Finite Group Databasebeta

← C2×#27161214 #38751889

Subgroups — C2×#27161214 #38751889

Order 25216.

1048 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1048 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×D4×D197C211yesmaximal, characteristic, complemented
2C22×D1576C211yesmaximal, characteristic, complemented
3C22×#15691343C211yesmaximal, characteristic, complemented
4C2×#20263301C211yesmaximal, characteristic, complemented
5#27161214C218yesmaximal, direct product factor
6C2×#20263246C211yesmaximal, characteristic, complemented
7C2×#20263238C211yesmaximal, characteristic, complemented
8C2×#102×C197C211yesmaximal, characteristic, complemented
9C24×D197—21no—
10C22×D788—21no—
11C22×D788C2211yescharacteristic
12C22×D788C2211yescharacteristic, complemented
13C22×C22⋊D197—21no—
14C2×D4×D197—24no—
15C2×D4×D197—22no—
16C2×D4×D197—24no—
17C2×D4×D197C2214yescomplemented
18C22×C4×D197C2211yescharacteristic
19C2×D1576—24no—
20C2×D1576C2214yescomplemented
21C2×#15691343—24no—
22C2×#15691343C2214yescomplemented
23C22×#13096802C2211yescharacteristic, complemented
24#20263301—22no—
25#20263301C2214yescomplemented
26C2×#15691335—21no—
27#20263246C2214yescomplemented
28#20263238C2214yescomplemented
29C22×D4×C197C2211yescharacteristic
30C22×C8×C197C2211yescharacteristic, complemented
31C2×C4⋊C4×C197C2211yescharacteristic, complemented
32#102×C197C2214yescomplemented
33C23×D197—28no—
34C23×D197—24no—
35C23×D197—21no—
36C23×D197—21no—
37C23×D197—44no—
38C23×D197—44no—
39C23×D197D411yescharacteristic
40C2×D788—24no—
41C2×D788—21no—
42C2×D788—22no—
43C2×D788—22no—
44C2×D788—22no—
45C2×D788—21no—
46C2×D788—44no—
47C2×D788D414yescomplemented
48C2×D788C2312yes—
49C2×D788C2312yescomplemented
50C2×C22⋊D197—28no—
51C2×C22⋊D197—24no—
52D4×D197—216no—
53D4×D197—28no—
54D4×D197—28no—
55C22×Dic197D411yescharacteristic
56C2×C4×D197—22no—
57C2×C4×D197—22no—
58C2×C4×D197D412yes—
59C2×C4×D197C2312yes—
60D1576—28no—
61#15691343—28no—
62C2×#13096802—22no—
63C2×#13096802C2312yescomplemented
64#15691335—24no—
65C24×C197—21no—
66C2×D4×C197—21no—
67C2×D4×C197—22no—
68C2×D4×C197D414yescomplemented
69C2×D4×C197C2312yes—
70C22×C4×C197—21no—
71C22×C4×C197C2311yescharacteristic
72C2×C8×C197—22no—
73C2×C8×C197C2312yescomplemented
74C4⋊C4×C197—21no—
75C4⋊C4×C197C2312yescomplemented
76C22×D197—22no—
77C22×D197—22no—
78C22×D197—216no—
79C22×D197—24no—
80C22×D197—22no—
81C22×D197—22no—
82C22×D197—48no—
83C22×D197—432no—
84C22×D197—42no—
85C22×D197—42no—
86C22×D197—48no—
87C22×D197—42no—
88C22×D197—42no—
89C22×D197—42no—
90C22×D197D814yescomplemented
91C22×D197C2×D412yes—
92D788—24no—
93D788—22no—
94D788—22no—
95D788—22no—
96D788—48no—
97D788—42no—
98D788C2×D414yescomplemented
99C22⋊D197—216no—
100C2×Dic197—22no—
101C2×Dic197C2×D412yes—
102C4×D197—24no—
103C4×D197—22no—
104C4×D197—22no—
