Finite Group Databasebeta

← #14534352

Subgroups — #14534352

Order 2176.

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

normal subgroups only

This group has 604 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×C4×D68C211yesmaximal, characteristic, complemented
2C2×#10587558C211yesmaximal, characteristic, complemented
3#11762749C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic, complemented
5#11762727C211yesmaximal, characteristic, complemented
6#11762719C211yesmaximal, characteristic
7#11762701C212yesmaximal, complemented
8#11762687C212yesmaximal, complemented
9#11762687C212yesmaximal, complemented
10#11762683C212yesmaximal, complemented
11C17×#383C211yesmaximal, characteristic, complemented
12C22×D68C2211yescharacteristic
13C22×D68C2212yes—
14C22×C4×D17C2211yescharacteristic
15C22×C4×D17C2212yes—
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×C4×Dic17C2211yescharacteristic
29C4×D68—24no—
30#10587558—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33C17×#95C2211yescharacteristic
34C17×#95C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2312yes—
38C23×D17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×D68—24no—
49C2×D68—22no—
50C2×D68—24no—
51C2×D68—24no—
52C2×D68—42no—
53C2×C4×D17—22no—
54C2×C4×D17—22no—
55C2×C4×D17—22no—
56C2×C4×D17—24no—
57C2×C4×D17—42no—
58C2×C4×D17—42no—
59C42×C17D414yescomplemented
60#63227—24no—
61#63227—22no—
62#63227—22no—
63#63227—24no—
64#63227—24no—
65#63227—42no—
66#63227—42no—
67C4⋊Dic17—22no—
68C4⋊Dic17—42no—
69#63225—22no—
70#63225—22no—
71C4×Dic17—22no—
72C23×C17C2411yescharacteristic
73C22×D17—22no—
74C22×D17—21no—
75C22×D17—21no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17#4112yes—
84C22×D17#4114yes—
85C2×C4×C17—21no—
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×C17—22no—
95C2×C4×C17C2×D412yes—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17#4112yes—
108D68—28no—
109D68—48no—
110C4×D17—28no—
111C4×D17—48no—
112#2598—171nomaximal, Sylow
113C22×C17C22×D411yescharacteristic
114C22×C17C2×#4111yescharacteristic
115C22×C17C2×#4112yes—
116C22×C17C2×#4111yescharacteristic
117C22×C17H(5,𝔽2)12yes—
118C2×D17—22no—
119C2×D17—24no—
120C2×D17—24no—
121C2×D17—24no—
122C2×D17—24no—
123C2×D17—42no—
124C2×D17—41no—
125C2×D17—42no—
126C2×D17—41no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—42no—
130C4×C17—22no—
131C4×C17—22no—
132C4×C17—22no—
133C4×C17—42no—
134C4×C17—42no—
135Dic17—22no—
136Dic17—41no—
137Dic17—42no—
138Dic17—41no—
139C2×C4×D4—171no—
140C2×#117—171no—
141#400—171no—
142#399—171no—
143#394—171no—
144#392—171no—
145#388—172no—
146#386—172no—
147#386—172no—
148#385—172no—
149#383D1711yescharacteristic, complemented
150C2×C17#52511yescharacteristic
151C2×C17#52511yescharacteristic
152C2×C17#53511yescharacteristic
153C2×C17#55112yes—
154C2×C17#55312yes—
155D17—42no—
156D17—44no—
157C22×D4—171no—
158C22×D4—172no—
159C23×C4—171no—
160C23×C4—172no—
161C4×D4—344no—
162#117—344no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—171no—
165C2×C4⋊C4—171no—
166C2×C4⋊C4—172no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C42—171no—
175C2×C42C2×D1711yescharacteristic
176#95—172no—
177#95—172no—
178#95C2×D1712yes—
179#95C2×D1712yes—
180#95C2×D1711yescharacteristic
181#95C2×D1711yescharacteristic
182C17#259811yescharacteristic, complemented, Sylow
183C24—172no—
184C24—171no—
185C2×D4—344no—
186C2×D4—344no—
187C2×D4—344no—
188C2×D4—342no—
189C2×D4—682no—
190C22×C4—171no—
191C22×C4—171no—
192C22×C4—172no—
193C22×C4—171no—
194C22×C4—342no—
195C22×C4—342no—
196C22×C4—342no—
197C22×C4—344no—
198C22×C4—682no—
199C22×C4—682no—
200C22×C4C22×D1712yes—
201C22×C4C22×D1712yes—
202C22×C4C22×D1711yescharacteristic
203C22×C4C22×D1711yescharacteristic
204C22×C4C22×D1711yescharacteristic
205C4⋊C4—342no—
206C4⋊C4—342no—
207C4⋊C4—342no—
208C4⋊C4—682no—
209C22⋊C4—344no—
210C22⋊C4—344no—
211C22⋊C4—342no—
212C22⋊C4—342no—
213C22⋊C4—344no—
214C22⋊C4—682no—
215C22⋊C4—682no—
216C42—342no—
217C42D6814yescomplemented
218C23—174no—
219C23—172no—
220C23—341no—
221C23—342no—
222C23—342no—
223C23—341no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23—342no—
228C23—342no—
229C23—342no—
230C23C23×D1711yescharacteristic
231D4—348no—
232D4—688no—
233C2×C4—21no—
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—22no—
243C2×C4—172no—
244C2×C4—348no—
245C2×C4—341no—
246C2×C4—341no—
247C2×C4—341no—
248C2×C4—341no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—688no—
255C2×C4C2×D6812yes—
256C2×C4C2×D6812yes—
257C2×C4C2×D6812yes—
258C22—344no—
259C22—344no—
260C22—344no—
261C22—342no—
262C22—344no—
263C22—682no—
264C22—682no—
265C22—682no—
266C22—682no—
267C22—681no—
268C22—682no—
269C22—681no—
270C22C22×D6811yescharacteristic
271C22#41×D1711yescharacteristic
272C22#41×D1712yes—
273C22#41×D1711yescharacteristic
274C22#1058769612yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—42no—
280C4—342no—
281C4—681no—
282C4—682no—
283C4—681no—
284C2—682no—
285C2—684no—
286C2#1176332111yescharacteristic
287C2#1176332111yescharacteristic
288C2#1176337811yescharacteristic
289C2#1176348412yes—
290C2#1176351012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#383C2
prime radical#11762719C2
hypercenter#383D17
hypocenterC17#2598
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×C4×D68C2semidirect
2. C2×#10587558C2semidirect
3. #11762749C2semidirect
4. #11762746C2semidirect
5. #11762727C2semidirect
7. #11762701C2semidirect
8. #11762687C2semidirect
9. #11762687C2semidirect
10. #11762683C2semidirect
11. C17×#383C2semidirect
59. C42×C17D4semidirect
149. #383D17semidirect
182. C17#2598semidirect
217. C42D68semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D682
C2×#105875582
#117627492
#117627462
#117627272
#117627192
#117627012
#117626872
#117626832
C17×#3832
#259817