Finite Group Databasebeta

← C2×#27161224 #38751913

Subgroups — C2×#27161224 #38751913

Order 25216.

812 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×#15691345C211yesmaximal, characteristic, complemented
2C22×#15691343C211yesmaximal, characteristic, complemented
3C2×#20263300C211yesmaximal, characteristic, complemented
4#27161224C218yesmaximal, direct product factor
5C2×#20263250C211yesmaximal, characteristic, complemented
6C2×#20263247C211yesmaximal, characteristic, complemented
7C2×C22⋊#13096802C211yesmaximal, characteristic, complemented
8C22×#41×C197C211yesmaximal, characteristic, complemented
9C22×D788C2211yescharacteristic
10C22×C22⋊D197—21no—
11C2×#15691345—24no—
12C2×#15691345C2214yescomplemented
13C2×#15691343—24no—
14C2×#15691343C2214yescomplemented
15C22×#13096802C2211yescharacteristic
16C22×#13096802C2211yescharacteristic
17#20263300—22no—
18#20263300C2214yescomplemented
19C2×C4⋊Dic197C2211yescharacteristic
20C2×#15691335—21no—
21#20263250C2214yescomplemented
22#20263247C2214yescomplemented
23C22⋊#13096802C2214yescomplemented
24C2×#41×C197—24no—
25C2×#41×C197—24no—
26C2×#41×C197—22no—
27C2×#41×C197C2214yescomplemented
28C22×Q8×C197C2211yescharacteristic
29C22×D4×C197—21no—
30C22×D4×C197C2211yescharacteristic
31C23×C4×C197—21no—
32C23×C4×C197C2211yescharacteristic
33C23×D197—21no—
34C2×D788—22no—
35C2×D788—22no—
36C2×D788—21no—
37C2×D788C2312yes—
38C2×C22⋊D197—24no—
39C2×C22⋊D197—44no—
40C22×Dic197—21no—
41#15691345—28no—
42#15691343—28no—
43C2×#13096802—22no—
44C2×#13096802—22no—
45C2×#13096802C2312yes—
46C2×#13096802C2312yes—
47C4⋊Dic197—21no—
48C4⋊Dic197C2312yes—
49#15691335—24no—
50C24×C197—21no—
51C24×C197D411yescharacteristic
52#41×C197—216no—
53#41×C197—28no—
54#41×C197—28no—
55C2×Q8×C197—22no—
56C2×Q8×C197—21no—
57C2×Q8×C197D414yescomplemented
58C2×Q8×C197C2312yes—
59C2×D4×C197—28no—
60C2×D4×C197—24no—
61C2×D4×C197—21no—
62C2×D4×C197—22no—
63C2×D4×C197D414yescomplemented
64C2×D4×C197C2312yes—
65C22×C4×C197—28no—
66C22×C4×C197—21no—
67C22×C4×C197—24no—
68C22×C4×C197—22no—
69C22×C4×C197—22no—
70C22×C4×C197D411yescharacteristic
71C22×C4×C197D412yes—
72C22×C4×C197C2312yes—
73C22×C4×C197C2311yescharacteristic
74C22×D197—22no—
75C22×D197—42no—
76C22×D197—42no—
77C22×D197—42no—
78D788—22no—
79D788—22no—
80D788—42no—
81C22⋊D197—48no—
82C2×Dic197—22no—
83C2×Dic197—42no—
84#13096802—22no—
85#13096802—22no—
86C23×C197—22no—
87C23×C197—24no—
88C23×C197—22no—
89C23×C197—22no—
90C23×C197—22no—
91C23×C197—42no—
92C23×C197—42no—
93C23×C197C2×D412yes—
94C23×C197C2×D411yescharacteristic
95Q8×C197—24no—
96Q8×C197—22no—
97Q8×C197C2×D414yescomplemented
98D4×C197—216no—
99D4×C197—24no—
100D4×C197—22no—
101D4×C197C2×D414yescomplemented
102C2×C4×C197—216no—
103C2×C4×C197—22no—
104C2×C4×C197—24no—
105C2×C4×C197—22no—
106C2×C4×C197—24no—
107C2×C4×C197—22no—
108C2×C4×C197—22no—
109C2×C4×C197C2×D412yes—
110C2×C4×C197C2×D411yescharacteristic
111C2×C4×C197C2×D412yes—
112C2×C4×C197C2×D412yes—
113C2×C4×C197C2411yescharacteristic
114C22×C197—22no—
115C22×C197—24no—
116C22×C197—22no—
117C22×C197—22no—
118C22×C197—42no—
119C22×C197—42no—
120C22×C197—44no—
121C22×C197—48no—
122C22×C197#12012yes—
123C22×C197#13512yes—
124C22×C197AGL(1,ℤ/8)12yes—
125C22×C197C22×D411yescharacteristic
126C2×D197—42no—
127C2×D197—42no—
128C2×D197—42no—
129C2×D197—84no—
130C4×C197—22no—
131C4×C197—24no—
132C4×C197#12012yes—
133C4×C197C22×D411yescharacteristic
134C4×C197C22×D411yescharacteristic
135Dic197—42no—
136C2×C197—42no—
137C2×C197—44no—
138C2×C197#44914yescomplemented
139C2×C197C2×#12011yescharacteristic
140C2×C197C2×#13511yescharacteristic
141C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
142D197—82no—
143C197C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1971nomaximal, Sylow
