Finite Group Databasebeta

← #36365387

Subgroups — #36365387

Order 22144.

462 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 141 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
1#26012198C211yesmaximal, characteristic, complemented
2#26011837C211yesmaximal, characteristic, complemented
3#26011807C212yesmaximal, complemented
4#26011781C211yesmaximal, characteristic, complemented
5#26011718C212yesmaximal, complemented
6#26011688C211yesmaximal, characteristic, complemented
7#26011642C212yesmaximal, complemented
8C4×#14583814C211yesmaximal, characteristic, complemented
9D4×#13063436C211yesmaximal, characteristic, complemented
10C2×#19149600C212yesmaximal, complemented
11D42×C173C211yesmaximal, characteristic
12C2×#14583814C2211yescharacteristic
13C2×#14583814C2212yescomplemented
14C22×#13063436C2212yescomplemented
15#19149679C2211yescharacteristic, complemented
16#19149670—21no—
17#19149658C2212yescomplemented
18#19149656—22no—
19C4×Dic346C2211yescharacteristic, complemented
20D4×Dic173—21no—
21C2×C4⋊Dic173C2212yescomplemented
22#19149610C2212yescomplemented
23#19149610C2211yescharacteristic, complemented
24C4⋊#13063436C2211yescharacteristic, complemented
25#19149602C2211yescharacteristic, complemented
26#19149600C2211yescharacteristic
27#19149600C2212yes—
28#19149600C2212yescomplemented
29#19149600C2212yescomplemented
30#19149600C2212yescomplemented
31C22⋊#13063436C2212yescomplemented
32C4×#13063436C2211yescharacteristic, complemented
33C22×D4×C173—21no—
34C22×D4×C173C2212yes—
35#127×C173C2211yescharacteristic
36#121×C173—21no—
37#121×C173C2212yes—
38#120×C173—22no—
39C4×D4×C173C2211yescharacteristic
40C4×D4×C173C2211yescharacteristic
41C2×Dic346—21no—
42C2×Dic346C2311yescharacteristic
43C22×Dic173—22no—
44#14583814—22no—
45#14583814—42no—
46C2×#13063436—42no—
47C2×#13063436C2311yescharacteristic
48C2×#13063436C2311yescharacteristic
49C2×#13063436C2312yes—
50C4⋊Dic173—22no—
51C4⋊Dic173—21no—
52C4⋊Dic173C2311yescharacteristic
53C4⋊Dic173C2312yes—
54#14583805—21no—
55#14583805—22no—
56C22⋊Dic173—22no—
57C4×Dic173—21no—
58C24×C173—22no—
59C2×D4×C173—24no—
60C2×D4×C173—22no—
61C2×D4×C173—21no—
62C2×D4×C173—22no—
63C2×D4×C173—21no—
64C2×D4×C173—42no—
65C2×D4×C173—44no—
66C2×D4×C173D411yescharacteristic
67C2×D4×C173C2312yes—
68C2×D4×C173C2311yescharacteristic
69C22×C4×C173—21no—
70C22×C4×C173C2312yes—
71C4⋊C4×C173D411yescharacteristic
72C4⋊C4×C173C2311yescharacteristic
73C22⋊C4×C173—21no—
74C22⋊C4×C173—22no—
75C22⋊C4×C173D412yes—
76C42×C173C2311yescharacteristic
77Dic346—21no—
78Dic346—21no—
79Dic346—81no—
80C2×Dic173—21no—
81C2×Dic173—22no—
82C2×Dic173—21no—
83C2×Dic173—82no—
84#13063436—21no—
85#13063436—21no—
86#13063436—22no—
87C23×C173—21no—
88C23×C173—22no—
89C23×C173—44no—
90C23×C173—44no—
91C23×C173—82no—
92C23×C173C2×D412yes—
93D4×C173—22no—
94D4×C173—21no—
95D4×C173—22no—
96D4×C173—44no—
97D4×C173—44no—
98D4×C173—42no—
99D4×C173—81no—
100D4×C173SD1614yescomplemented
101D4×C173#4112yes—
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173—21no—
105C2×C4×C173—42no—
106C2×C4×C173C2×D411yescharacteristic
107C2×C4×C173C2×D412yes—
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2411yescharacteristic
110C22×C173—22no—
111C22×C173—22no—
112C22×C173—22no—
113C22×C173—41no—
114C22×C173—41no—
115C22×C173—42no—
116C22×C173—42no—
117C22×C173—42no—
118C22×C173—88no—
119C22×C173C2×SD1614yes—
120C22×C173C22×D411yescharacteristic
121C4×C173—22no—
122C4×C173—21no—
123C4×C173—41no—
124C4×C173C2×SD1612yes—
125C4×C173C2×#4111yescharacteristic
126C4×C173H(5,𝔽2)11yescharacteristic
127Dic173—41no—
128Dic173—41no—
129Dic173—42no—
130C2×C173—24no—
131C2×C173—42no—
132C2×C173—82no—
133C2×C173#54611yescharacteristic
