Finite Group Databasebeta

← #36366703

Subgroups — #36366703

Order 22144.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#26012293C211yesmaximal, characteristic, complemented
2#26012292C211yesmaximal, characteristic, complemented
3#26011929C211yesmaximal, characteristic, complemented
4#26011928C212yesmaximal, complemented
5#26011925C212yesmaximal, complemented
6#26011922C212yesmaximal, complemented
7#26011921C211yesmaximal, characteristic, complemented
8C2×#19149620C211yesmaximal, characteristic, complemented
9D16×D173C212yesmaximal, complemented
10#26011909C211yesmaximal, characteristic, complemented
11#509×C173C211yesmaximal, characteristic, complemented
12#19149725—21no—
13#19149724—21no—
14#19149706—22no—
15#19149705—21no—
16#19149704—21no—
17#19149703—22no—
18#19149702C2212yescomplemented
19#19149701C2211yescharacteristic, complemented
20#19149698C2212yescomplemented
21C2×D1384C2211yescharacteristic, complemented
22D8×D173C2212yescomplemented
23D8×D173C2212yescomplemented
24#19149685C2211yescharacteristic
25#19149625C2211yescharacteristic, complemented
26#19149624C2212yescomplemented
27#19149623C2211yescharacteristic, complemented
28#19149622C2211yescharacteristic, complemented
29D2768C2212yescomplemented
30#19149620C2211yescharacteristic, complemented
31#19149620C2212yescomplemented
32#19149620C2212yescomplemented
33#19149618C2212yescomplemented
34C2×#14583801C2211yescharacteristic, complemented
35C16×D173C2212yescomplemented
36#135×C173C2211yescharacteristic, complemented
37C2×D8×C173C2211yescharacteristic, complemented
38SD32×C173C2212yescomplemented
39D16×C173C2212yescomplemented
40M32×C173C2211yescharacteristic, complemented
41#14583827—22no—
42#14583826—21no—
43#14583826—41no—
44#14583826D411yescharacteristic
45#14583825—22no—
46#14583825—41no—
47C2×D692—21no—
48C2×D692—41no—
49C2×C22⋊D173—42no—
50D4×D173—22no—
51D4×D173—22no—
52D4×D173—41no—
53D4×D173—42no—
54Dic692C2311yescharacteristic
55#14583815—21no—
56#14583814—22no—
57#14583813—22no—
58D1384—21no—
59D1384C2312yes—
60D1384C2311yescharacteristic
61#14583811—22no—
62#14583811—22no—
63#14583810D411yescharacteristic
64#14583809D412yes—
65C8×D173C2312yes—
66#14583801C2311yescharacteristic
67#14583801C2311yescharacteristic
68#41×C173—21no—
69C2×D4×C173—21no—
70Q16×C173C2311yescharacteristic
71SD16×C173—21no—
72D8×C173—21no—
73D8×C173C2312yes—
74D8×C173C2311yescharacteristic
75C2×C8×C173C2311yescharacteristic
76C16×C173C2312yes—
77C22×D173—41no—
78C22×D173—42no—
79C22×D173—42no—
80Dic346—21no—
81Dic346D811yescharacteristic
82D692—21no—
83D692—22no—
84D692—42no—
85D692—41no—
86D692D811yescharacteristic
87C22⋊D173—41no—
88C22⋊D173—42no—
89C22⋊D173—42no—
90C22⋊D173—42no—
91C22⋊D173D812yes—
92C2×Dic173—42no—
93C4×D173—41no—
94C4×D173—42no—
95C4×D173C2×D412yes—
96#13063436C2×D412yes—
97C23×C173—41no—
98Q8×C173—21no—
99D4×C173—21no—
100D4×C173—22no—
101D4×C173—41no—
102D4×C173—41no—
103C2×C4×C173—41no—
104C2×C4×C173C2×D411yescharacteristic
105C8×C173C2×D411yescharacteristic
106C8×C173C2411yescharacteristic
107C22×C173—41no—
108C22×C173—42no—
109C22×C173—161no—
110C22×C173C2×D811yescharacteristic
111C2×D173—41no—
112C2×D173—42no—
113C2×D173—162no—
114C2×D173—162no—
115C2×D173—161no—
116C2×D173C2×D812yes—
117C4×C173—41no—
118C4×C173C2×D811yescharacteristic
119C4×C173C22×D411yescharacteristic
120Dic173—41no—
121Dic173C2×D812yes—
122C2×C173—21no—
123C2×C173—81no—
124C2×C173—82no—
125C2×C173C22×D811yescharacteristic
126D173—22no—
127D173—81no—
128D173—82no—
129C173#341511yescharacteristic, complemented, Sylow
130#3415—1731nomaximal, Sylow
131#576—1731no—
132#576—1731no—
133#509—1732no—
134#509—1732no—
135#509—1731no—
