Finite Group Databasebeta

← C2×#27161245 #38752034

Subgroups — C2×#27161245 #38752034

Order 25216.

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

normal subgroups only

This group has 844 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, complemented
2C2×#20263352C211yesmaximal, characteristic, complemented
3C2×#20263322C211yesmaximal, characteristic, complemented
4#27161245C218yesmaximal, direct product factor
5C2×#20263253C211yesmaximal, characteristic, complemented
6C2×#20263252C211yesmaximal, characteristic, complemented
7C2×#20263242C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
9#20263372—22no—
10#20263372—22no—
11#20263372—22no—
12#20263372C2214yescomplemented
13C2×#15691359—21no—
14C2×#15691358—21no—
15C2×#15691358C2211yescharacteristic
16C22×D788—21no—
17C22×D788C2211yescharacteristic
18C22×C22⋊D197—22no—
19C2×D4×D197—22no—
20C2×D4×D197—21no—
21#20263352—22no—
22#20263352C2214yescomplemented
23C2×#15691346—21no—
24C2×#15691343—21no—
25C2×#15691342C2211yescharacteristic, complemented
26#20263322—22no—
27#20263322C2214yescomplemented
28C2×C4×Dic197C2211yescharacteristic, complemented
29#20263253C2214yescomplemented
30#20263252C2214yescomplemented
31#20263242C2214yescomplemented
32C2×#41×C197C2211yescharacteristic
33C22×D4×C197C2211yescharacteristic, complemented
34AGL(1,ℤ/8)×C197—22no—
35AGL(1,ℤ/8)×C197C2214yescomplemented
36C2×SD16×C197—21no—
37C2×D8×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#15691358D412yescomplemented
48#15691358C2312yes—
49C2×Dic394D411yescharacteristic, complemented
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, complemented
59C2×D788C2312yes—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—28no—
62C2×C22⋊D197—21no—
63C2×C22⋊D197—48no—
64D4×D197—28no—
65D4×D197—24no—
66D4×D197—28no—
67D4×D197—24no—
68C22×Dic197—21no—
69C2×C4×D197—22no—
70C2×C4×D197—21no—
71#15691346—24no—
72#15691343—24no—
73#15691342—21no—
74#15691342C2312yescomplemented
75C2×#13096802—21no—
76C4×Dic197—21no—
77C4×Dic197C2312yescomplemented
78C24×C197—21no—
79#41×C197—21no—
80#41×C197—21no—
81#41×C197D412yescomplemented
82#41×C197C2312yes—
83C2×Q8×C197D411yescharacteristic, complemented
84C2×D4×C197—21no—
85C2×D4×C197—21no—
86C2×D4×C197—21no—
87C2×D4×C197—21no—
88C2×D4×C197—21no—
89C2×D4×C197—21no—
90C2×D4×C197D411yescharacteristic, complemented
91C2×D4×C197C2312yescomplemented
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94SD16×C197—24no—
95D8×C197—24no—
96M16×C197—21no—
97M16×C197C2312yescomplemented
98C2×C8×C197—21no—
99C22×D197—22no—
100C22×D197—21no—
101C22×D197—28no—
102C22×D197—22no—
103C22×D197—44no—
104C22×D197—84no—
105C22×D197C2×D414yes—
106Dic394—21no—
107Dic394C2×D412yescomplemented
108D788—28no—
109D788—22no—
110D788—22no—
111D788—21no—
112D788—22no—
113D788—22no—
114D788C2×D412yescomplemented
115C22⋊D197—28no—
116C22⋊D197—22no—
117C22⋊D197—22no—
118C22⋊D197—416no—
119C2×Dic197—24no—
120C2×Dic197—21no—
121C2×Dic197—22no—
122C4×D197—28no—
123C4×D197—22no—
124C4×D197—22no—
125#13096802—22no—
126C23×C197—21no—
127C23×C197—22no—
128C23×C197—44no—
