Finite Group Databasebeta

← #36363359

Subgroups — #36363359

Order 22144.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#26012044C211yesmaximal, characteristic, complemented
2C2×#19149641C211yesmaximal, characteristic
3#26011794C212yesmaximal, complemented
4#26011790C212yesmaximal, complemented
5#26011780C212yesmaximal, complemented
6#26011770C212yesmaximal, complemented
7#26011531C211yesmaximal, characteristic, complemented
8#26011514C211yesmaximal, characteristic, complemented
9#26011497C211yesmaximal, characteristic
10#26011495C211yesmaximal, characteristic
11#428×C173C211yesmaximal, characteristic, complemented
12C22×D692C2211yescharacteristic
13C22×C4×D173—21no—
14C2×#14583810C2211yescharacteristic
15#19149678C2211yescharacteristic, complemented
16#19149678C2211yescharacteristic, complemented
17#19149670C2211yescharacteristic, complemented
18#19149670C2211yescharacteristic, complemented
19#19149656—21no—
20#19149656—21no—
21C4×Dic346—21no—
22C4×Dic346—21no—
23#19149641—21no—
24#19149641—21no—
25#19149641C2212yes—
26#19149641C2212yes—
27#19149638C2211yescharacteristic
28C2×#14583803—21no—
29C2×C4×Dic173C2211yescharacteristic
30#19149612C2212yescomplemented
31#19149610C2212yescomplemented
32#19149608C2212yescomplemented
33#19149608C2212yescomplemented
34#19149601C2212yes—
35#19149601C2212yes—
36#19149599C2212yes—
37#19149599C2212yes—
38C2×M16×C173C2211yescharacteristic
39#117×C173C2211yescharacteristic
40C2×C4⋊C4×C173C2211yescharacteristic
41#107×C173C2212yescomplemented
42#106×C173C2212yescomplemented
43C23×D173—21no—
44C2×Dic346—21no—
45C2×Dic346—21no—
46C2×D692—21no—
47C2×D692—22no—
48C2×D692Q814yescomplemented
49C2×D692C2312yes—
50C22×Dic173D411yescharacteristic
51C22×Dic173D411yescharacteristic
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173—44no—
55#14583810—22no—
56C2×#13063436C2312yes—
57C4⋊Dic173—21no—
58C4⋊Dic173—21no—
59C4⋊Dic173C2311yescharacteristic
60C4⋊Dic173C2311yescharacteristic
61#14583805—21no—
62#14583805—21no—
63#14583805—21no—
64#14583805—21no—
65C22⋊Dic173—21no—
66#14583803—22no—
67#14583803—22no—
68C4×Dic173—21no—
69C4×Dic173D411yescharacteristic
70C4×Dic173D411yescharacteristic
71C4×Dic173C2311yescharacteristic
72C4×Dic173C2311yescharacteristic
73C22×C4×C173—21no—
74C22×C4×C173C2311yescharacteristic
75M16×C173—22no—
76C2×C8×C173C2312yes—
77C4⋊C4×C173—21no—
78C4⋊C4×C173C2311yescharacteristic
79C4⋊C4×C173C2311yescharacteristic
80C4⋊C4×C173C2312yes—
81C22⋊C4×C173—21no—
82C42×C173—21no—
83C22×D173—24no—
84C22×D173—22no—
85C22×D173—42no—
86C22×D173—42no—
87Dic346—42no—
88Dic346—42no—
89D692—24no—
90D692—22no—
91D692C2×Q814yes—
92C2×Dic173—22no—
93C2×Dic173—21no—
94C2×Dic173—21no—
95C2×Dic173—41no—
96C2×Dic173C2×D411yescharacteristic
97C2×Dic173C2×D411yescharacteristic
98C2×Dic173C2×D411yescharacteristic
99C2×Dic173C2×D411yescharacteristic
100C4×D173—44no—
101C4×D173—44no—
102#13063436—22no—
103C23×C173C2×D411yescharacteristic
104C2×C4×C173—22no—
105C2×C4×C173—21no—
106C2×C4×C173—21no—
107C2×C4×C173—42no—
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2×Q812yes—
110C2×C4×C173#4112yes—
111C2×C4×C173C2411yescharacteristic
112C8×C173—22no—
113C22×C173—21no—
114C22×C173—21no—
115C22×C173C22⋊Q812yes—
116C22×C173C22×D411yescharacteristic
117C2×D173—24no—
118C2×D173—48no—
119C2×D173—42no—
120C2×D173—42no—
121C4×C173—42no—
122C4×C173—41no—
123C4×C173—41no—
124C4×C173C22⋊Q812yes—
125C4×C173C22×Q811yescharacteristic
126C4×C173C2×#4111yescharacteristic
127Dic173—21no—
128Dic173—21no—
129Dic173—21no—
130Dic173—41no—
131Dic173—41no—
132Dic173AGL(1,ℤ/8)12yes—
133C2×C173—22no—
134C2×C173C2×C22⋊Q811yescharacteristic
