Finite Group Databasebeta

← #14544794

Subgroups — #14544794

Order 2176.

386 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 68 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#11763876C211yesmaximal, characteristic
2D17⋊#99C211yesmaximal, characteristic
3#11763752C212yesmaximal, complemented
4#11763686C212yesmaximal, complemented
5#123×D17C211yesmaximal, characteristic
6C2×D4×D17C2211yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×C22⋊#595C2212yes—
10C2×C22⋊#595C2212yes—
11D4×#595—24no—
12#10587677—21no—
13C17×#123C411yescharacteristic, complemented
14#10587621C411yescharacteristic, complemented
15#10587615C411yescharacteristic, complemented
16#10587580—21no—
17C4⋊C4×D17—21no—
18#10587572C411yescharacteristic, complemented
19#10587569—21no—
20C22⋊C4×D17—21no—
21C22⋊C4×D17C2211yescharacteristic
22#10587551—21no—
23(C2×#595)⋊C4—24no—
24#10587539—21no—
25C23×D17D411yescharacteristic
26C23×D17C2311yescharacteristic
27C22×#595—22no—
28C22×#595—22no—
29C22×#595D414yes—
30C2×D4×C17C2×C411yescharacteristic
31C22×C4×C17C2×C411yescharacteristic, complemented
32C2×C22⋊D17C2×C411yescharacteristic
33C2×C22⋊D17D411yescharacteristic
34C22×Dic17C2×C411yescharacteristic, complemented
35D4×D17—22no—
36D4×D17—22no—
37C2×D68D411yescharacteristic
38C2×C4×D17—22no—
39C2×C4×D17—21no—
40C2×C4×D17—21no—
41C2×C4×D17—21no—
42C2×C4×D17—22no—
43C2×C4×D17D411yescharacteristic
44C22⋊#595—22no—
45C22⋊#595—22no—
46C22⋊#595—22no—
47C22⋊#595—22no—
48C22⋊#595—22no—
49C4⋊#595—22no—
50C4×#595—22no—
51C4⋊C4×C17—21no—
52C22⋊C4×C17—21no—
53C22⋊C4×C17C2×C411yescharacteristic, complemented
54C22⋊Dic17—21no—
55C22⋊Dic17C2×C411yescharacteristic, complemented
56#63227—21no—
57#63227—21no—
58#63227—21no—
59C4⋊Dic17—21no—
60#63225—21no—
61#63225—21no—
62C23×C17C22⋊C411yescharacteristic
63C23×C17C22×C411yescharacteristic
64C22×D17—21no—
65C22×D17—21no—
66C22×D17—42no—
67C22×D17—42no—
68C22×D17C22⋊C411yescharacteristic
69C22×D17C2×D412yes—
70C22×D17C2×D411yescharacteristic
71C22×D17C2×D411yescharacteristic
72C22×D17#4112yes—
73C2×#595—24no—
74C2×#595—44no—
75C2×#595—44no—
76C2×#595—42no—
77C2×#595—82no—
78D4×C17—22no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17C22⋊C411yescharacteristic
83C22⋊D17—22no—
84C22⋊D17—22no—
85C2×Dic17—21no—
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17C22⋊C411yescharacteristic
89D68—22no—
90C4×D17—21no—
91C4×D17—21no—
92C4×D17—42no—
93C4×D17—41no—
94C4×D17—42no—
95C4×D17—41no—
96#1890—171nomaximal, Sylow
97C22×C17—21no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17C2×C22⋊C411yescharacteristic
101C22×C17C4×D412yes—
102C2×D17—22no—
103C2×D17—24no—
104C2×D17—21no—
105C2×D17—41no—
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—41no—
109C2×D17—81no—
110C2×D17#12011yescharacteristic
111C2×D17#12112yes—
112C2×D17#12311yescharacteristic
113#595—44no—
114#595—82no—
115C4×C17—21no—
116C4×C17—41no—
117C4×C17—41no—
118Dic17—21no—
119Dic17—41no—
120Dic17—41no—
121C2×#123—171no—
122#518—171no—
123C2×#99—171no—
124#381—172no—