145C22×#41D19711yescharacteristic, complemented
146C22×SD16—1971no—
147C22×D8—1971no—
148C2×#121—1971no—
149#449—1978no—
150C2×#103—1971no—
151C2×#102—1971no—
152C2×C22⋊C8—1971no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—22no—
156C2×#41C2×D19714yescomplemented
157C22×Q8C2×D19711yescharacteristic
158C22×D4—21no—
159C22×D4—3941no—
160C22×D4—1971no—
161C22×D4C2×D19711yescharacteristic
162C23×C4—21no—
163C23×C4C2×D19711yescharacteristic
164C2×SD16—3944no—
165C2×SD16—1974no—
166C2×D8—3944no—
167C2×D8—1974no—
168C22×C8—1971no—
169C22×C8—1971no—
170#121—3942no—
171#121—1974no—
172C2×C4⋊C4—1971no—
173C2×C22⋊C4—3941no—
174#103—1974no—
175#102—1974no—
176C22⋊C8—1974no—
177C24—21no—
178C24—3941no—
179C24C22⋊D19711yescharacteristic
180#41—216no—
181#41—28no—
182#41—28no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8C22⋊D19714yescomplemented
186C2×Q8C22×D19712yes—
187C2×D4—28no—
188C2×D4—24no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—7884no—
192C2×D4—3944no—
193C2×D4—3941no—
194C2×D4—3942no—
195C2×D4—3942no—
196C2×D4—1972no—
197C2×D4C22⋊D19714yescomplemented
198C2×D4C22×D19712yes—
199C22×C4—28no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—3941no—
205C22×C4C22⋊D19712yes—
206C22×C4C22⋊D19711yescharacteristic
207C22×C4C22×D19712yes—
208C22×C4C22×D19711yescharacteristic
209SD16—3948no—
210D8—3948no—
211C2×C8—3942no—
212C2×C8—3942no—
213C2×C8—1972no—
214C2×C8—1972no—
215C4⋊C4—3941no—
216C4⋊C4—1972no—
217C22⋊C4—3944no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—7882no—
226C23—7882no—
227C23—7882no—
228C23—3942no—
229C23C2×C22⋊D19712yes—
230C23C2×C22⋊D19711yescharacteristic
231Q8—24no—
232Q8—22no—
233Q8C2×C22⋊D19714yescomplemented
234D4—216no—
235D4—24no—
236D4—22no—
237D4—7888no—
238D4—7882no—
239D4—3942no—
240D4—3942no—
241D4C2×C22⋊D19714yescomplemented
242C2×C4—216no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—7882no—
250C2×C4—3942no—
251C2×C4C2×C22⋊D19712yes—
252C2×C4C2×C22⋊D19711yescharacteristic
253C2×C4C2×C22⋊D19712yes—
254C2×C4C2×C22⋊D19712yes—
255C2×C4C23×D19711yescharacteristic
256C8—3942no—
257C8—3942no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—48no—
266C22—7882no—
267C22—7882no—
268C22—7882no—
269C22—15764no—
270C22#2026329512yes—
271C22#2026334612yes—
272C22#2026335112yes—
273C22C22×C22⋊D19711yescharacteristic
274C4—22no—
275C4—24no—
276C4—7882no—
277C4#2026329512yes—
278C4C22×C22⋊D19711yescharacteristic
279C4C22×C22⋊D19711yescharacteristic
280C2—42no—
281C2—44no—
282C2—15762no—
283C2#2716122414yesdirect product factor
284C2C2×#2026329511yescharacteristic
285C2C2×#2026334611yescharacteristic
286C2C2×#2026335111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×#41×C197C2
socleC23×C197C2×D4
hypercenterC22×#41D197
hypocenterC197C2×#449
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#15691345C2semidirect
2. C22×#15691343C2semidirect
3. C2×#20263300C2semidirect
4. #27161224C2direct (N × H)
5. C2×#20263250C2semidirect
6. C2×#20263247C2semidirect
7. C2×C22⋊#13096802C2semidirect
8. C22×#41×C197C2semidirect
12. C2×#15691345C22semidirect
14. C2×#15691343C22semidirect
18. #20263300C22semidirect
21. #20263250C22semidirect
22. #20263247C22semidirect
23. C22⋊#13096802C22semidirect
27. C2×#41×C197C22semidirect
57. C2×Q8×C197D4semidirect
63. C2×D4×C197D4semidirect
97. Q8×C197C2×D4semidirect
101. D4×C197C2×D4semidirect
138. C2×C197#449semidirect
143. C197C2×#449semidirect
145. C22×#41D197semidirect
156. C2×#41C2×D197semidirect
185. C2×Q8C22⋊D197semidirect
197. C2×D4C22⋊D197semidirect
233. Q8C2×C22⋊D197semidirect
241. D4C2×C22⋊D197semidirect
283. C2#27161224direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913452
C22×#156913432
C2×#202633002
#271612242
C2×#202632502
C2×#202632472
C2×C22⋊#130968022
C22×#41×C1972
C2×#449197