Finite Group Databasebeta

← #36365348

Subgroups — #36365348

Order 22144.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012304C211yesmaximal, characteristic, complemented
2#26012297C212yesmaximal, complemented
3#26012292C211yesmaximal, characteristic, complemented
4#26011763C211yesmaximal, characteristic, complemented
5#26011758C211yesmaximal, characteristic, complemented
6#26011751C212yesmaximal, complemented
7#26011668C212yesmaximal, complemented
8#26011663C212yesmaximal, complemented
9#26011609C211yesmaximal, characteristic, complemented
10#26011609C211yesmaximal, characteristic, complemented
11#495×C173C211yesmaximal, characteristic, complemented
12#19149727C2212yescomplemented
13#19149724C2212yes—
14#19149711—22no—
15#19149710C2212yes—
16#19149710C2212yescomplemented
17#19149708—22no—
18#19149706—22no—
19#19149705C2212yes—
20#19149705C2212yescomplemented
21#19149703—22no—
22#19149702—21no—
23#19149701C2211yescharacteristic
24#19149701C2211yescharacteristic
25#19149700—22no—
26#19149698—21no—
27Q16×D173—21no—
28SD16×D173—22no—
29D8×D173—21no—
30#19149685C2211yescharacteristic
31#19149685C2212yescomplemented
32#19149685C2211yescharacteristic, complemented
33#19149615C2211yescharacteristic, complemented
34#19149615C2211yescharacteristic, complemented
35#19149607C2211yescharacteristic
36#19149607C2211yescharacteristic, complemented
37#19149607C2212yescomplemented
38#19149598C2212yescomplemented
39#19149596C2212yescomplemented
40C2×Q16×C173C2211yescharacteristic, complemented
41C2×SD16×C173C2212yescomplemented
42C2×D8×C173C2211yescharacteristic
43#124×C173C2212yescomplemented
44C4×C8×C173C2211yescharacteristic, complemented
45#14583827—22no—
46#14583827—22no—
47#14583826—42no—
48#14583826C2311yescharacteristic
49#14583826C2312yes—
50#14583826C2311yescharacteristic
51#14583825—22no—
52#14583825—22no—
53Q8×D173—22no—
54Q8×D173—22no—
55C2×C22⋊D173—42no—
56D4×D173—22no—
57D4×D173—22no—
58Dic692D411yescharacteristic
59Dic692D411yescharacteristic
60#14583816—22no—
61#14583816—22no—
62#14583815D411yescharacteristic
63#14583815D411yescharacteristic
64#14583815D412yes—
65#14583814—22no—
66#14583814—22no—
67#14583813—22no—
68#14583813—22no—
69D1384D411yescharacteristic
70D1384D411yescharacteristic
71#14583811—22no—
72#14583811—22no—
73#14583810C2311yescharacteristic
74#14583810C2312yes—
75#14583810C2311yescharacteristic
76#14583809—21no—
77#14583809—22no—
78#14583809—21no—
79C8×D173—21no—
80C8×D173—22no—
81C8×D173—21no—
82C2×Q8×C173C2312yes—
83C2×D4×C173C2312yes—
84Q16×C173—21no—
85Q16×C173—21no—
86SD16×C173—22no—
87SD16×C173—22no—
88D8×C173—21no—
89D8×C173—21no—
90C2×C8×C173C2311yescharacteristic
91C2×C8×C173C2311yescharacteristic
92C22⋊C4×C173—42no—
93C42×C173C2311yescharacteristic
94C22×D173—42no—
95Dic346—42no—
96Dic346C2×D411yescharacteristic
97Dic346C2×D412yes—
98Dic346C2×D411yescharacteristic
99D692—42no—
100D692C2×D411yescharacteristic
101D692C2×D412yes—
102D692C2×D411yescharacteristic
103C22⋊D173—21no—
104C22⋊D173—22no—
105C22⋊D173—21no—
106C22⋊D173—42no—
107C22⋊D173—42no—
108C2×Dic173—42no—
109C4×D173—21no—
110C4×D173—22no—
111C4×D173—21no—
112C4×D173—42no—
113C4×D173—42no—
114#13063436—21no—
115#13063436—22no—
116#13063436—21no—
117C23×C173—22no—
118Q8×C173—22no—
119Q8×C173—22no—
120D4×C173—22no—
121D4×C173—22no—
122C2×C4×C173—22no—
123C2×C4×C173—21no—
124C2×C4×C173C2411yescharacteristic
125C8×C173C2×D411yescharacteristic
126C8×C173C2×D411yescharacteristic
127C8×C173C2×D411yescharacteristic
128C8×C173C2×D411yescharacteristic
129C22×C173—42no—
130C22×C173—82no—
131C22×C173H(5,𝔽2)11yescharacteristic
132C2×D173—21no—
133C2×D173—22no—
134C2×D173—21no—
135C2×D173—162no—
136C4×C173—41no—
137C4×C173—42no—
138C4×C173C22×D411yescharacteristic
139C4×C173C22×D411yescharacteristic
140Dic173—21no—
141Dic173—22no—
142Dic173—21no—
143C2×C173—21no—
144C2×C173—82no—
145C2×C173D4211yescharacteristic
146D173—41no—
147D173—42no—
148D173—41no—
