Finite Group Databasebeta

← #36365163

Subgroups — #36365163

Order 22144.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19149704C212yesmaximal, complemented
2D4×D692C211yesmaximal, characteristic, complemented
3#26011865C211yesmaximal, characteristic, complemented
4#26011739C212yesmaximal, complemented
5#26011693C211yesmaximal, characteristic, complemented
6#26011686C211yesmaximal, characteristic, complemented
7#26011641C212yesmaximal, complemented
8#26011636C212yesmaximal, complemented
9C4×#14583811C211yesmaximal, characteristic, complemented
10#26011569C211yesmaximal, characteristic, complemented
11#547×C173C211yesmaximal, characteristic, complemented
12C22×D692C2212yes—
13C2×D4×D173—21no—
14#19149704—28no—
15C2×#14583813C2212yes—
16C2×#14583813C2212yescomplemented
17C2×#14583813C2212yescomplemented
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2212yescomplemented
20C2×#14583810C2212yescomplemented
21#19149676C2211yescharacteristic
22#19149654—21no—
23#19149653C2212yes—
24#19149649—22no—
25C4×D692C2211yescharacteristic
26#19149612C2211yescharacteristic, complemented
27C4⋊#13063436C2211yescharacteristic, complemented
28#19149603C2212yescomplemented
29#19149600C2211yescharacteristic, complemented
30#19149599C2211yescharacteristic
31#19149599C2212yescomplemented
32C22⋊#13063436C2212yescomplemented
33C4×#13063436C2211yescharacteristic, complemented
34C2×#41×C173C2212yescomplemented
35C4⋊Q8×C173C2211yescharacteristic, complemented
36#123×C173—22no—
37C22⋊Q8×C173C2212yescomplemented
38#121×C173—21no—
39C4×D4×C173C2211yescharacteristic
40C4×D4×C173C2211yescharacteristic, complemented
41C2×C4⋊C4×C173—21no—
42C23×D173—22no—
43C2×D692—24no—
44C2×D692—22no—
45C2×D692—21no—
46C2×D692—21no—
47C2×D692—42no—
48C2×D692C2312yes—
49C2×D692C2311yescharacteristic
50C2×C22⋊D173—22no—
51D4×D173—44no—
52C2×C4×D173—21no—
53#14583813—24no—
54#14583813—24no—
55#14583813—42no—
56#14583811—24no—
57#14583811D414yescomplemented
58#14583810—24no—
59C2×#13063436C2311yescharacteristic
60C2×#13063436C2312yes—
61C2×#13063436C2311yescharacteristic
62C4⋊Dic173C2311yescharacteristic
63#14583803—21no—
64#14583803—22no—
65#41×C173—24no—
66#41×C173—42no—
67C2×Q8×C173C2312yes—
68C2×D4×C173—22no—
69C2×D4×C173D411yescharacteristic
70C2×D4×C173C2311yescharacteristic
71C22×C4×C173—22no—
72C22×C4×C173—21no—
73C22×C4×C173C2312yes—
74C4⋊C4×C173—21no—
75C4⋊C4×C173—21no—
76C4⋊C4×C173—22no—
77C4⋊C4×C173D411yescharacteristic
78C4⋊C4×C173C2311yescharacteristic
79C22⋊C4×C173—22no—
80C22⋊C4×C173—21no—
81C22⋊C4×C173D412yes—
82C42×C173C2311yescharacteristic
83C22×D173—21no—
84C22×D173—22no—
85C22×D173—44no—
86C22×D173—44no—
87C22×D173—82no—
88D692—22no—
89D692—21no—
90D692—22no—
91D692—44no—
92D692—42no—
93D692—81no—
94D692C2×D412yes—
95C22⋊D173—44no—
96C2×Dic173—21no—
97C4×D173—42no—
98#13063436—22no—
99#13063436—21no—
100#13063436C2×D412yescomplemented
101C23×C173—21no—
102C23×C173C2×D412yes—
103Q8×C173—22no—
104Q8×C173—22no—
105D4×C173—22no—
106D4×C173—21no—
107D4×C173—44no—
108D4×C173C2×D412yescomplemented
109C2×C4×C173—22no—
110C2×C4×C173—22no—
111C2×C4×C173—21no—
112C2×C4×C173—22no—
113C2×C4×C173—44no—
114C2×C4×C173—42no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2411yescharacteristic
119C22×C173—22no—
120C22×C173—22no—
121C22×C173—22no—
122C22×C173—22no—
123C22×C173—41no—
124C22×C173—41no—
125C22×C173C22×D411yescharacteristic
126C2×D173—22no—
127C2×D173—41no—
128C2×D173—41no—
129C2×D173—42no—
130C2×D173—42no—
131C2×D173—42no—
132C2×D173—88no—
133C4×C173—22no—
134C4×C173—21no—
135C4×C173—42no—
136C4×C173—41no—
137C4×C173AGL(1,ℤ/8)12yescomplemented
138C4×C173C22×D411yescharacteristic
139C4×C173H(5,𝔽2)11yescharacteristic
140Dic173—41no—
141C2×C173—42no—
142C2×C173—42no—
143C2×C173D4211yescharacteristic
144C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C173#57711yescharacteristic
146D173—42no—
147D173—82no—
148C173#327211yescharacteristic, complemented, Sylow
149#3272—1731nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1732no—
151#547D17311yescharacteristic, complemented
