Finite Group Databasebeta

← C2×#27161268 #38752099

Subgroups — C2×#27161268 #38752099

Order 25216.

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

normal subgroups only

This group has 794 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
2C2×#20263309C212yesmaximal, complemented
3#27161268C218yesmaximal, direct product factor
4C2×#20263241C211yesmaximal, characteristic, complemented
5C2×#20263240C212yesmaximal, complemented
6C2×#123×C197C211yesmaximal, characteristic, complemented
7#20263371—24no—
8#20263371—22no—
9#20263371C2214yes—
10C2×#15691358—21no—
11C2×#15691357—22no—
12C22×C22⋊D197—21no—
13C22×C22⋊D197C2212yes—
14C2×D4×D197—22no—
15#20263309—24no—
16#20263309C2218yescomplemented
17C2×#15691337—22no—
18C2×C22⋊Dic197C2212yescomplemented
19C2×#15691335—22no—
20#20263241C2214yescomplemented
21#20263240C2218yescomplemented
22C22×D4×C197C2211yescharacteristic
23C23×C4×C197C2211yescharacteristic, complemented
24#123×C197—22no—
25#123×C197C2214yescomplemented
26C2×C4⋊C4×C197—21no—
27C2×C22⋊C4×C197—21no—
28C2×C22⋊C4×C197C2211yescharacteristic, complemented
29C23×D197—21no—
30C23×D197D412yes—
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—22no—
40C2×C22⋊D197—22no—
41C2×C22⋊D197—24no—
42C2×C22⋊D197—22no—
43C2×C22⋊D197—22no—
44C2×C22⋊D197D414yes—
45C2×C22⋊D197C2314yes—
46D4×D197—28no—
47D4×D197—28no—
48C22×Dic197—21no—
49C22×Dic197—22no—
50C22×Dic197D412yes—
51C2×C4×D197—22no—
52#15691337—28no—
53C22⋊Dic197—22no—
54C22⋊Dic197C2314yescomplemented
55#15691335—28no—
56C24×C197D411yescharacteristic
57C24×C197C2311yescharacteristic
58C2×D4×C197—22no—
59C2×D4×C197—22no—
60C2×D4×C197D412yes—
61C2×D4×C197C2312yes—
62C22×C4×C197—21no—
63C22×C4×C197—21no—
64C22×C4×C197—21no—
65C22×C4×C197—22no—
66C22×C4×C197—22no—
67C22×C4×C197D411yescharacteristic
68C22×C4×C197C2312yescomplemented
69C4⋊C4×C197—24no—
70C22⋊C4×C197—24no—
71C22⋊C4×C197—21no—
72C22⋊C4×C197C2312yescomplemented
73C22×D197—22no—
74C22×D197—24no—
75C22×D197—22no—
76C22×D197—44no—
77C22×D197—82no—
78C22×D197C2×D414yes—
79Dic394—24no—
80D788—24no—
81C22⋊D197—28no—
82C22⋊D197—24no—
83C22⋊D197—24no—
84C22⋊D197—24no—
85C22⋊D197—24no—
86C2×Dic197—24no—
87C2×Dic197—22no—
88C2×Dic197—22no—
89C2×Dic197—24no—
90C2×Dic197—44no—
91C2×Dic197C2×D414yes—
92C4×D197—28no—
93C23×C197—21no—
94C23×C197—21no—
95C23×C197—42no—
96C23×C197—42no—
97C23×C197C2×D412yes—
98C23×C197C2×D412yes—
99C23×C197C2×D411yescharacteristic
100C23×C197C2×D412yes—
101C23×C197C2×D411yescharacteristic
102C23×C197C2411yescharacteristic
103D4×C197—24no—
104D4×C197—24no—
105C2×C4×C197—21no—
106C2×C4×C197—21no—
107C2×C4×C197—22no—
108C2×C4×C197—22no—
109C2×C4×C197—24no—
110C2×C4×C197—42no—
111C2×C4×C197—41no—
112C2×C4×C197—42no—
113C2×C4×C197—41no—
114C2×C4×C197C2×D412yes—
115C22×C197—22no—
116C22×C197—24no—
117C22×C197—22no—
118C22×C197—42no—
119C22×C197—41no—
120C22×C197—41no—
121C22×C197—42no—
122C22×C197—42no—
123C22×C197—81no—
124C22×C197#12011yescharacteristic
125C22×C197#12012yes—
126C22×C197#12011yescharacteristic
127C22×C197C22×D412yes—
128C22×C197C22×D411yescharacteristic
129C2×D197—24no—
130C2×D197—44no—
131C2×D197—44no—
132C2×D197—82no—
133C2×D197—84no—
134C4×C197—22no—
135C4×C197—42no—
136C4×C197—42no—
137Dic197—24no—
138Dic197—44no—
139C2×C197—22no—
140C2×C197—21no—
141C2×C197—22no—
142C2×C197—21no—
143C2×C197—42no—
144C2×C197#45812yescomplemented
145C2×C197C2×#12011yescharacteristic
146D197—44no—
147C197C2×#45811yescharacteristic, complemented, Sylow
148C2×#458—1971nomaximal, Sylow
149C2×H(5,𝔽2)—1971no—
150C2×#123—1972no—