136#509D17311yescharacteristic, complemented
137#508—1731no—
138#508—1732no—
139C2×D16—1731no—
140C2×D16—1732no—
141#504—1731no—
142H(5,𝔽2)—3461no—
143H(5,𝔽2)—3461no—
144AGL(1,ℤ/8)—3462no—
145AGL(1,ℤ/8)—3462no—
146AGL(1,ℤ/8)—3461no—
147AGL(1,ℤ/8)—3461no—
148#135—1732no—
149#135—1732no—
150#135—1731no—
151#135C2×D17311yescharacteristic, complemented
152C2×D8—1732no—
153C2×D8—1732no—
154C2×D8—1731no—
155C2×D8C2×D17311yescharacteristic, complemented
156#131—1731no—
157Q32—1731no—
158SD32—1731no—
159SD32—1731no—
160SD32—1732no—
161SD32C2×D17312yescomplemented
162D16—1732no—
163D16—1731no—
164D16—1732no—
165D16—1732no—
166D16C2×D17312yescomplemented
167M32—1732no—
168M32C2×D17311yescharacteristic, complemented
169C2×C16—1731no—
170C2×C16—1732no—
171#41—21no—
172#41—3462no—
173#41—3462no—
174#41—3461no—
175#41—1731no—
176#41—6921no—
177#41—6921no—
178C2×D4—21no—
179C2×D4—3462no—
180C2×D4—3462no—
181C2×D4—3461no—
182C2×D4—6921no—
183C2×D4—6922no—
184C2×D4—6921no—
185C2×D4—6922no—
186Q16—1731no—
187Q16C22×D17311yescharacteristic
188SD16—21no—
189SD16—3462no—
190SD16—3462no—
191SD16—3461no—
192D8—21no—
193D8—3462no—
194D8—3462no—
195D8—3461no—
196D8—1732no—
197D8—1731no—
198D8C22×D17312yes—
199D8C22×D17311yescharacteristic
200M16—1731no—
201M16—1732no—
202C2×C8—1732no—
203C2×C8C22×D17311yescharacteristic
204C16—1731no—
205C16—1731no—
206C16C22×D17312yes—
207C23—41no—
208C23—6922no—
209C23—6922no—
210C23—6921no—
211Q8—21no—
212Q8—3461no—
213Q8—1731no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—3461no—
219D4—3462no—
220D4—1732no—
221D4—1731no—
222D4—6922no—
223D4—6922no—
224D4—6922no—
225D4—6922no—
226D4—6921no—
227D4—6921no—
228C2×C4—41no—
229C2×C4—1732no—
230C2×C4—6922no—
231C2×C4—6922no—
232C2×C4—6921no—
233C2×C4D4×D17311yescharacteristic
234C8—1732no—
235C8D4×D17311yescharacteristic
236C8C23×D17311yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—1732no—
241C22—6922no—
242C22—6921no—
243C22—27681no—
244C22—27682no—
245C22—27682no—
246C22D8×D17311yescharacteristic
247C4—41no—
248C4—1732no—
249C4—6921no—
250C4D8×D17311yescharacteristic
251C4C2×D4×D17311yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—3462no—
256C2—13842no—
257C2—13841no—
258C2C2×D8×D17311yescharacteristic
subgroupG/H
centerC2C2×D8×D173
commutator, cosocleC8×C173C24
FrattiniC8C23×D173
Fitting#509×C173C2
socleC2×C173C22×D8
hypercenter#509D173
hypocenterC173#3415
LCSG ▷ C8×C173 ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #26012293C2semidirect
2. #26012292C2semidirect
3. #26011929C2semidirect
4. #26011928C2semidirect
5. #26011925C2semidirect
6. #26011922C2semidirect
7. #26011921C2semidirect
8. C2×#19149620C2semidirect
9. D16×D173C2semidirect
10. #26011909C2semidirect
11. #509×C173C2semidirect
18. #19149702C22semidirect
19. #19149701C22semidirect
20. #19149698C22semidirect
21. C2×D1384C22semidirect
22. D8×D173C22semidirect
23. D8×D173C22semidirect
25. #19149625C22semidirect
26. #19149624C22semidirect
27. #19149623C22semidirect
28. #19149622C22semidirect
29. D2768C22semidirect
30. #19149620C22semidirect
31. #19149620C22semidirect
32. #19149620C22semidirect
33. #19149618C22semidirect
34. C2×#14583801C22semidirect
35. C16×D173C22semidirect
36. #135×C173C22semidirect
37. C2×D8×C173C22semidirect
38. SD32×C173C22semidirect
39. D16×C173C22semidirect
40. M32×C173C22semidirect
129. C173#3415semidirect
136. #509D173semidirect
151. #135C2×D173semidirect
155. C2×D8C2×D173semidirect
161. SD32C2×D173semidirect
166. D16C2×D173semidirect
168. M32C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122932
#260122922
#260119292
#260119282
#260119252
#260119222
#260119212
C2×#191496202
D16×D1732
#260119092
#509×C1732
#3415173