Finite Group Databasebeta

← #14539851

Subgroups — #14539851

Order 2176.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#11763420C211yesmaximal, characteristic, complemented
2#11763377C211yesmaximal, characteristic
3#11763183C211yesmaximal, characteristic, complemented
4#11763174C211yesmaximal, characteristic, complemented
5#11763163C211yesmaximal, characteristic, complemented
6#11763033C212yesmaximal, complemented
7#11763007C212yesmaximal, complemented
8#11762970C212yesmaximal, complemented
9#11762958C212yesmaximal, complemented
10#11762871C211yesmaximal, characteristic, complemented
11C17×#497C211yesmaximal, characteristic, complemented
12C2×#63256C2211yescharacteristic
13C2×D4×D17C2211yescharacteristic
14C2×Q16×C17C2212yescomplemented
15C2×SD16×C17C2212yes—
16C17×C4⋊Q8C2211yescharacteristic, complemented
17C17×#124C2211yescharacteristic
18#10587633C2211yescharacteristic
19#10587632C2212yes—
20Q8×Dic17C2212yescomplemented
21C2×#63230C2212yes—
22C2×#63230C2212yescomplemented
23#10587626C2211yescharacteristic
24#10587584—22no—
25#10587575—22no—
26#10587570—22no—
27#10587568—22no—
28#10587560C2211yescharacteristic
29#10587558C2211yescharacteristic
30#10587556C2211yescharacteristic, complemented
31C17×#97C2211yescharacteristic, complemented
32#10587506C2212yescomplemented
33#10587503C2212yes—
34#10587492C2212yes—
35#10587492C2212yescomplemented
36#10587491C2212yes—
37#10587485C2212yescomplemented
38C4×#3622C2211yescharacteristic, complemented
39C23×D17—21no—
40C2×Q8×C17C2311yescharacteristic
41C2×Q8×C17C2312yes—
42C2×D4×C17C2311yescharacteristic
43C2×C22⋊D17—21no—
44C2×C22⋊D17—22no—
45#63256—42no—
46D4×D17—42no—
47C2×D68—21no—
48C2×D68—22no—
49C2×D68C2311yescharacteristic
50C2×D68C2311yescharacteristic
51C2×C4×D17—21no—
52C2×C4×D17C2311yescharacteristic
53Q16×C17—42no—
54SD16×C17—42no—
55C2×C8×C17C2312yes—
56C4⋊C4×C17—22no—
57C22⋊C4×C17—22no—
58C42×C17C2311yescharacteristic
59#63230—42no—
60#63230—42no—
61#63227—22no—
62#63227—22no—
63#63227D412yes—
64C4⋊Dic17—22no—
65C4⋊Dic17C2312yes—
66#63225—22no—
67#63225—22no—
68#63225D412yes—
69C4×Dic17—22no—
70C4×Dic17C2311yescharacteristic
71C2×#3622C2312yes—
72C23×C17—21no—
73C22×D17—21no—
74C22×D17—21no—
75C22×D17—42no—
76C22×D17—82no—
77C22×D17C2×D411yescharacteristic
78Q8×C17—22no—
79Q8×C17—22no—
80Q8×C17—22no—
81D4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2411yescharacteristic
86C22⋊D17—42no—
87C22⋊D17—82no—
88C2×Dic17—22no—
89C2×Dic17C2×D411yescharacteristic
90C2×Dic17C2×D412yes—
91D68—22no—
92D68—22no—
93D68—42no—
94D68—82no—
95C4×D17—22no—
96C4×D17—42no—
97C8×C17—22no—
98#3622—22no—
99#3222—171nomaximal, Sylow
100C22×C17—41no—
101C22×C17—42no—
102C22×C17C22×D411yescharacteristic
103C2×D17—21no—
104C2×D17—22no—
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—42no—
108C2×D17—42no—
109C2×D17—84no—
110C4×C17—22no—
111C4×C17—41no—
112C4×C17—42no—
113C4×C17H(5,𝔽2)12yes—
114Dic17—21no—
115Dic17—22no—
116Dic17—42no—
117#541—171no—
118#535—171no—
119#497D1711yescharacteristic, complemented
120#492—171no—
121#490—171no—
122#488—171no—
123#464—172no—
124#460—172no—
125#452—172no—
126#450—172no—
127#433—171no—
