Finite Group Databasebeta

← #31771282

Subgroups — #31771282

Order 17216.

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#22575448C211yesmaximal, characteristic, complemented
2#22575448C211yesmaximal, characteristic, complemented
3#22575442C211yesmaximal, characteristic, complemented
4#22575438C211yesmaximal, characteristic, complemented
5#22575438C211yesmaximal, characteristic, complemented
6#22575437C211yesmaximal, characteristic, complemented
7#22575429C211yesmaximal, characteristic, complemented
8#22575423C211yesmaximal, characteristic
9#22575415C211yesmaximal, characteristic, complemented
10C4×C22⋊D269C211yesmaximal, characteristic, complemented
11#22575411C211yesmaximal, characteristic, complemented
12#22575409C211yesmaximal, characteristic, complemented
13C2×#16864379C211yesmaximal, characteristic, complemented
14C22⋊C4×D269C211yesmaximal, characteristic, complemented
15C22⋊Q8×C269C211yesmaximal, characteristic, complemented
16C23×D269C2211yescharacteristic
17C2×D1076C2211yescharacteristic
18C2×D1076C2211yescharacteristic
19C2×C22⋊D269C2211yescharacteristic
20C22×Dic269C2211yescharacteristic
21C2×C4×D269C2211yescharacteristic
22C2×C4×D269C2211yescharacteristic
23C2×C4×D269C2211yescharacteristic, complemented
24C4⋊Dic269C2211yescharacteristic, complemented
25C4⋊Dic269C2211yescharacteristic, complemented
26#16864381C2211yescharacteristic, complemented
27#16864381C2211yescharacteristic, complemented
28#16864381C2211yescharacteristic, complemented
29C22⋊Dic269C2211yescharacteristic
30#16864379C2211yescharacteristic
31#16864379C2211yescharacteristic
32#16864379C2211yescharacteristic
33#16864379C2211yescharacteristic
34#16864379C2211yescharacteristic
35#16864379C2211yescharacteristic
36#16864379C2211yescharacteristic
37#16864379C2211yescharacteristic
38#16864379C2211yescharacteristic, complemented
39#16864379C2211yescharacteristic, complemented
40#16864379C2211yescharacteristic, complemented
41C4×Dic269C2211yescharacteristic, complemented
42C4×Dic269C2211yescharacteristic, complemented
43C4×Dic269C2211yescharacteristic, complemented
44C2×Q8×C269C2211yescharacteristic, complemented
45C22×C4×C269C2211yescharacteristic
46C4⋊C4×C269C2211yescharacteristic, complemented
47C4⋊C4×C269C2211yescharacteristic, complemented
48C4⋊C4×C269C2211yescharacteristic, complemented
49C22⋊C4×C269C2211yescharacteristic
50C22⋊C4×C269C2211yescharacteristic
51C22×D269—22no—
52C22×D269—22no—
53C22×D269—41no—
54C22×D269C2311yescharacteristic
55C22×D269C2311yescharacteristic
56C22×D269C2311yescharacteristic
57D1076—22no—
58D1076—41no—
59C22⋊D269—22no—
60C2×Dic269—22no—
61C2×Dic269C2311yescharacteristic
62C2×Dic269C2311yescharacteristic
63C2×Dic269C2311yescharacteristic
64C2×Dic269C2311yescharacteristic
65C2×Dic269C2311yescharacteristic
66C4×D269—22no—
67C4×D269—41no—
68C4×D269—41no—
69C23×C269C2311yescharacteristic
70Q8×C269—41no—
71C2×C4×C269—41no—
72C2×C4×C269C2311yescharacteristic
73C2×C4×C269C2311yescharacteristic
74C2×C4×C269C2311yescharacteristic
75C2×C4×C269C2311yescharacteristic
76C2×C4×C269C2311yescharacteristic
77C2×C4×C269C2311yescharacteristic
78C22×C269—21no—
79C22×C269—21no—
80C22×C269#4112yes—
81C22×C269C2411yescharacteristic
82C2×D269—21no—
83C2×D269—21no—
84C2×D269—21no—
85C2×D269—21no—
86C2×D269—21no—
87C2×D269—21no—
88C2×D269—21no—
89C2×D269—21no—
90C2×D269—44no—
91C2×D269#4112yes—
92C4×C269—21no—
93C4×C269—21no—
94C4×C269—21no—
95C4×C269—21no—
96C4×C269—21no—
97C4×C269—21no—
98Dic269—21no—
99Dic269—21no—
100Dic269—21no—
101Dic269—21no—
102Dic269—21no—
103C2×C269—22no—
104C2×C269C2×#4111yescharacteristic
105C2×C269C2×#4111yescharacteristic
106C2×C269H(5,𝔽2)11yescharacteristic
