Finite Group Databasebeta

← C2×#26011635 #36362763

Subgroups — C2×#26011635 #36362763

Order 22144.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14583827C211yesmaximal, characteristic, complemented
2C22×#14583811C211yesmaximal, characteristic, complemented
3C2×#19149650C211yesmaximal, characteristic, complemented
4#26011635C218yesmaximal, direct product factor
5C2×#19149603C211yesmaximal, characteristic, complemented
6C2×#19149601C211yesmaximal, characteristic, complemented
7C2×#19149591C211yesmaximal, characteristic, complemented
8C22×SD16×C173C211yesmaximal, characteristic, complemented
9C2×#14583827—24no—
10C2×#14583827—24no—
11C2×#14583827—22no—
12C2×#14583827C2214yescomplemented
13C22×D692—21no—
14C22×D692C2211yescharacteristic
15C22×C22⋊D173—21no—
16C22×C4×D173—21no—
17C22×C4×D173C2211yescharacteristic
18C2×#14583811—24no—
19C2×#14583811C2214yescomplemented
20C22×#13063436C2211yescharacteristic
21#19149650—22no—
22#19149650C2214yescomplemented
23C2×C4⋊Dic173C2211yescharacteristic
24C2×C22⋊Dic173—21no—
25#19149603C2214yescomplemented
26#19149601C2214yescomplemented
27#19149591C2214yescomplemented
28C22×Q8×C173C2211yescharacteristic
29C22×D4×C173C2211yescharacteristic
30C2×SD16×C173—24no—
31C2×SD16×C173C2214yescomplemented
32C22×C8×C173C2211yescharacteristic
33C23×D173—21no—
34C23×D173D411yescharacteristic
35#14583827—216no—
36#14583827—28no—
37#14583827—28no—
38C2×D692—28no—
39C2×D692—24no—
40C2×D692—21no—
41C2×D692—22no—
42C2×D692D414yescomplemented
43C2×D692C2312yes—
44C2×C22⋊D173—24no—
45C2×C22⋊D173—44no—
46C22×Dic173—21no—
47C22×Dic173D411yescharacteristic
48C2×C4×D173—28no—
49C2×C4×D173—24no—
50C2×C4×D173—22no—
51C2×C4×D173—22no—
52C2×C4×D173D412yes—
53C2×C4×D173C2312yes—
54#14583811—28no—
55C2×#13063436—22no—
56C2×#13063436C2312yes—
57C4⋊Dic173—21no—
58C4⋊Dic173C2312yes—
59C22⋊Dic173—24no—
60C24×C173—21no—
61C2×Q8×C173—22no—
62C2×Q8×C173—21no—
63C2×Q8×C173D414yescomplemented
64C2×Q8×C173C2312yes—
65C2×D4×C173—22no—
66C2×D4×C173—22no—
67C2×D4×C173—21no—
68C2×D4×C173C2312yes—
69C22×C4×C173—21no—
70C22×C4×C173C2311yescharacteristic
71SD16×C173—28no—
72C2×C8×C173—22no—
73C2×C8×C173C2312yes—
74C22×D173—22no—
75C22×D173—24no—
76C22×D173—22no—
77C22×D173—22no—
78C22×D173—22no—
79C22×D173—42no—
80C22×D173—42no—
81C22×D173C2×D412yes—
82D692—216no—
83D692—24no—
84D692—22no—
85D692C2×D414yescomplemented
86C22⋊D173—48no—
87C2×Dic173—22no—
88C2×Dic173—22no—
89C2×Dic173—42no—
90C2×Dic173C2×D412yes—
91C4×D173—216no—
92C4×D173—24no—
93C4×D173—22no—
94C4×D173—22no—
95#13063436—22no—
96C23×C173—22no—
97C23×C173—42no—
98C23×C173—42no—
99C23×C173—42no—
100C23×C173C2×D411yescharacteristic
101Q8×C173—24no—
102Q8×C173—22no—
103Q8×C173C2×D414yescomplemented
104D4×C173—22no—
105D4×C173—22no—
106D4×C173—42no—
107C2×C4×C173—24no—
108C2×C4×C173—22no—
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2×D412yes—
111C2×C4×C173C2×D412yes—
112C2×C4×C173C2411yescharacteristic
113C8×C173—22no—
114C22×C173—42no—
115C22×C173—42no—
116C22×C173—42no—
117C22×C173—84no—
118C22×C173#12012yes—
119C22×C173#13512yes—
120C22×C173AGL(1,ℤ/8)12yes—
121C22×C173C22×D411yescharacteristic
122C2×D173—22no—
123C2×D173—24no—
124C2×D173—22no—
125C2×D173—22no—
126C2×D173—42no—
127C2×D173—42no—
128C2×D173—44no—
129C2×D173—48no—
130C4×C173—24no—
131C4×C173#12012yes—
132C4×C173C22×D411yescharacteristic
133C4×C173C22×D411yescharacteristic
134Dic173—22no—
135Dic173—42no—
136C2×C173—82no—
137C2×C173#44914yescomplemented
138C2×C173C2×#12011yescharacteristic
139C2×C173C2×#13511yescharacteristic
140C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
141D173—42no—
142D173—44no—
143C173C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1731nomaximal, Sylow
145C22×#41—1731no—
