Finite Group Databasebeta

← #26012167

Subgroups — #26012167

Order 11072.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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#19149672C211yesmaximal, characteristic, complemented
2#19149672C211yesmaximal, characteristic, complemented
3#19149665C211yesmaximal, characteristic, complemented
4#19149663C211yesmaximal, characteristic, complemented
5#19149662C211yesmaximal, characteristic, complemented
6#19149661C211yesmaximal, characteristic, complemented
7#19149658C211yesmaximal, characteristic, complemented
8#19149656C211yesmaximal, characteristic, complemented
9#19149648C211yesmaximal, characteristic
10C4×C22⋊D173C211yesmaximal, characteristic, complemented
11#19149639C211yesmaximal, characteristic, complemented
12#19149638C211yesmaximal, characteristic, complemented
13#19149635C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic173C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C173C211yesmaximal, characteristic, complemented
16C2×Dic346C2211yescharacteristic, complemented
17C2×C22⋊D173C2211yescharacteristic
18C2×C22⋊D173C2211yescharacteristic
19C2×C22⋊D173C2211yescharacteristic
20C22×Dic173C2211yescharacteristic
21C22×Dic173C2211yescharacteristic
22C2×C4×D173C2211yescharacteristic, complemented
23C4⋊Dic173C2211yescharacteristic, complemented
24C4⋊Dic173C2211yescharacteristic, complemented
25C4⋊Dic173C2211yescharacteristic, complemented
26#14583805C2211yescharacteristic, complemented
27#14583805C2211yescharacteristic, complemented
28#14583805C2211yescharacteristic, complemented
29#14583805C2211yescharacteristic, complemented
30#14583805C2211yescharacteristic, complemented
31C22⋊Dic173C2211yescharacteristic
32C22⋊Dic173C2211yescharacteristic
33C22⋊Dic173C2211yescharacteristic
34C22⋊Dic173C2211yescharacteristic
35C22⋊Dic173C2211yescharacteristic
36C22⋊Dic173C2211yescharacteristic
37C22⋊Dic173C2211yescharacteristic
38#14583803C2211yescharacteristic, complemented
39#14583803C2211yescharacteristic, complemented
40#14583803C2211yescharacteristic, complemented
41C4×Dic173C2211yescharacteristic, complemented
42C4×Dic173C2211yescharacteristic, complemented
43C4×Dic173C2211yescharacteristic, complemented
44C24×C173C2211yescharacteristic
45C22×C4×C173C2211yescharacteristic
46C22×C4×C173C2211yescharacteristic
47C22⋊C4×C173C2211yescharacteristic
48C22⋊C4×C173C2211yescharacteristic
49C22⋊C4×C173C2211yescharacteristic
50C22⋊C4×C173C2211yescharacteristic
51C22×D173C2311yescharacteristic
52Dic346—41no—
53C22⋊D173—22no—
54C22⋊D173—22no—
55C22⋊D173—41no—
56C2×Dic173—22no—
57C2×Dic173—41no—
58C2×Dic173C2311yescharacteristic
59C2×Dic173C2311yescharacteristic
60C2×Dic173C2311yescharacteristic
61C2×Dic173C2311yescharacteristic
62C2×Dic173C2311yescharacteristic
63C2×Dic173C2311yescharacteristic
64C2×Dic173C2311yescharacteristic
65C4×D173—41no—
66C23×C173—22no—
67C23×C173—22no—
68C23×C173—41no—
69C23×C173C2311yescharacteristic
70C23×C173C2311yescharacteristic
71C23×C173C2311yescharacteristic
72C2×C4×C173—22no—
73C2×C4×C173—41no—
74C2×C4×C173C2311yescharacteristic
75C2×C4×C173C2311yescharacteristic
76C2×C4×C173C2311yescharacteristic
77C2×C4×C173C2311yescharacteristic
78C22×C173—21no—
79C22×C173—21no—
80C22×C173—21no—
81C22×C173—21no—
82C22×C173—21no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173—44no—
86C22×C173#4112yes—
87C22×C173#4112yes—
88C22×C173C2411yescharacteristic
89C2×D173—21no—
90C2×D173—21no—
91C2×D173—21no—
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—21no—