129C23×C197—41no—
130C23×C197—41no—
131C23×C197C2×D411yescharacteristic
132Q8×C197—21no—
133Q8×C197C2×D412yescomplemented
134D4×C197—22no—
135D4×C197—22no—
136D4×C197—21no—
137D4×C197—22no—
138D4×C197—22no—
139D4×C197—41no—
140D4×C197—41no—
141D4×C197C2×D412yescomplemented
142C2×C4×C197—21no—
143C2×C4×C197—22no—
144C2×C4×C197—22no—
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2411yescharacteristic
151C8×C197—22no—
152C22×C197—22no—
153C22×C197—22no—
154C22×C197—44no—
155C22×C197—42no—
156C22×C197—42no—
157C22×C197—82no—
158C22×C197—82no—
159C22×C197#12011yescharacteristic
160C22×C197#12011yescharacteristic
161C22×C197C22×D411yescharacteristic
162C2×D197—24no—
163C2×D197—22no—
164C2×D197—44no—
165C2×D197—42no—
166C2×D197—44no—
167C2×D197—82no—
168C2×D197—88no—
169C4×C197—22no—
170C4×C197#12011yescharacteristic
171C4×C197#12011yescharacteristic
172C4×C197C22×D411yescharacteristic
173C4×C197C22×D411yescharacteristic
174Dic197—24no—
175Dic197—22no—
176C2×C197—21no—
177C2×C197—21no—
178C2×C197—42no—
179C2×C197—82no—
180C2×C197#45312yescomplemented
181C2×C197C2×#12011yescharacteristic
182D197—44no—
183D197—42no—
184C197C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—1971nomaximal, Sylow
186C2×H(5,𝔽2)—1971no—
187C2×AGL(1,ℤ/8)—1971no—
188C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
189C2×#127—1971no—
190#453—1978no—
191C2×#104—1971no—
192C2×#104—1971no—
193C2×#100—1971no—
194H(5,𝔽2)—3942no—
195H(5,𝔽2)—3942no—
196H(5,𝔽2)—3942no—
197H(5,𝔽2)—1974no—
198C2×#41—3941no—
199C2×#41—3941no—
200C2×#41—1971no—
201C2×#41C2×D19711yescharacteristic
202C22×D4—3941no—
203C22×D4—3942no—
204C22×D4—3942no—
205C22×D4—3941no—
206C22×D4—1971no—
207C22×D4C2×D19711yescharacteristic, complemented
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—3942no—
210AGL(1,ℤ/8)—1974no—
211AGL(1,ℤ/8)C2×D19714yescomplemented
212C2×SD16—21no—
213C2×SD16—3941no—
214C2×D8—21no—
215C2×D8—3941no—
216C2×M16—1971no—
217C2×M16C2×D19711yescharacteristic, complemented
218#127—3942no—
219#127—1974no—
220C2×C42—1971no—
221#104—1974no—
222#104—1974no—
223#100—1974no—
224C24—21no—
225C24—3942no—
226C24—1972no—
227#41—21no—
228#41—21no—
229#41—3944no—
230#41—3944no—
231#41—3944no—
232#41—3944no—
233#41—3941no—
234#41—3941no—
235#41—1972no—
236#41—1972no—
237#41C22⋊D19712yescomplemented
238#41C22×D19712yes—
239C2×Q8—1971no—
240C2×Q8C22⋊D19711yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—7888no—
248C2×D4—3942no—
249C2×D4—3944no—
250C2×D4—3944no—
251C2×D4—3948no—
252C2×D4—3944no—
253C2×D4—3948no—
254C2×D4—3948no—
255C2×D4—3942no—
256C2×D4—3941no—
257C2×D4—3944no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1972no—
263C2×D4—1971no—
264C2×D4—1972no—
265C2×D4C22⋊D19711yescharacteristic, complemented
266C2×D4C22×D19712yescomplemented
267C22×C4—21no—
268C22×C4—3942no—
269C22×C4—3941no—
270C22×C4—3941no—
271C22×C4C22×D19711yescharacteristic
272SD16—24no—
273SD16—3944no—
274D8—24no—
275D8—3944no—
276M16—21no—
277M16—3941no—
278M16—1972no—
279M16C22×D19712yescomplemented
280C2×C8—21no—
281C2×C8—3941no—
282C42—3941no—
