Finite Group Databasebeta

← #14534950

Subgroups — #14534950

Order 2176.

434 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587584C211yesmaximal, characteristic
2C2×#10587576C211yesmaximal, characteristic, complemented
3#11762748C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic
5#11762741C212yesmaximal, complemented
6#11762735C211yesmaximal, characteristic
7#11762731C212yesmaximal
8#11762705C212yesmaximal, complemented
9#11762698C212yesmaximal
10#11762660C211yesmaximal, characteristic, complemented
11C17×#398C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×#63227C2211yescharacteristic
16C2×#63227C2212yes—
17C2×#63227C2211yescharacteristic
18C2×#63227C2211yescharacteristic
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2211yescharacteristic
26C2×#63225C2212yes—
27C2×C4×Dic17C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587584—24no—
30#10587576—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C4⋊C4×C17—22no—
49C4⋊C4×C17—22no—
50C4⋊C4×C17—42no—
51#63227—22no—
52#63227—24no—
53#63227—22no—
54#63227—22no—
55#63227—42no—
56C4⋊Dic17—22no—
57C4⋊Dic17—24no—
58C4⋊Dic17—24no—
59C4⋊Dic17—22no—
60#63225—24no—
61#63225—24no—
62#63225—22no—
63C4×Dic17—42no—
64C4×Dic17D414yes—
65C23×C17C2411yescharacteristic
66C22×D17—21no—
67C22×D17—22no—
68C22×D17—21no—
69C22×D17—22no—
70C22×D17—21no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17C2×D412yes—
83C2×C4×C17#4114yes—
84C2×Dic17—22no—
85C2×Dic17—21no—
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17—21no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17C2×D412yes—
96C2×Dic17C2×D412yes—
97#2664—171nomaximal, Sylow
98C22×C17C22×D411yescharacteristic
99C22×C17C2×#4112yes—
100C22×C17H(5,𝔽2)11yescharacteristic
101C22×C172−1+412yes—
102C22×C172−1+411yescharacteristic
103C2×D17—41no—
104C2×D17—41no—
105C2×D17—41no—
106C2×D17—42no—
107C2×D17—42no—
108C4×C17—22no—
109C4×C17—24no—
110C4×C17—41no—
111C4×C17—42no—
112C4×C17—41no—
113Dic17—22no—
114Dic17—22no—
115Dic17—42no—
116Dic17—41no—
117Dic17—42no—
118C2×#126—171no—
119C2×#125—171no—
120#400—171no—
121#399—171no—
122#398—172no—
123#398D1711yescharacteristic, complemented
124#397—171no—
125#396—172no—
126#389—172no—
127#388—172no—
128C4×C4⋊C4—171no—
129C2×C17#53211yescharacteristic
130C2×C17#53611yescharacteristic
131C2×C17#53711yescharacteristic
132C2×C17#53912yes—
133C2×C17#54112yes—
134D17—81no—
135#126—344no—
136#125—344no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C4⋊C4C2×D1712yes—
147C2×C22⋊C4—171no—
148C2×C22⋊C4—171no—
149C2×C22⋊C4—171no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C42—171no—
153C2×C42—172no—
154#95—171no—
155#95—172no—
156#95—172no—
157#95—172no—
158#95C2×D1711yescharacteristic
159#95C2×D1712yes—
160C17#266411yescharacteristic, complemented, Sylow
161C24—171no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—172no—
165C22×C4—171no—
166C22×C4—172no—
167C22×C4C22×D1711yescharacteristic
168C22×C4C22×D1711yescharacteristic
169C22×C4C22×D1712yes—
170C22×C4C22×D1711yescharacteristic
171C22×C4C22×D1712yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—344no—
176C4⋊C4—344no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4—342no—
180C4⋊C4—344no—
181C4⋊C4—344no—
182C22⋊C4—342no—
183C22⋊C4—342no—
184C22⋊C4—342no—
185C22⋊C4—344no—
186C22⋊C4—682no—
187C42—174no—
188C42—682no—
189C23—341no—
190C23—341no—
191C23—342no—
192C23—342no—
193C23—341no—
194C23C23×D1711yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—172no—
207C2×C4—172no—
208C2×C4—341no—
209C2×C4—342no—
210C2×C4—341no—
211C2×C4—341no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4D4×D1712yes—
220C2×C4#6325614yes—
221C22—681no—
222C22—681no—
223C22—681no—
224C22—682no—
225C22—682no—
226C22C2×D4×D1711yescharacteristic
227C22#1058769111yescharacteristic
228C22C2×#6325612yes—
229C22#1058769712yes—
230C22#1058769711yescharacteristic
231C4—22no—
232C4—24no—
233C4—41no—
234C4—41no—
235C4—42no—
236C4—342no—
237C4—342no—
238C4—681no—
239C4—682no—
240C4—682no—
241C2—1361no—
242C2#1176336011yescharacteristic
243C2#1176338111yescharacteristic
244C2#1176338811yescharacteristic
245C2#1176340812yes—
246C2#1176342212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC23×D17C23
hypercenter#398D17
hypocenterC17#2664
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#10587576C2semidirect
3. #11762748C2semidirect
5. #11762741C2semidirect
8. #11762705C2semidirect
10. #11762660C2semidirect
11. C17×#398C2semidirect
123. #398D17semidirect
160. C17#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#105875842
C2×#105875762
#117627482
#117627462
#117627412
#117627352
#117627312
#117627052
#117626982
#117626602
C17×#3982
#266417