Finite Group Databasebeta

← #14534351

Subgroups — #14534351

Order 2176.

604 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 135 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×#10587579C211yesmaximal, characteristic, complemented
2C2×#10587563C211yesmaximal, characteristic, complemented
3#11762747C211yesmaximal, characteristic, complemented
4#11762727C211yesmaximal, characteristic, complemented
5#11762718C211yesmaximal, characteristic
6#11762699C212yesmaximal, complemented
7#11762686C212yesmaximal, complemented
8#11762685C212yesmaximal, complemented
9#11762680C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2212yes—
13C23×Dic17C2211yescharacteristic
14C22×D68C2211yescharacteristic
15C22×C4×D17C2212yes—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63225C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2211yescharacteristic
29#10587579—24no—
30#10587563—24no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2312yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—42no—
46C22×Dic17—22no—
47C22×Dic17—22no—
48C22×Dic17—22no—
49C22×Dic17C2312yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
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—42no—
61C22⋊C4×C17—22no—
62C22⋊C4×C17—42no—
63C22⋊Dic17—22no—
64#63227—24no—
65#63227—24no—
66#63227—24no—
67#63227—42no—
68#63225—22no—
69#63225—22no—
70C4×Dic17—22no—
71C4×Dic17D414yescomplemented
72C23×C17—21no—
73C23×C17—21no—
74C23×C17—22no—
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—22no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17C2×D412yes—
95C22⋊D17—48no—
96C2×Dic17—28no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17C2×D412yes—
108C2×Dic17C2×D412yes—
109C2×Dic17#4112yes—
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×#4112yes—
121C22×C17C2×#4111yescharacteristic
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—42no—
132C4×C17—41no—
133C4×C17—42no—
134Dic17—22no—
135Dic17—22no—
136Dic17—22no—
137Dic17—41no—
138Dic17—42no—
139C2×C4×D4—171no—
140C2×#117—171no—
141#400—171no—
142#399D1711yescharacteristic, complemented
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—171no—
160C23×C4—172no—
161C4×D4—344no—
162#117—344no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4C2×D1711yescharacteristic
166C2×C4⋊C4C2×D1712yes—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1711yescharacteristic
173C2×C22⋊C4C2×D1712yes—
174C2×C42—171no—
175C2×C42—171no—
176#95—172no—
177#95—172no—
178#95—171no—
179#95—172no—
180#95—172no—
181#95C2×D1711yescharacteristic
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—171no—
191C22×C4—171no—
192C22×C4—171no—
193C22×C4—172no—
194C22×C4—171no—
195C22×C4—344no—
196C22×C4—342no—
197C22×C4—342no—
198C22×C4—342no—
199C22×C4—682no—
200C22×C4—682no—
201C22×C4C22×D1712yes—
202C22×C4C22×D1712yes—
203C22×C4C22×D1711yescharacteristic
204C22×C4C22×D1711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—342no—
208C4⋊C4—342no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—344no—
212C22⋊C4—344no—
213C22⋊C4—344no—
214C22⋊C4—342no—
215C22⋊C4—682no—
216C42—174no—
217C42—342no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—174no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23—342no—
228C23—342no—
229C23#6325612yes—
230C23C23×D1711yescharacteristic
231D4—348no—
232D4—688no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—172no—
243C2×C4—172no—
244C2×C4—172no—
245C2×C4—348no—
246C2×C4—341no—
247C2×C4—342no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4—341no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—688no—
257C2×C4D4×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×D4×D1711yescharacteristic
271C22#1058769112yes—
272C22C2×#6325611yescharacteristic
273C22#41×D1712yes—
274C22#41×D1711yescharacteristic
275C4—22no—
276C4—42no—
277C4—42no—
278C4—41no—
279C4—342no—
280C4—342no—
281C4—342no—
282C4—682no—
283C4—681no—
284C2—42no—
285C2—684no—
286C2#1176331711yescharacteristic
287C2#1176332211yescharacteristic
288C2#1176337211yescharacteristic
289C2#1176348612yes—
290C2#1176350412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radical#11762718C2
hypercenter#399D17
hypocenterC17#2598
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587579C2semidirect
2. C2×#10587563C2semidirect
3. #11762747C2semidirect
4. #11762727C2semidirect
6. #11762699C2semidirect
7. #11762686C2semidirect
8. #11762685C2semidirect
9. #11762680C2semidirect
10. #11762676C2semidirect
11. C17×#399C2semidirect
71. C4×Dic17D4semidirect
142. #399D17semidirect
182. C17#2598semidirect

Maximal subgroups

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

subgroupindex
C2×#105875792
C2×#105875632
#117627472
#117627272
#117627182
#117626992
#117626862
#117626852
#117626802
#117626762
C17×#3992
#259817