149C173#329011yescharacteristic, complemented, Sylow
150#3290—1731nomaximal, Sylow
151#578—1731no—
152#577—1732no—
153#576—1731no—
154#495D17311yescharacteristic, complemented
155#473—1731no—
156#472—1731no—
157#471—1732no—
158#455—1732no—
159#454—1732no—
160#443—1731no—
161#443—1731no—
1622−1+4—1732no—
163H(5,𝔽2)—1732no—
164#137—3462no—
165#137—3462no—
166#137—1732no—
167#137—1732no—
168AGL(1,ℤ/8)—3462no—
169AGL(1,ℤ/8)—3462no—
170AGL(1,ℤ/8)—1732no—
171AGL(1,ℤ/8)—1732no—
172#135—3461no—
173#135—3462no—
174#135—3461no—
175#135—1731no—
176#135—1731no—
177C2×Q16—3461no—
178C2×Q16C2×D17311yescharacteristic, complemented
179C2×SD16—3462no—
180C2×SD16C2×D17312yescomplemented
181C2×D8—3461no—
182C2×D8C2×D17311yescharacteristic
183#131—1731no—
184#131—1732no—
185#131—1731no—
186#124C2×D17312yescomplemented
187#108—1731no—
188#108—1731no—
189#104—1731no—
190#104—1731no—
191#104—1732no—
192#101—1732no—
193#100—1732no—
194C4×C8C2×D17311yescharacteristic, complemented
195#41—3462no—
196#41—3462no—
197#41—3462no—
198#41—3462no—
199#41—1731no—
200#41—1732no—
201#41—1731no—
202#41—6922no—
203C2×Q8—3462no—
204C2×Q8—3462no—
205C2×Q8C22×D17312yes—
206C2×D4—3462no—
207C2×D4—3462no—
208C2×D4—6922no—
209C2×D4C22×D17312yes—
210Q16—21no—
211Q16—21no—
212Q16—3462no—
213Q16—3462no—
214Q16—1731no—
215Q16—1731no—
216SD16—22no—
217SD16—22no—
218SD16—3462no—
219SD16—3462no—
220SD16—3462no—
221SD16—3462no—
222SD16—1731no—
223SD16—1731no—
224SD16—1732no—
225D8—21no—
226D8—21no—
227D8—3462no—
228D8—3462no—
229D8—1731no—
230D8—1731no—
231M16—3461no—
232M16—3462no—
233M16—3461no—
234M16—1731no—
235M16—1732no—
236M16—1731no—
237C2×C8—3461no—
238C2×C8—3462no—
239C2×C8—3461no—
240C2×C8C22×D17311yescharacteristic
241C2×C8C22×D17311yescharacteristic
242C22⋊C4—42no—
243C42C22×D17311yescharacteristic
244C23—22no—
245C23—6922no—
246Q8—22no—
247Q8—22no—
248Q8—1731no—
249Q8—1732no—
250Q8—1731no—
251Q8—6922no—
252D4—22no—
253D4—22no—
254D4—3461no—
255D4—3462no—
256D4—3461no—
257D4—1731no—
258D4—1732no—
259D4—1731no—
260D4—6922no—
261D4—6922no—
262D4—6922no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—3461no—
266C2×C4—3462no—
267C2×C4—3461no—
268C2×C4—6922no—
269C2×C4—6922no—
270C2×C4—6922no—
271C2×C4C23×D17311yescharacteristic
272C8—3461no—
273C8—3462no—
274C8—3461no—
275C8D4×D17311yescharacteristic
276C8D4×D17311yescharacteristic
277C8D4×D17311yescharacteristic
278C8D4×D17311yescharacteristic
279C22—42no—
280C22—82no—
281C22—3461no—
282C22—3462no—
283C22—3461no—
284C22—27682no—
285C22#1914972411yescharacteristic
286C4—41no—
287C4—42no—
288C4—3461no—
289C4—3462no—
290C4—3461no—
291C4C2×D4×D17311yescharacteristic
292C4C2×D4×D17311yescharacteristic
293C2—21no—
294C2—82no—
295C2—6921no—
296C2—6922no—
297C2—6921no—
298C2#2601212611yescharacteristic
subgroupG/H
centerC2#26012126
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#495×C173C2
socleC2×C173D42
hypercenter#495D173
hypocenterC173#3290
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #26012304C2semidirect
2. #26012297C2semidirect
3. #26012292C2semidirect
4. #26011763C2semidirect
5. #26011758C2semidirect
6. #26011751C2semidirect
7. #26011668C2semidirect
8. #26011663C2semidirect
9. #26011609C2semidirect
10. #26011609C2semidirect
11. #495×C173C2semidirect
12. #19149727C22semidirect
16. #19149710C22semidirect
20. #19149705C22semidirect
31. #19149685C22semidirect
32. #19149685C22semidirect
33. #19149615C22semidirect
34. #19149615C22semidirect
36. #19149607C22semidirect
37. #19149607C22semidirect
38. #19149598C22semidirect
39. #19149596C22semidirect
40. C2×Q16×C173C22semidirect
41. C2×SD16×C173C22semidirect
43. #124×C173C22semidirect
44. C4×C8×C173C22semidirect
149. C173#3290semidirect
154. #495D173semidirect
178. C2×Q16C2×D173semidirect
180. C2×SD16C2×D173semidirect
186. #124C2×D173semidirect
194. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123042
#260122972
#260122922
#260117632
#260117582
#260117512
#260116682
#260116632
#260116092
#495×C1732
#3290173