Finite Group Databasebeta

← #14532667

Subgroups — #14532667

Order 2176.

660 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#10587580C212yesmaximal, complemented
2C2×#10587567C212yesmaximal, complemented
3#11762725C212yesmaximal, complemented
4#11762718C211yesmaximal, characteristic
5#11762705C212yesmaximal, complemented
6#11762701C212yesmaximal, complemented
7#11762681C211yesmaximal, characteristic, complemented
8#11762680C212yesmaximal, complemented
9C17×#399C211yesmaximal, characteristic, complemented
10C22×C22⋊D17C2212yes—
11C23×Dic17C2211yescharacteristic
12C22×D68C2211yescharacteristic
13C22×C4×D17C2212yes—
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2211yescharacteristic
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2212yes—
27#10587580—28no—
28#10587567—28no—
29C17×#95C2211yescharacteristic
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32#10587502C2212yes—
33C24×C17C2311yescharacteristic
34C23×D17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C2×C22⋊D17—24no—
40C2×C22⋊D17—24no—
41C2×C22⋊D17—24no—
42C22×Dic17—24no—
43C22×Dic17—41no—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C2×D68—24no—
48C2×D68—41no—
49C2×C4×D17—24no—
50C2×C4×D17—24no—
51C2×C4×D17—42no—
52C4⋊C4×C17—24no—
53C4⋊C4×C17—41no—
54C22⋊C4×C17—24no—
55C22⋊C4×C17—41no—
56C22⋊Dic17—22no—
57#63227—24no—
58#63227—24no—
59#63227—24no—
60#63227D418yescomplemented
61C4⋊Dic17—24no—
62#63225—24no—
63#63225—24no—
64C4×Dic17—24no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—21no—
68C23×C17—21no—
69C23×C17#4112yes—
70C23×C17C2411yescharacteristic
71C22×D17—22no—
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17C2×D414yes—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17C2×D414yes—
88C22⋊D17—216no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—44no—
97C2×Dic17C2×D414yes—
98C2×Dic17#4114yes—
99D68—44no—
100C4×D17—48no—
101#2455—171nomaximal, Sylow
102C22×C17—24no—
103C22×C17—22no—
104C22×C17—41no—
105C22×C17—42no—
106C22×C17—41no—
107C22×C17C22×D412yes—
108C22×C17C2×#4111yescharacteristic
109C22×C17C2×#4112yes—
110C22×C17H(5,𝔽2)11yescharacteristic
111C22×C172−1+411yescharacteristic
112C2×D17—24no—
113C2×D17—24no—
114C2×D17—24no—
115C2×D17—42no—
116C2×D17—42no—
117C2×D17—42no—
118C2×D17—42no—
119C4×C17—24no—
120C4×C17—42no—
121C4×C17—41no—
122C4×C17—41no—
123Dic17—24no—
124Dic17—24no—
125Dic17—42no—
126C2×#123—172no—
127C2×C4×D4—172no—
128#399D1711yescharacteristic, complemented
129#394—172no—
130#392—171no—
131#389—172no—
132#388—172no—
133#385—172no—
134#385—171no—
135C2×C17—42no—
136C2×C17#52512yes—
137C2×C17#53911yescharacteristic
138C2×C17D4211yescharacteristic
139C2×C17#54712yes—
140C2×C17#55911yescharacteristic
141D17—44no—
142C22×D4—172no—
143C22×D4—171no—
144C23×C4—171no—
145C23×C4—172no—
146#123—348no—
147C4×D4—348no—
148C2×C4⋊C4—172no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4C2×D1712yes—
152C2×C4⋊C4C2×D1711yescharacteristic
153C2×C22⋊C4—172no—
154C2×C22⋊C4—171no—
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C22⋊C4—172no—
158C2×C22⋊C4C2×D1712yes—
159C2×C22⋊C4C2×D1711yescharacteristic
160C2×C42—172no—
161#95—172no—
162#95—172no—
163#95—171no—
164#95C2×D1711yescharacteristic
165C17#245511yescharacteristic, complemented, Sylow
166C24—172no—
167C24C22×D1711yescharacteristic
168C2×D4—344no—
169C2×D4—344no—
170C2×D4—344no—
171C2×D4—344no—
172C2×D4—681no—
173C22×C4—172no—
174C22×C4—172no—
175C22×C4—172no—
176C22×C4—344no—
177C22×C4—344no—
178C22×C4—344no—
179C22×C4—682no—
180C22×C4—681no—
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1712yes—
183C22×C4C22×D1711yescharacteristic
184C22×C4C22×D1712yes—
185C4⋊C4—24no—
186C4⋊C4—41no—
187C4⋊C4—344no—
188C4⋊C4—344no—
189C4⋊C4—344no—
190C22⋊C4—24no—
191C22⋊C4—41no—
192C22⋊C4—178no—
193C22⋊C4—344no—
194C22⋊C4—344no—
195C22⋊C4—342no—
196C22⋊C4—344no—
197C42—344no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23—21no—
202C23—174no—
203C23—342no—
204C23—342no—
205C23—342no—
206C23—342no—
207C23—342no—
208C23—342no—
209C23#6325612yes—
210C23C23×D1711yescharacteristic
211D4—3416no—
212D4—684no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—174no—
223C2×C4—174no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—688no—
232C2×C4—684no—
233C2×C4D4×D1714yes—
234C22—24no—
235C22—22no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22—344no—
240C22—344no—
241C22—344no—
242C22—682no—
243C22—682no—
244C22—682no—
245C22—682no—
246C22C2×D4×D1712yes—
247C22#1058769111yescharacteristic
248C22C2×#6325611yescharacteristic
249C22#41×D1712yes—
250C22#1058769711yescharacteristic
251C4—24no—
252C4—41no—
253C4—42no—
254C4—41no—
255C4—344no—
256C4—344no—
257C4—682no—
258C2—42no—
259C2—684no—
260C2#1176332212yes—
261C2#1176340811yescharacteristic
262C2#1176344411yescharacteristic
263C2#1176345712yes—
264C2#1176354311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radical#11762718C2
hypercenter#399D17
hypocenterC17#2455
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587580C2semidirect
2. C2×#10587567C2semidirect
3. #11762725C2semidirect
5. #11762705C2semidirect
6. #11762701C2semidirect
7. #11762681C2semidirect
8. #11762680C2semidirect
9. C17×#399C2semidirect
60. #63227D4semidirect
128. #399D17semidirect
165. C17#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#105875802
C2×#105875672
#117627252
#117627182
#117627052
#117627012
#117626812
#117626802
C17×#3992
#245517