Finite Group Databasebeta

← C2×#27161267 #38752097

Subgroups — C2×#27161267 #38752097

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×#20263296C212yesmaximal, complemented
3C2×#20263294C211yesmaximal, characteristic, complemented
4#27161267C218yesmaximal, direct product factor
5C2×#20263240C212yesmaximal, complemented
6C2×#99×C197C211yesmaximal, characteristic, complemented
7C24×D197C2211yescharacteristic, complemented
8#20263371—24no—
9#20263371—22no—
10#20263371C2214yescomplemented
11C2×#15691358—21no—
12C2×#15691357—22no—
13C22×D788—22no—
14C22×C22⋊D197—21no—
15C22×C22⋊D197—21no—
16C22×C22⋊D197C2212yes—
17C2×D4×D197—22no—
18#20263296—24no—
19#20263296C2218yescomplemented
20#20263294—22no—
21#20263294C2214yescomplemented
22C2×C22⋊Dic197C2211yescharacteristic, complemented
23C2×#15691335—22no—
24C2×#15691335—21no—
25#20263240C2218yescomplemented
26C22×D4×C197C2211yescharacteristic
27C2×C22⋊C4×C197C2212yescomplemented
28#99×C197C2214yescomplemented
29C23×D197—21no—
30C23×D197—21no—
31C23×D197—22no—
32C23×D197—21no—
33C23×D197—22no—
34C23×D197—44no—
35C23×D197D412yescomplemented
36C23×D197C2312yescomplemented
37#15691358—24no—
38#15691358—24no—
39#15691357—28no—
40#15691357—28no—
41C2×Dic394—21no—
42C2×D788—21no—
43C2×D788—28no—
44C2×D788—48no—
45C2×C22⋊D197—22no—
46C2×C22⋊D197—24no—
47C2×C22⋊D197—22no—
48C2×C22⋊D197—22no—
49C2×C22⋊D197—22no—
50C2×C22⋊D197—22no—
51C2×C22⋊D197—24no—
52C2×C22⋊D197—24no—
53C2×C22⋊D197—44no—
54C2×C22⋊D197D414yescomplemented
55C2×C22⋊D197C2314yes—
56D4×D197—28no—
57D4×D197—28no—
58C22×Dic197—21no—
59C22×Dic197—21no—
60C22×Dic197D412yescomplemented
61C2×C4×D197—22no—
62C22⋊Dic197—21no—
63C22⋊Dic197C2312yescomplemented
64#15691335—28no—
65#15691335—24no—
66C24×C197D411yescharacteristic, complemented
67C24×C197C2311yescharacteristic
68C2×D4×C197—22no—
69C2×D4×C197—22no—
70C2×D4×C197D412yescomplemented
71C2×D4×C197C2312yes—
72C22×C4×C197—22no—
73C22×C4×C197D411yescharacteristic, complemented
74C22⋊C4×C197—22no—
75C22⋊C4×C197C2314yescomplemented
76C22×D197—24no—
77C22×D197—28no—
78C22×D197—22no—
79C22×D197—22no—
80C22×D197—24no—
81C22×D197—24no—
82C22×D197—22no—
83C22×D197—44no—
84C22×D197—44no—
85C22×D197—48no—
86C22×D197—44no—
87C22×D197—42no—
88C22×D197—41no—
89C22×D197—42no—
90C22×D197—41no—
91C22×D197—44no—
92C22×D197—82no—
93C22×D197C2×D414yescomplemented
94Dic394—24no—
95D788—24no—
96D788—416no—
97C22⋊D197—28no—
98C22⋊D197—24no—
99C22⋊D197—24no—
100C22⋊D197—24no—
101C22⋊D197—24no—
102C22⋊D197—48no—
103C2×Dic197—24no—
104C2×Dic197—22no—
105C2×Dic197—22no—
106C2×Dic197—22no—
107C2×Dic197—42no—
108C2×Dic197C2×D414yescomplemented
109C4×D197—28no—
110C23×C197—21no—
111C23×C197—21no—
112C23×C197—42no—
113C23×C197—42no—
114C23×C197C2×D412yes—