135C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C173#57511yescharacteristic
137D173—44no—
138C173#307711yescharacteristic, complemented, Sylow
139#3077—1731nomaximal, Sylow
140#533—1731no—
141C2×C4×D4—1731no—
142#478—1732no—
143#478—1732no—
144#476—1732no—
145#474—1732no—
146#428—1731no—
147#428D17311yescharacteristic, complemented
148#423—1731no—
149#418—1731no—
150#418—1731no—
151C22×D4—1731no—
152C23×C4—3461no—
153C2×M16—1731no—
154C2×M16C2×D17311yescharacteristic
155C4⋊Q8—1731no—
156C4⋊Q8—1731no—
157#125—1731no—
158#125—1731no—
159C22⋊Q8—3461no—
160C22⋊Q8—3461no—
161C4×Q8—3461no—
162C4×Q8—3461no—
163C4×D4—3461no—
164C4×D4—3461no—
165C4×D4—1732no—
166C4×D4—1732no—
167#117—1731no—
168#117C2×D17311yescharacteristic
169C2×C4⋊C4C2×D17311yescharacteristic
170C2×C22⋊C4—3461no—
171C2×C42—1731no—
172#107—1732no—
173#107C2×D17312yescomplemented
174#106—1732no—
175#106C2×D17312yescomplemented
176C4⋊C8—1732no—
177C4⋊C8—1732no—
178#102—1732no—
179#102—1732no—
180#102—1732no—
181#102—1732no—
182C24—3461no—
183C2×Q8—3461no—
184C2×Q8—3461no—
185C2×D4—3461no—
186C2×D4—3462no—
187C2×D4—1734no—
188C2×D4—1732no—
189C22×C4—21no—
190C22×C4—3462no—
191C22×C4—3462no—
192C22×C4—1731no—
193C22×C4—1731no—
194C22×C4—6924no—
195C22×C4C22×D17311yescharacteristic
196M16—22no—
197M16—3462no—
198C2×C8—1732no—
199C2×C8C22×D17312yes—
200C4⋊C4—21no—
201C4⋊C4—3461no—
202C4⋊C4—3461no—
203C4⋊C4—3461no—
204C4⋊C4—3461no—
205C4⋊C4—3461no—
206C4⋊C4—3461no—
207C4⋊C4—1731no—
208C4⋊C4—1731no—
209C4⋊C4C22×D17311yescharacteristic
210C4⋊C4C22×D17311yescharacteristic
211C4⋊C4C22×D17312yes—
212C22⋊C4—21no—
213C22⋊C4—3462no—
214C22⋊C4—3462no—
215C22⋊C4—3461no—
216C42—21no—
217C42—3461no—
218C42—1731no—
219C42—1731no—
220C42—1731no—
221C42—1731no—
222C23—3464no—
223C23—3462no—
224C23—6922no—
225C23—6922no—
226C23D4×D17311yescharacteristic
227Q8—6922no—
228Q8—6922no—
229D4—3464no—
230D4—3462no—
231D4—1734no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—3462no—
237C2×C4—3461no—
238C2×C4—3461no—
239C2×C4—1731no—
240C2×C4—1731no—
241C2×C4—1731no—
242C2×C4—1731no—
243C2×C4—6924no—
244C2×C4—6924no—
245C2×C4—6921no—
246C2×C4D4×D17311yescharacteristic
247C2×C4Q8×D17312yes—
248C2×C4#1458382612yes—
249C2×C4C23×D17311yescharacteristic
250C8—22no—
251C8—3462no—
252C22—21no—
253C22—21no—
254C22—3464no—
255C22—6928no—
256C22—6922no—
257C22—6922no—
258C22#1914965712yes—
259C22C2×D4×D17311yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—3461no—
264C4—3461no—
265C4—3461no—
266C4—1732no—
267C4—6921no—
268C4—6921no—
269C4#1914965712yes—
270C4C2×Q8×D17311yescharacteristic
271C4C2×#1458382611yescharacteristic
272C2—22no—
273C2—6924no—
274C2C2×#1914965711yescharacteristic
275C2AGL(1,ℤ/8)×D17311yescharacteristic
276C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#428×C173C2
socleC22×C173C22×D4
prime radicalC22×D692C22
hypercenter#428D173
hypocenterC173#3077
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #26012044C2semidirect
3. #26011794C2semidirect
4. #26011790C2semidirect
5. #26011780C2semidirect
6. #26011770C2semidirect
7. #26011531C2semidirect
8. #26011514C2semidirect
11. #428×C173C2semidirect
15. #19149678C22semidirect
16. #19149678C22semidirect
17. #19149670C22semidirect
18. #19149670C22semidirect
30. #19149612C22semidirect
31. #19149610C22semidirect
32. #19149608C22semidirect
33. #19149608C22semidirect
41. #107×C173C22semidirect
42. #106×C173C22semidirect
48. C2×D692Q8semidirect
138. C173#3077semidirect
147. #428D173semidirect
173. #107C2×D173semidirect
175. #106C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120442
C2×#191496412
#260117942
#260117902
#260117802
#260117702
#260115312
#260115142
#260114972
#260114952
#428×C1732
#3077173