Finite Group Databasebeta

← #31774574

Subgroups — #31774574

Order 17344.

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#22576155C211yesmaximal, characteristic, complemented
2#22576155C211yesmaximal, characteristic, complemented
3#22576149C211yesmaximal, characteristic, complemented
4#22576145C211yesmaximal, characteristic, complemented
5#22576145C211yesmaximal, characteristic, complemented
6#22576144C211yesmaximal, characteristic, complemented
7#22576136C211yesmaximal, characteristic, complemented
8#22576130C211yesmaximal, characteristic
9#22576122C211yesmaximal, characteristic, complemented
10C4×C22⋊D271C211yesmaximal, characteristic, complemented
11#22576118C211yesmaximal, characteristic, complemented
12#22576116C211yesmaximal, characteristic, complemented
13C2×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D271C211yesmaximal, characteristic, complemented
15C22⋊Q8×C271C211yesmaximal, characteristic, complemented
16C23×D271C2211yescharacteristic
17C2×D1084C2211yescharacteristic
18C2×D1084C2211yescharacteristic
19C2×C22⋊D271C2211yescharacteristic
20C22×Dic271C2211yescharacteristic
21C2×C4×D271C2211yescharacteristic
22C2×C4×D271C2211yescharacteristic
23C2×C4×D271C2211yescharacteristic, complemented
24C4⋊Dic271C2211yescharacteristic, complemented
25C4⋊Dic271C2211yescharacteristic, complemented
26Dic271⋊C4C2211yescharacteristic, complemented
27Dic271⋊C4C2211yescharacteristic, complemented
28Dic271⋊C4C2211yescharacteristic, complemented
29C22⋊Dic271C2211yescharacteristic
30(C2×D271)⋊C4C2211yescharacteristic
31(C2×D271)⋊C4C2211yescharacteristic
32(C2×D271)⋊C4C2211yescharacteristic
33(C2×D271)⋊C4C2211yescharacteristic
34(C2×D271)⋊C4C2211yescharacteristic
35(C2×D271)⋊C4C2211yescharacteristic
36(C2×D271)⋊C4C2211yescharacteristic
37(C2×D271)⋊C4C2211yescharacteristic
38(C2×D271)⋊C4C2211yescharacteristic, complemented
39(C2×D271)⋊C4C2211yescharacteristic, complemented
40(C2×D271)⋊C4C2211yescharacteristic, complemented
41C4×Dic271C2211yescharacteristic, complemented
42C4×Dic271C2211yescharacteristic, complemented
43C4×Dic271C2211yescharacteristic, complemented
44C2×Q8×C271C2211yescharacteristic, complemented
45C22×C4×C271C2211yescharacteristic
46C4⋊C4×C271C2211yescharacteristic, complemented
47C4⋊C4×C271C2211yescharacteristic, complemented
48C4⋊C4×C271C2211yescharacteristic, complemented
49C22⋊C4×C271C2211yescharacteristic
50C22⋊C4×C271C2211yescharacteristic
51C22×D271—22no—
52C22×D271—22no—
53C22×D271—41no—
54C22×D271C2311yescharacteristic
55C22×D271C2311yescharacteristic
56C22×D271C2311yescharacteristic
57D1084—22no—
58D1084—41no—
59C22⋊D271—22no—
60C2×Dic271—22no—
61C2×Dic271C2311yescharacteristic
62C2×Dic271C2311yescharacteristic
63C2×Dic271C2311yescharacteristic
64C2×Dic271C2311yescharacteristic
65C2×Dic271C2311yescharacteristic
66C4×D271—22no—
67C4×D271—41no—
68C4×D271—41no—
69C23×C271C2311yescharacteristic
70Q8×C271—41no—
71C2×C4×C271—41no—
72C2×C4×C271C2311yescharacteristic
73C2×C4×C271C2311yescharacteristic
74C2×C4×C271C2311yescharacteristic
75C2×C4×C271C2311yescharacteristic
76C2×C4×C271C2311yescharacteristic
77C2×C4×C271C2311yescharacteristic
78C22×C271—21no—
79C22×C271—21no—
80C22×C271#4112yes—
81C22×C271C2411yescharacteristic
82C2×D271—21no—
83C2×D271—21no—
84C2×D271—21no—
85C2×D271—21no—
86C2×D271—21no—
87C2×D271—21no—
88C2×D271—21no—
89C2×D271—21no—
90C2×D271—44no—
91C2×D271#4112yes—
92C4×C271—21no—
93C4×C271—21no—
94C4×C271—21no—
95C4×C271—21no—
96C4×C271—21no—
97C4×C271—21no—
98Dic271—21no—
99Dic271—21no—
100Dic271—21no—
101Dic271—21no—
102Dic271—21no—
103C2×C271—22no—
104C2×C271C2×#4111yescharacteristic
105C2×C271C2×#4111yescharacteristic
106C2×C271H(5,𝔽2)11yescharacteristic
107D271—22no—
