Finite Group Databasebeta

← C2×#27161255 #38752040

Subgroups — C2×#27161255 #38752040

Order 25216.

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

normal subgroups only

This group has 760 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×#20263372C211yesmaximal, characteristic
2C2×#20263357C211yesmaximal, characteristic, complemented
3C2×#20263315C211yesmaximal, characteristic, complemented
4#27161255C218yesmaximal, direct product factor
5C2×#20263253C211yesmaximal, characteristic, complemented
6C2×#20263252C211yesmaximal, characteristic, complemented
7C2×#20263242C211yesmaximal, characteristic, complemented
8C2×#137×C197C211yesmaximal, characteristic, complemented
9#20263372—22no—
10#20263372—22no—
11#20263372—22no—
12#20263372C2214yes—
13C2×#15691359—21no—
14C2×#15691358—21no—
15C2×#15691358C2211yescharacteristic
16C22×D788—21no—
17C22×D788C2211yescharacteristic
18C2×D4×D197—22no—
19C2×D4×D197—21no—
20#20263357—22no—
21#20263357C2214yescomplemented
22C2×#15691348—21no—
23C2×#15691345—21no—
24C2×#15691342C2211yescharacteristic, complemented
25#20263315—22no—
26#20263315C2214yescomplemented
27C2×#15691335—22no—
28C2×C4×Dic197C2211yescharacteristic, complemented
29#20263253C2214yescomplemented
30#20263252C2214yescomplemented
31#20263242C2214yescomplemented
32C2×#41×C197C2211yescharacteristic
33C22×Q8×C197C2211yescharacteristic, complemented
34#137×C197—22no—
35#137×C197C2214yescomplemented
36C2×Q16×C197—21no—
37C2×SD16×C197—21no—
38C2×M16×C197C2211yescharacteristic, complemented
39C23×D197—22no—
40C23×D197D412yes—
41#15691359—24no—
42#15691359—24no—
43#15691358—24no—
44#15691358—24no—
45#15691358—21no—
46#15691358—21no—
47#15691358D412yes—
48#15691358C2312yes—
49C2×Dic394D411yescharacteristic
50C2×D788—22no—
51C2×D788—24no—
52C2×D788—24no—
53C2×D788—21no—
54C2×D788—21no—
55C2×D788—21no—
56C2×D788—21no—
57C2×D788D412yes—
58C2×D788D411yescharacteristic
59C2×D788C2312yes—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—21no—
62D4×D197—28no—
63D4×D197—24no—
64D4×D197—28no—
65D4×D197—24no—
66C22×Dic197—21no—
67C2×C4×D197—22no—
68C2×C4×D197—21no—
69#15691348—24no—
70#15691345—24no—
71#15691342—21no—
72#15691342C2312yescomplemented
73C2×#13096802—21no—
74#15691335—28no—
75C4×Dic197—21no—
76C4×Dic197C2312yescomplemented
77#41×C197—21no—
78#41×C197—21no—
79#41×C197D412yes—
80#41×C197C2312yes—
81C2×Q8×C197—21no—
82C2×Q8×C197—21no—
83C2×Q8×C197—21no—
84C2×Q8×C197—21no—
85C2×Q8×C197—21no—
86C2×Q8×C197D411yescharacteristic
87C2×Q8×C197C2312yescomplemented
88C2×D4×C197—21no—
89C2×D4×C197D411yescharacteristic
90C22×C4×C197—21no—
91C22×C4×C197—21no—
92C22×C4×C197C2311yescharacteristic
93Q16×C197—24no—
94SD16×C197—24no—
95M16×C197—21no—
96M16×C197C2312yescomplemented
97C2×C8×C197—21no—
98C22×D197—22no—
99C22×D197—21no—
100C22×D197—28no—
101C22×D197—22no—
102C22×D197—44no—
103C22×D197—84no—
104C22×D197C2×D414yes—
105Dic394—21no—
106Dic394C2×D412yes—
