Finite Group Databasebeta

← C2×#27161265 #38752095

Subgroups — C2×#27161265 #38752095

Order 25216.

1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20263371C211yesmaximal, characteristic, complemented
2C2×#20263295C212yesmaximal, complemented
3#27161265C218yesmaximal, direct product factor
4C2×#20263241C211yesmaximal, characteristic, complemented
5C2×#20263240C212yesmaximal, complemented
6C2×#120×C197C211yesmaximal, characteristic, complemented
7#20263371—24no—
8#20263371—22no—
9#20263371C2214yescomplemented
10C2×#15691358—21no—
11C2×#15691357—22no—
12C22×C22⋊D197—22no—
13C22×C22⋊D197—21no—
14C22×C22⋊D197C2212yes—
15C2×D4×D197—22no—
16#20263295—24no—
17#20263295C2218yescomplemented
18C2×C22⋊Dic197—22no—
19C2×C22⋊Dic197C2212yescomplemented
20#20263241C2214yescomplemented
21#20263240C2218yescomplemented
22C25×C197C2211yescharacteristic, complemented
23C22×D4×C197—21no—
24C22×D4×C197C2211yescharacteristic
25#120×C197—22no—
26#120×C197C2214yescomplemented
27C2×C22⋊C4×C197—21no—
28C2×C22⋊C4×C197C2211yescharacteristic, complemented
29C23×D197—21no—
30C23×D197D412yescomplemented
31#15691358—24no—
32#15691358—24no—
33#15691357—28no—
34#15691357—28no—
35C2×Dic394—21no—
36C2×D788—21no—
37C2×C22⋊D197—22no—
38C2×C22⋊D197—24no—
39C2×C22⋊D197—28no—
40C2×C22⋊D197—22no—
41C2×C22⋊D197—22no—
42C2×C22⋊D197—22no—
43C2×C22⋊D197—22no—
44C2×C22⋊D197—24no—
45C2×C22⋊D197—48no—
46C2×C22⋊D197D414yescomplemented
47C2×C22⋊D197C2314yes—
48D4×D197—28no—
49D4×D197—28no—
50C22×Dic197—21no—
51C22×Dic197—22no—
52C22×Dic197D412yescomplemented
53C2×C4×D197—22no—
54C22⋊Dic197—28no—
55C22⋊Dic197—22no—
56C22⋊Dic197C2314yescomplemented
57C24×C197—21no—
58C24×C197—22no—
59C24×C197—21no—
60C24×C197—22no—
61C24×C197—44no—
62C24×C197D411yescharacteristic, complemented
63C24×C197C2311yescharacteristic
64C24×C197C2312yescomplemented
65C2×D4×C197—22no—
66C2×D4×C197—22no—
67C2×D4×C197—24no—
68C2×D4×C197—44no—
69C2×D4×C197D412yescomplemented
70C2×D4×C197C2312yes—
71C22×C4×C197—21no—
72C22×C4×C197D411yescharacteristic, complemented
73C22⋊C4×C197—21no—
74C22⋊C4×C197—24no—
75C22⋊C4×C197C2312yescomplemented
76C22×D197—22no—
77C22×D197—24no—
78C22×D197—22no—
79C22×D197—44no—
80C22×D197—82no—
81C22×D197C2×D414yescomplemented
82Dic394—24no—
83D788—24no—
84C22⋊D197—28no—
85C22⋊D197—24no—
86C22⋊D197—24no—
87C22⋊D197—24no—
88C22⋊D197—24no—
89C22⋊D197—416no—
90C2×Dic197—24no—
91C2×Dic197—22no—
92C2×Dic197—22no—
93C2×Dic197—24no—
94C2×Dic197—44no—
95C2×Dic197C2×D414yescomplemented
96C4×D197—28no—
97C23×C197—24no—
98C23×C197—28no—
99C23×C197—21no—
100C23×C197—22no—
101C23×C197—24no—
102C23×C197—21no—
103C23×C197—42no—
104C23×C197—44no—
105C23×C197—42no—
106C23×C197—44no—
107C23×C197—44no—
108C23×C197—48no—
109C23×C197—42no—
110C23×C197—41no—
111C23×C197—42no—
112C23×C197—41no—
113C23×C197C2×D412yes—