107D269—22no—
108D269—41no—
109D269—41no—
110C269#55111yescharacteristic, complemented, Sylow
111#551—2691nomaximal, Sylow
112#126—2691no—
113#126—2691no—
114#124—2691no—
115#123—2691no—
116#123—2691no—
117#123—2691no—
118C22⋊Q8—2691no—
119C22⋊Q8D26911yescharacteristic, complemented
120#120—2691no—
121C4×D4—2691no—
122C4×D4—2691no—
123#117—2691no—
124#117—2691no—
125C2×C22⋊C4—2691no—
126C2×C22⋊C4—2691no—
127C24—2691no—
128C2×Q8C2×D26911yescharacteristic, complemented
129C2×D4—2691no—
130C2×D4—2691no—
131C2×D4—2691no—
132C22×C4—2691no—
133C22×C4—2691no—
134C22×C4—2691no—
135C22×C4—2691no—
136C22×C4C2×D26911yescharacteristic
137C4⋊C4—2691no—
138C4⋊C4—2691no—
139C4⋊C4—2691no—
140C4⋊C4—2691no—
141C4⋊C4—2691no—
142C4⋊C4C2×D26911yescharacteristic, complemented
143C4⋊C4C2×D26911yescharacteristic, complemented
144C4⋊C4C2×D26911yescharacteristic, complemented
145C22⋊C4—2691no—
146C22⋊C4—2691no—
147C22⋊C4—2691no—
148C22⋊C4—2691no—
149C22⋊C4—2691no—
150C22⋊C4—2691no—
151C22⋊C4—2691no—
152C22⋊C4—2691no—
153C22⋊C4—2691no—
154C22⋊C4—2691no—
155C22⋊C4—2691no—
156C22⋊C4—2691no—
157C22⋊C4C2×D26911yescharacteristic
158C22⋊C4C2×D26911yescharacteristic
159C42—2691no—
160C42—2691no—
161C42—2691no—
162C23—2691no—
163C23—2691no—
164C23—2691no—
165C23—5382no—
166C23—5382no—
167C23—10761no—
168C23C22×D26911yescharacteristic
169Q8—41no—
170D4—5382no—
171D4—5382no—
172D4—10761no—
173C2×C4—41no—
174C2×C4—2691no—
175C2×C4—2691no—
176C2×C4—2691no—
177C2×C4—2691no—
178C2×C4—2691no—
179C2×C4—5382no—
180C2×C4—5382no—
181C2×C4—10761no—
182C2×C4—10761no—
183C2×C4C22×D26911yescharacteristic
184C2×C4C22×D26911yescharacteristic
185C2×C4C22×D26911yescharacteristic
186C2×C4C22×D26911yescharacteristic
187C2×C4C22×D26911yescharacteristic
188C2×C4C22×D26911yescharacteristic
189C22—21no—
190C22—21no—
191C22—2692no—
192C22—5381no—
193C22—5381no—
194C22—5381no—
195C22—5381no—
196C22—5381no—
197C22—5381no—
198C22—5381no—
199C22—5381no—
200C22—10764no—
201C22#1686440312yes—
202C22C23×D26911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—5381no—
210C4—5381no—
211C4—5381no—
212C4—5381no—
213C4—5381no—
214C2—22no—
215C2—5382no—
216C2—10761no—
217C2—10761no—
218C2#41×D26911yescharacteristic
219C2C2×#1686440311yescharacteristic
220C2#2257549811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D269
commutator, socle, cosocleC22×C269C24
FittingC22⋊Q8×C269C2
prime radical#22575423C2
hypercenterC22⋊Q8D269
hypocenterC269#551
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22575448C2semidirect
2. #22575448C2semidirect
3. #22575442C2semidirect
4. #22575438C2semidirect
5. #22575438C2semidirect
6. #22575437C2semidirect
7. #22575429C2semidirect
9. #22575415C2semidirect
10. C4×C22⋊D269C2semidirect
11. #22575411C2semidirect
12. #22575409C2semidirect
13. C2×#16864379C2semidirect
14. C22⋊C4×D269C2semidirect
15. C22⋊Q8×C269C2semidirect
23. C2×C4×D269C22semidirect
24. C4⋊Dic269C22semidirect
25. C4⋊Dic269C22semidirect
26. #16864381C22semidirect
27. #16864381C22semidirect
28. #16864381C22semidirect
38. #16864379C22semidirect
39. #16864379C22semidirect
40. #16864379C22semidirect
41. C4×Dic269C22semidirect
42. C4×Dic269C22semidirect
43. C4×Dic269C22semidirect
44. C2×Q8×C269C22semidirect
46. C4⋊C4×C269C22semidirect
47. C4⋊C4×C269C22semidirect
48. C4⋊C4×C269C22semidirect
110. C269#551semidirect
119. C22⋊Q8D269semidirect
128. C2×Q8C2×D269semidirect
142. C4⋊C4C2×D269semidirect
143. C4⋊C4C2×D269semidirect
144. C4⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
#225754482
#225754422
#225754382
#225754372
#225754292
#225754232
#225754152
C4×C22⋊D2692
#225754112
#225754092
C2×#168643792
C22⋊C4×D2692
C22⋊Q8×C2692
#551269