Finite Group Databasebeta

← #14537246

Subgroups — #14537246

Order 2176.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#10587697C211yesmaximal, characteristic
2#11763286C212yesmaximal, complemented
3#11762783C211yesmaximal, characteristic, complemented
4#11762782C212yesmaximal, complemented
5#11762781C218yesmaximal, complemented
6C17×#410C211yesmaximal, characteristic, complemented
7C2×#41×C17C2211yescharacteristic
8#10587697—24no—
9#10587697—24no—
10C2×Q8×D17C412yescomplemented
11C2×#63254C412yescomplemented
12C2×#63254C2214yes—
13C2×#63252C412yescomplemented
14C2×#63252C2211yescharacteristic
15C22×Dic34C412yescomplemented
16C22×Dic34C2211yescharacteristic
17C17×#117C2212yes—
18#10587612C2212yes—
19#10587573—24no—
20#10587563C2218yes—
21C4×Dic34—24no—
22C17×#99C2214yes—
23#10587505C2214yes—
24#10587477C2218yes—
25#41×C17—22no—
26#41×C17—22no—
27C2×Q8×C17C2×C411yescharacteristic
28C2×D4×C17C2×C411yescharacteristic
29C2×D4×C17C2312yes—
30C22×C4×C17C2×C412yes—
31C22×C4×C17C2311yescharacteristic
32C2×C22⋊D17C2×C414yes—
33C2×C22⋊D17C2312yes—
34C22×Dic17C2×C414yes—
35C22×Dic17C2312yes—
36Q8×D17—28no—
37#63254—216no—
38#63254—28no—
39#63252—22no—
40#63252—22no—
41#63252—24no—
42#63252—24no—
43C2×D68C2×C411yescharacteristic
44C2×C4×D17C2×C414yes—
45C2×C4×D17C2×C412yes—
46C2×Dic34—24no—
47C2×Dic34—24no—
48C2×Dic34C2×C411yescharacteristic
49C2×Dic34C2×C414yes—
50C2×Dic34C2×C412yes—
51C2×Dic34C2×C412yes—
52C2×Dic34D414yes—
53C2×Dic34D414yes—
54C4⋊C4×C17—22no—
55C22⋊C4×C17C2314yes—
56C42×C17—22no—
57C22⋊Dic17C2314yes—
58C4⋊Dic17—22no—
59#63225—28no—
60C4×Dic17—22no—
61C4×Dic17—28no—
62C23×C17C22×C412yes—
63C23×C17C2411yescharacteristic
64C22×D17C22×C412yes—
65Q8×C17—22no—
66D4×C17—24no—
67D4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—24no—
71C2×C4×C17C22×C412yes—
72C2×C4×C17C22×C411yescharacteristic
73C2×C4×C17C22×C411yescharacteristic
74C2×C4×C17C2×D412yes—
75C2×C4×C17C2×D412yes—
76C22⋊D17—24no—
77C22⋊D17—28no—
78C2×Dic17—28no—
79C2×Dic17—24no—
80C2×Dic17C22×C412yes—
81C2×Dic17C22×C414yes—
82C2×Dic17C2×D418yes—
83D68—21no—
84D68—21no—
85C4×D17—24no—
86C4×D17—28no—
87Dic34—21no—
88Dic34—21no—
89Dic34—28no—
90Dic34C22⋊C414yes—
91Dic34C22⋊C414yes—
92Dic34C22⋊C418yes—
93#2884—171nomaximal, Sylow
94C22×C17—22no—
95C22×C17—41no—
96C22×C17—42no—
97C22×C17C23×C411yescharacteristic
98C22×C17C22×D412yes—
99C2×D17—22no—
100C2×D17—42no—
101C4×C17—22no—
102C4×C17—44no—
103C4×C17C2×C22⋊C411yescharacteristic
104C4×C17C2×C22⋊C411yescharacteristic
105C4×C17C2×C22⋊C412yes—
106Dic17—22no—
107Dic17—44no—
108Dic17C2×C22⋊C418yes—
109C2×2−1+4—171no—
110#519—172no—
111#410—171no—
112#410—172no—
113#410—178no—
114#410D1711yescharacteristic, complemented
115C2×C17—22no—