125#342—172no—
126C2×C17—21no—
127C2×C17—22no—
128C2×C17—41no—
129C2×C17#38611yescharacteristic
130D17—22no—
131D17—21no—
132D17—41no—
133D17#45812yes—
134C22×D4—171no—
135C23×C4—171no—
136#123—171no—
137#123—171no—
138#123—171no—
139#123—341no—
140#123—341no—
141#123#59511yescharacteristic, complemented
142C4×D4—344no—
143#117—341no—
144C2×C4⋊C4—341no—
145C2×C22⋊C4—171no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—341no—
149C2×C22⋊C4—341no—
150#99—341no—
151#99—341no—
152#95—344no—
153C17#189011yescharacteristic, complemented, Sylow
154C24—171no—
155C24—171no—
156C2×D4—171no—
157C2×D4—171no—
158C2×D4—171no—
159C2×D4—342no—
160C2×D4—342no—
161C2×D4C2×#59511yescharacteristic
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—174no—
165C22×C4—342no—
166C22×C4—341no—
167C22×C4—341no—
168C22×C4—341no—
169C22×C4—342no—
170C22×C4—342no—
171C22×C4—342no—
172C22×C4C2×#59511yescharacteristic, complemented
173C4⋊C4—21no—
174C4⋊C4—341no—
175C4⋊C4—341no—
176C4⋊C4—342no—
177C4⋊C4—341no—
178C22⋊C4—21no—
179C22⋊C4—171no—
180C22⋊C4—341no—
181C22⋊C4—341no—
182C22⋊C4—341no—
183C22⋊C4—342no—
184C22⋊C4—342no—
185C22⋊C4—342no—
186C22⋊C4—341no—
187C22⋊C4—342no—
188C22⋊C4—342no—
189C22⋊C4C2×#59511yescharacteristic, complemented
190C42—342no—
191C23—172no—
192C23—171no—
193C23—172no—
194C23—171no—
195C23—171no—
196C23—341no—
197C23—341no—
198C23—682no—
199C23—682no—
200C23C22⋊#59511yescharacteristic
201C23C22×#59511yescharacteristic
202D4—22no—
203D4—342no—
204D4—342no—
205D4—342no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—171no—
210C2×C4—341no—
211C2×C4—341no—
212C2×C4—344no—
213C2×C4—342no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4—681no—
217C2×C4—681no—
218C2×C4—682no—
219C2×C4—682no—
220C2×C4—684no—
221C2×C4—682no—
222C2×C4—684no—
223C2×C4—1362no—
224C2×C4C22⋊#59511yescharacteristic
225C22—21no—
226C22—41no—
227C22—41no—
228C22—171no—
229C22—171no—
230C22—172no—
231C22—344no—
232C22—342no—
233C22—341no—
234C22—681no—
235C22—682no—
236C22—681no—
237C22—682no—
238C22—1361no—
239C22D4×#59512yes—
240C22C2×C22⋊#59511yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—341no—
245C4—681no—
246C4—684no—
247C4—681no—
248C4—1362no—
249C2—21no—
250C2—22no—
251C2—41no—
252C2—172no—
253C2—342no—
254C2—341no—
255C2—681no—
256C2#1176375711yescharacteristic
subgroupG/H
centerC2#11763757
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC17×#123C4
socleC2×C17#386
prime radicalC2×D4×D17C22
cosocleC23×D17C23
hypercenter#123#595
hypocenterC17#1890
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #11763752C2semidirect
4. #11763686C2semidirect
13. C17×#123C4semidirect
14. #10587621C4semidirect
15. #10587615C4semidirect
18. #10587572C4semidirect
31. C22×C4×C17C2×C4semidirect
34. C22×Dic17C2×C4semidirect
53. C22⋊C4×C17C2×C4semidirect
55. C22⋊Dic17C2×C4semidirect
141. #123#595semidirect
153. C17#1890semidirect
172. C22×C4C2×#595semidirect
189. C22⋊C4C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117638762
D17⋊#992
#117637522
#117636862
#123×D172
#189017