107D788—28no—
108D788—22no—
109D788—22no—
110D788—21no—
111D788—22no—
112D788—22no—
113D788C2×D412yes—
114C22⋊D197—28no—
115C22⋊D197—22no—
116C22⋊D197—22no—
117C2×Dic197—21no—
118C2×Dic197—22no—
119C2×Dic197—42no—
120C4×D197—28no—
121C4×D197—22no—
122C4×D197—22no—
123#13096802—22no—
124C23×C197—21no—
125C23×C197C2×D411yescharacteristic
126Q8×C197—21no—
127Q8×C197—22no—
128Q8×C197—22no—
129Q8×C197—41no—
130Q8×C197—41no—
131Q8×C197C2×D412yes—
132D4×C197—22no—
133D4×C197—22no—
134D4×C197—21no—
135D4×C197C2×D412yes—
136C2×C4×C197—21no—
137C2×C4×C197—22no—
138C2×C4×C197—22no—
139C2×C4×C197—22no—
140C2×C4×C197—41no—
141C2×C4×C197—41no—
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C8×C197—22no—
149C22×C197—22no—
150C22×C197—22no—
151C22×C197—42no—
152C22×C197#12011yescharacteristic
153C22×C197#12011yescharacteristic
154C22×C197C22×D411yescharacteristic
155C2×D197—24no—
156C2×D197—22no—
157C2×D197—44no—
158C2×D197—42no—
159C2×D197—44no—
160C2×D197—82no—
161C2×D197—88no—
162C4×C197—22no—
163C4×C197—42no—
164C4×C197#12011yescharacteristic
165C4×C197#12011yescharacteristic
166C4×C197C22×D411yescharacteristic
167C4×C197C22×D411yescharacteristic
168Dic197—22no—
169Dic197—42no—
170C2×C197—21no—
171C2×C197—21no—
172C2×C197—42no—
173C2×C197#45512yescomplemented
174C2×C197C2×#12011yescharacteristic
175D197—44no—
176D197—42no—
177C197C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1971nomaximal, Sylow
179C2×H(5,𝔽2)—1971no—
180C2×#137—1971no—
181C2×#137D19711yescharacteristic, complemented
182C2×#124—1971no—
183#455—1978no—
184C2×#104—1971no—
185C2×#104—1971no—
186C2×#100—1971no—
187H(5,𝔽2)—3942no—
188H(5,𝔽2)—3942no—
189H(5,𝔽2)—3942no—
190H(5,𝔽2)—1974no—
191C2×#41—3941no—
192C2×#41—3941no—
193C2×#41—1971no—
194C2×#41C2×D19711yescharacteristic
195C22×Q8C2×D19711yescharacteristic, complemented
196C22×D4—3941no—
197C22×D4—3942no—
198C22×D4—3941no—
199C22×D4—1971no—
200#137—22no—
201#137—3942no—
202#137—1974no—
203#137C2×D19714yescomplemented
204C2×Q16—21no—
205C2×Q16—3941no—
206C2×SD16—21no—
207C2×SD16—3941no—
208C2×M16—1971no—
209C2×M16C2×D19711yescharacteristic, complemented
210#124—3942no—
211#124—1974no—
212C2×C22⋊C4—3942no—
213C2×C42—1971no—
214#104—1974no—
215#104—1974no—
216#100—1974no—
217C24—3942no—
218C24—1972no—
219#41—21no—
220#41—21no—
221#41—3944no—
222#41—3944no—
223#41—3944no—
224#41—3944no—
225#41—3941no—
226#41—3941no—
227#41—1972no—
228#41—1972no—
229#41C22⋊D19712yes—
230#41C22×D19712yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—1971no—
237C2×Q8C22⋊D19711yescharacteristic
238C2×Q8C22×D19712yescomplemented
239C2×D4—21no—
240C2×D4—3942no—
241C2×D4—3944no—
242C2×D4—3944no—
243C2×D4—3948no—
244C2×D4—3944no—
245C2×D4—3948no—
246C2×D4—3942no—
247C2×D4—3941no—
248C2×D4—3944no—
249C2×D4—3941no—
250C2×D4—3941no—
251C2×D4—3941no—
252C2×D4—3941no—
253C2×D4—1972no—
254C2×D4—1971no—
255C2×D4—1972no—
256C2×D4C22⋊D19711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3942no—
