Finite Group Databasebeta

← C2×#27161248 #38752037

Subgroups — C2×#27161248 #38752037

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×#20263354C211yesmaximal, characteristic, complemented
3C2×#20263351C211yesmaximal, characteristic, complemented
4C2×#20263323C211yesmaximal, characteristic, complemented
5#27161248C218yesmaximal, direct product factor
6C2×#20263254C211yesmaximal, characteristic, complemented
7C2×#20263242C211yesmaximal, characteristic, complemented
8C2×#104×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×D788—22no—
18C22×D788C2211yescharacteristic
19C22×D788C2211yescharacteristic, complemented
20C2×D4×D197—22no—
21C2×D4×D197—21no—
22#20263354—22no—
23#20263354C2214yescomplemented
24#20263351—22no—
25#20263351C2214yescomplemented
26C2×#15691347—21no—
27C2×#15691345—21no—
28C2×D1576—21no—
29C2×#15691343—21no—
30C2×#15691342C2211yescharacteristic, complemented
31#20263323—22no—
32#20263323C2214yescomplemented
33#20263254C2214yescomplemented
34#20263242C2214yescomplemented
35C2×#41×C197C2211yescharacteristic
36C2×M16×C197C2211yescharacteristic, complemented
37C2×C42×C197C2211yescharacteristic, complemented
38#104×C197C2214yescomplemented
39C23×D197—22no—
40C23×D197—21no—
41C23×D197D412yes—
42#15691359—24no—
43#15691359—24no—
44#15691358—24no—
45#15691358—24no—
46#15691358—21no—
47#15691358—21no—
48#15691358D412yescomplemented
49#15691358C2312yes—
50C2×Dic394D411yescharacteristic, complemented
51C2×D788—22no—
52C2×D788—24no—
53C2×D788—24no—
54C2×D788—28no—
55C2×D788—21no—
56C2×D788—21no—
57C2×D788—21no—
58C2×D788—21no—
59C2×D788—21no—
60C2×D788—21no—
61C2×D788—21no—
62C2×D788—21no—
63C2×D788—21no—
64C2×D788—48no—
65C2×D788D412yes—
66C2×D788D411yescharacteristic, complemented
67C2×D788C2312yes—
68C2×D788C2312yescomplemented
69C2×C22⋊D197—22no—
70C2×C22⋊D197—21no—
71D4×D197—28no—
72D4×D197—24no—
73D4×D197—28no—
74D4×D197—24no—
75C2×C4×D197—22no—
76C2×C4×D197—21no—
77#15691347—24no—
78#15691345—24no—
79D1576—24no—
80#15691343—24no—
81#15691342—21no—
82#15691342C2312yescomplemented
83C2×#13096802—21no—
84#41×C197—21no—
85#41×C197—21no—
86#41×C197D412yescomplemented
87#41×C197C2312yes—
88C2×Q8×C197D411yescharacteristic, complemented
89C2×D4×C197—21no—
90C2×D4×C197D411yescharacteristic, complemented
91C22×C4×C197—21no—
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94M16×C197—21no—
95M16×C197C2312yescomplemented
96C2×C8×C197—21no—
97C42×C197—21no—
98C42×C197C2312yescomplemented
99C22×D197—22no—
100C22×D197—21no—
101C22×D197—28no—
102C22×D197—22no—
103C22×D197—22no—
104C22×D197—44no—
105C22×D197—44no—
106C22×D197—41no—
107C22×D197—41no—
108C22×D197—84no—
109C22×D197C2×D414yes—
110Dic394—21no—
111Dic394C2×D412yescomplemented
112D788—28no—
113D788—22no—
114D788—22no—
115D788—21no—
116D788—22no—
117D788—22no—
118D788—22no—
119D788—22no—
120D788—416no—
121D788—41no—
122D788—41no—
123D788C2×D412yescomplemented
124C22⋊D197—28no—
125C22⋊D197—22no—
126C22⋊D197—22no—
127C2×Dic197—21no—