146C22×SD16D17311yescharacteristic, complemented
147C22×D8—1731no—
148C2×#121—1731no—
149#449—1738no—
150C2×#103—1731no—
151C2×#102—1731no—
152C2×C22⋊C8—1731no—
153C2×#41—3464no—
154C2×#41—3464no—
155C2×#41—3462no—
156C2×#41—1734no—
157C22×Q8C2×D17311yescharacteristic
158C22×D4—3461no—
159C22×D4—3461no—
160C22×D4—1731no—
161C22×D4C2×D17311yescharacteristic
162C23×C4—3461no—
163C23×C4—1731no—
164C2×SD16—24no—
165C2×SD16C2×D17314yescomplemented
166C2×D8—3464no—
167C2×D8—1734no—
168C22×C8—1731no—
169C22×C8C2×D17311yescharacteristic
170#121—3462no—
171#121—1734no—
172C2×C4⋊C4—1731no—
173C2×C22⋊C4—3461no—
174#103—1734no—
175#102—1734no—
176C22⋊C8—1734no—
177C24—21no—
178C24—3461no—
179C24—1731no—
180#41—34616no—
181#41—3468no—
182#41—3468no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8C22⋊D17314yescomplemented
186C2×Q8C22×D17312yes—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4—3468no—
191C2×D4—3464no—
192C2×D4—3464no—
193C2×D4—3461no—
194C2×D4—3462no—
195C2×D4—1734no—
196C2×D4—1732no—
197C2×D4—6924no—
198C2×D4C22×D17312yes—
199C22×C4—21no—
200C22×C4—3468no—
201C22×C4—3464no—
202C22×C4—3462no—
203C22×C4—3462no—
204C22×C4—3461no—
205C22×C4—1731no—
206C22×C4—1732no—
207C22×C4—1732no—
208C22×C4C22×D17311yescharacteristic
209SD16—28no—
210D8—3468no—
211C2×C8—22no—
212C2×C8—3462no—
213C2×C8—1732no—
214C2×C8C22×D17312yes—
215C4⋊C4—3461no—
216C4⋊C4—1732no—
217C22⋊C4—3464no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—3462no—
223C23—3464no—
224C23—3462no—
225C23—3462no—
226C23—3462no—
227C23—1732no—
228C23—6922no—
229C23—6922no—
230C23D4×D17311yescharacteristic
231Q8—24no—
232Q8—22no—
233Q8C2×C22⋊D17314yescomplemented
234D4—22no—
235D4—22no—
236D4—42no—
237D4—34616no—
238D4—3464no—
239D4—3462no—
240D4—1734no—
241D4—6928no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—34616no—
245C2×C4—3462no—
246C2×C4—3464no—
247C2×C4—3462no—
248C2×C4—3462no—
249C2×C4—3462no—
250C2×C4—1732no—
251C2×C4—6922no—
252C2×C4D4×D17311yescharacteristic
253C2×C4D4×D17312yes—
254C2×C4C2×C22⋊D17312yes—
255C2×C4C23×D17311yescharacteristic
256C8—22no—
257C8—3462no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—84no—
262C22—3462no—
263C22—3464no—
264C22—3462no—
265C22—3462no—
266C22—6922no—
267C22—6922no—
268C22—6924no—
269C22—6928no—
270C22#1914964712yes—
271C22#1914970012yes—
272C22#1914970612yes—
273C22C2×D4×D17311yescharacteristic
274C4—24no—
275C4—3462no—
276C4—6922no—
277C4#1914964712yes—
278C4C2×D4×D17311yescharacteristic
279C4C22×C22⋊D17311yescharacteristic
280C2—82no—
281C2—6922no—
282C2—6924no—
283C2#2601163514yesdirect product factor
284C2C2×#1914964711yescharacteristic
285C2C2×#1914970011yescharacteristic
286C2C2×#1914970611yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×SD16×C173C2
socleC23×C173C2×D4
hypercenterC22×SD16D173
hypocenterC173C2×#449
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C22×#14583811C2semidirect
3. C2×#19149650C2semidirect
4. #26011635C2direct (N × H)
5. C2×#19149603C2semidirect
6. C2×#19149601C2semidirect
7. C2×#19149591C2semidirect
8. C22×SD16×C173C2semidirect
12. C2×#14583827C22semidirect
19. C2×#14583811C22semidirect
22. #19149650C22semidirect
25. #19149603C22semidirect
26. #19149601C22semidirect
27. #19149591C22semidirect
31. C2×SD16×C173C22semidirect
42. C2×D692D4semidirect
63. C2×Q8×C173D4semidirect
85. D692C2×D4semidirect
103. Q8×C173C2×D4semidirect
137. C2×C173#449semidirect
143. C173C2×#449semidirect
146. C22×SD16D173semidirect
165. C2×SD16C2×D173semidirect
185. C2×Q8C22⋊D173semidirect
233. Q8C2×C22⋊D173semidirect
283. C2#26011635direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838272
C22×#145838112
C2×#191496502
#260116352
C2×#191496032
C2×#191496012
C2×#191495912
C22×SD16×C1732
C2×#449173