Finite Group Databasebeta

← #31774479

Subgroups — #31774479

Order 17344.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#22576155C211yesmaximal, characteristic, complemented
2#22576153C211yesmaximal, characteristic, complemented
3#22576147C211yesmaximal, characteristic, complemented
4#22576146C211yesmaximal, characteristic, complemented
5#22576143C211yesmaximal, characteristic, complemented
6#22576141C211yesmaximal, characteristic, complemented
7#22576138C211yesmaximal, characteristic, complemented
8#22576132C211yesmaximal, characteristic, complemented
9#22576131C211yesmaximal, characteristic, complemented
10#22576128C211yesmaximal, characteristic, complemented
11#22576128C211yesmaximal, characteristic, complemented
12C4×C22⋊D271C211yesmaximal, characteristic, complemented
13(C22⋊D271)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D271C211yesmaximal, characteristic, complemented
15#121×C271C211yesmaximal, characteristic, complemented
16C23×D271C2211yescharacteristic
17C2×Dic542C2211yescharacteristic, complemented
18C2×C22⋊D271C2211yescharacteristic
19C2×C22⋊D271C2211yescharacteristic
20C2×C22⋊D271C2211yescharacteristic
21C2×C22⋊D271C2211yescharacteristic
22C22×Dic271C2211yescharacteristic, complemented
23C2×C4×D271C2211yescharacteristic
24C2×C4×D271C2211yescharacteristic
25C4⋊Dic271C2211yescharacteristic, complemented
26Dic271⋊C4C2211yescharacteristic, complemented
27Dic271⋊C4C2211yescharacteristic, complemented
28Dic271⋊C4C2211yescharacteristic, complemented
29Dic271⋊C4C2211yescharacteristic, complemented
30C22⋊Dic271C2211yescharacteristic
31C22⋊Dic271C2211yescharacteristic
32C22⋊Dic271C2211yescharacteristic
33C22⋊Dic271C2211yescharacteristic, complemented
34C22⋊Dic271C2211yescharacteristic, complemented
35C22⋊Dic271C2211yescharacteristic, complemented
36(C2×D271)⋊C4C2211yescharacteristic
37(C2×D271)⋊C4C2211yescharacteristic
38(C2×D271)⋊C4C2211yescharacteristic
39(C2×D271)⋊C4C2211yescharacteristic
40(C2×D271)⋊C4C2211yescharacteristic
41(C2×D271)⋊C4C2211yescharacteristic
42C4×Dic271C2211yescharacteristic, complemented
43C4×Dic271C2211yescharacteristic, complemented
44C2×D4×C271C2211yescharacteristic
45C2×D4×C271C2211yescharacteristic
46C2×D4×C271C2211yescharacteristic, complemented
47C22×C4×C271C2211yescharacteristic, complemented
48C4⋊C4×C271C2211yescharacteristic, complemented
49C22⋊C4×C271C2211yescharacteristic
50C22⋊C4×C271C2211yescharacteristic, complemented
51C22×D271—22no—
52C22×D271—41no—
53C22×D271—41no—
54C22×D271C2311yescharacteristic
55C22×D271C2311yescharacteristic
56Dic542—41no—
57C22⋊D271—22no—
58C22⋊D271—22no—
59C22⋊D271—41no—
60C22⋊D271—41no—
61C2×Dic271—41no—
62C2×Dic271C2311yescharacteristic
63C2×Dic271C2311yescharacteristic
64C2×Dic271C2311yescharacteristic
65C2×Dic271C2311yescharacteristic
66C2×Dic271C2311yescharacteristic
67C2×Dic271C2311yescharacteristic
68C4×D271—41no—
69C4×D271—41no—
70C23×C271C2311yescharacteristic
71C23×C271C2311yescharacteristic
72C23×C271C2311yescharacteristic
73D4×C271—41no—
74D4×C271—41no—
75D4×C271—41no—
76C2×C4×C271—41no—
77C2×C4×C271C2311yescharacteristic
78C2×C4×C271C2311yescharacteristic
79C2×C4×C271C2311yescharacteristic
80C2×C4×C271C2311yescharacteristic
81C22×C271—21no—
82C22×C271—21no—
83C22×C271—21no—
84C22×C271—21no—
85C22×C271—21no—
86C22×C271—21no—
87C22×C271—21no—
88C22×C271—21no—
89C22×C271—21no—
90C22×C271C2411yescharacteristic
91C2×D271—21no—
92C2×D271—21no—
93C2×D271—21no—
94C2×D271—21no—
95C2×D271—21no—
96C2×D271—44no—
97C2×D271#4112yes—
98C4×C271—21no—
99C4×C271—21no—
100C4×C271—21no—
101C4×C271—21no—
102Dic271—21no—
103Dic271—21no—
104Dic271—21no—
105Dic271—21no—
106Dic271—21no—
107Dic271—21no—
108C2×C271—41no—
109C2×C271—41no—
110C2×C271—41no—
111C2×C271C2×#4111yescharacteristic