95C4×C173—21no—
96Dic173—21no—
97Dic173—21no—
98Dic173—21no—
99Dic173—21no—
100Dic173—21no—
101Dic173—21no—
102Dic173—21no—
103C2×C173—22no—
104C2×C173—22no—
105C2×C173—41no—
106C2×C173C2×#4111yescharacteristic
107C2×C173C2×#4111yescharacteristic
108C2×C173H(5,𝔽2)11yescharacteristic
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⋊Q8—1731no—
120#120—1731no—
121C4×D4—1731no—
122C4×D4—1731no—
123#117—1731no—
124#117—1731no—
125C2×C22⋊C4—1731no—
126C2×C22⋊C4D17311yescharacteristic, complemented
127C24C2×D17311yescharacteristic
128C2×Q8—1731no—
129C2×D4—1731no—
130C2×D4—1731no—
131C2×D4—1731no—
132C22×C4—1731no—
133C22×C4—1731no—
134C22×C4—1731no—
135C22×C4C2×D17311yescharacteristic
136C22×C4C2×D17311yescharacteristic
137C4⋊C4—1731no—
138C4⋊C4—1731no—
139C4⋊C4—1731no—
140C4⋊C4—1731no—
141C4⋊C4—1731no—
142C4⋊C4—1731no—
143C4⋊C4—1731no—
144C4⋊C4—1731no—
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⋊C4C2×D17311yescharacteristic
156C22⋊C4C2×D17311yescharacteristic
157C22⋊C4C2×D17311yescharacteristic
158C22⋊C4C2×D17311yescharacteristic
159C42—1731no—
160C42—1731no—
161C42—1731no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—1731no—
166C23C22×D17311yescharacteristic
167C23C22×D17311yescharacteristic
168C23C22×D17311yescharacteristic
169Q8—6921no—
170D4—3462no—
171D4—3462no—
172D4—6921no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—3462no—
176C2×C4—1731no—
177C2×C4—1731no—
178C2×C4—1731no—
179C2×C4—1731no—
180C2×C4—1731no—
181C2×C4—1731no—
182C2×C4—1731no—
183C2×C4—6921no—
184C2×C4—6921no—
185C2×C4C22×D17311yescharacteristic
186C2×C4C22×D17311yescharacteristic
187C2×C4C22×D17311yescharacteristic
188C2×C4C22×D17311yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—3461no—
198C22—3461no—
199C22—3461no—
200C22#1458382512yes—
201C22#1458382612yes—
202C22C23×D17311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—3461no—
208C4—3461no—
209C4—3461no—
210C4—3461no—
211C4—3461no—
212C4—3461no—
213C4—3461no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—6921no—
218C2C2×#1458382511yescharacteristic
219C2C2×#1458382611yescharacteristic
220C2#1914972411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D173
commutator, socle, cosocleC22×C173C24
FittingC2×C22⋊C4×C173C2
prime radical#19149648C2
hypercenterC2×C22⋊C4D173
hypocenterC173#551
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #19149672C2semidirect
2. #19149672C2semidirect
3. #19149665C2semidirect
4. #19149663C2semidirect
5. #19149662C2semidirect
6. #19149661C2semidirect
7. #19149658C2semidirect
8. #19149656C2semidirect
10. C4×C22⋊D173C2semidirect
11. #19149639C2semidirect
12. #19149638C2semidirect
13. #19149635C2semidirect
14. C2×C22⋊Dic173C2semidirect
15. C2×C22⋊C4×C173C2semidirect
16. C2×Dic346C22semidirect
22. C2×C4×D173C22semidirect
23. C4⋊Dic173C22semidirect
24. C4⋊Dic173C22semidirect
25. C4⋊Dic173C22semidirect
26. #14583805C22semidirect
27. #14583805C22semidirect
28. #14583805C22semidirect
29. #14583805C22semidirect
30. #14583805C22semidirect
38. #14583803C22semidirect
39. #14583803C22semidirect
40. #14583803C22semidirect
41. C4×Dic173C22semidirect
42. C4×Dic173C22semidirect
43. C4×Dic173C22semidirect
110. C173#551semidirect
126. C2×C22⋊C4D173semidirect

Maximal subgroups

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

subgroupindex
#191496722
#191496652
#191496632
#191496622
#191496612
#191496582
#191496562
#191496482
C4×C22⋊D1732
#191496392
#191496382
#191496352
C2×C22⋊Dic1732
C2×C22⋊C4×C1732
#551173