⋊(C5⋊D5)C2×D4semidirect
105. #6045C2×D4semidirect
109. C22×C5×D4C2×D5semidirect
118. C5×#127C2×D5semidirect
119. C2×C42×C5C2×D5semidirect
159. C2×C5×D4C22×D5semidirect
179. C42×C5C22×D5semidirect
187. C2×#127C5⋊D5semidirect
195. C2×C52#453semidirect
242. C22×D4C2×C5⋊D5semidirect
249. #127C2×C5⋊D5semidirect
250. C2×C42C2×C5⋊D5semidirect
253. C52C2×#453semidirect
299. C2×D4C22×C5⋊D5semidirect
310. C42C22×C5⋊D5semidirect
315. C2×C5#65366semidirect
353. C5C2×#65366semidirect
378. C2#13102940direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117175172
C2×C52×#1272
C2×#117172522
C2×#117167002
#131029402
C2×#117166722
C2×#653665