Finite Group Databasebeta

← #14534350

Subgroups — #14534350

Order 2176.

604 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 145 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×C4×D68C211yesmaximal, characteristic, complemented
2#11762748C211yesmaximal, characteristic, complemented
3#11762745C211yesmaximal, characteristic, complemented
4#11762726C211yesmaximal, characteristic, complemented
5#11762718C211yesmaximal, characteristic
6#11762700C212yesmaximal, complemented
7#11762688C212yesmaximal, complemented
8#11762685C212yesmaximal, complemented
9#11762680C212yesmaximal, complemented
10#11762677C211yesmaximal, characteristic, complemented
11C2×C17×#117C211yesmaximal, characteristic, complemented
12C23×C4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C22×D68C2211yescharacteristic
15C22×C4×D17C2212yes—
16C17×#117—24no—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C42×C17C2211yescharacteristic
22C2×C42×C17C2211yescharacteristic
23C2×C22⋊Dic17C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2211yescharacteristic
29C2×#63225C2212yes—
30C4×D68—24no—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2312yes—
39C22×C4×C17—22no—
40C22×C4×C17—22no—
41C22×C4×C17—22no—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C2×D68—24no—
55C2×D68—22no—
56C2×C4×D17—24no—
57C2×C4×D17—42no—
58C2×C4×D17—42no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17—22no—
61C22⋊C4×C17—22no—
62C22⋊C4×C17—22no—
63C42×C17—22no—
64C42×C17D414yescomplemented
65C22⋊Dic17—42no—
66#63227—24no—
67#63227—24no—
68#63227—24no—
69#63227—42no—
70C4⋊Dic17—22no—
71#63225—42no—
72C23×C17—21no—
73C23×C17—22no—
74C23×C17—21no—
75C23×C17—22no—
76C23×C17#4112yes—
77C23×C17C2411yescharacteristic
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17#4114yes—
85C2×C4×C17—28no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C2×C4×C17C2×D412yes—
99C2×C4×C17#4112yes—
100C22⋊D17—48no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110D68—28no—
111C4×D17—48no—
112#2598—171nomaximal, Sylow
113C22×C17—22no—
114C22×C17—24no—
115C22×C17—41no—
116C22×C17—41no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4111yescharacteristic
120C22×C17C2×#4111yescharacteristic
121C22×C17C2×#4112yes—
122C22×C17H(5,𝔽2)12yes—
123C2×D17—24no—
124C2×D17—24no—
125C2×D17—24no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—42no—
130C4×C17—22no—
131C4×C17—22no—
132C4×C17—22no—
133C4×C17—22no—
134C4×C17—41no—
135C4×C17—41no—
136Dic17—42no—
137Dic17—42no—
138Dic17—42no—
139C2×C4×D4—171no—
140C2×#117D1711yescharacteristic, complemented
141#400—171no—
142#399—171no—
143#394—171no—
144#392—171no—
145#388—172no—
146#386—172no—
147#386—172no—
148#385—172no—
149#383—171no—
150C2×C17—42no—
151C2×C17#52511yescharacteristic
152C2×C17#52511yescharacteristic
153C2×C17#53511yescharacteristic
154C2×C17#55112yes—
155C2×C17#55312yes—
156D17—44no—
157C22×D4—172no—
158C22×D4—171no—
159C23×C4—172no—
160C23×C4C2×D1711yescharacteristic
161C4×D4—344no—
162#117—24no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—172no—
165C2×C4⋊C4C2×D1711yescharacteristic
166C2×C4⋊C4C2×D1711yescharacteristic
167C2×C22⋊C4—172no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4C2×D1711yescharacteristic
174C2×C42C2×D1711yescharacteristic
175C2×C42C2×D1711yescharacteristic
176#95—172no—
177#95—172no—
178#95—171no—
179#95—171no—
180#95—172no—
181#95—172no—
182C17#259811yescharacteristic, complemented, Sylow
183C24—172no—
184C24C22×D1711yescharacteristic
185C2×D4—344no—
186C2×D4—344no—
187C2×D4—344no—
188C2×D4—342no—
189C2×D4—682no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—172no—
194C22×C4—172no—
195C22×C4—172no—
196C22×C4—344no—
197C22×C4—682no—
198C22×C4—682no—
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1711yescharacteristic
201C22×C4C22×D1711yescharacteristic
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C22×C4C22×D1711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—342no—
208C4⋊C4—682no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—344no—
212C22⋊C4—344no—
213C22⋊C4—344no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C42—22no—
217C42C22⋊D1714yescomplemented
218C23—21no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—174no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23—342no—
228C23—342no—
229C23#6325212yes—
230C23C23×D1711yescharacteristic
231D4—348no—
232D4—688no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—688no—
254C2×C4#6325212yes—
255C2×C4C2×C22⋊D1712yes—
256C2×C4C2×C22⋊D1712yes—
257C2×C4C2×C22⋊D1712yes—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—344no—
264C22—344no—
265C22—344no—
266C22—682no—
267C22—682no—
268C22—682no—
269C22—682no—
270C22C2×#6325211yescharacteristic
271C22C2×#6325211yescharacteristic
272C22#41×D1712yes—
273C22#1058769612yes—
274C22C22×C22⋊D1711yescharacteristic
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—682no—
282C4—682no—
283C4—682no—
284C2—42no—
285C2—684no—
286C2#1176331611yescharacteristic
287C2#1176332011yescharacteristic
288C2#1176337311yescharacteristic
289C2#1176348312yes—
290C2#1176350312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#117C2
prime radical#11762718C2
hypercenterC2×#117D17
hypocenterC17#2598
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C4×D68C2semidirect
2. #11762748C2semidirect
3. #11762745C2semidirect
4. #11762726C2semidirect
6. #11762700C2semidirect
7. #11762688C2semidirect
8. #11762685C2semidirect
9. #11762680C2semidirect
10. #11762677C2semidirect
11. C2×C17×#117C2semidirect
64. C42×C17D4semidirect
140. C2×#117D17semidirect
182. C17#2598semidirect
217. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D682
#117627482
#117627452
#117627262
#117627182
#117627002
#117626882
#117626852
#117626802
#117626772
C2×C17×#1172
#259817