151C2×#123D19711yescharacteristic, complemented
152#458—1978no—
153C2×#99—1971no—
154C2×#99—1972no—
155H(5,𝔽2)—3944no—
156H(5,𝔽2)—3942no—
157H(5,𝔽2)—1974no—
158C2×#41—3942no—
159C2×#41—3941no—
160C22×D4—3942no—
161C22×D4—3941no—
162C22×D4—1972no—
163C22×D4C2×D19711yescharacteristic
164C23×C4C2×D19711yescharacteristic, complemented
165#123—22no—
166#123—3944no—
167#123—1978no—
168#123C2×D19714yescomplemented
169C2×C4⋊C4—21no—
170C2×C4⋊C4—3942no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—3942no—
173C2×C22⋊C4—1972no—
174C2×C22⋊C4C2×D19711yescharacteristic, complemented
175#99—1974no—
176#99—1978no—
177C24—3941no—
178C24—1972no—
179C24C22⋊D19711yescharacteristic
180C24C22×D19711yescharacteristic
181#41—3944no—
182#41—3948no—
183#41—3948no—
184#41—3944no—
185C2×Q8—3941no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—3941no—
189C2×D4—3948no—
190C2×D4—3948no—
191C2×D4—3942no—
192C2×D4—3944no—
193C2×D4—3942no—
194C2×D4—3942no—
195C2×D4—3942no—
196C2×D4—3944no—
197C2×D4—3942no—
198C2×D4—1974no—
199C2×D4—1974no—
200C2×D4C22⋊D19712yes—
201C2×D4C22×D19712yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—3942no—
208C22×C4—3942no—
209C22×C4—3941no—
210C22×C4—1972no—
211C22×C4C22⋊D19711yescharacteristic
212C22×C4C22×D19712yescomplemented
213C4⋊C4—24no—
214C4⋊C4—3948no—
215C22⋊C4—24no—
216C22⋊C4—21no—
217C22⋊C4—3948no—
218C22⋊C4—3942no—
219C22⋊C4—1974no—
220C22⋊C4C22×D19712yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—7884no—
226C23—15762no—
227C23—3944no—
228C23—3942no—
229C23—3942no—
230C23—1974no—
231C23D4×D19712yes—
232C23D4×D19712yes—
233C23C2×C22⋊D19711yescharacteristic
234C23C2×C22⋊D19712yes—
235C23C2×C22⋊D19711yescharacteristic
236C23C23×D19711yescharacteristic
237Q8—3944no—
238D4—24no—
239D4—24no—
240D4—3944no—
241D4—3948no—
242D4—3944no—
243D4—3944no—
244D4—3944no—
245D4—3944no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—7884no—
256C2×C4—3948no—
257C2×C4—3944no—
258C2×C4—3944no—
259C2×C4—3942no—
260C2×C4—3942no—
261C2×C4—1974no—
262C2×C4C2×C22⋊D19712yes—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—7884no—
273C22—7884no—
274C22—15762no—
275C22—15764no—
276C22—3944no—
277C22#2026329411yescharacteristic
278C22#2026329412yes—
279C22#2026329411yescharacteristic
280C22C2×D4×D19712yes—
281C22C22×C22⋊D19711yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—7884no—
286C4—3944no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—7884no—
293C2#2716126812yesdirect product factor
294C2C2×#2026329411yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC23×C197C24
FrattiniC23C23×D197
FittingC2×#123×C197C2
socleC22×C197#120
prime radicalC2×#20263371C2
hypercenterC2×#123D197
hypocenterC197C2×#458
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#20263309C2semidirect
3. #27161268C2direct (N × H)
4. C2×#20263241C2semidirect
5. C2×#20263240C2semidirect
6. C2×#123×C197C2semidirect
16. #20263309C22semidirect
18. C2×C22⋊Dic197C22semidirect
20. #20263241C22semidirect
21. #20263240C22semidirect
23. C23×C4×C197C22semidirect
25. #123×C197C22semidirect
28. C2×C22⋊C4×C197C22semidirect
54. C22⋊Dic197C23semidirect
68. C22×C4×C197C23semidirect
72. C22⋊C4×C197C23semidirect
144. C2×C197#458semidirect
147. C197C2×#458semidirect
151. C2×#123D197semidirect
164. C23×C4C2×D197semidirect
168. #123C2×D197semidirect
174. C2×C22⋊C4C2×D197semidirect
212. C22×C4C22×D197semidirect
220. C22⋊C4C22×D197semidirect
293. C2#27161268direct (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×#202633092
#271612682
C2×#202632412
C2×#202632402
C2×#123×C1972
C2×#458197