115C23×C197C2×D412yes—
116C23×C197C2×D411yescharacteristic
117C23×C197C2×D411yescharacteristic
118C23×C197C2×D412yescomplemented
119C23×C197C2411yescharacteristic
120D4×C197—24no—
121D4×C197—24no—
122C2×C4×C197—21no—
123C2×C4×C197—21no—
124C2×C4×C197—24no—
125C2×C4×C197—44no—
126C2×C4×C197C2×D412yescomplemented
127C22×C197—24no—
128C22×C197—22no—
129C22×C197—22no—
130C22×C197—42no—
131C22×C197—41no—
132C22×C197—41no—
133C22×C197—42no—
134C22×C197—42no—
135C22×C197—81no—
136C22×C197#12011yescharacteristic
137C22×C197#12012yes—
138C22×C197#12011yescharacteristic
139C22×C197C22×D412yes—
140C22×C197C22×D411yescharacteristic
141C2×D197—24no—
142C2×D197—28no—
143C2×D197—24no—
144C2×D197—42no—
145C2×D197—44no—
146C2×D197—48no—
147C2×D197—44no—
148C2×D197—42no—
149C2×D197—44no—
150C2×D197—44no—
151C2×D197—44no—
152C2×D197—82no—
153C2×D197—84no—
154C4×C197—22no—
155C4×C197—44no—
156Dic197—24no—
157Dic197—42no—
158C2×C197—21no—
159C2×C197—22no—
160C2×C197—22no—
161C2×C197—21no—
162C2×C197—42no—
163C2×C197B(4,𝔽2)12yescomplemented
164C2×C197C2×#12011yescharacteristic
165D197—44no—
166D197—44no—
167C197C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—1971nomaximal, Sylow
169C2×H(5,𝔽2)—1971no—
170C2×#120—1971no—
171C2×#120—1972no—
172B(4,𝔽2)—1978no—
173C2×#99—1972no—
174C2×#99D19711yescharacteristic, complemented
175C25—1971no—
176H(5,𝔽2)—3944no—
177H(5,𝔽2)—3942no—
178H(5,𝔽2)—1974no—
179C2×#41—3942no—
180C2×#41—3941no—
181C22×D4—3942no—
182C22×D4—3942no—
183C22×D4—3941no—
184C22×D4—3941no—
185C22×D4—1972no—
186C22×D4C2×D19711yescharacteristic
187#120—3944no—
188#120—3942no—
189#120—1974no—
190#120—1978no—
191C2×C22⋊C4—3942no—
192C2×C22⋊C4—3941no—
193C2×C22⋊C4—1971no—
194C2×C22⋊C4C2×D19712yescomplemented
195#99—1978no—
196#99C2×D19714yescomplemented
197C24—7884no—
198C24—3941no—
199C24—3942no—
200C24—3941no—
201C24—3942no—
202C24—3941no—
203C24—1972no—
204C24—1972no—
205C24D78811yescharacteristic, complemented
206C24C22×D19711yescharacteristic
207#41—3944no—
208#41—3948no—
209#41—3948no—
210#41—3944no—
211C2×Q8—3941no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—7888no—
215C2×D4—7884no—
216C2×D4—3948no—
217C2×D4—3941no—
218C2×D4—3948no—
219C2×D4—3948no—
220C2×D4—3944no—
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×D4D78812yescomplemented
231C2×D4C22×D19712yes—
232C22×C4—22no—
233C22×C4—3942no—
234C22×C4—3941no—
235C22×C4—3941no—
236C22×C4—1972no—
237C22×C4D78811yescharacteristic, complemented
238C22⋊C4—22no—
239C22⋊C4—3948no—
240C22⋊C4—3941no—
241C22⋊C4—3944no—
242C22⋊C4—1972no—
243C22⋊C4C22×D19714yescomplemented
244C23—21no—
245C23—21no—
246C23—42no—
247C23—42no—
248C23—7884no—
249C23—7882no—
250C23—7884no—
251C23—7888no—
252C23—7884no—
253C23—7884no—
254C23—7881no—
255C23—7882no—
