Finite Group Databasebeta

← #14545383

Subgroups — #14545383

Order 2176.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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
1#11763876C212yesmaximal, complemented
2#11763789C212yesmaximal, complemented
3#11763788C212yesmaximal, complemented
4D17⋊#99C214yesmaximal, complemented
5C2×#10587551C214yesmaximal, complemented
6C2×#10587691C211yesmaximal, characteristic
7C22×D4×C17C411yescharacteristic, complemented
8C22×C22⋊D17C412yescomplemented
9C22×C22⋊D17C2212yes—
10#10587691—28no—
11C2×#63254C412yescomplemented
12C2×#63254C411yescharacteristic, complemented
13C2×#63254C2211yescharacteristic
14C2×D4×D17C411yescharacteristic, complemented
15C2×D4×D17C2212yes—
16C2×D4×D17C2211yescharacteristic
17C2×#63252C411yescharacteristic, complemented
18C2×#63252C2211yescharacteristic
19C2×C22⋊#595C2218yescomplemented
20D4×#595—28no—
21#10587677C2214yescomplemented
22#10587551C2214yescomplemented
23#10587551C2218yescomplemented
24#10587539C2214yescomplemented
25C24×C17C2×C412yescomplemented
26C23×D17C2×C412yescomplemented
27C23×D17C2312yes—
28C22×#595—28no—
29C2×D4×C17—24no—
30C2×D4×C17C2×C411yescharacteristic
31C2×D4×C17C2×C411yescharacteristic, complemented
32C2×D4×C17C2×C412yescomplemented
33C22×C4×C17C2×C411yescharacteristic, complemented
34C2×C22⋊D17—28no—
35C2×C22⋊D17—28no—
36C2×C22⋊D17C2×C412yes—
37C2×C22⋊D17C2×C412yescomplemented
38C2×C22⋊D17C2×C411yescharacteristic, complemented
39C2×C22⋊D17C2×C412yescomplemented
40C2×C22⋊D17C2×C412yescomplemented
41C2×C22⋊D17C2311yescharacteristic
42C2×C22⋊D17C2312yes—
43C22×Dic17C2×C412yescomplemented
44C22×Dic17C2×C412yescomplemented
45#63254—24no—
46#63254—28no—
47#63254—24no—
48D4×D17—28no—
49D4×D17—24no—
50D4×D17D418yes—
51#63252—24no—
52#63252—24no—
53C2×D68C2×C411yescharacteristic
54C2×D68C2311yescharacteristic
55C2×C4×D17C2×C412yescomplemented
56C2×C4×D17C2×C411yescharacteristic, complemented
57C2×C4×D17C2311yescharacteristic
58C2×Dic34C2×C411yescharacteristic, complemented
59C2×Dic34C2×C411yescharacteristic, complemented
60C22⋊#595—28no—
61C22⋊#595C2318yes—
62C4⋊#595—24no—
63C4×#595—24no—
64C23×C17—24no—
65C23×C17—82no—
66C23×C17C22×C412yes—
67C23×C17C22×C411yescharacteristic
68C23×C17C22×C412yes—
69C22×D17—24no—
70C22×D17—82no—
71C22×D17—82no—
72C22×D17C22×C411yescharacteristic
73C22×D17C22×C412yes—
74C22×D17C2×D418yes—
75C22×D17C2411yescharacteristic
76C2×#595—28no—
77C2×#595—88no—
78D4×C17—24no—
79D4×C17—22no—
80D4×C17C22⋊C414yes—
81C2×C4×C17—22no—
82C2×C4×C17C22×C411yescharacteristic
83C2×C4×C17C22×C411yescharacteristic
84C22⋊D17—24no—
85C22⋊D17—24no—
86C22⋊D17—24no—
87C22⋊D17—22no—
88C22⋊D17—24no—
89C22⋊D17—22no—
90C22⋊D17C22⋊C418yes—
91C2×Dic17—24no—
92C2×Dic17—24no—
93C2×Dic17C22×C412yes—
94C2×Dic17C22×C411yescharacteristic
95C2×Dic17C22×C411yescharacteristic
96D68—22no—
97D68C22⋊C414yes—
98C4×D17—24no—
99C4×D17—22no—
100C4×D17C2×D414yes—
101Dic34—22no—
102Dic34—22no—
103#2883—171nomaximal, Sylow
104C22×C17—22no—
105C22×C17—21no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—41no—
109C22×C17—84no—
110C22×C17C2×C22⋊C414yes—
111C22×C17C23×C411yescharacteristic
112C2×D17—21no—
113C2×D17—48no—
114C2×D17—42no—
115C2×D17—41no—
116C2×D17—41no—
117C2×D17—84no—
118C2×D17C2×C22⋊C414yes—
119C2×D17C22×D412yes—
120#595—48no—
121C4×C17—21no—
122C4×C17C2×C22⋊C412yes—
123Dic17—22no—
124Dic17—21no—
125Dic17C2×C22⋊C412yes—
126C2×H(5,𝔽2)—171no—
127#518—172no—
128#410—172no—
129#410—172no—
130C2×#99—174no—
131C2×#99—174no—
132C2×C17—21no—
133C2×C17—24no—
134C2×C17—42no—
135C2×C17—41no—
136C2×C17C22×C22⋊C411yescharacteristic
137D17—22no—
