Finite Group Databasebeta

← #14529284

Subgroups — #14529284

Order 2176.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#11763609C211yesmaximal, characteristic
2C4×#63252C211yesmaximal, characteristic, complemented
3#11762827C211yesmaximal, characteristic, complemented
4C2×#10587476C211yesmaximal, characteristic, complemented
5#11762782C211yesmaximal, characteristic, complemented
6#11762581C211yesmaximal, characteristic, complemented
7C17×#339C211yesmaximal, characteristic, complemented
8C2×#63252C2211yescharacteristic
9C2×#63252C2211yescharacteristic
10C2×M16×C17C2211yescharacteristic
11C17×#117C2211yescharacteristic
12C2×C42×C17C2211yescharacteristic
13C4×C22⋊D17—22no—
14#10587612C2211yescharacteristic
15C2×#63223C2211yescharacteristic
16#10587595—21no—
17#10587595C412yescomplemented
18#10587594—21no—
19#10587594C412yescomplemented
20#10587590—22no—
21C4×D68—21no—
22#10587558—21no—
23C42×D17—21no—
24C4×Dic34—21no—
25#10587477—22no—
26#10587476—22no—
27#10587476—21no—
28#10587476—21no—
29C22×C4×C17—21no—
30C22×C4×C17C2311yescharacteristic
31C2×C22⋊D17—21no—
32C2×C22⋊D17—21no—
33#63252—21no—
34#63252—21no—
35#63252—21no—
36#63252—21no—
37#63252—21no—
38#63252—21no—
39#63252C2×C412yes—
40#63252D412yes—
41C2×D68C2×C411yescharacteristic, complemented
42C2×D68D411yescharacteristic
43C2×C4×D17—21no—
44C2×C4×D17—22no—
45C2×C4×D17—21no—
46C2×Dic34C2×C411yescharacteristic, complemented
47C2×Dic34D411yescharacteristic
48M16×C17—21no—
49M16×C17C2×C412yescomplemented
50C2×C8×C17—21no—
51C4⋊C4×C17—21no—
52C22⋊C4×C17—21no—
53C42×C17—21no—
54C42×C17—21no—
55C42×C17D411yescharacteristic
56C42×C17D411yescharacteristic
57C22⋊Dic17D412yes—
58#63227—22no—
59C4⋊Dic17—21no—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17—21no—
63#63223—21no—
64#63223—21no—
65C2×#3622—21no—
66Dic68—22no—
67D136—22no—
68#63219—22no—
69#63219—22no—
70C23×C17C2×D411yescharacteristic
71C22×D17—21no—
72C22×D17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—42no—
76C2×C4×C17—41no—
77C2×C4×C17C22×C411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17#4111yescharacteristic
82C2×C4×C17#4111yescharacteristic
83C22⋊D17—22no—
84C22⋊D17—22no—
85C22⋊D17—41no—
86C22⋊D17—41no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—22no—
90D68—21no—
91D68—21no—
92D68—21no—
93D68C22⋊C412yes—
94C4×D17—22no—
95C4×D17—22no—
96C4×D17—44no—
97C4×D17—41no—
98C4×D17—41no—
99Dic34—21no—
100Dic34—21no—
101Dic34—21no—
102Dic34C22⋊C412yes—
103C8×C17—22no—
104#3622—41no—
105#1906—171nomaximal, Sylow
106C22×C17—41no—
107C22×C17C4×D411yescharacteristic
108C22×C17#12011yescharacteristic
109C22×C17#12111yescharacteristic
110C2×D17—22no—
111C2×D17—42no—
112C2×D17—41no—
113C2×D17—81no—
114C4×C17—22no—
115C4×C17—41no—
116C4×C17—81no—
117C4×C17C2×C22⋊C411yescharacteristic
118C4×C17C4×D411yescharacteristic
119C4×C17#12111yescharacteristic
120C4×C17#12311yescharacteristic
121Dic17—22no—
122Dic17—42no—
123Dic17—41no—
124#575—171no—
125C4×#41—171no—
126#421—171no—
127C2×#104—171no—
128#410—171no—
129#344—171no—
130#339D1711yescharacteristic, complemented
131C2×C17—21no—
