Finite Group Databasebeta

← #14534719

Subgroups — #14534719

Order 2176.

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

normal subgroups only

This group has 500 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×#10587574C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic
3#11762736C211yesmaximal, characteristic, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic, complemented
6#11762722C212yesmaximal, complemented
7#11762721C212yesmaximal, complemented
8#11762713C212yesmaximal, complemented
9#11762709C212yesmaximal
10#11762654C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C22×Dic34C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×#63227C2211yescharacteristic
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×#63225C2212yes—
26C2×C4×Dic17C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587574—24no—
29#10587571—24no—
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—22no—
44C2×C22⋊D17—24no—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—24no—
52C2×Dic34—24no—
53C2×Dic34—42no—
54C2×Dic34—42no—
55C4⋊C4×C17—22no—
56C22⋊C4×C17—22no—
57C22⋊C4×C17—42no—
58C22⋊Dic17—22no—
59C22⋊Dic17—42no—
60#63227—22no—
61#63227—22no—
62#63227—42no—
63#63227—42no—
64C4⋊Dic17—22no—
65#63225—24no—
66C4×Dic17—24no—
67C4×Dic17D414yes—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17C2411yescharacteristic
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—21no—
77C22×D17—21no—
78C22×D17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17C2×D412yes—
89C22⋊D17—44no—
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17C2×D412yes—
101C2×Dic17C2×D412yes—
102C2×Dic17#4114yes—
103Dic34—48no—
104Dic34—44no—
105#2640—171nomaximal, Sylow
106C22×C17—41no—
107C22×C17—41no—
108C22×C17—41no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)11yescharacteristic
114C22×C17H(5,𝔽2)11yescharacteristic
115C22×C172−1+412yes—
116C2×D17—41no—
117C2×D17—41no—
118C2×D17—41no—
119C2×D17—42no—
120C2×D17—42no—
121C4×C17—22no—
122C4×C17—41no—
123C4×C17—42no—
124C4×C17—42no—
125Dic17—22no—
126Dic17—41no—
127Dic17—42no—
128Dic17—42no—
129Dic17#12414yes—
130C2×C4⋊Q8—171no—
131C2×#124—171no—
132#397—171no—
133#397—171no—
134#397—171no—
135#397D1711yescharacteristic, complemented
136#393—172no—
137#393—172no—
138#391—172no—
139#390—172no—
140#376—171no—
141C2×C17—81no—
142C2×C17C2×#12411yescharacteristic
143C2×C17#53511yescharacteristic
144C2×C17#53611yescharacteristic
145C2×C17#54212yes—
146C2×C17#54212yes—
147D17—81no—
148C22×Q8—171no—
149C22×Q8—172no—
150C22×D4—171no—
151C4⋊Q8—344no—
152#124—344no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4C2×D1711yescharacteristic
156C2×C22⋊C4—171no—
157C2×C22⋊C4—171no—
158C2×C22⋊C4—171no—
159C2×C22⋊C4—172no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4C2×D1711yescharacteristic
163C2×C22⋊C4C2×D1712yes—
164C2×C42—171no—
165C2×C42—172no—
166#95—172no—
167#95—171no—
168#95—172no—
169#95—172no—
170#95—172no—
171#95C2×D1711yescharacteristic
172#95C2×D1712yes—
173C17#264011yescharacteristic, complemented, Sylow
174C24—171no—
175C24C22×D1711yescharacteristic
176C2×Q8—342no—
177C2×Q8—344no—
178C2×Q8—344no—
179C2×Q8—682no—
180C2×Q8—682no—
181C2×D4—342no—
182C2×D4—344no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—172no—
186C22×C4—171no—
187C22×C4—172no—
188C22×C4C22×D1712yes—
189C22×C4C22×D1711yescharacteristic
190C22×C4C22×D1712yes—
191C22×C4C22×D1711yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—344no—
194C4⋊C4—342no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—342no—
198C22⋊C4—342no—
199C22⋊C4—342no—
200C22⋊C4—682no—
201C22⋊C4—682no—
202C22⋊C4—682no—
203C42—174no—
204C42—344no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—341no—
211C23—341no—
212C23—342no—
213C23—341no—
214C23—342no—
215C23C23×D1711yescharacteristic
216Q8—684no—
217Q8—688no—
218D4—684no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—172no—
229C2×C4—174no—
230C2×C4—172no—
231C2×C4—341no—
232C2×C4—341no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4D4×D1712yes—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—681no—
248C22—681no—
249C22—681no—
250C22—682no—
251C22—682no—
252C22C2×D4×D1711yescharacteristic
253C22#1058769111yescharacteristic
254C22#41×D1712yes—
255C22#1058769611yescharacteristic
256C22#1058769712yes—
257C4—22no—
258C4—41no—
259C4—42no—
260C4—42no—
261C4—174no—
262C4—342no—
263C4—681no—
264C4—682no—
265C4—682no—
266C2—81no—
267C2—1361no—
268C2#124×D1711yescharacteristic
269C2#1176337511yescharacteristic
270C2#1176338111yescharacteristic
271C2#1176342512yes—
272C2#1176342712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC22×C22⋊D17C22
hypercenter#397D17
hypocenterC17#2640
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
3. #11762736C2semidirect
4. #11762735C2semidirect
5. #11762734C2semidirect
6. #11762722C2semidirect
7. #11762721C2semidirect
8. #11762713C2semidirect
10. #11762654C2semidirect
11. C17×#397C2semidirect
135. #397D17semidirect
173. C17#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875712
#117627362
#117627352
#117627342
#117627222
#117627212
#117627132
#117627092
#117626542
C17×#3972
#264017