112C2×C271H(5,𝔽2)11yescharacteristic
113C2×C271H(5,𝔽2)11yescharacteristic
114D271—22no—
115D271—41no—
116C271#53811yescharacteristic, complemented, Sylow
117#538—2711nomaximal, Sylow
118#126—2711no—
119#126—2711no—
120#124—2711no—
121#124—2711no—
122#123—2711no—
123#123—2711no—
124C22⋊Q8—2711no—
125#121—2711no—
126#121—2711no—
127#121D27111yescharacteristic, complemented
128#120—2711no—
129#120—2711no—
130C4×D4—2711no—
131C4×D4—2711no—
132C2×C22⋊C4—2711no—
133C24—2711no—
134C2×Q8—2711no—
135C2×D4—2711no—
136C2×D4—2711no—
137C2×D4—2711no—
138C2×D4—2711no—
139C2×D4C2×D27111yescharacteristic
140C2×D4C2×D27111yescharacteristic
141C2×D4C2×D27111yescharacteristic, complemented
142C22×C4—2711no—
143C22×C4—2711no—
144C22×C4—2711no—
145C22×C4C2×D27111yescharacteristic, complemented
146C4⋊C4—2711no—
147C4⋊C4—2711no—
148C4⋊C4—2711no—
149C4⋊C4—2711no—
150C4⋊C4—2711no—
151C4⋊C4C2×D27111yescharacteristic, complemented
152C22⋊C4—2711no—
153C22⋊C4—2711no—
154C22⋊C4—2711no—
155C22⋊C4—2711no—
156C22⋊C4—2711no—
157C22⋊C4—2711no—
158C22⋊C4—2711no—
159C22⋊C4—2711no—
160C22⋊C4—2711no—
161C22⋊C4—2711no—
162C22⋊C4—2711no—
163C22⋊C4—2711no—
164C22⋊C4C2×D27111yescharacteristic
165C22⋊C4C2×D27111yescharacteristic, complemented
166C42—2711no—
167C42—2711no—
168C23—2711no—
169C23—2711no—
170C23—5422no—
171C23—10841no—
172C23—10841no—
173C23C22×D27111yescharacteristic
174C23C22×D27111yescharacteristic
175C23C22×D27111yescharacteristic
176Q8—10841no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—5422no—
181D4—5422no—
182D4—10841no—
183D4—10841no—
184C2×C4—41no—
185C2×C4—2711no—
186C2×C4—2711no—
187C2×C4—2711no—
188C2×C4—2711no—
189C2×C4—2711no—
190C2×C4—2711no—
191C2×C4—10841no—
192C2×C4—10841no—
193C2×C4—10841no—
194C2×C4C22×D27111yescharacteristic
195C2×C4C22×D27111yescharacteristic
196C2×C4C22×D27111yescharacteristic
197C2×C4C22×D27111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—2712no—
208C22—5421no—
209C22—5421no—
210C22—5421no—
211C22—5421no—
212C22—5421no—
213C22—10844no—
214C22C23×D27111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—5421no—
220C4—5421no—
221C4—5421no—
222C4—5421no—
223C4—5421no—
224C4—5421no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—5422no—
229C2—10841no—
230C2#41×D27111yescharacteristic
231C2#2257620511yescharacteristic
232C2#2257620511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D271
commutator, socle, cosocleC22×C271C24
Fitting#121×C271C2
hypercenter#121D271
hypocenterC271#538
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22576155C2semidirect
2. #22576153C2semidirect
3. #22576147C2semidirect
4. #22576146C2semidirect
5. #22576143C2semidirect
6. #22576141C2semidirect
7. #22576138C2semidirect
8. #22576132C2semidirect
9. #22576131C2semidirect
10. #22576128C2semidirect
11. #22576128C2semidirect
12. C4×C22⋊D271C2semidirect
13. (C22⋊D271)⋊C4C2semidirect
14. C22⋊C4×D271C2semidirect
15. #121×C271C2semidirect
17. C2×Dic542C22semidirect
22. C22×Dic271C22semidirect
25. C4⋊Dic271C22semidirect
26. Dic271⋊C4C22semidirect
27. Dic271⋊C4C22semidirect
28. Dic271⋊C4C22semidirect
29. Dic271⋊C4C22semidirect
33. C22⋊Dic271C22semidirect
34. C22⋊Dic271C22semidirect
35. C22⋊Dic271C22semidirect
42. C4×Dic271C22semidirect
43. C4×Dic271C22semidirect
46. C2×D4×C271C22semidirect
47. C22×C4×C271C22semidirect
48. C4⋊C4×C271C22semidirect
50. C22⋊C4×C271C22semidirect
116. C271#538semidirect
127. #121D271semidirect
141. C2×D4C2×D271semidirect
145. C22×C4C2×D271semidirect
151. C4⋊C4C2×D271semidirect
165. C22⋊C4C2×D271semidirect

Maximal subgroups

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

subgroupindex
#225761552
#225761532
#225761472
#225761462
#225761432
#225761412
#225761382
#225761322
#225761312
#225761282
C4×C22⋊D2712
(C22⋊D271)⋊C42
C22⋊C4×D2712
#121×C2712
#538271