128C4×D197—28no—
129C4×D197—22no—
130C4×D197—22no—
131#13096802—22no—
132C23×C197—21no—
133C23×C197C2×D411yescharacteristic
134Q8×C197—21no—
135Q8×C197C2×D412yescomplemented
136D4×C197—22no—
137D4×C197—22no—
138D4×C197—21no—
139D4×C197C2×D412yescomplemented
140C2×C4×C197—24no—
141C2×C4×C197—21no—
142C2×C4×C197—22no—
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—42no—
155C22×C197#12011yescharacteristic
156C22×C197#12011yescharacteristic
157C22×C197C22×D411yescharacteristic
158C2×D197—24no—
159C2×D197—22no—
160C2×D197—44no—
161C2×D197—42no—
162C2×D197—44no—
163C2×D197—44no—
164C2×D197—42no—
165C2×D197—82no—
166C2×D197—88no—
167C2×D197—82no—
168C2×D197—82no—
169C4×C197—24no—
170C4×C197—22no—
171C4×C197#12011yescharacteristic
172C4×C197#12011yescharacteristic
173C4×C197C22×D411yescharacteristic
174C4×C197C22×D411yescharacteristic
175Dic197—22no—
176C2×C197—21no—
177C2×C197—21no—
178C2×C197—42no—
179C2×C197#45312yescomplemented
180C2×C197C2×#12011yescharacteristic
181D197—44no—
182D197—42no—
183D197—82no—
184C197C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—1971nomaximal, Sylow
186C2×H(5,𝔽2)—1971no—
187C2×AGL(1,ℤ/8)—1971no—
188C2×AGL(1,ℤ/8)—1971no—
189C2×#127—1971no—
190#453—1978no—
191C2×#104—1971no—
192C2×#104D19711yescharacteristic, complemented
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—3942no—
203C22×D4—3941no—
204C22×D4—3942no—
205C22×D4—3941no—
206C22×D4—1971no—
207C22×D4—1971no—
208AGL(1,ℤ/8)—3942no—
209AGL(1,ℤ/8)—3942no—
210AGL(1,ℤ/8)—1974no—
211AGL(1,ℤ/8)—1974no—
212C2×SD16—3941no—
213C2×SD16—3941no—
214C2×D8—3941no—
215C2×D8—3941no—
216C2×M16—1971no—
217C2×M16C2×D19711yescharacteristic, complemented
218#127—3942no—
219#127—1974no—
220C2×C42C2×D19711yescharacteristic, complemented
221#104—1974no—
222#104C2×D19714yescomplemented
223#100—1974no—
224C24—3942no—
225C24—3941no—
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#41D78812yescomplemented
238#41C22×D19712yes—
239C2×Q8—1971no—
240C2×Q8D78811yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—7888no—
243C2×D4—3948no—
244C2×D4—3942no—
245C2×D4—3944no—
246C2×D4—3944no—
247C2×D4—3948no—
248C2×D4—3944no—
249C2×D4—3948no—
250C2×D4—3942no—
251C2×D4—3941no—
252C2×D4—3944no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—3941no—
256C2×D4—3941no—
257C2×D4—3941no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—3941no—
261C2×D4—3941no—
262C2×D4—1972no—
263C2×D4—1971no—
264C2×D4—1972no—
265C2×D4—1972no—
266C2×D4D78811yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—3942no—
270C22×C4—3941no—
271C22×C4C22×D19711yescharacteristic
272SD16—3944no—
273SD16—3944no—
274D8—3944no—
275D8—3944no—
276M16—21no—
277M16—3941no—
278M16—1972no—
279M16C22×D19712yescomplemented
280C2×C8—21no—
281C2×C8—3941no—
282C42—21no—
283C42C22×D19712yescomplemented
284C23—21no—
285C23—7884no—
286C23—7884no—
287C23—7881no—
288C23—7881no—
289C23—15764no—
290C23—3948no—
291C23—3942no—
292C23—3942no—
293C23—3941no—
294C23—3942no—
295C23—1974no—