256C23—7881no—
257C23—15762no—
258C23—3948no—
259C23—3944no—
260C23—3944no—
261C23—3942no—
262C23—3944no—
263C23—3942no—
264C23—3942no—
265C23—1974no—
266C23D4×D19712yes—
267C23D4×D19712yes—
268C23C2×D78811yescharacteristic
269C23C2×D78811yescharacteristic
270C23C2×D78812yescomplemented
271C23C23×D19711yescharacteristic
272Q8—3944no—
273D4—24no—
274D4—24no—
275D4—78816no—
276D4—7888no—
277D4—3944no—
278D4—3948no—
279D4—3944no—
280D4—3944no—
281D4—3944no—
282D4—3944no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—44no—
287C2×C4—7882no—
288C2×C4—3948no—
289C2×C4—3942no—
290C2×C4—3944no—
291C2×C4—3942no—
292C2×C4—3942no—
293C2×C4—1974no—
294C2×C4C2×D78812yescomplemented
295C22—24no—
296C22—22no—
297C22—22no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—81no—
304C22—7884no—
305C22—7884no—
306C22—7882no—
307C22—7884no—
308C22—7888no—
309C22—7884no—
310C22—7884no—
311C22—7882no—
312C22—15762no—
313C22—15764no—
314C22—3948no—
315C22—3944no—
316C22—3944no—
317C22#2026329611yescharacteristic
318C22#2026329612yes—
319C22#2026329611yescharacteristic
320C22C2×D4×D19712yes—
321C22C22×D78811yescharacteristic
322C4—22no—
323C4—44no—
324C4—7882no—
325C4—3944no—
326C2—22no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—42no—
331C2—7884no—
332C2—7884no—
333C2#2716126712yesdirect product factor
334C2C2×#2026329611yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
FittingC2×#99×C197C2
socleC22×C197#120
hypercenterC2×#99D197
hypocenterC197C2×B(4,𝔽2)
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#20263371C2semidirect
2. C2×#20263296C2semidirect
3. C2×#20263294C2semidirect
4. #27161267C2direct (N × H)
5. C2×#20263240C2semidirect
6. C2×#99×C197C2semidirect
7. C24×D197C22semidirect
10. #20263371C22semidirect
19. #20263296C22semidirect
21. #20263294C22semidirect
22. C2×C22⋊Dic197C22semidirect
25. #20263240C22semidirect
27. C2×C22⋊C4×C197C22semidirect
28. #99×C197C22semidirect
35. C23×D197D4semidirect
36. C23×D197C23semidirect
54. C2×C22⋊D197D4semidirect
60. C22×Dic197D4semidirect
63. C22⋊Dic197C23semidirect
66. C24×C197D4semidirect
70. C2×D4×C197D4semidirect
73. C22×C4×C197D4semidirect
75. C22⋊C4×C197C23semidirect
93. C22×D197C2×D4semidirect
108. C2×Dic197C2×D4semidirect
118. C23×C197C2×D4semidirect
126. C2×C4×C197C2×D4semidirect
163. C2×C197B(4,𝔽2)semidirect
167. C197C2×B(4,𝔽2)semidirect
174. C2×#99D197semidirect
194. C2×C22⋊C4C2×D197semidirect
196. #99C2×D197semidirect
205. C24D788semidirect
230. C2×D4D788semidirect
237. C22×C4D788semidirect
243. C22⋊C4C22×D197semidirect
270. C23C2×D788semidirect
294. C2×C4C2×D788semidirect
333. C2#27161267direct (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×#202632962
C2×#202632942
#271612672
C2×#202632402
C2×#99×C1972
C2×B(4,𝔽2)197