Finite Group Databasebeta

← #14534345

Subgroups — #14534345

Order 2176.

604 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 139 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×D4×Dic17C211yesmaximal, characteristic, complemented
2C2×#10587578C211yesmaximal, characteristic, complemented
3#11762748C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic, complemented
5#11762717C211yesmaximal, characteristic
6#11762702C212yesmaximal, complemented
7#11762686C212yesmaximal, complemented
8#11762685C212yesmaximal, complemented
9#11762684C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C23×Dic17C2212yes—
15C22×C4×D17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21D4×Dic17—24no—
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×C4⋊Dic17C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587578—24no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2312yes—
38C23×D17C2311yescharacteristic
39C2×D4×C17—24no—
40C2×D4×C17—22no—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—42no—
48C22×Dic17—24no—
49C22×Dic17—42no—
50C22×Dic17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C2×C4×D17—22no—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C4⋊C4×C17—22no—
60C22⋊C4×C17—24no—
61C22⋊C4×C17—24no—
62C22⋊Dic17—24no—
63C22⋊Dic17—42no—
64#63227—22no—
65#63227—22no—
66#63227—42no—
67C4⋊Dic17—22no—
68C4⋊Dic17—22no—
69C4⋊Dic17—42no—
70C4×Dic17—22no—
71C4×Dic17D414yescomplemented
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17#4114yes—
79C23×C17C2411yescharacteristic
80C22×D17—21no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17#4112yes—
85D4×C17—28no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
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×C17C2×D412yes—
95C2×C4×C17#4112yes—
96C22⋊D17—48no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—22no—
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—48no—
109C2×Dic17C2×D412yes—
110C2×Dic17C2×D412yes—
111C4×D17—28no—
112#2598—171nomaximal, Sylow
113C22×C17—24no—
114C22×C17—24no—
115C22×C17—24no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17—42no—
120C22×C17C22×D411yescharacteristic
121C22×C17C2×#4112yes—
122C22×C17C2×#4111yescharacteristic
123C22×C17C2×#4111yescharacteristic
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—41no—
133C4×C17—42no—
134Dic17—22no—
135Dic17—22no—
136Dic17—41no—
137Dic17—42no—
138Dic17—42no—
139C2×C4×D4—171no—
140C2×#117—171no—
141#400—171no—
142#399—171no—
143#394D1711yescharacteristic, complemented
144#392—171no—
145#388—172no—
146#386—172no—
147#386—172no—
148#385—172no—
149#383—171no—
150C2×C17—44no—
151C2×C17#52511yescharacteristic
152C2×C17#52511yescharacteristic
153C2×C17#53511yescharacteristic
154C2×C17#55112yes—
155C2×C17#55312yes—
156D17—42no—
157C22×D4—172no—
158C22×D4C2×D1711yescharacteristic
159C23×C4—172no—
160C23×C4—171no—
161C4×D4—344no—
162#117—344no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—172no—
166C2×C4⋊C4C2×D1711yescharacteristic
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—171no—
176#95—172no—
177#95—172no—
178#95—171no—
179#95—172no—
180#95—172no—
181#95C2×D1711yescharacteristic
182C17#259811yescharacteristic, complemented, Sylow
183C24—171no—
184C24C22×D1712yes—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—344no—
188C2×D4—344no—
189C2×D4—682no—
190C22×C4—171no—
191C22×C4—171no—
192C22×C4—172no—
193C22×C4—172no—
194C22×C4—171no—
195C22×C4—342no—
196C22×C4—342no—
197C22×C4—344no—
198C22×C4—342no—
199C22×C4—682no—
200C22×C4—682no—
201C22×C4C22×D1712yes—
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C22×C4C22×D1711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—342no—
207C4⋊C4—342no—
208C4⋊C4—682no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—342no—
212C22⋊C4—344no—
213C22⋊C4—342no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C42—174no—
217C42—342no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—172no—
225C23—341no—
226C23—341no—
227C23—342no—
228C23—342no—
229C23#6325414yes—
230C23C23×D1711yescharacteristic
231D4—28no—
232D4—688no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—172no—
242C2×C4—172no—
243C2×C4—348no—
244C2×C4—341no—
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—342no—
254C2×C4—342no—
255C2×C4—688no—
256C2×C4D4×D1712yes—
257C2×C4#6325412yes—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—344no—
266C22—342no—
267C22—682no—
268C22—681no—
269C22—681no—
270C22C2×D4×D1711yescharacteristic
271C22C2×#6325412yes—
272C22C2×#6325411yescharacteristic
273C22#41×D1711yescharacteristic
274C22#1058769612yes—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—41no—
279C4—342no—
280C4—342no—
281C4—682no—
282C4—682no—
283C4—681no—
284C2—44no—
285C2—682no—
286C2#1176331511yescharacteristic
287C2#1176331911yescharacteristic
288C2#1176337711yescharacteristic
289C2#1176348212yes—
290C2#1176351112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radical#11762717C2
hypercenter#394D17
hypocenterC17#2598
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic17C2semidirect
2. C2×#10587578C2semidirect
3. #11762748C2semidirect
4. #11762746C2semidirect
6. #11762702C2semidirect
7. #11762686C2semidirect
8. #11762685C2semidirect
9. #11762684C2semidirect
10. #11762676C2semidirect
11. C17×#394C2semidirect
71. C4×Dic17D4semidirect
143. #394D17semidirect
182. C17#2598semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic172
C2×#105875782
#117627482
#117627462
#117627172
#117627022
#117626862
#117626852
#117626842
#117626762
C17×#3942
#259817