Finite Group Databasebeta

← #26012133

Subgroups — #26012133

Order 11072.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#14583826C211yesmaximal, characteristic, complemented
2C2×D4×D173C211yesmaximal, characteristic, complemented
3#19149665C211yesmaximal, characteristic, complemented
4#19149664C211yesmaximal, characteristic, complemented
5#19149657C211yesmaximal, characteristic, complemented
6#19149654C211yesmaximal, characteristic, complemented
7#19149652C211yesmaximal, characteristic, complemented
8#19149651C211yesmaximal, characteristic, complemented
9#19149649C211yesmaximal, characteristic, complemented
10#19149649C211yesmaximal, characteristic, complemented
11#19149641C211yesmaximal, characteristic, complemented
12#19149639C211yesmaximal, characteristic, complemented
13C2×#14583803C211yesmaximal, characteristic, complemented
14C22⋊C4×D173C211yesmaximal, characteristic, complemented
15#123×C173C211yesmaximal, characteristic, complemented
16C23×D173C2211yescharacteristic
17C23×D173C2211yescharacteristic
18#14583826—24no—
19C2×Dic346C2211yescharacteristic, complemented
20C2×D692C2211yescharacteristic
21C2×D692C2211yescharacteristic
22C2×D692C2211yescharacteristic, complemented
23C2×D692C2211yescharacteristic, complemented
24C2×C22⋊D173C2211yescharacteristic
25C2×C22⋊D173C2211yescharacteristic
26C2×C22⋊D173C2211yescharacteristic, complemented
27C2×C22⋊D173C2211yescharacteristic, complemented
28D4×D173—24no—
29C22×Dic173C2211yescharacteristic, complemented
30C2×C4×D173C2211yescharacteristic
31C2×C4×D173C2211yescharacteristic, complemented
32C2×C4×D173C2211yescharacteristic, complemented
33C2×C4×D173C2211yescharacteristic, complemented
34#14583805C2211yescharacteristic, complemented
35#14583805C2211yescharacteristic, complemented
36C22⋊Dic173C2211yescharacteristic, complemented
37#14583803C2211yescharacteristic
38#14583803C2211yescharacteristic
39#14583803C2211yescharacteristic
40#14583803C2211yescharacteristic, complemented
41#14583803C2211yescharacteristic, complemented
42#14583803C2211yescharacteristic, complemented
43#14583803C2211yescharacteristic, complemented
44#14583803C2211yescharacteristic, complemented
45C4×Dic173C2211yescharacteristic, complemented
46C2×D4×C173C2211yescharacteristic
47C22×C4×C173C2211yescharacteristic, complemented
48C4⋊C4×C173C2211yescharacteristic, complemented
49C4⋊C4×C173C2211yescharacteristic, complemented
50C22⋊C4×C173C2211yescharacteristic, complemented
51C22⋊C4×C173C2211yescharacteristic, complemented
52C22⋊C4×C173C2211yescharacteristic, complemented
53C22×D173—22no—
54C22×D173—22no—
55C22×D173—22no—
56C22×D173—22no—
57C22×D173—41no—
58C22×D173—41no—
59C22×D173C2311yescharacteristic
60C22×D173C2311yescharacteristic
61C22×D173C2311yescharacteristic
62C22×D173C2311yescharacteristic
63Dic346—22no—
64D692—22no—
65D692—22no—
66D692—22no—
67D692—41no—
68C22⋊D173—22no—
69C22⋊D173—22no—
70C22⋊D173—41no—
71C22⋊D173D414yescomplemented
72C2×Dic173—41no—
73C2×Dic173C2311yescharacteristic
74C2×Dic173C2311yescharacteristic
75C2×Dic173C2311yescharacteristic
76C2×Dic173C2311yescharacteristic
77C4×D173—22no—
78C4×D173—22no—
79C4×D173—22no—
80C4×D173—41no—
81C23×C173C2311yescharacteristic
82C23×C173C2311yescharacteristic
83D4×C173—22no—
84C2×C4×C173—22no—
85C2×C4×C173C2311yescharacteristic
86C2×C4×C173C2311yescharacteristic
87C2×C4×C173C2311yescharacteristic
88C2×C4×C173C2311yescharacteristic
89C2×C4×C173C2311yescharacteristic
90C22×C173—21no—
91C22×C173—21no—
92C22×C173—21no—
93C22×C173—21no—
94C22×C173—21no—
95C22×C173C2×D412yes—
96C22×C173C2411yescharacteristic
97C2×D173—21no—
98C2×D173—21no—
99C2×D173—21no—
100C2×D173—21no—
101C2×D173—21no—
102C2×D173—21no—
103C2×D173—21no—
104C2×D173—21no—
105C2×D173—21no—
106C2×D173—21no—
107C2×D173—44no—
108C2×D173—44no—
109C2×D173C2×D412yes—
110C2×D173#4112yescomplemented
111C4×C173—21no—
112C4×C173—21no—
113C4×C173—21no—
114C4×C173—21no—
115C4×C173—21no—
116Dic173—21no—
117Dic173—21no—
118Dic173—21no—
119Dic173C2×D412yes—
120C2×C173—22no—
121C2×C173—41no—
122C2×C173C22×D411yescharacteristic
123C2×C173C2×#4111yescharacteristic
124C2×C173H(5,𝔽2)11yescharacteristic
125D173—22no—
126D173—22no—
127D173—41no—
128D173—41no—