116C2×C17—21no—
117C2×C17—42no—
118C2×C17C22×C22⋊C411yescharacteristic
119D17—42no—
1202−1+4—344no—
1212−1+4—344no—
122C2×#41—172no—
123C2×#41—174no—
124C2×#41—172no—
125C2×#41—171no—
126C2×#41C2×D1711yescharacteristic
127C22×Q8—172no—
128C22×Q8—172no—
129C22×Q8—171no—
130C4×Q8—344no—
131C4×Q8—344no—
132#117—172no—
133#117—178no—
134#117C2×D1712yes—
135#99—174no—
136#99—178no—
137#99C2×D1714yes—
138C17#288411yescharacteristic, complemented, Sylow
139#41—22no—
140#41—22no—
141#41—344no—
142#41—344no—
143#41—3416no—
144#41—348no—
145#41—342no—
146#41—342no—
147C2×Q8—172no—
148C2×Q8—172no—
149C2×Q8—174no—
150C2×Q8—171no—
151C2×Q8—174no—
152C2×Q8—174no—
153C2×Q8—344no—
154C2×Q8—344no—
155C2×Q8—348no—
156C2×Q8C4×D1711yescharacteristic
157C2×D4—174no—
158C2×D4—172no—
159C2×D4—171no—
160C2×D4C4×D1711yescharacteristic
161C2×D4C22×D1712yes—
162C22×C4—172no—
163C22×C4—174no—
164C22×C4—174no—
165C22×C4—172no—
166C22×C4C4×D1712yes—
167C22×C4C22×D1711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—348no—
170C4⋊C4—342no—
171C22⋊C4—174no—
172C22⋊C4C22×D1714yes—
173C42—22no—
174C42—348no—
175C42—342no—
176C23—172no—
177C23C2×C4×D1712yes—
178C23C23×D1711yescharacteristic
179Q8—22no—
180Q8—174no—
181Q8—174no—
182Q8—178no—
183Q8—348no—
184Q8—341no—
185Q8—341no—
186D4—24no—
187D4—22no—
188D4—348no—
189D4—344no—
190D4—341no—
191D4—341no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—174no—
196C2×C4—178no—
197C2×C4—172no—
198C2×C4—348no—
199C2×C4—348no—
200C2×C4—344no—
201C2×C4—344no—
202C2×C4C2×C4×D1712yes—
203C2×C4C2×C4×D1711yescharacteristic
204C2×C4C2×C4×D1711yescharacteristic
205C2×C4D4×D1712yes—
206C2×C4D4×D1712yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—342no—
211C22—682no—
212C22C22×C4×D1711yescharacteristic
213C22C2×D4×D1712yes—
214C4—22no—
215C4—44no—
216C4—178no—
217C4—342no—
218C4—684no—
219C4C22⋊C4×D1711yescharacteristic
220C4C22⋊C4×D1711yescharacteristic
221C4C22⋊C4×D1712yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—682no—
226C2C2×C22⋊C4×D1711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D17
commutatorC22×C17C23×C4
FrattiniC23C23×D17
FittingC17×#410C2
socleC2×C17C22×C22⋊C4
prime radicalC2×#10587697C2
cosocleC23×C17C24
hypercenter#410D17
hypocenterC17#2884
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #11763286C2semidirect
3. #11762783C2semidirect
4. #11762782C2semidirect
5. #11762781C2semidirect
6. C17×#410C2semidirect
10. C2×Q8×D17C4semidirect
11. C2×#63254C4semidirect
13. C2×#63252C4semidirect
15. C22×Dic34C4semidirect
114. #410D17semidirect
138. C17#2884semidirect

Maximal subgroups

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

subgroupindex
C2×#105876972
#117632862
#117627832
#117627822
#117627812
C17×#4102
#288417