152D42—1731no—
153#492—1731no—
154#468—1732no—
155#461—1731no—
156#459—1731no—
157#450—1732no—
158#449—1732no—
159C4×D8—1731no—
160#436—1731no—
161C2×#41C2×D17312yescomplemented
162C22×D4—3461no—
163C22×D4—1732no—
164AGL(1,ℤ/8)—3468no—
165C2×SD16—1732no—
166C2×SD16—1732no—
167C2×SD16—1732no—
168C2×D8—1732no—
169C2×D8—1731no—
170C2×M16—1732no—
171C4⋊Q8C2×D17311yescharacteristic, complemented
172#127—1731no—
173#123—22no—
174C22⋊Q8C2×D17312yescomplemented
175#121—21no—
176#121—3461no—
177#121—1732no—
178#120—3462no—
179C4×D4—1731no—
180C4×D4C2×D17311yescharacteristic
181C4×D4C2×D17311yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1731no—
184C4⋊C8—1731no—
185#103—1732no—
186#102—1731no—
187#102—1731no—
188#102—1732no—
189C22⋊C8—1732no—
190C4×C8—1731no—
191C24—3462no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D17312yes—
195C2×D4—22no—
196C2×D4—3464no—
197C2×D4—3462no—
198C2×D4—3461no—
199C2×D4—3462no—
200C2×D4—3461no—
201C2×D4—1732no—
202C2×D4—1731no—
203C2×D4—6922no—
204C2×D4—6924no—
205C2×D4C22⋊D17311yescharacteristic
206C2×D4C22×D17311yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—3461no—
210C22×C4C22×D17312yes—
211SD16—3464no—
212SD16—3464no—
213SD16—6922no—
214D8—3464no—
215D8—1734no—
216M16—3464no—
217C2×C8—1731no—
218C2×C8—1732no—
219C2×C8—1731no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1731no—
224C4⋊C4C22⋊D17311yescharacteristic
225C4⋊C4C22×D17311yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—3461no—
229C22⋊C4—3462no—
230C22⋊C4C22⋊D17312yes—
231C42C22×D17311yescharacteristic
232C23—21no—
233C23—3461no—
234C23—3462no—
235C23—13842no—
236C23—6924no—
237C23—6924no—
238C23C2×C22⋊D17312yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—3462no—
245D4—3461no—
246D4—3462no—
247D4—13841no—
248D4—1732no—
249D4—6924no—
250D4—6924no—
251D4—6922no—
252D4D4×D17312yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3461no—
260C2×C4—6922no—
261C2×C4C2×C22⋊D17311yescharacteristic
262C2×C4C2×C22⋊D17312yes—
263C2×C4C2×C22⋊D17311yescharacteristic
264C2×C4C23×D17311yescharacteristic
265C8—3462no—
266C8—3461no—
267C8—1732no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—3462no—
275C22—13848no—
276C22—6922no—
277C22—6922no—
278C22—6922no—
279C22—6921no—
280C22—6921no—
281C22C22×C22⋊D17311yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—6921no—
287C4#1914970412yescomplemented
288C4C2×D4×D17311yescharacteristic
289C4#1914972411yescharacteristic
290C2—42no—
291C2—42no—
292C2—13842no—
293C2—6922no—
294C2D4×C22⋊D17311yescharacteristic
295C2C2×#1914970411yescharacteristic
296C2#2601229911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#547×C173C2
socleC22×C173C22×D4
hypercenter#547D173
hypocenterC173#3272
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19149704C2semidirect
2. D4×D692C2semidirect
3. #26011865C2semidirect
4. #26011739C2semidirect
5. #26011693C2semidirect
6. #26011686C2semidirect
7. #26011641C2semidirect
8. #26011636C2semidirect
9. C4×#14583811C2semidirect
10. #26011569C2semidirect
11. #547×C173C2semidirect
16. C2×#14583813C22semidirect
17. C2×#14583813C22semidirect
19. C2×#14583811C22semidirect
20. C2×#14583810C22semidirect
26. #19149612C22semidirect
27. C4⋊#13063436C22semidirect
28. #19149603C22semidirect
29. #19149600C22semidirect
31. #19149599C22semidirect
32. C22⋊#13063436C22semidirect
33. C4×#13063436C22semidirect
34. C2×#41×C173C22semidirect
35. C4⋊Q8×C173C22semidirect
37. C22⋊Q8×C173C22semidirect
40. C4×D4×C173C22semidirect
57. #14583811D4semidirect
100. #13063436C2×D4semidirect
108. D4×C173C2×D4semidirect
137. C4×C173AGL(1,ℤ/8)semidirect
148. C173#3272semidirect
151. #547D173semidirect
161. C2×#41C2×D173semidirect
171. C4⋊Q8C2×D173semidirect
174. C22⋊Q8C2×D173semidirect
181. C4×D4C2×D173semidirect
252. D4D4×D173semidirect
287. C4#19149704semidirect

Maximal subgroups

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

subgroupindex
C2×#191497042
D4×D6922
#260118652
#260117392
#260116932
#260116862
#260116412
#260116362
C4×#145838112
#260115692
#547×C1732
#3272173