Finite Group Databasebeta

← #14539849

Subgroups — #14539849

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#11763372C211yesmaximal, characteristic
3#11763201C211yesmaximal, characteristic, complemented
4#11763173C211yesmaximal, characteristic, complemented
5#11763164C211yesmaximal, characteristic, complemented
6#11763029C212yesmaximal, complemented
7#11763007C212yesmaximal, complemented
8#11762968C212yesmaximal, complemented
9#11762958C212yesmaximal, complemented
10#11762868C211yesmaximal, characteristic, complemented
11C17×#492C211yesmaximal, characteristic, complemented
12C2×#63254C2211yescharacteristic
13C2×D4×D17C2211yescharacteristic
14C2×SD16×C17C2212yes—
15C2×SD16×C17C2212yescomplemented
16C17×C4⋊Q8C2211yescharacteristic, complemented
17C17×#127C2211yescharacteristic
18#10587632C2212yes—
19Q8×Dic17C2212yescomplemented
20C2×#63231C2212yescomplemented
21C2×#63230C2212yes—
22#10587626C2211yescharacteristic
23#10587625—22no—
24#10587623—22no—
25#10587622C2211yescharacteristic
26#10587584—22no—
27#10587575—22no—
28#10587561C2211yescharacteristic
29#10587558C2211yescharacteristic
30#10587556C2211yescharacteristic, complemented
31C4×C8×C17C2211yescharacteristic, complemented
32#10587503C2212yes—
33#10587503C2212yescomplemented
34#10587492C2212yes—
35#10587491C2212yes—
36#10587487C2212yescomplemented
37#10587485C2212yescomplemented
38#10587474C2211yescharacteristic, complemented
39C23×D17—21no—
40C2×Q8×C17C2312yes—
41C2×D4×C17—22no—
42C2×D4×C17C2311yescharacteristic
43C2×D4×C17C2311yescharacteristic
44C2×C22⋊D17—21no—
45C2×C22⋊D17—21no—
46C2×C22⋊D17—22no—
47C22×Dic17—21no—
48#63254—42no—
49D4×D17—42no—
50C2×D68C2311yescharacteristic
51C2×C4×D17C2311yescharacteristic
52C2×Dic34C2311yescharacteristic
53SD16×C17—42no—
54SD16×C17—42no—
55C2×C8×C17C2312yes—
56C4⋊C4×C17—22no—
57C42×C17C2311yescharacteristic
58C22⋊Dic17—22no—
59#63231—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—
73C23×C17—21no—
74C22×D17—21no—
75C22×D17—42no—
76C22×D17—82no—
77C22×D17C2×D411yescharacteristic
78Q8×C17—22no—
79Q8×C17—22no—
80D4×C17—22no—
81D4×C17—22no—
82D4×C17—82no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2411yescharacteristic
86C22⋊D17—42no—
87C22⋊D17—42no—
88C22⋊D17—82no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—42no—
92C2×Dic17C2×D412yes—
93C2×Dic17C2×D411yescharacteristic
94D68—22no—
95C4×D17—22no—
96Dic34—22no—
97C8×C17—22no—
98#3622—22no—
99#3222—171nomaximal, Sylow
100C22×C17—41no—
101C22×C17—41no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17C22×D411yescharacteristic
105C2×D17—21no—
106C2×D17—22no—
107C2×D17—41no—
108C2×D17—42no—
109C2×D17—84no—
110C4×C17—22no—
111C4×C17—42no—
112C4×C17H(5,𝔽2)12yes—
113Dic17—21no—
114Dic17—22no—
115Dic17—41no—
116Dic17—42no—
117#541—171no—
118#535—171no—
119#497—171no—
120#492D1711yescharacteristic, complemented
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—81no—
130C2×C17#53511yescharacteristic
131C2×C17#57712yes—
132D17—41no—
133D17—81no—
134C2×#41—171no—
135C22×D4—171no—
136C2×Q16—172no—
137C2×SD16—172no—
138C2×SD16C2×D1712yes—
139C2×SD16C2×D1712yescomplemented
140C4⋊Q8C2×D1711yescharacteristic, complemented
141#127—171no—
142#127C2×D1711yescharacteristic
143#126—342no—
144#125—171no—
145#125—342no—
146#124—171no—
147#124—171no—
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#97—171no—
160C4×C8C2×D1711yescharacteristic, complemented
161C17#322211yescharacteristic, complemented, Sylow
162C24—341no—
163#41—682no—
164C2×Q8—171no—
165C2×Q8C22×D1712yes—
166C2×D4—22no—
167C2×D4—171no—
168C2×D4—341no—
169C2×D4—342no—
170C2×D4—341no—
171C2×D4—682no—
172C2×D4C22×D1711yescharacteristic
173C2×D4C22×D1711yescharacteristic
174C22×C4—171no—
175C22×C4—341no—
176Q16—682no—
177SD16—42no—
178SD16—42no—
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—172no—
189C22⋊C4—342no—
190C22⋊C4—342no—
191C22⋊C4—342no—
192C42—171no—
193C42—342no—
194C42C22×D1711yescharacteristic
195C23—21no—
196C23—21no—
197C23—171no—
198C23—341no—
199C23—682no—
200C23—1362no—
201Q8—22no—
202Q8—22no—
203Q8—342no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—342no—
208D4—682no—
209D4—682no—
210D4—1362no—
211C2×C4—22no—
212C2×C4—172no—
213C2×C4—171no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—341no—
217C2×C4—682no—
218C2×C4D4×D1712yes—
219C2×C4C23×D1711yescharacteristic
220C8—22no—
221C8—342no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—342no—
227C22—341no—
228C22—682no—
229C22—681no—
230C22—1364no—
231C22C2×D4×D1711yescharacteristic
232C4—22no—
233C4—42no—
234C4—342no—
235C4—341no—
236C4—682no—
237C4—681no—
238C4#1058769112yes—
239C2—81no—
240C2—81no—
241C2—681no—
242C2—1361no—
243C2#1176337211yescharacteristic
244C2#1176361512yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#492C2
socleC22×C17C22×D4
prime radical#11763372C2
hypercenter#492D17
hypocenterC17#3222
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #11763420C2semidirect
3. #11763201C2semidirect
4. #11763173C2semidirect
5. #11763164C2semidirect
6. #11763029C2semidirect
7. #11763007C2semidirect
8. #11762968C2semidirect
9. #11762958C2semidirect
10. #11762868C2semidirect
11. C17×#492C2semidirect
15. C2×SD16×C17C22semidirect
16. C17×C4⋊Q8C22semidirect
19. Q8×Dic17C22semidirect
20. C2×#63231C22semidirect
30. #10587556C22semidirect
31. C4×C8×C17C22semidirect
33. #10587503C22semidirect
36. #10587487C22semidirect
37. #10587485C22semidirect
38. #10587474C22semidirect
120. #492D17semidirect
139. C2×SD16C2×D17semidirect
140. C4⋊Q8C2×D17semidirect
160. C4×C8C2×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
#117633722
#117632012
#117631732
#117631642
#117630292
#117630072
#117629682
#117629582
#117628682
C17×#4922
#322217