Finite Group Databasebeta

← #26012168

Subgroups — #26012168

Order 11072.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#19149674C211yesmaximal, characteristic, complemented
2#19149674C211yesmaximal, characteristic, complemented
3#19149668C211yesmaximal, characteristic, complemented
4#19149664C211yesmaximal, characteristic, complemented
5#19149664C211yesmaximal, characteristic, complemented
6#19149663C211yesmaximal, characteristic, complemented
7#19149655C211yesmaximal, characteristic, complemented
8#19149649C211yesmaximal, characteristic
9#19149641C211yesmaximal, characteristic, complemented
10C4×C22⋊D173C211yesmaximal, characteristic, complemented
11#19149637C211yesmaximal, characteristic, complemented
12#19149635C211yesmaximal, characteristic, complemented
13C2×#14583803C211yesmaximal, characteristic, complemented
14C22⋊C4×D173C211yesmaximal, characteristic, complemented
15C22⋊Q8×C173C211yesmaximal, characteristic, complemented
16C23×D173C2211yescharacteristic
17C2×D692C2211yescharacteristic
18C2×D692C2211yescharacteristic
19C2×C22⋊D173C2211yescharacteristic
20C22×Dic173C2211yescharacteristic
21C2×C4×D173C2211yescharacteristic
22C2×C4×D173C2211yescharacteristic
23C2×C4×D173C2211yescharacteristic, complemented
24C4⋊Dic173C2211yescharacteristic, complemented
25C4⋊Dic173C2211yescharacteristic, complemented
26#14583805C2211yescharacteristic, complemented
27#14583805C2211yescharacteristic, complemented
28#14583805C2211yescharacteristic, complemented
29C22⋊Dic173C2211yescharacteristic
30#14583803C2211yescharacteristic
31#14583803C2211yescharacteristic
32#14583803C2211yescharacteristic
33#14583803C2211yescharacteristic
34#14583803C2211yescharacteristic
35#14583803C2211yescharacteristic
36#14583803C2211yescharacteristic
37#14583803C2211yescharacteristic
38#14583803C2211yescharacteristic, complemented
39#14583803C2211yescharacteristic, complemented
40#14583803C2211yescharacteristic, complemented
41C4×Dic173C2211yescharacteristic, complemented
42C4×Dic173C2211yescharacteristic, complemented
43C4×Dic173C2211yescharacteristic, complemented
44C2×Q8×C173C2211yescharacteristic, complemented
45C22×C4×C173C2211yescharacteristic
46C4⋊C4×C173C2211yescharacteristic, complemented
47C4⋊C4×C173C2211yescharacteristic, complemented
48C4⋊C4×C173C2211yescharacteristic, complemented
49C22⋊C4×C173C2211yescharacteristic
50C22⋊C4×C173C2211yescharacteristic
51C22×D173—22no—
52C22×D173—22no—
53C22×D173—41no—
54C22×D173C2311yescharacteristic
55C22×D173C2311yescharacteristic
56C22×D173C2311yescharacteristic
57D692—22no—
58D692—41no—
59C22⋊D173—22no—
60C2×Dic173—22no—
61C2×Dic173C2311yescharacteristic
62C2×Dic173C2311yescharacteristic
63C2×Dic173C2311yescharacteristic
64C2×Dic173C2311yescharacteristic
65C2×Dic173C2311yescharacteristic
66C4×D173—22no—
67C4×D173—41no—
68C4×D173—41no—
69C23×C173C2311yescharacteristic
70Q8×C173—41no—
71C2×C4×C173—41no—
72C2×C4×C173C2311yescharacteristic
73C2×C4×C173C2311yescharacteristic
74C2×C4×C173C2311yescharacteristic
75C2×C4×C173C2311yescharacteristic
76C2×C4×C173C2311yescharacteristic
77C2×C4×C173C2311yescharacteristic
78C22×C173—21no—
79C22×C173—21no—
80C22×C173#4112yes—
81C22×C173C2411yescharacteristic
82C2×D173—21no—
83C2×D173—21no—
84C2×D173—21no—
85C2×D173—21no—
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—21no—
89C2×D173—21no—
90C2×D173—44no—
91C2×D173#4112yes—
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—21no—
97C4×C173—21no—
98Dic173—21no—
99Dic173—21no—
100Dic173—21no—
101Dic173—21no—
102Dic173—21no—
103C2×C173—22no—
104C2×C173C2×#4111yescharacteristic
105C2×C173C2×#4111yescharacteristic