129C173#54611yescharacteristic, complemented, Sylow
130#546—1731nomaximal, Sylow
131C2×#41—1731no—
132C22×D4—1731no—
133#124—1731no—
134#123—1731no—
135#123D17311yescharacteristic, complemented
136C22⋊Q8—1731no—
137#121—1731no—
138#121—1731no—
139#121—1731no—
140#120—1731no—
141#120—1731no—
142C4×D4—1731no—
143C4×D4—1731no—
144C2×C22⋊C4—1731no—
145C2×C22⋊C4—1731no—
146C24—1731no—
147C24—1731no—
148#41—3464no—
149C2×Q8—1731no—
150C2×D4—3464no—
151C2×D4—1731no—
152C2×D4—1731no—
153C2×D4—1731no—
154C2×D4—1731no—
155C2×D4—1731no—
156C2×D4—1731no—
157C2×D4—1731no—
158C2×D4—1731no—
159C2×D4C2×D17311yescharacteristic
160C22×C4—1731no—
161C22×C4—1731no—
162C22×C4—1731no—
163C22×C4—1731no—
164C22×C4—1731no—
165C22×C4C2×D17311yescharacteristic, complemented
166C4⋊C4—1731no—
167C4⋊C4—1731no—
168C4⋊C4C2×D17311yescharacteristic, complemented
169C4⋊C4C2×D17311yescharacteristic, complemented
170C22⋊C4—1731no—
171C22⋊C4—1731no—
172C22⋊C4—1731no—
173C22⋊C4—1731no—
174C22⋊C4—1731no—
175C22⋊C4—1731no—
176C22⋊C4—1731no—
177C22⋊C4—1731no—
178C22⋊C4—1731no—
179C22⋊C4C2×D17311yescharacteristic, complemented
180C22⋊C4C2×D17311yescharacteristic, complemented
181C22⋊C4C2×D17311yescharacteristic, complemented
182C42—1731no—
183C23—3462no—
184C23—3462no—
185C23—3462no—
186C23—3462no—
187C23—1731no—
188C23—1731no—
189C23—1731no—
190C23—1731no—
191C23—6921no—
192C23—6921no—
193C23C22×D17311yescharacteristic
194C23C22×D17311yescharacteristic
195Q8—3462no—
196D4—22no—
197D4—3462no—
198D4—3462no—
199D4—3462no—
200D4—3462no—
201D4—3462no—
202D4—1734no—
203D4—6921no—
204D4—6921no—
205C2×C4—22no—
206C2×C4—3462no—
207C2×C4—3462no—
208C2×C4—3462no—
209C2×C4—1731no—
210C2×C4—1731no—
211C2×C4—1731no—
212C2×C4—1731no—
213C2×C4—6921no—
214C2×C4—6921no—
215C2×C4C22×D17311yescharacteristic
216C2×C4C22×D17311yescharacteristic
217C2×C4C22×D17311yescharacteristic
218C2×C4C22×D17311yescharacteristic
219C2×C4C22×D17311yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—3461no—
226C22—3461no—
227C22—3461no—
228C22—3461no—
229C22—3461no—
230C22—3461no—
231C22—3461no—
232C22—3461no—
233C22—3461no—
234C22—3461no—
235C22—1732no—
236C22—1732no—
237C22—6924no—
238C22—6924no—
239C22D4×D17312yes—
240C22C23×D17311yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—3461no—
247C4—3461no—
248C4—3461no—
249C4—1732no—
250C2—22no—
251C2—41no—
252C2—3462no—
253C2—3462no—
254C2—6921no—
255C2—6921no—
256C2C2×D4×D17311yescharacteristic
257C2#41×D17311yescharacteristic
258C2#1914972511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
Fitting#123×C173C2
hypercenter#123D173
hypocenterC173#546
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×#14583826C2semidirect
2. C2×D4×D173C2semidirect
3. #19149665C2semidirect
4. #19149664C2semidirect
5. #19149657C2semidirect
6. #19149654C2semidirect
7. #19149652C2semidirect
8. #19149651C2semidirect
9. #19149649C2semidirect
10. #19149649C2semidirect
11. #19149641C2semidirect
12. #19149639C2semidirect
13. C2×#14583803C2semidirect
14. C22⋊C4×D173C2semidirect
15. #123×C173C2semidirect
19. C2×Dic346C22semidirect
22. C2×D692C22semidirect
23. C2×D692C22semidirect
26. C2×C22⋊D173C22semidirect
27. C2×C22⋊D173C22semidirect
29. C22×Dic173C22semidirect
31. C2×C4×D173C22semidirect
32. C2×C4×D173C22semidirect
33. C2×C4×D173C22semidirect
34. #14583805C22semidirect
35. #14583805C22semidirect
36. C22⋊Dic173C22semidirect
40. #14583803C22semidirect
41. #14583803C22semidirect
42. #14583803C22semidirect
43. #14583803C22semidirect
44. #14583803C22semidirect
45. C4×Dic173C22semidirect
47. C22×C4×C173C22semidirect
48. C4⋊C4×C173C22semidirect
49. C4⋊C4×C173C22semidirect
50. C22⋊C4×C173C22semidirect
51. C22⋊C4×C173C22semidirect
52. C22⋊C4×C173C22semidirect
71. C22⋊D173D4semidirect
110. C2×D173#41semidirect
129. C173#546semidirect
135. #123D173semidirect
165. C22×C4C2×D173semidirect
168. C4⋊C4C2×D173semidirect
169. C4⋊C4C2×D173semidirect
179. C22⋊C4C2×D173semidirect
180. C22⋊C4C2×D173semidirect
181. C22⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#145838262
C2×D4×D1732
#191496652
#191496642
#191496572
#191496542
#191496522
#191496512
#191496492
#191496412
#191496392
C2×#145838032
C22⋊C4×D1732
#123×C1732
#546173