Finite Group Databasebeta

← C2×#27161216 #38751898

Subgroups — C2×#27161216 #38751898

Order 25216.

792 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 173 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×Q8×D197C211yesmaximal, characteristic, complemented
2C22×#15691346C211yesmaximal, characteristic, complemented
3C2×#20263297C211yesmaximal, characteristic
4#27161216C218yesmaximal, direct product factor
5C2×#20263251C211yesmaximal, characteristic, complemented
6C2×#20263250C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C22×SD16×C197C211yesmaximal, characteristic, complemented
9C22×Dic394—21no—
10C22×Dic394C2211yescharacteristic
11C2×Q8×D197—24no—
12C2×Q8×D197—22no—
13C2×Q8×D197—24no—
14C2×Q8×D197C2214yescomplemented
15C22×C22⋊D197—21no—
16C22×C4×D197—21no—
17C22×C4×D197C2211yescharacteristic
18C2×#15691346—24no—
19C2×#15691346C2214yescomplemented
20C22×#13096802C2211yescharacteristic
21#20263297—22no—
22#20263297C2214yes—
23C2×C4⋊Dic197C2211yescharacteristic
24C2×C22⋊Dic197—21no—
25#20263251C2214yescomplemented
26#20263250C2214yescomplemented
27#20263238C2214yescomplemented
28C22×Q8×C197C2211yescharacteristic
29C22×D4×C197C2211yescharacteristic
30C2×SD16×C197—24no—
31C2×SD16×C197C2214yescomplemented
32C22×C8×C197C2211yescharacteristic
33C23×D197D411yescharacteristic
34C2×Dic394—28no—
35C2×Dic394—24no—
36C2×Dic394—21no—
37C2×Dic394—22no—
38C2×Dic394D414yescomplemented
39C2×Dic394C2312yes—
40Q8×D197—216no—
41Q8×D197—28no—
42Q8×D197—28no—
43C2×C22⋊D197—24no—
44C2×C22⋊D197—44no—
45C22×Dic197—21no—
46C22×Dic197—21no—
47C22×Dic197D411yescharacteristic
48C2×C4×D197—28no—
49C2×C4×D197—24no—
50C2×C4×D197—22no—
51C2×C4×D197—22no—
52C2×C4×D197D412yes—
53C2×C4×D197C2312yes—
54#15691346—28no—
55C2×#13096802—22no—
56C2×#13096802C2312yes—
57C4⋊Dic197—21no—
58C4⋊Dic197C2312yes—
59C22⋊Dic197—24no—
60C24×C197—21no—
61C2×Q8×C197—22no—
62C2×Q8×C197—21no—
63C2×Q8×C197D414yescomplemented
64C2×Q8×C197C2312yes—
65C2×D4×C197—22no—
66C2×D4×C197—22no—
67C2×D4×C197—21no—
68C2×D4×C197C2312yes—
69C22×C4×C197—21no—
70C22×C4×C197C2311yescharacteristic
71SD16×C197—28no—
72C2×C8×C197—22no—
73C2×C8×C197C2312yes—
74C22×D197—22no—
75C22×D197—22no—
76C22×D197SD1614yescomplemented
77C22×D197C2×D412yes—
78Dic394—216no—
79Dic394—24no—
80Dic394—22no—
81Dic394C2×D414yescomplemented
82C22⋊D197—48no—
83C2×Dic197—24no—
84C2×Dic197—22no—
85C2×Dic197—22no—
86C2×Dic197—22no—
87C2×Dic197—42no—
88C2×Dic197C2×D412yes—
89C4×D197—216no—
90C4×D197—24no—
91C4×D197—22no—
92C4×D197—22no—
93#13096802—22no—
94C23×C197—22no—
95C23×C197—42no—
96C23×C197—42no—
97C23×C197—42no—
98C23×C197C2×D411yescharacteristic
99Q8×C197—24no—
100Q8×C197—22no—
101Q8×C197C2×D414yescomplemented
102D4×C197—22no—
103D4×C197—22no—
104D4×C197—42no—
105C2×C4×C197—24no—
106C2×C4×C197—22no—
107C2×C4×C197C2×D411yescharacteristic
108C2×C4×C197C2×D412yes—
109C2×C4×C197C2×D412yes—
110C2×C4×C197C2411yescharacteristic
111C8×C197—22no—
112C22×C197—42no—
113C22×C197—42no—
114C22×C197—42no—
115C22×C197—84no—
116C22×C197#12012yes—
117C22×C197C2×SD1612yes—
118C22×C197#13712yes—
119C22×C197C22×D411yescharacteristic
120C2×D197—22no—
121C2×D197—22no—
122C2×D197—28no—
123C2×D197C2×SD1614yescomplemented
124C4×C197—24no—
125C4×C197#12012yes—
126C4×C197C22×D411yescharacteristic
127C4×C197C22×D411yescharacteristic
128Dic197—24no—
129Dic197—22no—
130Dic197—42no—
131C2×C197—82no—
132C2×C197#44814yescomplemented
133C2×C197C2×#12011yescharacteristic
134C2×C197C22×SD1611yescharacteristic
135C2×C197C2×#13711yescharacteristic
136D197—24no—
137C197C2×#44811yescharacteristic, complemented, Sylow
138C2×#448—1971nomaximal, Sylow
139C23×Q8—1971no—
140C22×SD16—1971no—
