Finite Group Databasebeta

← #14536786

Subgroups — #14536786

Order 2176.

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

normal subgroups only

This group has 578 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×#10587616C211yesmaximal, characteristic, complemented
2#11762748C211yesmaximal, characteristic, complemented
3#11762745C211yesmaximal, characteristic, complemented
4#11762731C212yesmaximal, complemented
5#11762726C211yesmaximal, characteristic, complemented
6#11762725C212yesmaximal, complemented
7#11762724C212yesmaximal, complemented
8#11762718C211yesmaximal, characteristic, complemented
9#11762717C212yesmaximal, complemented
10#11762669C211yesmaximal, characteristic, complemented
11C2×C17×#123C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2212yes—
15C22×C22⋊D17C2212yes—
16C22×D68C2211yescharacteristic
17C17×#123—24no—
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25#10587616—24no—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63227C2212yes—
29C2×#63227C2212yes—
30C2×C4⋊Dic17C2211yescharacteristic
31C2×C4⋊Dic17C2212yes—
32C2×#63225C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C24×C17C2311yescharacteristic
39C23×D17C2312yes—
40C2×D4×C17—22no—
41C2×D4×C17—42no—
42C22×C4×C17—42no—
43C22×C4×C17D414yescomplemented
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—24no—
50C2×C22⋊D17—42no—
51C2×C22⋊D17—42no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C2×C22⋊D17—42no—
55C22×Dic17C2312yes—
56C22×Dic17C2312yes—
57C22×Dic17C2312yes—
58C2×D68—22no—
59C2×D68—42no—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—22no—
62C22⋊C4×C17—22no—
63C22⋊C4×C17—22no—
64C22⋊C4×C17—22no—
65C22⋊Dic17—42no—
66C22⋊Dic17—42no—
67#63227—24no—
68#63227—42no—
69#63227—42no—
70#63227—42no—
71C4⋊Dic17—22no—
72C4⋊Dic17—42no—
73#63225—42no—
74C23×C17—21no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—21no—
78C23×C17—21no—
79C23×C17—22no—
80C23×C17—21no—
81C23×C17—22no—
82C23×C17—21no—
83C23×C17C2×D412yes—
84C23×C17C2411yescharacteristic
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—22no—
92D4×C17—44no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—44no—
103C2×C4×C17C2×D412yes—
104C2×C4×C17C2×D412yes—
105C22⋊D17—48no—
106C22⋊D17—48no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—22no—
116D68—44no—
117#2813—171nomaximal, Sylow
118C22×C17—22no—
119C22×C17—22no—
120C22×C17—22no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17—41no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17C22×D411yescharacteristic
129C22×C17H(5,𝔽2)12yes—
130C22×C17H(5,𝔽2)12yes—
131C22×C17H(5,𝔽2)11yescharacteristic
132C22×C172−1+411yescharacteristic
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C4×C17—22no—
141C4×C17—22no—
142C4×C17—41no—
143C4×C17—41no—
144C4×C17—41no—
145Dic17—42no—
146Dic17—42no—
147Dic17—42no—
148C2×#123D1711yescharacteristic, complemented
149C2×#121—171no—
150#400—171no—
151#399—171no—
152#396—172no—
153#394—171no—
154#394—172no—
155#394—172no—
156#392—172no—
157#392—171no—
158#380—171no—
159C2×C17—42no—
160C2×C17—81no—
161C2×C17#53411yescharacteristic
162C2×C17#53511yescharacteristic
163C2×C17#53711yescharacteristic
164C2×C17#56012yes—
165C2×C17#56212yes—
166D17—82no—
167C22×D4—172no—
168C22×D4—172no—
169C22×D4—171no—
170C22×D4C2×D1711yescharacteristic
171C23×C4C2×D1711yescharacteristic
172#123—24no—
173#121—344no—
174C2×C4⋊C4—171no—
175C2×C4⋊C4—172no—
176C2×C4⋊C4—172no—
177C2×C4⋊C4C2×D1711yescharacteristic
178C2×C4⋊C4C2×D1711yescharacteristic
179C2×C22⋊C4—172no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—172no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4C2×D1711yescharacteristic
186C2×C22⋊C4C2×D1711yescharacteristic
187C2×C22⋊C4C2×D1711yescharacteristic
188#95—171no—
189#95—171no—
190#95—172no—
191#95—172no—
192C17#281311yescharacteristic, complemented, Sylow
193C24—172no—
194C24C22×D1711yescharacteristic
195C24C22×D1711yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—344no—
199C2×D4—342no—
200C2×D4—682no—
201C2×D4—682no—
202C2×D4—682no—
203C2×D4—682no—
204C2×D4—682no—
205C2×D4—682no—
206C22×C4—42no—
207C22×C4—172no—
208C22×C4—172no—
209C22×C4—172no—
210C22×C4C22⋊D1714yescomplemented
211C22×C4C22×D1711yescharacteristic
212C22×C4C22×D1711yescharacteristic
213C22×C4C22×D1711yescharacteristic
214C22×C4C22×D1711yescharacteristic
215C22×C4C22×D1711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—342no—
219C4⋊C4—682no—
220C4⋊C4—682no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—344no—
225C22⋊C4—682no—
226C22⋊C4—682no—
227C22⋊C4—682no—
228C22⋊C4—682no—
229C22⋊C4—682no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—342no—
240C23—342no—
241C23—342no—
242C23—342no—
243C23—342no—
244C23—342no—
245C23—342no—
246C23C2×C22⋊D1712yes—
247C23C23×D1711yescharacteristic
248D4—44no—
249D4—684no—
250D4—688no—
251D4—688no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4C2×C22⋊D1712yes—
272C2×C4C2×C22⋊D1712yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—682no—
284C22—682no—
285C22—682no—
286C22—682no—
287C22—682no—
288C22—682no—
289C22—682no—
290C22#1058769112yes—
291C22#1058769611yescharacteristic
292C22#1058769612yes—
293C22#1058769711yescharacteristic
294C22C22×C22⋊D1711yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—682no—
301C4—682no—
302C4—682no—
303C2—42no—
304C2—81no—
305C2—1362no—
306C2#1176336811yescharacteristic
307C2#1176337311yescharacteristic
308C2#1176338711yescharacteristic
309C2#1176354912yes—
310C2#1176355612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
hypercenterC2×#123D17
hypocenterC17#2813
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#10587616C2semidirect
2. #11762748C2semidirect
3. #11762745C2semidirect
4. #11762731C2semidirect
5. #11762726C2semidirect
6. #11762725C2semidirect
7. #11762724C2semidirect
8. #11762718C2semidirect
9. #11762717C2semidirect
10. #11762669C2semidirect
11. C2×C17×#123C2semidirect
43. C22×C4×C17D4semidirect
148. C2×#123D17semidirect
192. C17#2813semidirect
210. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876162
#117627482
#117627452
#117627312
#117627262
#117627252
#117627242
#117627182
#117627172
#117626692
C2×C17×#1232
#281317