114C23×C197C2×D412yes—
115C23×C197C2×D411yescharacteristic
116C23×C197C2×D411yescharacteristic
117C23×C197C2×D412yescomplemented
118C23×C197C2411yescharacteristic
119D4×C197—24no—
120D4×C197—24no—
121D4×C197—48no—
122C2×C4×C197—21no—
123C2×C4×C197—21no—
124C2×C4×C197—22no—
125C2×C4×C197—42no—
126C2×C4×C197C2×D412yescomplemented
127C22×C197—22no—
128C22×C197—24no—
129C22×C197—24no—
130C22×C197—28no—
131C22×C197—22no—
132C22×C197—42no—
133C22×C197—41no—
134C22×C197—41no—
135C22×C197—42no—
136C22×C197—44no—
137C22×C197—42no—
138C22×C197—48no—
139C22×C197—44no—
140C22×C197—42no—
141C22×C197—44no—
142C22×C197—42no—
143C22×C197—81no—
144C22×C197#12011yescharacteristic
145C22×C197#12012yes—
146C22×C197#12011yescharacteristic
147C22×C197C22×D412yes—
148C22×C197C22×D411yescharacteristic
149C2×D197—24no—
150C2×D197—44no—
151C2×D197—44no—
152C2×D197—82no—
153C2×D197—84no—
154C4×C197—22no—
155C4×C197—42no—
156Dic197—24no—
157Dic197—44no—
158C2×C197—22no—
159C2×C197—21no—
160C2×C197—22no—
161C2×C197—21no—
162C2×C197—42no—
163C2×C197—44no—
164C2×C197B(4,𝔽2)12yescomplemented
165C2×C197C2×#12011yescharacteristic
166D197—44no—
167C197C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—1971nomaximal, Sylow
169C2×H(5,𝔽2)—1971no—
170C2×#120—1972no—
171C2×#120D19711yescharacteristic, complemented
172B(4,𝔽2)—1978no—
173C2×#99—1971no—
174C2×#99—1972no—
175C25C2×D19711yescharacteristic, complemented
176H(5,𝔽2)—3944no—
177H(5,𝔽2)—3942no—
178H(5,𝔽2)—1974no—
179C2×#41—3942no—
180C2×#41—3941no—
181C22×D4—21no—
182C22×D4—3942no—
183C22×D4—3942no—
184C22×D4—3941no—
185C22×D4—1972no—
186C22×D4C2×D19711yescharacteristic
187#120—22no—
188#120—3944no—
189#120—1978no—
190#120C2×D19714yescomplemented
191C2×C22⋊C4—21no—
192C2×C22⋊C4—3942no—
193C2×C22⋊C4—1972no—
194C2×C22⋊C4C2×D19711yescharacteristic, complemented
195#99—1974no—
196#99—1978no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—21no—
201C24—44no—
202C24—3941no—
203C24—1972no—
204C24C22⋊D19711yescharacteristic, complemented
205C24C22×D19711yescharacteristic
206C24C22×D19712yescomplemented
207#41—3944no—
208#41—3948no—
209#41—3948no—
210#41—3944no—
211C2×Q8—3941no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—44no—
216C2×D4—7888no—
217C2×D4—3941no—
218C2×D4—3948no—
219C2×D4—3948no—
220C2×D4—3948no—
221C2×D4—3942no—
222C2×D4—3944no—
223C2×D4—3942no—
224C2×D4—3942no—
225C2×D4—3942no—
226C2×D4—3942no—
227C2×D4—3944no—
228C2×D4—1974no—
229C2×D4—1974no—
230C2×D4C22⋊D19712yescomplemented
231C2×D4C22×D19712yes—
232C22×C4—21no—
233C22×C4—3942no—
234C22×C4—3942no—
235C22×C4—3941no—
236C22×C4—1972no—
237C22×C4C22⋊D19711yescharacteristic, complemented
238C22⋊C4—21no—
239C22⋊C4—24no—
240C22⋊C4—3942no—
241C22⋊C4—3948no—
242C22⋊C4—1974no—
243C22⋊C4C22×D19712yescomplemented
244C23—24no—
245C23—28no—
246C23—21no—
247C23—24no—