105#13096802—22no—
106C23×C197—24no—
107C23×C197—22no—
108C23×C197—42no—
109C23×C197—42no—
110C23×C197C2×D411yescharacteristic
111D4×C197—24no—
112D4×C197—22no—
113D4×C197C2×D414yescomplemented
114C2×C4×C197—22no—
115C2×C4×C197—42no—
116C2×C4×C197C2×D411yescharacteristic
117C2×C4×C197C2×D412yes—
118C2×C4×C197C2×D412yes—
119C2×C4×C197C2411yescharacteristic
120C8×C197—22no—
121C22×C197—24no—
122C22×C197—42no—
123C22×C197—42no—
124C22×C197—48no—
125C22×C197#12012yes—
126C22×C197C2×D812yes—
127C22×C197AGL(1,ℤ/8)12yes—
128C22×C197C22×D411yescharacteristic
129C2×D197—22no—
130C2×D197—22no—
131C2×D197—28no—
132C2×D197—24no—
133C2×D197—432no—
134C2×D197—42no—
135C2×D197—42no—
136C2×D197—48no—
137C2×D197—42no—
138C2×D197—42no—
139C2×D197—42no—
140C2×D197—84no—
141C2×D197C2×D814yescomplemented
142C4×C197—42no—
143C4×C197#12012yes—
144C4×C197C22×D411yescharacteristic
145C4×C197C22×D411yescharacteristic
146Dic197—22no—
147C2×C197—44no—
148C2×C197#44714yescomplemented
149C2×C197C2×#12011yescharacteristic
150C2×C197C22×D811yescharacteristic
151C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
152D197—24no—
153D197—44no—
154D197—82no—
155C197C2×#44711yescharacteristic, complemented, Sylow
156C2×#447—1971nomaximal, Sylow
157C23×D4—1971no—
158C22×D8—1971no—
159C22×D8—1971no—
160C2×#121—1971no—
161#447—1978no—
162C2×#102—1971no—
163C2×#102D19711yescharacteristic, complemented
164C2×C22⋊C8—1971no—
165C25—3941no—
166C22×D4—3941no—
167C22×D4—3944no—
168C22×D4—3944no—
169C22×D4—3942no—
170C22×D4—3941no—
171C22×D4—1974no—
172C22×D4—1971no—
173C22×D4—1971no—
174C22×D4C2×D19711yescharacteristic
175C23×C4—1971no—
176C2×D8—3944no—
177C2×D8—3944no—
178C2×D8—1974no—
179C2×D8—1974no—
180C22×C8—1971no—
181C22×C8C2×D19711yescharacteristic, complemented
182#121—3942no—
183#121—1974no—
184C2×C4⋊C4C2×D19711yescharacteristic, complemented
185C2×C22⋊C4—3941no—
186#102—1974no—
187#102C2×D19714yescomplemented
188C22⋊C8—1974no—
189C24—21no—
190C24—7884no—
191C24—7884no—
192C24—3948no—
193C24—3944no—
194C24—3941no—
195C24—3941no—
196C24—1971no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—7884no—
200C2×D4—3944no—
201C2×D4—39416no—
202C2×D4—3948no—
203C2×D4—3948no—
204C2×D4—3948no—
205C2×D4—3944no—
206C2×D4—3941no—
207C2×D4—3942no—
208C2×D4—3941no—
209C2×D4—3942no—
210C2×D4—3942no—
211C2×D4—1974no—
212C2×D4—1972no—
213C2×D4—1972no—
214C2×D4D78814yescomplemented
215C2×D4C22×D19712yes—
216C22×C4—21no—
217C22×C4—3942no—
218C22×C4—3942no—
219C22×C4—1971no—
220C22×C4—1972no—
221C22×C4—1972no—
222C22×C4C22×D19711yescharacteristic
223D8—3948no—
224D8—3948no—
225C2×C8—22no—
226C2×C8—3942no—
227C2×C8—1972no—
228C2×C8C22×D19712yescomplemented
229C4⋊C4—21no—
230C4⋊C4C22×D19712yescomplemented
231C22⋊C4—3944no—
232C23—24no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—7888no—
237C23—78832no—
