Finite Group Databasebeta

← #14534598

Subgroups — #14534598

Order 2176.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×#10587583C211yesmaximal, characteristic
2C2×C4⋊C4×D17C211yesmaximal, characteristic
3#11762747C211yesmaximal, characteristic, complemented
4#11762743C212yesmaximal, complemented
5#11762741C212yesmaximal, complemented
6#11762730C211yesmaximal, characteristic, complemented
7#11762729C211yesmaximal, characteristic, complemented
8#11762683C212yesmaximal
9#11762671C212yesmaximal
10#11762668C211yesmaximal, characteristic
11C17×#400C211yesmaximal, characteristic, complemented
12C22×C4×D17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2212yes—
23C2×#63225C2211yescharacteristic
24C2×#63225C2211yescharacteristic
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27#10587583—24no—
28C4⋊C4×D17—24no—
29C17×#95C2211yescharacteristic
30C17×#95C2212yes—
31C17×#95C2212yes—
32#10587502C2212yes—
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×Dic17C2312yes—
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C2×C4×D17—22no—
49C2×C4×D17—22no—
50C2×C4×D17—22no—
51C2×C4×D17—42no—
52C2×C4×D17—42no—
53C2×C4×D17D414yes—
54C2×C4×D17Q814yescomplemented
55C2×Dic34—22no—
56C2×Dic34—42no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17—22no—
59#63227—24no—
60#63227—24no—
61C4⋊Dic17—22no—
62C4⋊Dic17—24no—
63C4⋊Dic17—42no—
64#63225—22no—
65#63225—22no—
66#63225—42no—
67#63225—42no—
68C23×C17C2411yescharacteristic
69C22×D17—22no—
70C22×D17—22no—
71C22×D17C2×D412yes—
72C22×D17C2×Q812yes—
73C22×D17#4112yes—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—24no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2×Q812yes—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17C2×D412yes—
98C2×Dic17C2×Q812yes—
99C4×D17—28no—
100C4×D17—44no—
101C4×D17—44no—
102Dic34—44no—
103#2627—171nomaximal, Sylow
104C22×C17C22×D411yescharacteristic
105C22×C17C22×Q811yescharacteristic
106C22×C17C2×#4111yescharacteristic
107C22×C17H(5,𝔽2)12yes—
108C22×C172−1+412yes—
109C2×D17—22no—
110C2×D17—22no—
111C2×D17—24no—
112C2×D17—24no—
113C2×D17C22⋊Q814yes—
114C4×C17—22no—
115C4×C17—22no—
116C4×C17—41no—
117C4×C17—42no—
118C4×C17—42no—
119Dic17—22no—
120Dic17—22no—
121Dic17—42no—
122Dic17—41no—
123Dic17—42no—
124C2×C22⋊Q8—171no—
125C22×C4⋊C4—171no—
126#400—171no—
127#400D1711yescharacteristic, complemented
128#398—172no—
129#398—172no—
130#395—171no—
131#395—171no—
132#385—172no—
133#381—172no—
134#380—171no—
135C2×C17C2×C22⋊Q811yescharacteristic
136C2×C17#53411yescharacteristic
137C2×C17#53611yescharacteristic
138C2×C17#53912yes—
139C2×C17#54412yes—
140D17—24no—
141C22×Q8—171no—
142C23×C4—171no—
143C23×C4—171no—
144C23×C4—171no—
145C22⋊Q8—344no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4—172no—
152C2×C4⋊C4—172no—
153C2×C4⋊C4—344no—
154C2×C4⋊C4C2×D1711yescharacteristic
155C2×C4⋊C4C2×D1711yescharacteristic
156C2×C22⋊C4—172no—
157C2×C22⋊C4—172no—
158#95—171no—
159#95—172no—
160#95—172no—
161#95—172no—
162#95—172no—
163#95C2×D1711yescharacteristic
164#95C2×D1712yes—
165#95C2×D1712yes—
166C17#262711yescharacteristic, complemented, Sylow
167C24—171no—
168C2×Q8—342no—
169C2×Q8—682no—
170C22×C4—171no—
171C22×C4—171no—
172C22×C4—174no—
173C22×C4—174no—
174C22×C4—172no—
175C22×C4—172no—
176C22×C4—171no—
177C22×C4—342no—
178C22×C4—342no—
179C22×C4—342no—
180C22×C4—682no—
181C22×C4—682no—
182C22×C4C22×D1712yes—
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C22×C4C22×D1711yescharacteristic
186C22×C4C22×D1711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—344no—
192C4⋊C4—342no—
193C4⋊C4—682no—
194C4⋊C4—682no—
195C4⋊C4—682no—
196C22⋊C4—344no—
197C22⋊C4—344no—
198C23—172no—
199C23—172no—
200C23—172no—
201C23—342no—
202C23—342no—
203C23C23×D1711yescharacteristic
204Q8—684no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—172no—
216C2×C4—172no—
217C2×C4—348no—
218C2×C4—341no—
219C2×C4—341no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—341no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—684no—
230C2×C4—684no—
231C2×C4D4×D1712yes—
232C2×C4Q8×D1712yes—
233C22—174no—
234C22—342no—
235C22—342no—
236C22—344no—
237C22—344no—
238C22C2×D4×D1711yescharacteristic
239C22C2×Q8×D1711yescharacteristic
240C22#41×D1711yescharacteristic
241C22#1058769612yes—
242C22#1058769712yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—342no—
249C4—342no—
250C4—681no—
251C4—682no—
252C4—682no—
253C2—344no—
254C2C22⋊Q8×D1711yescharacteristic
255C2#1176337011yescharacteristic
256C2#1176337911yescharacteristic
257C2#1176340012yes—
258C2#1176343912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC23×D17C23
hypercenter#400D17
hypocenterC17#2627
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #11762747C2semidirect
4. #11762743C2semidirect
5. #11762741C2semidirect
6. #11762730C2semidirect
7. #11762729C2semidirect
11. C17×#400C2semidirect
54. C2×C4×D17Q8semidirect
127. #400D17semidirect
166. C17#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×C4⋊C4×D172
#117627472
#117627432
#117627412
#117627302
#117627292
#117626832
#117626712
#117626682
C17×#4002
#262717