Finite Group Databasebeta

← #14534713

Subgroups — #14534713

Order 2176.

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

normal subgroups only

This group has 576 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×#10587632C211yesmaximal, characteristic, complemented
2C2×#10587570C211yesmaximal, characteristic
3#11762750C212yesmaximal, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic, complemented
6#11762724C212yesmaximal, complemented
7#11762720C211yesmaximal, characteristic, complemented
8#11762687C212yesmaximal, complemented
9#11762685C212yesmaximal, complemented
10#95×D17C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C22×D68C2211yescharacteristic
16C22×C4×D17C2212yes—
17C22×C4×D17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2212yes—
21#10587632—24no—
22C2×#63227C2211yescharacteristic
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2211yescharacteristic
29#10587570—24no—
30C17×#95C2211yescharacteristic
31C17×#95C2212yes—
32#10587502C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C2×Q8×C17—22no—
41C2×Q8×C17—42no—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—22no—
47C2×C22⋊D17—24no—
48C2×C22⋊D17—22no—
49C2×C22⋊D17—42no—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C2×D68—22no—
55C2×D68—24no—
56C2×C4×D17—24no—
57C2×C4×D17—42no—
58C2×C4×D17—42no—
59C2×C4×D17—42no—
60C2×C4×D17D414yes—
61C22⋊C4×C17—22no—
62C22⋊C4×C17—42no—
63C22⋊Dic17—42no—
64#63227—24no—
65#63227—22no—
66#63227—42no—
67#63227—42no—
68C4⋊Dic17—22no—
69#63225—22no—
70#63225—24no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17—21no—
74C23×C17—22no—
75C23×C17—21no—
76C23×C17C2411yescharacteristic
77C22×D17—22no—
78C22×D17—21no—
79C22×D17—21no—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—21no—
84C22×D17—21no—
85C22×D17C2×D412yes—
86C22×D17#4114yes—
87Q8×C17—44no—
88C2×C4×C17—22no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17C2×D412yes—
98C22⋊D17—44no—
99C22⋊D17—44no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17C2×D412yes—
110D68—44no—
111C4×D17—44no—
112C4×D17—48no—
113#2639—171nomaximal, Sylow
114C22×C17—41no—
115C22×C17—41no—
116C22×C17—41no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17C22×D411yescharacteristic
120C22×C17C2×#4112yes—
121C22×C17H(5,𝔽2)12yes—
122C22×C17H(5,𝔽2)11yescharacteristic
123C22×C172−1+411yescharacteristic
124C2×D17—22no—
125C2×D17—22no—
126C2×D17—24no—
127C2×D17—24no—
128C2×D17—41no—
129C2×D17—41no—
130C2×D17—41no—
131C2×D17—42no—
132C2×D17—42no—
133C2×D17#12414yes—
134C4×C17—22no—
135C4×C17—41no—
136C4×C17—42no—
137C4×C17—42no—
138Dic17—22no—
139Dic17—41no—
140Dic17—42no—
141Dic17—42no—
142C2×C22⋊Q8—171no—
143C2×#121—171no—
144#400—172no—
145#397—171no—
146#397—171no—
147#394—172no—
148#393—171no—
149#393D1711yescharacteristic, complemented
150#386—172no—
151#386—172no—
152C2×#95—171no—
153C2×C17—81no—
154C2×C17C2×#12411yescharacteristic
155C2×C17#53411yescharacteristic
156C2×C17#53711yescharacteristic
157C2×C17#53812yes—
158C2×C17#53912yes—
159D17—24no—
160D17—81no—
161C22×Q8C2×D1711yescharacteristic
162C22×D4—171no—
163C22×D4—171no—
164C22×D4—171no—
165C23×C4—171no—
166C23×C4—172no—
167C22⋊Q8—344no—
168#121—344no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—172no—
171C2×C4⋊C4—171no—
172C2×C22⋊C4—171no—
173C2×C22⋊C4—172no—
174C2×C22⋊C4—172no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—172no—
177C2×C22⋊C4C2×D1711yescharacteristic
178C2×C22⋊C4C2×D1712yes—
179#95—171no—
180#95—172no—
181#95—172no—
182#95—172no—
183#95—172no—
184#95C2×D1711yescharacteristic
185#95C2×D1712yes—
186C17#263911yescharacteristic, complemented, Sylow
187C24—171no—
188C24—171no—
189C24C22×D1711yescharacteristic
190C2×Q8—22no—
191C2×Q8—42no—
192C2×D4—342no—
193C2×D4—344no—
194C2×D4—342no—
195C2×D4—342no—
196C2×D4—344no—
197C2×D4—682no—
198C22×C4—171no—
199C22×C4—174no—
200C22×C4—172no—
201C22×C4—171no—
202C22×C4—172no—
203C22×C4—344no—
204C22×C4—682no—
205C22×C4—682no—
206C22×C4—682no—
207C22×C4C22×D1712yes—
208C22×C4C22×D1711yescharacteristic
209C22×C4C22×D1712yes—
210C22×C4C22×D1711yescharacteristic
211C4⋊C4—342no—
212C4⋊C4—344no—
213C4⋊C4—342no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—344no—
217C22⋊C4—342no—
218C22⋊C4—682no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—172no—
227C23—174no—
228C23—342no—
229C23—341no—
230C23—341no—
231C23—341no—
232C23—342no—
233C23—341no—
234C23—342no—
235C23—341no—
236C23C23×D1711yescharacteristic
237Q8—44no—
238D4—684no—
239D4—684no—
240D4—684no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—172no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—684no—
261C2×C4—688no—
262C2×C4D4×D1712yes—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—174no—
269C22—342no—
270C22—342no—
271C22—344no—
272C22—344no—
273C22—681no—
274C22—681no—
275C22—681no—
276C22—682no—
277C22—682no—
278C22C2×D4×D1711yescharacteristic
279C22#1058769112yes—
280C22#1058769411yescharacteristic
281C22#41×D1712yes—
282C22#1058769611yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—42no—
287C4—342no—
288C4—681no—
289C4—682no—
290C4—682no—
291C2—81no—
292C2—344no—
293C2—1361no—
294C2#124×D1711yescharacteristic
295C2#1176336911yescharacteristic
296C2#1176338911yescharacteristic
297C2#1176339312yes—
298C2#1176340512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC2×#10587570C2
hypercenter#393D17
hypocenterC17#2639
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#10587632C2semidirect
3. #11762750C2semidirect
4. #11762736C2semidirect
5. #11762734C2semidirect
6. #11762724C2semidirect
7. #11762720C2semidirect
8. #11762687C2semidirect
9. #11762685C2semidirect
10. #95×D17C2semidirect
11. C17×#393C2semidirect
149. #393D17semidirect
186. C17#2639semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
C2×#105875702
#117627502
#117627362
#117627342
#117627242
#117627202
#117626872
#117626852
#95×D172
C17×#3932
#263917