Finite Group Databasebeta

← #14534587

Subgroups — #14534587

Order 2176.

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

normal subgroups only

This group has 464 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×#10587582C211yesmaximal, characteristic
2C2×#10587578C211yesmaximal, characteristic
3#11762749C211yesmaximal, characteristic, complemented
4#11762748C211yesmaximal, characteristic, complemented
5#11762740C211yesmaximal, characteristic, complemented
6#11762738C212yesmaximal
7#11762735C212yesmaximal
8#11762713C212yesmaximal, complemented
9#11762704C212yesmaximal, complemented
10#11762668C211yesmaximal, characteristic
11C17×#398C211yesmaximal, characteristic, complemented
12C22×C4×D17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2212yes—
17C2×#63227C2211yescharacteristic
18C2×#63227C2212yes—
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2212yes—
26C2×C4×Dic17C2212yes—
27#10587582—24no—
28#10587578—24no—
29C17×#95C2212yes—
30C17×#95C2211yescharacteristic
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×C4×D17—24no—
49C2×C4×D17D414yes—
50C2×Dic34—22no—
51C2×Dic34—24no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—42no—
55#63227—24no—
56#63227—22no—
57#63227—22no—
58#63227—42no—
59C4⋊Dic17—22no—
60#63225—22no—
61#63225—22no—
62#63225—24no—
63#63225—42no—
64C4×Dic17—24no—
65C23×C17C2411yescharacteristic
66C22×D17—22no—
67C22×D17—21no—
68C22×D17—22no—
69C22×D17—21no—
70C22×D17C2×D412yes—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D412yes—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—24no—
92C2×Dic17—22no—
93C2×Dic17—21no—
94C2×Dic17C2×D412yes—
95C2×Dic17#4114yes—
96C4×D17—28no—
97Dic34—44no—
98#2625—171nomaximal, Sylow
99C22×C17C22×D411yescharacteristic
100C22×C17C2×#4112yes—
101C22×C17H(5,𝔽2)11yescharacteristic
102C22×C172−1+412yes—
103C22×C172−1+411yescharacteristic
104C2×D17—22no—
105C2×D17—22no—
106C2×D17—22no—
107C2×D17—42no—
108C2×D17—42no—
109C4×C17—22no—
110C4×C17—41no—
111C4×C17—42no—
112C4×C17—41no—
113C4×C17—42no—
114Dic17—41no—
115Dic17—42no—
116Dic17—41no—
117Dic17—42no—
118Dic17#12314yes—
119C2×C22⋊Q8—171no—
120C2×#117—171no—
121#400—171no—
122#400—171no—
123#398—171no—
124#398D1711yescharacteristic, complemented
125#397—172no—
126#397—172no—
127#391—172no—
128#389—172no—
129#380—171no—
130C2×C17C2×#12311yescharacteristic
131C2×C17#53611yescharacteristic
132C2×C17#53711yescharacteristic
133C2×C17#54112yes—
134C2×C17#54212yes—
135D17—42no—
136C22×Q8—171no—
137C23×C4—171no—
138C22⋊Q8—344no—
139#117—344no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C4⋊C4C2×D1712yes—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—171no—
150C2×C22⋊C4—171no—
151C2×C22⋊C4—172no—
152C2×C42—172no—
153#95—171no—
154#95—172no—
155#95—172no—
156#95—172no—
157#95—172no—
158#95—172no—
159#95C2×D1711yescharacteristic
160#95C2×D1712yes—
161C17#262511yescharacteristic, complemented, Sylow
162C24—171no—
163C2×Q8—342no—
164C2×Q8—344no—
165C22×C4—171no—
166C22×C4—174no—
167C22×C4—172no—
168C22×C4—172no—
169C22×C4—171no—
170C22×C4—171no—
171C22×C4—344no—
172C22×C4C22×D1712yes—
173C22×C4C22×D1711yescharacteristic
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1712yes—
176C22×C4C22×D1711yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—342no—
181C4⋊C4—344no—
182C4⋊C4—342no—
183C4⋊C4—342no—
184C4⋊C4—682no—
185C22⋊C4—344no—
186C22⋊C4—342no—
187C22⋊C4—342no—
188C22⋊C4—682no—
189C42—344no—
190C23—172no—
191C23—341no—
192C23—342no—
193C23—342no—
194C23—341no—
195C23C23×D1711yescharacteristic
196Q8—684no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—172no—
210C2×C4—174no—
211C2×C4—348no—
212C2×C4—342no—
213C2×C4—341no—
214C2×C4—341no—
215C2×C4—342no—
216C2×C4—341no—
217C2×C4—344no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—341no—
222C2×C4D4×D1712yes—
223C22—342no—
224C22—342no—
225C22—342no—
226C22—682no—
227C22—682no—
228C22C2×D4×D1711yescharacteristic
229C22#1058769111yescharacteristic
230C22#1058769412yes—
231C22#41×D1712yes—
232C22#1058769711yescharacteristic
233C4—22no—
234C4—41no—
235C4—42no—
236C4—41no—
237C4—42no—
238C4—174no—
239C4—681no—
240C4—682no—
241C4—681no—
242C4—682no—
243C2—682no—
244C2#123×D1711yescharacteristic
245C2#1176338411yescharacteristic
246C2#1176338811yescharacteristic
247C2#1176341812yes—
248C2#1176342912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC23×D17C23
hypercenter#398D17
hypocenterC17#2625
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #11762749C2semidirect
4. #11762748C2semidirect
5. #11762740C2semidirect
8. #11762713C2semidirect
9. #11762704C2semidirect
11. C17×#398C2semidirect
124. #398D17semidirect
161. C17#2625semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
C2×#105875782
#117627492
#117627482
#117627402
#117627382
#117627352
#117627132
#117627042
#117626682
C17×#3982
#262517