134C2×C173C22×SD1611yescharacteristic
135C2×C173#57611yescharacteristic
136C173#329511yescharacteristic, complemented, Sylow
137#3295—1731nomaximal, Sylow
138#554—1731no—
139D42D17311yescharacteristic
140#486—1731no—
141#481—1732no—
142#476—1731no—
143#465—1732no—
144#460—1731no—
145#450—1732no—
146C4×SD16—1731no—
147C8×D4—1731no—
148C2×#102—1732no—
149C22×D4—21no—
150C22×D4C2×D17312yes—
151C2×SD16—1731no—
152C2×SD16—1732no—
153C22×C8—1732no—
154C4⋊Q8—1731no—
155#127C2×D17311yescharacteristic
156#125—3461no—
157C22⋊Q8—3462no—
158C22⋊Q8—1732no—
159#121—21no—
160#121C2×D17312yes—
161#120—22no—
162C4×Q8—1731no—
163C4×D4—3461no—
164C4×D4C2×D17311yescharacteristic
165C4×D4C2×D17311yescharacteristic
166C2×C4⋊C4—1732no—
167#106—1731no—
168#106—1732no—
169C4⋊C8—1731no—
170#103—1731no—
171#102—1731no—
172#102—1732no—
173#102—1732no—
174#102—1732no—
175#102—1732no—
176C22⋊C8—1732no—
177C4×C8—1731no—
178C24—22no—
179C2×Q8—3461no—
180C2×Q8—1731no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4C22⋊D17311yescharacteristic
189C2×D4C22×D17312yes—
190C2×D4C22×D17311yescharacteristic
191C22×C4—21no—
192C22×C4—3462no—
193C22×C4C22×D17312yes—
194SD16—3462no—
195SD16—6922no—
196C2×C8—1731no—
197C2×C8—1731no—
198C2×C8—1732no—
199C2×C8—6922no—
200C4⋊C4—3462no—
201C4⋊C4—3461no—
202C4⋊C4—3461no—
203C4⋊C4—3462no—
204C4⋊C4—1731no—
205C4⋊C4—1732no—
206C4⋊C4C22⋊D17311yescharacteristic
207C4⋊C4C22×D17311yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—3462no—
211C22⋊C4C22⋊D17312yes—
212C42—3461no—
213C42C22×D17311yescharacteristic
214C23—21no—
215C23—22no—
216C23—44no—
217C23—44no—
218C23—82no—
219C23C2×C22⋊D17312yes—
220Q8—3461no—
221Q8—3461no—
222Q8—13841no—
223D4—22no—
224D4—21no—
225D4—22no—
226D4—44no—
227D4—44no—
228D4—42no—
229D4—81no—
230D4#1458381414yescomplemented
231D4#1458382512yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3461no—
237C2×C4—3461no—
238C2×C4—3462no—
239C2×C4—13842no—
240C2×C4C2×C22⋊D17311yescharacteristic
241C2×C4C2×C22⋊D17312yes—
242C2×C4C2×C22⋊D17311yescharacteristic
243C2×C4C23×D17311yescharacteristic
244C8—3461no—
245C8—3461no—
246C8—3462no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—88no—
256C22C2×#1458381414yes—
257C22C22×C22⋊D17311yescharacteristic
258C4—21no—
259C4—22no—
260C4—41no—
261C4—6921no—
262C4—6922no—
263C4—6921no—
264C4C2×#1458381412yes—
265C4C2×#1458382511yescharacteristic
266C4#1914972411yescharacteristic
267C2—24no—
268C2—42no—
269C2—82no—
270C2#2601213511yescharacteristic
271C2C22×#1458381411yescharacteristic
272C2#2601229411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radicalD42×C173C2
socleC22×C173C22×D4
hypercenterD42D173
hypocenterC173#3295
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #26012198C2semidirect
2. #26011837C2semidirect
3. #26011807C2semidirect
4. #26011781C2semidirect
5. #26011718C2semidirect
6. #26011688C2semidirect
7. #26011642C2semidirect
8. C4×#14583814C2semidirect
9. D4×#13063436C2semidirect
10. C2×#19149600C2semidirect
13. C2×#14583814C22semidirect
14. C22×#13063436C22semidirect
15. #19149679C22semidirect
17. #19149658C22semidirect
19. C4×Dic346C22semidirect
21. C2×C4⋊Dic173C22semidirect
22. #19149610C22semidirect
23. #19149610C22semidirect
24. C4⋊#13063436C22semidirect
25. #19149602C22semidirect
28. #19149600C22semidirect
29. #19149600C22semidirect
30. #19149600C22semidirect
31. C22⋊#13063436C22semidirect
32. C4×#13063436C22semidirect
100. D4×C173SD16semidirect
136. C173#3295semidirect
230. D4#14583814semidirect

Maximal subgroups

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

subgroupindex
#260121982
#260118372
#260118072
#260117812
#260117182
#260116882
#260116422
C4×#145838142
D4×#130634362
C2×#191496002
D42×C1732
#3295173