Finite Group Databasebeta

← #14533257

Subgroups — #14533257

Order 2176.

544 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 161 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×C4⋊C4×D17C211yesmaximal, characteristic
3#11762741C211yesmaximal, characteristic, complemented
4#11762740C211yesmaximal, characteristic, complemented
5#11762739C211yesmaximal, characteristic, complemented
6#11762713C212yesmaximal, complemented
7#11762683C212yesmaximal
8#11762680C212yesmaximal
9#11762678C212yesmaximal, complemented
10#11762656C211yesmaximal, characteristic
11C17×#398C211yesmaximal, characteristic, complemented
12C22×C4×D17C2211yescharacteristic
13C22×C4×D17C2212yes—
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2212yes—
18C2×#63227C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2212yes—
29#10587582—24no—
30C4⋊C4×D17—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C2×C4×D17—24no—
49C2×C4×D17—24no—
50C2×C4×D17—42no—
51C2×C4×D17—42no—
52C2×C4×D17D414yes—
53C2×Dic34—22no—
54C2×Dic34—24no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—22no—
57C4⋊C4×C17—42no—
58#63227—22no—
59#63227—22no—
60#63227Q818yescomplemented
61C4⋊Dic17—22no—
62C4⋊Dic17—42no—
63#63225—22no—
64#63225—24no—
65#63225—22no—
66#63225—42no—
67#63225—42no—
68C4×Dic17—24no—
69C23×C17C2411yescharacteristic
70C22×D17—22no—
71C22×D17—21no—
72C22×D17—21no—
73C22×D17C2×D412yes—
74C22×D17C2×Q814yes—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×Q814yes—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17C2×D412yes—
99C2×Dic17C2×Q814yes—
100C2×Dic17#4114yes—
101C4×D17—28no—
102C4×D17—48no—
103Dic34—44no—
104#2502—171nomaximal, Sylow
105C22×C17C22×D411yescharacteristic
106C22×C17C22×Q812yes—
107C22×C17C2×#4112yes—
108C22×C17H(5,𝔽2)11yescharacteristic
109C22×C17H(5,𝔽2)11yescharacteristic
110C2×D17—22no—
111C2×D17—22no—
112C2×D17—24no—
113C2×D17—24no—
114C2×D17C4⋊Q814yes—
115C4×C17—22no—
116C4×C17—24no—
117C4×C17—41no—
118C4×C17—42no—
119C4×C17—41no—
120Dic17—24no—
121Dic17—41no—
122Dic17—42no—
123Dic17—41no—
124Dic17#12314yes—
125C2×C22⋊Q8—171no—
126C22×C4⋊C4—171no—
127#398—171no—
128#398—171no—
129#398—171no—
130#398D1711yescharacteristic, complemented
131#391—172no—
132#385—172no—
133#385—172no—
134#384—172no—
135C4×C22⋊C4—171no—
136C2×C17C2×#12311yescharacteristic
137C2×C17C2×C4⋊Q811yescharacteristic
138C2×C17#53511yescharacteristic
139C2×C17#55412yes—
140C2×C17#55412yes—
141D17—24no—
142C22×Q8—171no—
143C23×C4—171no—
144C23×C4—172no—
145C22⋊Q8—344no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—344no—
154C2×C4⋊C4C2×D1711yescharacteristic
155C2×C4⋊C4C2×D1711yescharacteristic
156C2×C4⋊C4C2×D1712yes—
157C2×C22⋊C4—172no—
158C2×C22⋊C4—171no—
159C2×C22⋊C4—171no—
160C2×C42—172no—
161#95—171no—
162#95—172no—
163#95—172no—
164#95—172no—
165#95C2×D1711yescharacteristic
166#95C2×D1712yes—
167C17#250211yescharacteristic, complemented, Sylow
168C24—171no—
169C2×Q8—342no—
170C2×Q8—344no—
171C22×C4—171no—
172C22×C4—172no—
173C22×C4—174no—
174C22×C4—171no—
175C22×C4—172no—
176C22×C4—171no—
177C22×C4—344no—
178C22×C4—344no—
179C22×C4—682no—
180C22×C4—682no—
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1712yes—
183C22×C4C22×D1711yescharacteristic
184C22×C4C22×D1711yescharacteristic
185C22×C4C22×D1712yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—344no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—342no—
193C4⋊C4—682no—
194C4⋊C4—682no—
195C4⋊C4—682no—
196C22⋊C4—178no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C42—344no—
200C23—172no—
201C23—174no—
202C23—341no—
203C23—342no—
204C23—341no—
205C23C23×D1711yescharacteristic
206Q8—684no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—172no—
219C2×C4—174no—
220C2×C4—174no—
221C2×C4—348no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—341no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—341no—
231C2×C4—342no—
232C2×C4—688no—
233C2×C4D4×D1712yes—
234C2×C4Q8×D1714yes—
235C22—174no—
236C22—344no—
237C22—344no—
238C22—342no—
239C22—342no—
240C22C2×D4×D1711yescharacteristic
241C22#1058769111yescharacteristic
242C22C2×Q8×D1712yes—
243C22#41×D1712yes—
244C22#1058769611yescharacteristic
245C4—24no—
246C4—22no—
247C4—41no—
248C4—42no—
249C4—41no—
250C4—174no—
251C4—344no—
252C4—681no—
253C4—682no—
254C4—681no—
255C2—344no—
256C2#123×D1711yescharacteristic
257C2C4⋊Q8×D1711yescharacteristic
258C2#1176337511yescharacteristic
259C2#1176351312yes—
260C2#1176351512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC23×D17C23
hypercenter#398D17
hypocenterC17#2502
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #11762741C2semidirect
4. #11762740C2semidirect
5. #11762739C2semidirect
6. #11762713C2semidirect
9. #11762678C2semidirect
11. C17×#398C2semidirect
60. #63227Q8semidirect
130. #398D17semidirect
167. C17#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#105875822
C2×C4⋊C4×D172
#117627412
#117627402
#117627392
#117627132
#117626832
#117626802
#117626782
#117626562
C17×#3982
#250217