Finite Group Databasebeta

← #14534683

Subgroups — #14534683

Order 2176.

562 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587562C211yesmaximal, characteristic, complemented
2C2×C4×D68C211yesmaximal, characteristic, complemented
3#11762749C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic, complemented
5#11762731C212yesmaximal, complemented
6#11762727C211yesmaximal, characteristic, complemented
7#11762726C212yesmaximal, complemented
8#11762719C211yesmaximal, characteristic
9#11762687C212yesmaximal, complemented
10#11762683C212yesmaximal, complemented
11C17×#382C211yesmaximal, characteristic, complemented
12C22×D68C2211yescharacteristic
13C22×D68C2212yes—
14C22×C4×D17C2212yes—
15C2×C4⋊C4×C17C2212yes—
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2211yescharacteristic
18C2×#63227C2211yescharacteristic
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2212yes—
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30#10587562—24no—
31C4×D68—24no—
32C17×#95C2211yescharacteristic
33C17×#95C2211yescharacteristic
34C17×#95C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2311yescharacteristic
38C23×D17C2312yes—
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—42no—
52C2×D68—42no—
53C2×C4×D17—24no—
54C2×C4×D17—42no—
55C2×C4×D17—42no—
56C4⋊C4×C17—42no—
57C42×C17D414yescomplemented
58#63227—24no—
59#63227—24no—
60#63227—22no—
61#63227—22no—
62#63227—22no—
63#63227—42no—
64#63227—42no—
65#63227—42no—
66C4⋊Dic17—22no—
67C4⋊Dic17—42no—
68#63225—22no—
69#63225—22no—
70#63225—22no—
71C23×C17C2411yescharacteristic
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—21no—
79C22×D17—22no—
80C22×D17—21no—
81C22×D17—22no—
82C22×D17—21no—
83C22×D17#4114yes—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
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×C17C2×D412yes—
95C2×C4×C17C2×D412yes—
96C2×C4×C17C2×D412yes—
97C2×Dic17—22no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107D68—28no—
108D68—48no—
109C4×D17—48no—
110#2636—171nomaximal, Sylow
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17H(5,𝔽2)11yescharacteristic
114C22×C17H(5,𝔽2)12yes—
115C22×C172−1+411yescharacteristic
116C2×D17—24no—
117C2×D17—24no—
118C2×D17—24no—
119C2×D17—42no—
120C2×D17—42no—
121C2×D17—42no—
122C2×D17—42no—
123C2×D17—41no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—42no—
127C2×D17—42no—
128C4×C17—22no—
129C4×C17—22no—
130C4×C17—22no—
131C4×C17—42no—
132C4×C17—42no—
133Dic17—41no—
134Dic17—42no—
135Dic17—41no—
136Dic17—41no—
137C2×#126—171no—
138C2×C4×D4—171no—
139#400—171no—
140#399—171no—
141#396—172no—
142#394—172no—
143#394—171no—
144#392—171no—
145#386—172no—
146#385—172no—
147#382D1711yescharacteristic, complemented
148C2×C17#52511yescharacteristic
149C2×C17#53511yescharacteristic
150C2×C17#53711yescharacteristic
151C2×C17#53812yes—
152C2×C17#53912yes—
153D17—44no—
154D17—81no—
155C22×D4—171no—
156C22×D4—172no—
157C23×C4—172no—
158#126—344no—
159C4×D4—344no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—171no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—172no—
165C2×C4⋊C4C2×D1712yes—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—171no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—172no—
174C2×C42C2×D1711yescharacteristic
175#95—172no—
176#95—172no—
177#95C2×D1712yes—
178#95C2×D1711yescharacteristic
179#95C2×D1711yescharacteristic
180C17#263611yescharacteristic, complemented, Sylow
181C24—172no—
182C24—171no—
183C2×D4—344no—
184C2×D4—344no—
185C2×D4—342no—
186C2×D4—682no—
187C2×D4—682no—
188C22×C4—171no—
189C22×C4—171no—
190C22×C4—172no—
191C22×C4—171no—
192C22×C4—344no—
193C22×C4—682no—
194C22×C4—682no—
195C22×C4C22×D1712yes—
196C22×C4C22×D1712yes—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C22×C4C22×D1711yescharacteristic
200C4⋊C4—42no—
201C4⋊C4—342no—
202C4⋊C4—342no—
203C4⋊C4—342no—
204C4⋊C4—342no—
205C4⋊C4—682no—
206C22⋊C4—344no—
207C22⋊C4—344no—
208C22⋊C4—342no—
209C22⋊C4—342no—
210C22⋊C4—342no—
211C22⋊C4—682no—
212C22⋊C4—682no—
213C22⋊C4—682no—
214C42D6814yescomplemented
215C23—174no—
216C23—342no—
217C23—342no—
218C23—342no—
219C23—342no—
220C23—342no—
221C23—342no—
222C23—341no—
223C23—341no—
224C23—341no—
225C23—342no—
226C23—342no—
227C23C23×D1711yescharacteristic
228D4—348no—
229D4—688no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—341no—
241C2×C4—341no—
242C2×C4—341no—
243C2×C4—342no—
244C2×C4—341no—
245C2×C4—341no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—688no—
251C2×C4C2×D6812yes—
252C2×C4C2×D6812yes—
253C2×C4C2×D6812yes—
254C22—344no—
255C22—344no—
256C22—344no—
257C22—682no—
258C22—682no—
259C22—682no—
260C22—682no—
261C22—681no—
262C22—681no—
263C22—681no—
264C22—682no—
265C22—682no—
266C22C22×D6811yescharacteristic
267C22#1058769111yescharacteristic
268C22#1058769411yescharacteristic
269C22#41×D1712yes—
270C22#1058769612yes—
271C4—22no—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—681no—
277C4—682no—
278C4—681no—
279C4—681no—
280C2—684no—
281C2—1361no—
282C2#1176332111yescharacteristic
283C2#1176337811yescharacteristic
284C2#1176339011yescharacteristic
285C2#1176339312yes—
286C2#1176340712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#382C2
prime radical#11762719C2
hypercenter#382D17
hypocenterC17#2636
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#10587562C2semidirect
2. C2×C4×D68C2semidirect
3. #11762749C2semidirect
4. #11762746C2semidirect
5. #11762731C2semidirect
6. #11762727C2semidirect
7. #11762726C2semidirect
9. #11762687C2semidirect
10. #11762683C2semidirect
11. C17×#382C2semidirect
57. C42×C17D4semidirect
147. #382D17semidirect
180. C17#2636semidirect
214. C42D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875622
C2×C4×D682
#117627492
#117627462
#117627312
#117627272
#117627262
#117627192
#117626872
#117626832
C17×#3822
#263617