128C2×C17—81no—
129C2×C17#53511yescharacteristic
130C2×C17#57712yes—
131D17—41no—
132D17—81no—
133D17—81no—
134C2×#41—171no—
135C22×D4—171no—
136C2×Q16C2×D1712yescomplemented
137C2×SD16—172no—
138C2×SD16—172no—
139C2×SD16C2×D1712yes—
140C4⋊Q8C2×D1711yescharacteristic, complemented
141#127—171no—
142#127—171no—
143#126—342no—
144#125—171no—
145#125—342no—
146#124—171no—
147#124C2×D1711yescharacteristic
148C22⋊Q8—172no—
149#121—342no—
150#120—342no—
151C4×Q8—172no—
152#117—171no—
153C4⋊C8—172no—
154#103—172no—
155#102—172no—
156#102—172no—
157#102—172no—
158C22⋊C8—172no—
159#97C2×D1711yescharacteristic, complemented
160C4×C8—171no—
161C17#322211yescharacteristic, complemented, Sylow
162C24—341no—
163#41—682no—
164C2×Q8C22×D1711yescharacteristic
165C2×Q8C22×D1712yes—
166C2×D4—171no—
167C2×D4—171no—
168C2×D4—341no—
169C2×D4—341no—
170C2×D4—342no—
171C2×D4—342no—
172C2×D4—682no—
173C2×D4C22×D1711yescharacteristic
174C22×C4—171no—
175C22×C4—341no—
176Q16—42no—
177SD16—42no—
178SD16—682no—
179SD16—682no—
180C2×C8—172no—
181C2×C8C22×D1712yes—
182C4⋊C4—22no—
183C4⋊C4—172no—
184C4⋊C4—172no—
185C4⋊C4—342no—
186C4⋊C4—342no—
187C4⋊C4—342no—
188C22⋊C4—22no—
189C22⋊C4—172no—
190C22⋊C4—342no—
191C22⋊C4—342no—
192C42—171no—
193C42—342no—
194C42C22×D1711yescharacteristic
195C23—21no—
196C23—171no—
197C23—341no—
198C23—341no—
199C23—682no—
200C23—1362no—
201Q8—22no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—342no—
206D4—342no—
207D4—682no—
208D4—682no—
209D4—1362no—
210D4—1362no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—172no—
214C2×C4—171no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—682no—
218C2×C4D4×D1712yes—
219C2×C4C23×D1711yescharacteristic
220C8—22no—
221C8—342no—
222C22—42no—
223C22—41no—
224C22—342no—
225C22—341no—
226C22—682no—
227C22—682no—
228C22—681no—
229C22—681no—
230C22—1364no—
231C22C2×D4×D1711yescharacteristic
232C4—22no—
233C4—41no—
234C4—42no—
235C4—342no—
236C4—341no—
237C4—682no—
238C4#1058769112yes—
239C2—81no—
240C2—681no—
241C2—1361no—
242C2—1361no—
243C2#1176337211yescharacteristic
244C2#1176361812yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#497C2
socleC22×C17C22×D4
prime radical#11763377C2
hypercenter#497D17
hypocenterC17#3222
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #11763420C2semidirect
3. #11763183C2semidirect
4. #11763174C2semidirect
5. #11763163C2semidirect
6. #11763033C2semidirect
7. #11763007C2semidirect
8. #11762970C2semidirect
9. #11762958C2semidirect
10. #11762871C2semidirect
11. C17×#497C2semidirect
14. C2×Q16×C17C22semidirect
16. C17×C4⋊Q8C22semidirect
20. Q8×Dic17C22semidirect
22. C2×#63230C22semidirect
30. #10587556C22semidirect
31. C17×#97C22semidirect
32. #10587506C22semidirect
35. #10587492C22semidirect
37. #10587485C22semidirect
38. C4×#3622C22semidirect
119. #497D17semidirect
136. C2×Q16C2×D17semidirect
140. C4⋊Q8C2×D17semidirect
159. #97C2×D17semidirect
161. C17#3222semidirect

Maximal subgroups

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

subgroupindex
#117634202
#117633772
#117631832
#117631742
#117631632
#117630332
#117630072
#117629702
#117629582
#117628712
C17×#4972
#322217