Finite Group Databasebeta

← #36365376

Subgroups — #36365376

Order 22144.

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

normal subgroups only

This group has 494 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#26012145C211yesmaximal, characteristic, complemented
2#26011837C211yesmaximal, characteristic, complemented
3#26011802C212yesmaximal, complemented
4#26011771C211yesmaximal, characteristic, complemented
5#26011725C212yesmaximal, complemented
6#26011683C211yesmaximal, characteristic, complemented
7#26011625C212yesmaximal, complemented
8C4×#14583811C211yesmaximal, characteristic, complemented
9D4×#13063436C211yesmaximal, characteristic, complemented
10C2×#19149600C212yesmaximal, complemented
11D42×C173C211yesmaximal, characteristic, complemented
12C2×#14583825—21no—
13C2×#14583811C2211yescharacteristic
14C2×#14583811C2212yescomplemented
15C22×#13063436C2212yescomplemented
16#19149679C2211yescharacteristic, complemented
17#19149663—22no—
18#19149659—21no—
19#19149653C2212yescomplemented
20C4×D692C2211yescharacteristic, complemented
21C2×C4⋊Dic173C2212yescomplemented
22#19149612C2212yescomplemented
23#19149612C2211yescharacteristic, complemented
24C4⋊#13063436C2211yescharacteristic, complemented
25#19149600C2211yescharacteristic
26#19149600C2212yes—
27#19149600C2212yescomplemented
28#19149600C2212yescomplemented
29#19149600C2212yescomplemented
30#19149599C2211yescharacteristic, complemented
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
41#14583825—44no—
42C2×Dic346—21no—
43C2×D692C2311yescharacteristic
44C2×C22⋊D173—22no—
45C22×Dic173—22no—
46C2×C4×D173—21no—
47#14583811—22no—
48#14583811—42no—
49C2×#13063436—42no—
50C2×#13063436C2311yescharacteristic
51C2×#13063436C2311yescharacteristic
52C2×#13063436C2312yes—
53C4⋊Dic173—22no—
54C4⋊Dic173—21no—
55C4⋊Dic173C2311yescharacteristic
56C4⋊Dic173C2312yes—
57#14583803—21no—
58#14583803—22no—
59C24×C173—22no—
60C2×D4×C173—24no—
61C2×D4×C173—22no—
62C2×D4×C173—21no—
63C2×D4×C173—22no—
64C2×D4×C173—21no—
65C2×D4×C173—42no—
66C2×D4×C173—44no—
67C2×D4×C173D411yescharacteristic
68C2×D4×C173C2312yes—
69C2×D4×C173C2311yescharacteristic
70C22×C4×C173—21no—
71C22×C4×C173C2312yes—
72C4⋊C4×C173D411yescharacteristic
73C4⋊C4×C173C2311yescharacteristic
74C22⋊C4×C173—21no—
75C22⋊C4×C173—22no—
76C22⋊C4×C173D412yes—
77C42×C173C2311yescharacteristic
78C22×D173—21no—
79Dic346—42no—
80D692—21no—
81D692—21no—
82C22⋊D173—44no—
83C2×Dic173—21no—
84C2×Dic173—22no—
85C2×Dic173—44no—
86C4×D173—42no—
87#13063436—21no—
88#13063436—21no—
89#13063436—22no—
90C23×C173—21no—
91C23×C173—22no—
92C23×C173—44no—
93C23×C173—44no—
94C23×C173—82no—
95C23×C173C2×D412yes—
96D4×C173—22no—
97D4×C173—21no—
98D4×C173—22no—
99D4×C173—44no—
100D4×C173—44no—
101D4×C173—42no—
102D4×C173—81no—
103D4×C173D814yescomplemented
104D4×C173#4112yescomplemented
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—21no—
108C2×C4×C173—42no—
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2×D412yes—
111C2×C4×C173C2×D411yescharacteristic
112C2×C4×C173C2411yescharacteristic
113C22×C173—22no—
114C22×C173—22no—
115C22×C173—22no—
116C22×C173—41no—
117C22×C173—41no—
118C22×C173—42no—
119C22×C173—42no—
120C22×C173—42no—
121C22×C173—88no—
122C22×C173C2×D814yes—
123C22×C173C22×D411yescharacteristic
124C2×D173—41no—
125C2×D173—41no—
126C2×D173—41no—
127C4×C173—22no—
128C4×C173—21no—
129C4×C173—41no—
130C4×C173C2×D812yes—
131C4×C173C2×#4111yescharacteristic
132C4×C173H(5,𝔽2)11yescharacteristic
133Dic173—41no—
134Dic173—42no—
135C2×C173—24no—
136C2×C173—42no—
137C2×C173—82no—
138C2×C173#54611yescharacteristic
139C2×C173C22×D811yescharacteristic
140C2×C173#57711yescharacteristic
141D173—81no—
142C173#329411yescharacteristic, complemented, Sylow