260C22×C4—3941no—
261C22×C4—3941no—
262C22×C4C22×D19711yescharacteristic
263Q16—24no—
264Q16—3944no—
265SD16—24no—
266SD16—3944no—
267M16—21no—
268M16—3941no—
269M16—1972no—
270M16C22×D19712yescomplemented
271C2×C8—21no—
272C2×C8—3941no—
273C22⋊C4—3948no—
274C42—3941no—
275C42—1972no—
276C23—21no—
277C23—7884no—
278C23—15764no—
279C23—3948no—
280C23—3942no—
281C23—3941no—
282C23—3942no—
283C23—1974no—
284C23D4×D19711yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—3941no—
291Q8—1972no—
292Q8C2×C22⋊D19712yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3948no—
297D4—3948no—
298D4—3942no—
299D4—3942no—
300D4—3942no—
301D4—3941no—
302D4—3942no—
303D4—3942no—
304D4—3942no—
305D4—1972no—
306D4C2×C22⋊D19712yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—7882no—
314C2×C4—3948no—
315C2×C4—3942no—
316C2×C4—3941no—
317C2×C4—3942no—
318C2×C4—3942no—
319C2×C4D4×D19711yescharacteristic
320C2×C4D4×D19711yescharacteristic
321C2×C4D4×D19711yescharacteristic
322C2×C4C2×C22⋊D19711yescharacteristic
323C2×C4C2×C22⋊D19711yescharacteristic
324C2×C4C23×D19711yescharacteristic
325C8—22no—
326C8—3942no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—7884no—
331C22—7884no—
332C22—7882no—
333C22—15762no—
334C22—15768no—
335C22—3944no—
336C22—3942no—
337C22#2026329411yescharacteristic
338C22#2026329411yescharacteristic
339C22C2×D4×D19711yescharacteristic
340C4—22no—
341C4—42no—
342C4—7882no—
343C4—3942no—
344C4#2026329411yescharacteristic
345C4#2026329411yescharacteristic
346C4C2×D4×D19711yescharacteristic
347C4C22×C22⋊D19711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—7884no—
352C2—7882no—
353C2#2716125512yesdirect product factor
354C2C2×#2026329411yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#137×C197C2
socleC22×C197#120
prime radicalC2×#20263372C2
hypercenterC2×#137D197
hypocenterC197C2×#455
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#20263357C2semidirect
3. C2×#20263315C2semidirect
4. #27161255C2direct (N × H)
5. C2×#20263253C2semidirect
6. C2×#20263252C2semidirect
7. C2×#20263242C2semidirect
8. C2×#137×C197C2semidirect
21. #20263357C22semidirect
24. C2×#15691342C22semidirect
26. #20263315C22semidirect
28. C2×C4×Dic197C22semidirect
29. #20263253C22semidirect
30. #20263252C22semidirect
31. #20263242C22semidirect
33. C22×Q8×C197C22semidirect
35. #137×C197C22semidirect
38. C2×M16×C197C22semidirect
72. #15691342C23semidirect
76. C4×Dic197C23semidirect
87. C2×Q8×C197C23semidirect
96. M16×C197C23semidirect
173. C2×C197#455semidirect
177. C197C2×#455semidirect
181. C2×#137D197semidirect
195. C22×Q8C2×D197semidirect
203. #137C2×D197semidirect
209. C2×M16C2×D197semidirect
238. C2×Q8C22×D197semidirect
270. M16C22×D197semidirect
353. C2#27161255direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202633722
C2×#202633572
C2×#202633152
#271612552
C2×#202632532
C2×#202632522
C2×#202632422
C2×#137×C1972
C2×#455197