132C2×C17—21no—
133C2×C17—21no—
134C2×C17#38611yescharacteristic
135D17—42no—
136D17—81no—
137C2×#41—171no—
138C2×#41—171no—
139#137—172no—
140#137—341no—
141AGL(1,ℤ/8)—172no—
142AGL(1,ℤ/8)—341no—
143#135—342no—
144C2×M16—171no—
145C2×M16C2×D1711yescharacteristic
146C4×Q8—341no—
147C4×D4—341no—
148C4×D4—342no—
149#117—171no—
150#117—341no—
151#117C2×D1711yescharacteristic
152C2×C42—341no—
153C2×C42C2×D1711yescharacteristic
154#104—342no—
155#104—341no—
156#104—341no—
157#99—342no—
158C17#190611yescharacteristic, complemented, Sylow
159#41—172no—
160#41—172no—
161#41—341no—
162#41—341no—
163#41—341no—
164#41—341no—
165#41—341no—
166#41—341no—
167C2×Q8—171no—
168C2×Q8—171no—
169C2×D4—171no—
170C2×D4—171no—
171C2×D4—341no—
172C2×D4—341no—
173C22×C4—21no—
174C22×C4—342no—
175C22×C4—341no—
176C22×C4—341no—
177C22×C4C22×D1711yescharacteristic
178Q16—342no—
179SD16—342no—
180SD16—342no—
181D8—342no—
182M16—21no—
183M16—341no—
184M16—341no—
185M16C4×D1712yescomplemented
186C2×C8—21no—
187C2×C8—341no—
188C4⋊C4—21no—
189C4⋊C4—342no—
190C4⋊C4—341no—
191C22⋊C4—21no—
192C22⋊C4—172no—
193C22⋊C4—342no—
194C42—21no—
195C42—21no—
196C42—342no—
197C42—341no—
198C42D6811yescharacteristic
199C42D6811yescharacteristic
200C23—341no—
201C23—341no—
202C23D4×D1711yescharacteristic
203Q8—172no—
204Q8—341no—
205Q8—341no—
206Q8—341no—
207D4—172no—
208D4—342no—
209D4—342no—
210D4—341no—
211D4—341no—
212D4—341no—
213D4—681no—
214D4—681no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—341no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—341no—
224C2×C4—684no—
225C2×C4—681no—
226C2×C4—681no—
227C2×C4C2×C4×D1711yescharacteristic
228C2×C4C2×D6811yescharacteristic
229C2×C4#6325211yescharacteristic
230C2×C4D4×D1711yescharacteristic
231C2×C4D4×D1711yescharacteristic
232C2×C4#6325611yescharacteristic
233C8—22no—
234C8—681no—
235C22—41no—
236C22—342no—
237C22—682no—
238C22—681no—
239C22—1361no—
240C22#1058756811yescharacteristic
241C22#1058757011yescharacteristic
242C22#1058757911yescharacteristic
243C4—22no—
244C4—41no—
245C4—81no—
246C4—342no—
247C4—682no—
248C4—681no—
249C4C4×D6811yescharacteristic
250C4C22⋊C4×D1711yescharacteristic
251C4#1058758011yescharacteristic
252C4#1058758111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—682no—
257C2—1361no—
258C2#1176268711yescharacteristic
subgroupG/H
centerC4#10587581
commutatorC2×C4×C17C22×C4
FrattiniC22×C4C22×D17
FittingC17×#339C2
socleC2×C17#386
prime radical#11763609C2
cosocleC22×C4×C17C23
hypercenter#339D17
hypocenterC17#1906
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C4×#63252C2semidirect
3. #11762827C2semidirect
4. C2×#10587476C2semidirect
5. #11762782C2semidirect
6. #11762581C2semidirect
7. C17×#339C2semidirect
17. #10587595C4semidirect
19. #10587594C4semidirect
41. C2×D68C2×C4semidirect
46. C2×Dic34C2×C4semidirect
49. M16×C17C2×C4semidirect
130. #339D17semidirect
158. C17#1906semidirect
185. M16C4×D17semidirect

Maximal subgroups

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

subgroupindex
#117636092
C4×#632522
#117628272
C2×#105874762
#117627822
#117625812
C17×#3392
#190617