143#3294—1731nomaximal, Sylow
144#547—1731no—
145D42D17311yescharacteristic, complemented
146#486—1731no—
147#480—1732no—
148#474—1731no—
149#466—1732no—
150#459—1731no—
151#447—1732no—
152C4×D8—1731no—
153C8×D4—1731no—
154C2×#102—1732no—
155C2×#41—3461no—
156C22×D4—21no—
157C22×D4C2×D17312yes—
158C2×D8—1731no—
159C2×D8—1732no—
160C22×C8—1732no—
161C4⋊Q8—1731no—
162#127C2×D17311yescharacteristic
163#123—3462no—
164C22⋊Q8—3461no—
165#121—21no—
166#121—1732no—
167#121C2×D17312yes—
168#120—22no—
169C4×D4—1731no—
170C4×D4C2×D17311yescharacteristic
171C4×D4C2×D17311yescharacteristic
172C2×C4⋊C4—1732no—
173#107—1731no—
174#107—1732no—
175C4⋊C8—1731no—
176#102—1731no—
177#102—1732no—
178#102—1732no—
179#102—1732no—
180#102—1732no—
181#102—1731no—
182C22⋊C8—1732no—
183C4×C8—1731no—
184C24—22no—
185#41—6924no—
186C2×Q8—3461no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—3462no—
195C2×D4—1731no—
196C2×D4C22⋊D17311yescharacteristic
197C2×D4C22×D17312yes—
198C2×D4C22×D17311yescharacteristic
199C22×C4—21no—
200C22×C4—3462no—
201C22×C4—3461no—
202C22×C4C22×D17312yes—
203D8—3462no—
204D8—6922no—
205C2×C8—1731no—
206C2×C8—1731no—
207C2×C8—1732no—
208C2×C8—6922no—
209C4⋊C4—3462no—
210C4⋊C4—3461no—
211C4⋊C4—1731no—
212C4⋊C4—1732no—
213C4⋊C4C22⋊D17311yescharacteristic
214C4⋊C4C22×D17311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—3461no—
218C22⋊C4—3462no—
219C22⋊C4C22⋊D17312yes—
220C42C22×D17311yescharacteristic
221C23—21no—
222C23—22no—
223C23—44no—
224C23—44no—
225C23—82no—
226C23—3461no—
227C23C2×C22⋊D17312yes—
228Q8—6922no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—3461no—
237D4—3461no—
238D4—6924no—
239D4#1458381114yescomplemented
240D4#1458382512yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—3461no—
246C2×C4—3462no—
247C2×C4—6924no—
248C2×C4—6922no—
249C2×C4C2×C22⋊D17311yescharacteristic
250C2×C4C2×C22⋊D17312yes—
251C2×C4C2×C22⋊D17311yescharacteristic
252C2×C4C23×D17311yescharacteristic
253C8—3461no—
254C8—3461no—
255C8—3462no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—88no—
265C22—6921no—
266C22—6921no—
267C22—6921no—
268C22C2×#1458381114yes—
269C22C22×C22⋊D17311yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—6921no—
274C4—6922no—
275C4C2×#1458381112yes—
276C4C2×#1458382511yescharacteristic
277C4#1914972411yescharacteristic
278C2—24no—
279C2—42no—
280C2—82no—
281C2—13841no—
282C2#2601213511yescharacteristic
283C2C22×#1458381111yescharacteristic
284C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingD42×C173C2
socleC22×C173C22×D4
hypercenterD42D173
hypocenterC173#3294
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. #26012145C2semidirect
2. #26011837C2semidirect
3. #26011802C2semidirect
4. #26011771C2semidirect
5. #26011725C2semidirect
6. #26011683C2semidirect
7. #26011625C2semidirect
8. C4×#14583811C2semidirect
9. D4×#13063436C2semidirect
10. C2×#19149600C2semidirect
11. D42×C173C2semidirect
14. C2×#14583811C22semidirect
15. C22×#13063436C22semidirect
16. #19149679C22semidirect
19. #19149653C22semidirect
20. C4×D692C22semidirect
21. C2×C4⋊Dic173C22semidirect
22. #19149612C22semidirect
23. #19149612C22semidirect
24. C4⋊#13063436C22semidirect
27. #19149600C22semidirect
28. #19149600C22semidirect
29. #19149600C22semidirect
30. #19149599C22semidirect
31. C22⋊#13063436C22semidirect
32. C4×#13063436C22semidirect
103. D4×C173D8semidirect
104. D4×C173#41semidirect
142. C173#3294semidirect
145. D42D173semidirect
239. D4#14583811semidirect
240. D4#14583825semidirect

Maximal subgroups

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

subgroupindex
#260121452
#260118372
#260118022
#260117712
#260117252
#260116832
#260116252
C4×#145838112
D4×#130634362
C2×#191496002
D42×C1732
#3294173