106C2×C173H(5,𝔽2)11yescharacteristic
107D173—22no—
108D173—41no—
109D173—41no—
110C173#55111yescharacteristic, complemented, Sylow
111#551—1731nomaximal, Sylow
112#126—1731no—
113#126—1731no—
114#124—1731no—
115#123—1731no—
116#123—1731no—
117#123—1731no—
118C22⋊Q8—1731no—
119C22⋊Q8D17311yescharacteristic, complemented
120#120—1731no—
121C4×D4—1731no—
122C4×D4—1731no—
123#117—1731no—
124#117—1731no—
125C2×C22⋊C4—1731no—
126C2×C22⋊C4—1731no—
127C24—1731no—
128C2×Q8C2×D17311yescharacteristic, complemented
129C2×D4—1731no—
130C2×D4—1731no—
131C2×D4—1731no—
132C22×C4—1731no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4—1731no—
136C22×C4C2×D17311yescharacteristic
137C4⋊C4—1731no—
138C4⋊C4—1731no—
139C4⋊C4—1731no—
140C4⋊C4—1731no—
141C4⋊C4—1731no—
142C4⋊C4C2×D17311yescharacteristic, complemented
143C4⋊C4C2×D17311yescharacteristic, complemented
144C4⋊C4C2×D17311yescharacteristic, complemented
145C22⋊C4—1731no—
146C22⋊C4—1731no—
147C22⋊C4—1731no—
148C22⋊C4—1731no—
149C22⋊C4—1731no—
150C22⋊C4—1731no—
151C22⋊C4—1731no—
152C22⋊C4—1731no—
153C22⋊C4—1731no—
154C22⋊C4—1731no—
155C22⋊C4—1731no—
156C22⋊C4—1731no—
157C22⋊C4C2×D17311yescharacteristic
158C22⋊C4C2×D17311yescharacteristic
159C42—1731no—
160C42—1731no—
161C42—1731no—
162C23—3462no—
163C23—3462no—
164C23—1731no—
165C23—1731no—
166C23—1731no—
167C23—6921no—
168C23C22×D17311yescharacteristic
169Q8—41no—
170D4—3462no—
171D4—3462no—
172D4—6921no—
173C2×C4—41no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—1731no—
179C2×C4—1731no—
180C2×C4—1731no—
181C2×C4—6921no—
182C2×C4—6921no—
183C2×C4C22×D17311yescharacteristic
184C2×C4C22×D17311yescharacteristic
185C2×C4C22×D17311yescharacteristic
186C2×C4C22×D17311yescharacteristic
187C2×C4C22×D17311yescharacteristic
188C2×C4C22×D17311yescharacteristic
189C22—21no—
190C22—21no—
191C22—3461no—
192C22—3461no—
193C22—3461no—
194C22—3461no—
195C22—3461no—
196C22—3461no—
197C22—3461no—
198C22—3461no—
199C22—1732no—
200C22—6924no—
201C22#1458382712yes—
202C22C23×D17311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—3461no—
210C4—3461no—
211C4—3461no—
212C4—3461no—
213C4—3461no—
214C2—22no—
215C2—3462no—
216C2—6921no—
217C2—6921no—
218C2#41×D17311yescharacteristic
219C2C2×#1458382711yescharacteristic
220C2#1914972411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
FittingC22⋊Q8×C173C2
prime radical#19149649C2
hypercenterC22⋊Q8D173
hypocenterC173#551
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #19149674C2semidirect
2. #19149674C2semidirect
3. #19149668C2semidirect
4. #19149664C2semidirect
5. #19149664C2semidirect
6. #19149663C2semidirect
7. #19149655C2semidirect
9. #19149641C2semidirect
10. C4×C22⋊D173C2semidirect
11. #19149637C2semidirect
12. #19149635C2semidirect
13. C2×#14583803C2semidirect
14. C22⋊C4×D173C2semidirect
15. C22⋊Q8×C173C2semidirect
23. C2×C4×D173C22semidirect
24. C4⋊Dic173C22semidirect
25. C4⋊Dic173C22semidirect
26. #14583805C22semidirect
27. #14583805C22semidirect
28. #14583805C22semidirect
38. #14583803C22semidirect
39. #14583803C22semidirect
40. #14583803C22semidirect
41. C4×Dic173C22semidirect
42. C4×Dic173C22semidirect
43. C4×Dic173C22semidirect
44. C2×Q8×C173C22semidirect
46. C4⋊C4×C173C22semidirect
47. C4⋊C4×C173C22semidirect
48. C4⋊C4×C173C22semidirect
110. C173#551semidirect
119. C22⋊Q8D173semidirect
128. C2×Q8C2×D173semidirect
142. C4⋊C4C2×D173semidirect
143. C4⋊C4C2×D173semidirect
144. C4⋊C4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#191496742
#191496682
#191496642
#191496632
#191496552
#191496492
#191496412
C4×C22⋊D1732
#191496372
#191496352
C2×#145838032
C22⋊C4×D1732
C22⋊Q8×C1732
#551173