Finite Group Databasebeta

← #14535147

Subgroups — #14535147

Order 2176.

668 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587569C212yesmaximal, complemented
2C2×#10587570C212yesmaximal, complemented
3#11762732C212yesmaximal, complemented
4#11762725C212yesmaximal, complemented
5#11762717C212yesmaximal, complemented
6#11762711C211yesmaximal, characteristic, complemented
7#11762707C211yesmaximal, characteristic, complemented
8#11762680C212yesmaximal, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C22×C4×D17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18C2×C42×C17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29#10587570—28no—
30#10587569—28no—
31#10587502C2212yes—
32#10587502C2212yes—
33C24×C17C2312yes—
34C23×D17C2312yes—
35C2×D4×C17—22no—
36C2×D4×C17—41no—
37C2×D4×C17—41no—
38C22×C4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C2×C22⋊D17—24no—
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—42no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17—42no—
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C2×D68—41no—
52C2×D68—41no—
53C2×D68D414yescomplemented
54C2×C4×D17—24no—
55C2×C4×D17—24no—
56C2×C4×D17—42no—
57C4⋊C4×C17—41no—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—24no—
60C42×C17—41no—
61C22⋊Dic17—24no—
62C22⋊Dic17—42no—
63#63227—24no—
64#63227—24no—
65#63227—24no—
66C4⋊Dic17—41no—
67C4⋊Dic17—41no—
68C4⋊Dic17D414yescomplemented
69#63225—24no—
70#63225—24no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17C2411yescharacteristic
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17C2×D414yes—
86D4×C17—44no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C2×C4×C17#4112yes—
99C22⋊D17—48no—
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×Dic17C2×D414yes—
110D68—44no—
111C4×D17—48no—
112#2682—171nomaximal, Sylow
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17C22×D411yescharacteristic
121C22×C17C22×D411yescharacteristic
122C22×C17C2×#4111yescharacteristic
123C22×C17H(5,𝔽2)11yescharacteristic
124C22×C17H(5,𝔽2)11yescharacteristic
125C22×C17H(5,𝔽2)11yescharacteristic
126C22×C172−1+411yescharacteristic
127C2×D17—24no—
128C2×D17—24no—
129C2×D17—24no—
130C2×D17—42no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17—42no—
134C4×C17—22no—
135C4×C17—22no—
136C4×C17—22no—
137C4×C17—42no—
138Dic17—24no—
139Dic17—42no—
140Dic17—42no—
141C2×#123—172no—
142C2×#121—172no—
143#396—172no—
144#394—172no—
145#392—172no—
146#390—171no—
147#390D1711yescharacteristic, complemented
148#389—171no—
149#385—172no—
150C2×C17—82no—
151C2×C17#53411yescharacteristic
152C2×C17#53711yescharacteristic
153C2×C17#53811yescharacteristic
154C2×C17D4211yescharacteristic
155C2×C17#54711yescharacteristic
156C2×C17#55011yescharacteristic
157C2×C17#56211yescharacteristic
158D17—44no—
159C22×D4—172no—
160C22×D4—172no—
161C22×D4—171no—
162C22×D4C2×D1711yescharacteristic
163C23×C4—172no—
164#123—348no—
165#121—348no—
166C2×C4⋊C4—171no—
167C2×C4⋊C4—172no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—171no—
171C2×C4⋊C4C2×D1711yescharacteristic
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4C2×D1712yes—
178C2×C22⋊C4C2×D1712yes—
179C2×C42C2×D1711yescharacteristic
180#95—172no—
181#95—172no—
182C17#268211yescharacteristic, complemented, Sylow
183C24—172no—
184C24C22×D1712yes—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—174no—
189C2×D4—344no—
190C2×D4—344no—
191C2×D4—344no—
192C2×D4—681no—
193C2×D4—681no—
194C2×D4—682no—
195C2×D4—682no—
196C2×D4—682no—
197C22×C4—172no—
198C22×C4—172no—
199C22×C4—172no—
200C22×C4—344no—
201C22×C4—344no—
202C22×C4—682no—
203C22×C4C22×D1712yes—
204C22×C4C22×D1711yescharacteristic
205C22×C4C22×D1711yescharacteristic
206C22×C4C22×D1711yescharacteristic
207C4⋊C4—41no—
208C4⋊C4—174no—
209C4⋊C4—344no—
210C4⋊C4—344no—
211C4⋊C4—681no—
212C4⋊C4—681no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—344no—
216C22⋊C4—344no—
217C22⋊C4—344no—
218C22⋊C4—344no—
219C22⋊C4—682no—
220C42—41no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—174no—
229C23—342no—
230C23—342no—
231C23—342no—
232C23—342no—
233C23—342no—
234C23—342no—
235C23C23×D1711yescharacteristic
236D4—44no—
237D4—684no—
238D4—688no—
239D4—688no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—174no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—688no—
259C2×C4#6325212yes—
260C2×C4D4×D1712yes—
261C2×C4D4×D1712yes—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—344no—
270C22—344no—
271C22—344no—
272C22—682no—
273C22—682no—
274C22—682no—
275C22—682no—
276C22C2×#6325211yescharacteristic
277C22C2×D4×D1711yescharacteristic
278C22C2×D4×D1711yescharacteristic
279C22#1058769111yescharacteristic
280C22#1058769111yescharacteristic
281C22#1058769611yescharacteristic
282C22#1058769711yescharacteristic
283C4—22no—
284C4—22no—
285C4—22no—
286C4—42no—
287C4—344no—
288C4—682no—
289C4—682no—
290C2—82no—
291C2—684no—
292C2#1176336711yescharacteristic
293C2#1176338611yescharacteristic
294C2#1176339711yescharacteristic
295C2#1176344411yescharacteristic
296C2#1176346411yescharacteristic
297C2#1176348011yescharacteristic
298C2#1176355311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
hypercenter#390D17
hypocenterC17#2682
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587569C2semidirect
2. C2×#10587570C2semidirect
3. #11762732C2semidirect
4. #11762725C2semidirect
5. #11762717C2semidirect
6. #11762711C2semidirect
7. #11762707C2semidirect
8. #11762680C2semidirect
9. C17×#390C2semidirect
53. C2×D68D4semidirect
68. C4⋊Dic17D4semidirect
147. #390D17semidirect
182. C17#2682semidirect

Maximal subgroups

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

subgroupindex
C2×#105875692
C2×#105875702
#117627322
#117627252
#117627172
#117627112
#117627072
#117626802
C17×#3902
#268217