238C23—7888no—
239C23—7882no—
240C23—7882no—
241C23—7882no—
242C23—7882no—
243C23—7882no—
244C23—39416no—
245C23—3942no—
246C23—3944no—
247C23—3942no—
248C23—3942no—
249C23—3942no—
250C23—1974no—
251C23—1972no—
252C23D4×D19711yescharacteristic
253D4—24no—
254D4—22no—
255D4—7888no—
256D4—7882no—
257D4—39416no—
258D4—3944no—
259D4—3942no—
260D4—3942no—
261D4—3942no—
262D4—1974no—
263D4C2×D78814yescomplemented
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—3942no—
267C2×C4—3944no—
268C2×C4—3942no—
269C2×C4—3942no—
270C2×C4—1972no—
271C2×C4D4×D19711yescharacteristic
272C2×C4D4×D19712yes—
273C2×C4C2×D78812yes—
274C2×C4C23×D19711yescharacteristic
275C8—22no—
276C8—3942no—
277C22—24no—
278C22—42no—
279C22—48no—
280C22—42no—
281C22—78832no—
282C22—7882no—
283C22—7888no—
284C22—7882no—
285C22—7882no—
286C22—7882no—
287C22—7882no—
288C22—15764no—
289C22—3944no—
290C22—3942no—
291C22—3948no—
292C22—3942no—
293C22—1974no—
294C22#2026329612yes—
295C22D8×D19712yes—
296C22#2026335312yes—
297C22C2×D4×D19711yescharacteristic
298C4—42no—
299C4—3942no—
300C4#2026329612yes—
301C4C2×D4×D19711yescharacteristic
302C4C22×D78811yescharacteristic
303C2—44no—
304C2—7884no—
305C2—15762no—
306C2—3944no—
307C2#2716121414yesdirect product factor
308C2C2×#2026329611yescharacteristic
309C2C2×D8×D19711yescharacteristic
310C2C2×#2026335311yescharacteristic
subgroupG/H
centerC23D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#102×C197C2
socleC23×C197C2×D4
hypercenterC2×#102D197
hypocenterC197C2×#447
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×D4×D197C2semidirect
2. C22×D1576C2semidirect
3. C22×#15691343C2semidirect
4. C2×#20263301C2semidirect
5. #27161214C2direct (N × H)
6. C2×#20263246C2semidirect
7. C2×#20263238C2semidirect
8. C2×#102×C197C2semidirect
12. C22×D788C22semidirect
17. C2×D4×D197C22semidirect
20. C2×D1576C22semidirect
22. C2×#15691343C22semidirect
23. C22×#13096802C22semidirect
25. #20263301C22semidirect
27. #20263246C22semidirect
28. #20263238C22semidirect
30. C22×C8×C197C22semidirect
31. C2×C4⋊C4×C197C22semidirect
32. #102×C197C22semidirect
47. C2×D788D4semidirect
49. C2×D788C23semidirect
63. C2×#13096802C23semidirect
68. C2×D4×C197D4semidirect
73. C2×C8×C197C23semidirect
75. C4⋊C4×C197C23semidirect
90. C22×D197D8semidirect
98. D788C2×D4semidirect
113. D4×C197C2×D4semidirect
141. C2×D197C2×D8semidirect
148. C2×C197#447semidirect
155. C197C2×#447semidirect
163. C2×#102D197semidirect
181. C22×C8C2×D197semidirect
184. C2×C4⋊C4C2×D197semidirect
187. #102C2×D197semidirect
214. C2×D4D788semidirect
228. C2×C8C22×D197semidirect
230. C4⋊C4C22×D197semidirect
263. D4C2×D788semidirect
307. C2#27161214direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1972
C22×D15762
C22×#156913432
C2×#202633012
#271612142
C2×#202632462
C2×#202632382
C2×#102×C1972
C2×#447197