Finite Group Databasebeta

← #14534473

Subgroups — #14534473

Order 2176.

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

normal subgroups only

This group has 576 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×#10587558C211yesmaximal, characteristic, complemented
2#11762728C211yesmaximal, characteristic, complemented
3#11762723C211yesmaximal, characteristic, complemented
4#11762720C211yesmaximal, characteristic, complemented
5#11762717C211yesmaximal, characteristic
6#11762709C212yesmaximal, complemented
7#11762703C212yesmaximal, complemented
8#11762697C212yesmaximal, complemented
9#11762685C212yesmaximal, complemented
10#11762655C211yesmaximal, characteristic, complemented
11C2×C17×#124C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17C2212yes—
15C22×C4×D17C2211yescharacteristic
16C17×#124—24no—
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C42×C17C2211yescharacteristic
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×C4×Dic17C2211yescharacteristic
30C2×C4×Dic17C2212yes—
31#10587558—24no—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2312yes—
38C23×D17C2311yescharacteristic
39C2×Q8×C17—22no—
40C2×Q8×C17—42no—
41C2×D4×C17—22no—
42C2×D4×C17—42no—
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C2×C22⋊D17—24no—
48C2×C22⋊D17—24no—
49C2×C22⋊D17—42no—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2312yes—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17—22no—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—24no—
60C42×C17D414yescomplemented
61C22⋊Dic17—24no—
62C22⋊Dic17—24no—
63C22⋊Dic17—42no—
64#63227—22no—
65#63227—22no—
66#63227—42no—
67#63225—22no—
68#63225—22no—
69#63225—42no—
70C4×Dic17—22no—
71C4×Dic17—24no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17C2411yescharacteristic
80C22×D17—21no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17#4112yes—
85Q8×C17—44no—
86D4×C17—44no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17C2×D412yes—
95C2×C4×C17C2×D412yes—
96C2×C4×C17C2×D412yes—
97C22⋊D17—48no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17#4112yes—
110C2×Dic17#4114yes—
111C4×D17—28no—
112#2610—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×C17C2×#4111yescharacteristic
122C22×C17C2×#4111yescharacteristic
123C22×C17C2×#4112yes—
124C22×C17H(5,𝔽2)12yes—
125C2×D17—22no—
126C2×D17—24no—
127C2×D17—41no—
128C2×D17—41no—
129C2×D17—42no—
130C4×C17—22no—
131C4×C17—22no—
132C4×C17—22no—
133C4×C17—42no—
134Dic17—22no—
135Dic17—24no—
136Dic17—41no—
137Dic17—41no—
138Dic17—42no—
139C2×#124D1711yescharacteristic, complemented
140C2×#117—171no—
141#395—171no—
142#394—171no—
143#393—171no—
144#392—171no—
145#390—172no—
146#388—172no—
147#388—172no—
148#386—172no—
149#376—171no—
150C2×C17—82no—
151C2×C17#52511yescharacteristic
152C2×C17#53211yescharacteristic
153C2×C17#53511yescharacteristic
154C2×C17#55112yes—
155C2×C17#55512yes—
156D17—42no—
157C22×Q8C2×D1711yescharacteristic
158C22×D4—172no—
159C22×D4C2×D1711yescharacteristic
160C23×C4—171no—
161#124—24no—
162#117—344no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—171no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1712yes—
173C2×C22⋊C4C2×D1712yes—
174C2×C42—171no—
175C2×C42—172no—
176C2×C42C2×D1711yescharacteristic
177#95—172no—
178#95—171no—
179#95—171no—
180#95—172no—
181#95—172no—
182C17#261011yescharacteristic, complemented, Sylow
183C24—171no—
184C24C22×D1712yes—
185C2×Q8—22no—
186C2×Q8—42no—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—344no—
190C2×D4—344no—
191C2×D4—682no—
192C22×C4—172no—
193C22×C4—171no—
194C22×C4—172no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—342no—
198C22×C4—342no—
199C22×C4—342no—
200C22×C4C22×D1712yes—
201C22×C4C22×D1711yescharacteristic
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C4⋊C4—342no—
205C4⋊C4—342no—
206C4⋊C4—682no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—344no—
210C22⋊C4—344no—
211C22⋊C4—342no—
212C22⋊C4—342no—
213C22⋊C4—682no—
214C22⋊C4—682no—
215C42—344no—
216C42—342no—
217C42C22⋊D1714yescomplemented
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—172no—
226C23—342no—
227C23—341no—
228C23—342no—
229C23—341no—
230C23C23×D1711yescharacteristic
231Q8—44no—
232D4—44no—
233D4—688no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—174no—
242C2×C4—172no—
243C2×C4—348no—
244C2×C4—342no—
245C2×C4—341no—
246C2×C4—342no—
247C2×C4—341no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4C2×C22⋊D1712yes—
256C2×C4C2×C22⋊D1712yes—
257C2×C4C2×C22⋊D1712yes—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—344no—
266C22—342no—
267C22—682no—
268C22—681no—
269C22—681no—
270C22#1058769112yes—
271C22#41×D1711yescharacteristic
272C22#41×D1712yes—
273C22#41×D1711yescharacteristic
274C22C22×C22⋊D1711yescharacteristic
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—344no—
280C4—342no—
281C4—682no—
282C4—681no—
283C4—681no—
284C2—82no—
285C2—682no—
286C2#1176331611yescharacteristic
287C2#1176335511yescharacteristic
288C2#1176337611yescharacteristic
289C2#1176348912yes—
290C2#1176352112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
prime radical#11762717C2
hypercenterC2×#124D17
hypocenterC17#2610
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#10587558C2semidirect
2. #11762728C2semidirect
3. #11762723C2semidirect
4. #11762720C2semidirect
6. #11762709C2semidirect
7. #11762703C2semidirect
8. #11762697C2semidirect
9. #11762685C2semidirect
10. #11762655C2semidirect
11. C2×C17×#124C2semidirect
60. C42×C17D4semidirect
139. C2×#124D17semidirect
182. C17#2610semidirect
217. C42C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875582
#117627282
#117627232
#117627202
#117627172
#117627092
#117627032
#117626972
#117626852
#117626552
C2×C17×#1242
#261017