283C42—1972no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—7884no—
290C23—15764no—
291C23—3948no—
292C23—3942no—
293C23—3941no—
294C23—3942no—
295C23—1974no—
296C23D4×D19711yescharacteristic
297Q8—21no—
298Q8—3941no—
299Q8—1972no—
300Q8C2×C22⋊D19712yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—78816no—
309D4—3948no—
310D4—3948no—
311D4—3942no—
312D4—3942no—
313D4—3942no—
314D4—3941no—
315D4—3942no—
316D4—3942no—
317D4—3942no—
318D4—1972no—
319D4C2×C22⋊D19712yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—3948no—
324C2×C4—3944no—
325C2×C4—3941no—
326C2×C4—3942no—
327C2×C4—3942no—
328C2×C4—3942no—
329C2×C4D4×D19711yescharacteristic
330C2×C4D4×D19711yescharacteristic
331C2×C4D4×D19711yescharacteristic
332C2×C4C2×C22⋊D19711yescharacteristic
333C2×C4C2×C22⋊D19711yescharacteristic
334C2×C4C23×D19711yescharacteristic
335C8—22no—
336C8—3942no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—7884no—
345C22—7884no—
346C22—7882no—
347C22—15762no—
348C22—15768no—
349C22—3944no—
350C22—3942no—
351C22#2026329411yescharacteristic
352C22#2026329411yescharacteristic
353C22C2×D4×D19711yescharacteristic
354C4—22no—
355C4—3944no—
356C4—3942no—
357C4#2026329411yescharacteristic
358C4#2026329411yescharacteristic
359C4C2×D4×D19711yescharacteristic
360C4C22×C22⋊D19711yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—7884no—
366C2—7882no—
367C2#2716124512yesdirect product factor
368C2C2×#2026329411yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197#120
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197C2×#453
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. C2×#20263352C2semidirect
3. C2×#20263322C2semidirect
4. #27161245C2direct (N × H)
5. C2×#20263253C2semidirect
6. C2×#20263252C2semidirect
7. C2×#20263242C2semidirect
8. C2×AGL(1,ℤ/8)×C197C2semidirect
12. #20263372C22semidirect
22. #20263352C22semidirect
25. C2×#15691342C22semidirect
27. #20263322C22semidirect
28. C2×C4×Dic197C22semidirect
29. #20263253C22semidirect
30. #20263252C22semidirect
31. #20263242C22semidirect
33. C22×D4×C197C22semidirect
35. AGL(1,ℤ/8)×C197C22semidirect
38. C2×M16×C197C22semidirect
47. #15691358D4semidirect
49. C2×Dic394D4semidirect
58. C2×D788D4semidirect
74. #15691342C23semidirect
77. C4×Dic197C23semidirect
81. #41×C197D4semidirect
83. C2×Q8×C197D4semidirect
90. C2×D4×C197D4semidirect
91. C2×D4×C197C23semidirect
97. M16×C197C23semidirect
107. Dic394C2×D4semidirect
114. D788C2×D4semidirect
133. Q8×C197C2×D4semidirect
141. D4×C197C2×D4semidirect
180. C2×C197#453semidirect
184. C197C2×#453semidirect
188. C2×AGL(1,ℤ/8)D197semidirect
207. C22×D4C2×D197semidirect
211. AGL(1,ℤ/8)C2×D197semidirect
217. C2×M16C2×D197semidirect
237. #41C22⋊D197semidirect
240. C2×Q8C22⋊D197semidirect
265. C2×D4C22⋊D197semidirect
266. C2×D4C22×D197semidirect
279. M16C22×D197semidirect
300. Q8C2×C22⋊D197semidirect
319. D4C2×C22⋊D197semidirect
367. C2#27161245direct (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×#202633522
C2×#202633222
#271612452
C2×#202632532
C2×#202632522
C2×#202632422
C2×AGL(1,ℤ/8)×C1972
C2×#453197