108D271—41no—
109D271—41no—
110C271#55111yescharacteristic, complemented, Sylow
111#551—2711nomaximal, Sylow
112#126—2711no—
113#126—2711no—
114#124—2711no—
115#123—2711no—
116#123—2711no—
117#123—2711no—
118C22⋊Q8—2711no—
119C22⋊Q8D27111yescharacteristic, complemented
120#120—2711no—
121C4×D4—2711no—
122C4×D4—2711no—
123#117—2711no—
124#117—2711no—
125C2×C22⋊C4—2711no—
126C2×C22⋊C4—2711no—
127C24—2711no—
128C2×Q8C2×D27111yescharacteristic, complemented
129C2×D4—2711no—
130C2×D4—2711no—
131C2×D4—2711no—
132C22×C4—2711no—
133C22×C4—2711no—
134C22×C4—2711no—
135C22×C4—2711no—
136C22×C4C2×D27111yescharacteristic
137C4⋊C4—2711no—
138C4⋊C4—2711no—
139C4⋊C4—2711no—
140C4⋊C4—2711no—
141C4⋊C4—2711no—
142C4⋊C4C2×D27111yescharacteristic, complemented
143C4⋊C4C2×D27111yescharacteristic, complemented
144C4⋊C4C2×D27111yescharacteristic, complemented
145C22⋊C4—2711no—
146C22⋊C4—2711no—
147C22⋊C4—2711no—
148C22⋊C4—2711no—
149C22⋊C4—2711no—
150C22⋊C4—2711no—
151C22⋊C4—2711no—
152C22⋊C4—2711no—
153C22⋊C4—2711no—
154C22⋊C4—2711no—
155C22⋊C4—2711no—
156C22⋊C4—2711no—
157C22⋊C4C2×D27111yescharacteristic
158C22⋊C4C2×D27111yescharacteristic
159C42—2711no—
160C42—2711no—
161C42—2711no—
162C23—2711no—
163C23—2711no—
164C23—2711no—
165C23—5422no—
166C23—5422no—
167C23—10841no—
168C23C22×D27111yescharacteristic
169Q8—41no—
170D4—5422no—
171D4—5422no—
172D4—10841no—
173C2×C4—41no—
174C2×C4—2711no—
175C2×C4—2711no—
176C2×C4—2711no—
177C2×C4—2711no—
178C2×C4—2711no—
179C2×C4—5422no—
180C2×C4—5422no—
181C2×C4—10841no—
182C2×C4—10841no—
183C2×C4C22×D27111yescharacteristic
184C2×C4C22×D27111yescharacteristic
185C2×C4C22×D27111yescharacteristic
186C2×C4C22×D27111yescharacteristic
187C2×C4C22×D27111yescharacteristic
188C2×C4C22×D27111yescharacteristic
189C22—21no—
190C22—21no—
191C22—2712no—
192C22—5421no—
193C22—5421no—
194C22—5421no—
195C22—5421no—
196C22—5421no—
197C22—5421no—
198C22—5421no—
199C22—5421no—
200C22—10844no—
201C22#1686465512yes—
202C22C23×D27111yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—5421no—
210C4—5421no—
211C4—5421no—
212C4—5421no—
213C4—5421no—
214C2—22no—
215C2—5422no—
216C2—10841no—
217C2—10841no—
218C2#41×D27111yescharacteristic
219C2C2×#1686465511yescharacteristic
220C2#2257620511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D271
commutator, socle, cosocleC22×C271C24
FittingC22⋊Q8×C271C2
prime radical#22576130C2
hypercenterC22⋊Q8D271
hypocenterC271#551
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22576155C2semidirect
2. #22576155C2semidirect
3. #22576149C2semidirect
4. #22576145C2semidirect
5. #22576145C2semidirect
6. #22576144C2semidirect
7. #22576136C2semidirect
9. #22576122C2semidirect
10. C4×C22⋊D271C2semidirect
11. #22576118C2semidirect
12. #22576116C2semidirect
13. C2×(C2×D271)⋊C4C2semidirect
14. C22⋊C4×D271C2semidirect
15. C22⋊Q8×C271C2semidirect
23. C2×C4×D271C22semidirect
24. C4⋊Dic271C22semidirect
25. C4⋊Dic271C22semidirect
26. Dic271⋊C4C22semidirect
27. Dic271⋊C4C22semidirect
28. Dic271⋊C4C22semidirect
38. (C2×D271)⋊C4C22semidirect
39. (C2×D271)⋊C4C22semidirect
40. (C2×D271)⋊C4C22semidirect
41. C4×Dic271C22semidirect
42. C4×Dic271C22semidirect
43. C4×Dic271C22semidirect
44. C2×Q8×C271C22semidirect
46. C4⋊C4×C271C22semidirect
47. C4⋊C4×C271C22semidirect
48. C4⋊C4×C271C22semidirect
110. C271#551semidirect
119. C22⋊Q8D271semidirect
128. C2×Q8C2×D271semidirect
142. C4⋊C4C2×D271semidirect
143. C4⋊C4C2×D271semidirect
144. C4⋊C4C2×D271semidirect

Maximal subgroups

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

subgroupindex
#225761552
#225761492
#225761452
#225761442
#225761362
#225761302
#225761222
C4×C22⋊D2712
#225761182
#225761162
C2×(C2×D271)⋊C42
C22⋊C4×D2712
C22⋊Q8×C2712
#551271