248C23—22no—
249C23—21no—
250C23—42no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—7884no—
261C23—15762no—
262C23—3944no—
263C23—3942no—
264C23—3942no—
265C23—1974no—
266C23D4×D19712yes—
267C23D4×D19712yes—
268C23C2×C22⋊D19711yescharacteristic
269C23C2×C22⋊D19711yescharacteristic
270C23C2×C22⋊D19712yescomplemented
271C23C23×D19711yescharacteristic
272Q8—3944no—
273D4—24no—
274D4—24no—
275D4—48no—
276D4—78816no—
277D4—3944no—
278D4—3948no—
279D4—3944no—
280D4—3944no—
281D4—3944no—
282D4—3944no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—7884no—
288C2×C4—3948no—
289C2×C4—3944no—
290C2×C4—3944no—
291C2×C4—3942no—
292C2×C4—3942no—
293C2×C4—1974no—
294C2×C4C2×C22⋊D19712yescomplemented
295C22—22no—
296C22—24no—
297C22—24no—
298C22—28no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—44no—
307C22—48no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—81no—
312C22—7884no—
313C22—7884no—
314C22—15762no—
315C22—15764no—
316C22—3944no—
317C22#2026329411yescharacteristic
318C22#2026329412yes—
319C22#2026329411yescharacteristic
320C22C2×D4×D19712yes—
321C22C22×C22⋊D19711yescharacteristic
322C4—22no—
323C4—42no—
324C4—7884no—
325C4—3944no—
326C2—21no—
327C2—22no—
328C2—21no—
329C2—22no—
330C2—44no—
331C2—42no—
332C2—7884no—
333C2#2716126512yesdirect product factor
334C2C2×#2026329411yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
FittingC2×#120×C197C2
socleC22×C197#120
hypercenterC2×#120D197
hypocenterC197C2×B(4,𝔽2)
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#20263371C2semidirect
2. C2×#20263295C2semidirect
3. #27161265C2direct (N × H)
4. C2×#20263241C2semidirect
5. C2×#20263240C2semidirect
6. C2×#120×C197C2semidirect
9. #20263371C22semidirect
17. #20263295C22semidirect
19. C2×C22⋊Dic197C22semidirect
20. #20263241C22semidirect
21. #20263240C22semidirect
22. C25×C197C22semidirect
26. #120×C197C22semidirect
28. C2×C22⋊C4×C197C22semidirect
30. C23×D197D4semidirect
46. C2×C22⋊D197D4semidirect
52. C22×Dic197D4semidirect
56. C22⋊Dic197C23semidirect
62. C24×C197D4semidirect
64. C24×C197C23semidirect
69. C2×D4×C197D4semidirect
72. C22×C4×C197D4semidirect
75. C22⋊C4×C197C23semidirect
81. C22×D197C2×D4semidirect
95. C2×Dic197C2×D4semidirect
117. C23×C197C2×D4semidirect
126. C2×C4×C197C2×D4semidirect
164. C2×C197B(4,𝔽2)semidirect
167. C197C2×B(4,𝔽2)semidirect
171. C2×#120D197semidirect
175. C25C2×D197semidirect
190. #120C2×D197semidirect
194. C2×C22⋊C4C2×D197semidirect
204. C24C22⋊D197semidirect
206. C24C22×D197semidirect
230. C2×D4C22⋊D197semidirect
237. C22×C4C22⋊D197semidirect
243. C22⋊C4C22×D197semidirect
270. C23C2×C22⋊D197semidirect
294. C2×C4C2×C22⋊D197semidirect
333. C2#27161265direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202633712
C2×#202632952
#271612652
C2×#202632412
C2×#202632402
C2×#120×C1972
C2×B(4,𝔽2)197