141C22×SD16D19711yescharacteristic, complemented
142C2×#121—1971no—
143#448—1978no—
144C2×#103—1971no—
145C2×#103—1971no—
146C2×C22⋊C8—1971no—
147C22×Q8—3941no—
148C22×Q8—3944no—
149C22×Q8—3944no—
150C22×Q8—3942no—
151C22×Q8—1974no—
152C22×Q8—1971no—
153C22×Q8C2×D19711yescharacteristic
154C22×D4—3941no—
155C22×D4C2×D19711yescharacteristic
156C23×C4—3941no—
157C23×C4—1971no—
158C2×SD16—24no—
159C2×SD16—3944no—
160C2×SD16—1974no—
161C2×SD16C2×D19714yescomplemented
162C22×C8—1971no—
163C22×C8C2×D19711yescharacteristic
164#121—3942no—
165#121—1974no—
166C2×C4⋊C4—1971no—
167C2×C22⋊C4—3941no—
168#103—1974no—
169#103—1974no—
170C22⋊C8—1974no—
171C24—21no—
172C24—1971no—
173C2×Q8—22no—
174C2×Q8—21no—
175C2×Q8—3948no—
176C2×Q8—3944no—
177C2×Q8—39416no—
178C2×Q8—3948no—
179C2×Q8—3948no—
180C2×Q8—3941no—
181C2×Q8—3942no—
182C2×Q8—1974no—
183C2×Q8—1972no—
184C2×Q8C22⋊D19714yescomplemented
185C2×Q8C22×D19712yes—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—7884no—
190C2×D4—3944no—
191C2×D4C22×D19712yes—
192C22×C4—21no—
193C22×C4—3948no—
194C22×C4—3944no—
195C22×C4—3941no—
196C22×C4—3941no—
197C22×C4—3942no—
198C22×C4—3942no—
199C22×C4—1971no—
200C22×C4—1972no—
201C22×C4—1972no—
202C22×C4C22×D19711yescharacteristic
203SD16—28no—
204SD16—3948no—
205C2×C8—22no—
206C2×C8—3942no—
207C2×C8—1972no—
208C2×C8C22×D19712yes—
209C4⋊C4—3941no—
210C4⋊C4—1972no—
211C22⋊C4—3944no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—3942no—
217C23—3942no—
218C23—1974no—
219C23—1972no—
220C23D4×D19711yescharacteristic
221Q8—24no—
222Q8—22no—
223Q8—39416no—
224Q8—3944no—
225Q8—3942no—
226Q8—1974no—
227Q8C2×C22⋊D19714yescomplemented
228D4—22no—
229D4—22no—
230D4—42no—
231D4—7888no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—7882no—
235C2×C4—39416no—
236C2×C4—3942no—
237C2×C4—3944no—
238C2×C4—3942no—
239C2×C4—3942no—
240C2×C4—3944no—
241C2×C4—3942no—
242C2×C4—3942no—
243C2×C4—1972no—
244C2×C4D4×D19711yescharacteristic
245C2×C4D4×D19712yes—
246C2×C4C2×C22⋊D19712yes—
247C2×C4C23×D19711yescharacteristic
248C8—22no—
249C8—3942no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—84no—
254C22—3942no—
255C22—3948no—
256C22—3942no—
257C22—1974no—
258C22#2026329412yes—
259C22SD16×D19712yes—
260C22#2026335512yes—
261C22C2×D4×D19711yescharacteristic
262C4—24no—
263C4—7882no—
264C4—3944no—
265C4—3942no—
266C4#2026329412yes—
267C4C2×D4×D19711yescharacteristic
268C4C22×C22⋊D19711yescharacteristic
269C2—82no—
270C2—3944no—
271C2#2716121614yesdirect product factor
272C2C2×#2026329411yescharacteristic
273C2C2×SD16×D19711yescharacteristic
274C2C2×#2026335511yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×SD16×C197C2
socleC23×C197C2×D4
prime radicalC2×#20263297C2
hypercenterC22×SD16D197
hypocenterC197C2×#448
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D197C2semidirect
2. C22×#15691346C2semidirect
4. #27161216C2direct (N × H)
5. C2×#20263251C2semidirect
6. C2×#20263250C2semidirect
7. C2×#20263238C2semidirect
8. C22×SD16×C197C2semidirect
14. C2×Q8×D197C22semidirect
19. C2×#15691346C22semidirect
25. #20263251C22semidirect
26. #20263250C22semidirect
27. #20263238C22semidirect
31. C2×SD16×C197C22semidirect
38. C2×Dic394D4semidirect
63. C2×Q8×C197D4semidirect
76. C22×D197SD16semidirect
81. Dic394C2×D4semidirect
101. Q8×C197C2×D4semidirect
123. C2×D197C2×SD16semidirect
132. C2×C197#448semidirect
137. C197C2×#448semidirect
141. C22×SD16D197semidirect
161. C2×SD16C2×D197semidirect
184. C2×Q8C22⋊D197semidirect
227. Q8C2×C22⋊D197semidirect
271. C2#27161216direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1972
C22×#156913462
C2×#202632972
#271612162
C2×#202632512
C2×#202632502
C2×#202632382
C22×SD16×C1972
C2×#448197