296C23D4×D19711yescharacteristic
297Q8—21no—
298Q8—3941no—
299Q8—1972no—
300Q8C2×D78812yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—78816no—
305D4—7881no—
306D4—7881no—
307D4—3948no—
308D4—3948no—
309D4—3942no—
310D4—3942no—
311D4—3942no—
312D4—3941no—
313D4—3942no—
314D4—3942no—
315D4—3942no—
316D4—3942no—
317D4—3942no—
318D4—1972no—
319D4C2×D78812yescomplemented
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—3948no—
326C2×C4—3941no—
327C2×C4—3942no—
328C2×C4—3942no—
329C2×C4D4×D19711yescharacteristic
330C2×C4D4×D19711yescharacteristic
331C2×C4D4×D19711yescharacteristic
332C2×C4C2×D78811yescharacteristic
333C2×C4C2×D78811yescharacteristic
334C2×C4C23×D19711yescharacteristic
335C8—22no—
336C8—3942no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—7884no—
341C22—7884no—
342C22—7884no—
343C22—7882no—
344C22—7882no—
345C22—15762no—
346C22—15762no—
347C22—15768no—
348C22—15762no—
349C22—3944no—
350C22—3942no—
351C22#2026329611yescharacteristic
352C22#2026329611yescharacteristic
353C22C2×D4×D19711yescharacteristic
354C4—24no—
355C4—22no—
356C4—3942no—
357C4#2026329611yescharacteristic
358C4#2026329611yescharacteristic
359C4C2×D4×D19711yescharacteristic
360C4C22×D78811yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—7884no—
365C2—7882no—
366C2—15762no—
367C2#2716124812yesdirect product factor
368C2C2×#2026329611yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#104×C197C2
socleC22×C197#120
hypercenterC2×#104D197
hypocenterC197C2×#453
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#20263372C2semidirect
2. C2×#20263354C2semidirect
3. C2×#20263351C2semidirect
4. C2×#20263323C2semidirect
5. #27161248C2direct (N × H)
6. C2×#20263254C2semidirect
7. C2×#20263242C2semidirect
8. C2×#104×C197C2semidirect
12. #20263372C22semidirect
19. C22×D788C22semidirect
23. #20263354C22semidirect
25. #20263351C22semidirect
30. C2×#15691342C22semidirect
32. #20263323C22semidirect
33. #20263254C22semidirect
34. #20263242C22semidirect
36. C2×M16×C197C22semidirect
37. C2×C42×C197C22semidirect
38. #104×C197C22semidirect
48. #15691358D4semidirect
50. C2×Dic394D4semidirect
66. C2×D788D4semidirect
68. C2×D788C23semidirect
82. #15691342C23semidirect
86. #41×C197D4semidirect
88. C2×Q8×C197D4semidirect
90. C2×D4×C197D4semidirect
95. M16×C197C23semidirect
98. C42×C197C23semidirect
111. Dic394C2×D4semidirect
123. D788C2×D4semidirect
135. Q8×C197C2×D4semidirect
139. D4×C197C2×D4semidirect
179. C2×C197#453semidirect
184. C197C2×#453semidirect
192. C2×#104D197semidirect
217. C2×M16C2×D197semidirect
220. C2×C42C2×D197semidirect
222. #104C2×D197semidirect
237. #41D788semidirect
240. C2×Q8D788semidirect
266. C2×D4D788semidirect
279. M16C22×D197semidirect
283. C42C22×D197semidirect
300. Q8C2×D788semidirect
319. D4C2×D788semidirect
367. C2#27161248direct (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×#202633542
C2×#202633512
C2×#202633232
#271612482
C2×#202632542
C2×#202632422
C2×#104×C1972
C2×#453197