Finite Group Databasebeta

← #14533096

Subgroups — #14533096

Order 2176.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×#10587579C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic, complemented
3#11762748C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic, complemented
5#11762744C211yesmaximal, characteristic, complemented
6#11762706C212yesmaximal, complemented
7#11762687C212yesmaximal
8#11762683C212yesmaximal, complemented
9#11762668C212yesmaximal, complemented
10#11762668C211yesmaximal, characteristic, complemented
11C17×#400C211yesmaximal, characteristic, complemented
12C22×D68C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×C4×D17C2212yes—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×#63227C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587579—24no—
30C4⋊C4×D17—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33C17×#95C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C23×D17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×D68—22no—
49C2×D68—22no—
50C2×D68Q814yes—
51C2×C4×D17—24no—
52C2×C4×D17—24no—
53C2×C4×D17—22no—
54C2×C4×D17—24no—
55C2×C4×D17—42no—
56C2×C4×D17—42no—
57C2×C4×D17D414yes—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—22no—
60#63227—22no—
61#63227—24no—
62#63227—22no—
63#63227—42no—
64#63227—42no—
65C4⋊Dic17—22no—
66C4⋊Dic17—42no—
67#63225—24no—
68#63225—24no—
69#63225—42no—
70C4×Dic17—22no—
71C23×C17C2411yescharacteristic
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—21no—
77C22×D17—22no—
78C22×D17C2×D412yes—
79C22×D17C2×Q812yes—
80C22×D17C2×Q812yes—
81C22×D17#4112yes—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
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×C17C2×D412yes—
94C2×C4×C17C2×Q812yes—
95C2×Dic17—22no—
96C2×Dic17—21no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17C2×D412yes—
104C2×Dic17#4114yes—
105D68—28no—
106C4×D17—28no—
107C4×D17—48no—
108C4×D17—44no—
109#2487—171nomaximal, Sylow
110C22×C17C22×D411yescharacteristic
111C22×C17C22×Q811yescharacteristic
112C22×C17C2×#4111yescharacteristic
113C22×C17C2×#4112yes—
114C22×C17H(5,𝔽2)12yes—
115C2×D17—22no—
116C2×D17—22no—
117C2×D17—24no—
118C2×D17—24no—
119C2×D17—22no—
120C2×D17—24no—
121C2×D17—41no—
122C2×D17—41no—
123C2×D17—42no—
124C2×D17C22⋊Q814yes—
125C4×C17—22no—
126C4×C17—22no—
127C4×C17—41no—
128C4×C17—42no—
129C4×C17—42no—
130Dic17—42no—
131Dic17—41no—
132Dic17—42no—
133Dic17#12314yes—
134C2×C4×D4—171no—
135C22×C4⋊C4—171no—
136#400—171no—
137#400D1711yescharacteristic, complemented
138#399—171no—
139#399—171no—
140#389—172no—
141#386—172no—
142#385—172no—
143#380—172no—
144#380—171no—
145C2×C17C2×C22⋊Q811yescharacteristic
146C2×C17C2×#12311yescharacteristic
147C2×C17#53511yescharacteristic
148C2×C17#55312yes—
149C2×C17#55412yes—
150D17—24no—
151D17—42no—
152C22×D4—171no—
153C23×C4—171no—
154C23×C4—171no—
155C23×C4—172no—
156C4×D4—344no—
157C2×C4⋊C4—172no—
158C2×C4⋊C4—171no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4—344no—
163C2×C4⋊C4C2×D1711yescharacteristic
164C2×C4⋊C4C2×D1711yescharacteristic
165C2×C22⋊C4—171no—
166C2×C22⋊C4—171no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C42—171no—
171#95—172no—
172#95—171no—
173#95—172no—
174#95C2×D1712yes—
175#95C2×D1711yescharacteristic
176#95C2×D1712yes—
177C17#248711yescharacteristic, complemented, Sylow
178C24—171no—
179C24—171no—
180C2×D4—174no—
181C2×D4—342no—
182C2×D4—342no—
183C22×C4—171no—
184C22×C4—174no—
185C22×C4—172no—
186C22×C4—172no—
187C22×C4—171no—
188C22×C4—344no—
189C22×C4—342no—
190C22×C4—344no—
191C22×C4—344no—
192C22×C4—682no—
193C22×C4—682no—
194C22×C4C22×D1711yescharacteristic
195C22×C4C22×D1712yes—
196C22×C4C22×D1712yes—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—344no—
202C4⋊C4—344no—
203C4⋊C4—342no—
204C4⋊C4—682no—
205C4⋊C4—682no—
206C22⋊C4—342no—
207C22⋊C4—344no—
208C22⋊C4—342no—
209C22⋊C4—682no—
210C22⋊C4—682no—
211C42—342no—
212C23—172no—
213C23—172no—
214C23—172no—
215C23—172no—
216C23—342no—
217C23—342no—
218C23—341no—
219C23—342no—
220C23—341no—
221C23—342no—
222C23C23×D1711yescharacteristic
223D4—348no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—172no—
236C2×C4—174no—
237C2×C4—348no—
238C2×C4—341no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—684no—
247C2×C4—688no—
248C2×C4D4×D1712yes—
249C2×C4Q8×D1712yes—
250C22—174no—
251C22—344no—
252C22—344no—
253C22—342no—
254C22—342no—
255C22—344no—
256C22—342no—
257C22—682no—
258C22—681no—
259C22—681no—
260C22C2×D4×D1711yescharacteristic
261C22C2×Q8×D1711yescharacteristic
262C22#41×D1711yescharacteristic
263C22#41×D1712yes—
264C22#1058769612yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—41no—
269C4—42no—
270C4—174no—
271C4—682no—
272C4—682no—
273C4—681no—
274C2—344no—
275C2—682no—
276C2C22⋊Q8×D1711yescharacteristic
277C2#123×D1711yescharacteristic
278C2#1176337711yescharacteristic
279C2#1176351012yes—
280C2#1176351512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC22×D68C22
hypercenter#400D17
hypocenterC17#2487
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D17C2semidirect
3. #11762748C2semidirect
4. #11762746C2semidirect
5. #11762744C2semidirect
6. #11762706C2semidirect
8. #11762683C2semidirect
9. #11762668C2semidirect
10. #11762668C2semidirect
11. C17×#400C2semidirect
137. #400D17semidirect
177. C17#2487semidirect

Maximal subgroups

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

subgroupindex
C2×#105875792
C2×C4⋊C4×D172
#117627482
#117627462
#117627442
#117627062
#117626872
#117626832
#117626682
C17×#4002
#248717