138D17—24no—
139D17—41no—
140H(5,𝔽2)—348no—
141C2×#41—171no—
142C2×#41—172no—
143C2×#41—171no—
144C2×#41—171no—
145C2×#41—171no—
146C22×D4—171no—
147C22×D4—172no—
148C22×D4—172no—
149C22×D4—171no—
150C22×D4—172no—
151C22×D4#59511yescharacteristic, complemented
152C4×D4—348no—
153#117—174no—
154C2×C22⋊C4—178no—
155#99—174no—
156#99—174no—
157#99—178no—
158C17#288311yescharacteristic, complemented, Sylow
159C24—172no—
160C24—172no—
161C24C2×#59512yescomplemented
162#41—344no—
163#41—344no—
164#41—344no—
165#41—348no—
166#41—344no—
167C2×Q8—171no—
168C2×Q8—171no—
169C2×D4—24no—
170C2×D4—172no—
171C2×D4—178no—
172C2×D4—171no—
173C2×D4—172no—
174C2×D4—172no—
175C2×D4—172no—
176C2×D4—172no—
177C2×D4—171no—
178C2×D4—171no—
179C2×D4—171no—
180C2×D4—344no—
181C2×D4—348no—
182C2×D4—348no—
183C2×D4—348no—
184C2×D4C2×#59511yescharacteristic
185C2×D4C2×#59512yescomplemented
186C2×D4C2×#59511yescharacteristic, complemented
187C22×C4—171no—
188C22×C4—172no—
189C22×C4—172no—
190C22×C4—172no—
191C22×C4—171no—
192C22×C4—348no—
193C22×C4C2×#59511yescharacteristic, complemented
194C4⋊C4—344no—
195C22⋊C4—178no—
196C22⋊C4—348no—
197C42—344no—
198C23—24no—
199C23—82no—
200C23—178no—
201C23—171no—
202C23—172no—
203C23—171no—
204C23—344no—
205C23—1362no—
206C23—1362no—
207C23C22×#59512yes—
208C23C22×#59511yescharacteristic
209C23C22×#59512yes—
210Q8—342no—
211Q8—342no—
212D4—24no—
213D4—22no—
214D4—178no—
215D4—174no—
216D4—342no—
217D4—342no—
218D4—342no—
219D4—344no—
220D4—344no—
221D4—344no—
222D4—344no—
223D4C22⋊#59514yes—
224C2×C4—22no—
225C2×C4—171no—
226C2×C4—174no—
227C2×C4—172no—
228C2×C4—171no—
229C2×C4—344no—
230C2×C4—344no—
231C2×C4—344no—
232C2×C4—342no—
233C2×C4—348no—
234C2×C4—1368no—
235C2×C4C22×#59511yescharacteristic
236C2×C4C22×#59511yescharacteristic
237C22—21no—
238C22—22no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—84no—
243C22—172no—
244C22—174no—
245C22—341no—
246C22—688no—
247C22—681no—
248C22—682no—
249C22—681no—
250C22—1364no—
251C22C2×C22⋊#59514yes—
252C22C23×#59511yescharacteristic
253C4—21no—
254C4—172no—
255C4—342no—
256C4—341no—
257C4—688no—
258C4C2×C22⋊#59512yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—42no—
263C2—342no—
264C2—344no—
265C2—681no—
266C2C22×C22⋊#59511yescharacteristic
subgroupG/H
centerC2C22×C22⋊#595
commutatorC22×C17C23×C4
FrattiniC22C23×#595
FittingC22×D4×C17C4
socleC2×C17C22×C22⋊C4
prime radicalC2×#10587691C2
cosocleC22×D17C24
hypercenterC22×D4#595
hypocenterC17#2883
LCSG ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×D4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #11763876C2semidirect
2. #11763789C2semidirect
3. #11763788C2semidirect
4. D17⋊#99C2semidirect
5. C2×#10587551C2semidirect
7. C22×D4×C17C4semidirect
8. C22×C22⋊D17C4semidirect
11. C2×#63254C4semidirect
12. C2×#63254C4semidirect
14. C2×D4×D17C4semidirect
17. C2×#63252C4semidirect
19. C2×C22⋊#595C22semidirect
21. #10587677C22semidirect
22. #10587551C22semidirect
23. #10587551C22semidirect
24. #10587539C22semidirect
25. C24×C17C2×C4semidirect
26. C23×D17C2×C4semidirect
31. C2×D4×C17C2×C4semidirect
32. C2×D4×C17C2×C4semidirect
33. C22×C4×C17C2×C4semidirect
37. C2×C22⋊D17C2×C4semidirect
38. C2×C22⋊D17C2×C4semidirect
39. C2×C22⋊D17C2×C4semidirect
40. C2×C22⋊D17C2×C4semidirect
43. C22×Dic17C2×C4semidirect
44. C22×Dic17C2×C4semidirect
55. C2×C4×D17C2×C4semidirect
56. C2×C4×D17C2×C4semidirect
58. C2×Dic34C2×C4semidirect
59. C2×Dic34C2×C4semidirect
151. C22×D4#595semidirect
158. C17#2883semidirect
161. C24C2×#595semidirect
185. C2×D4C2×#595semidirect
186. C2×D4C2×#595semidirect
193. C22×C4C2×#595semidirect

Maximal subgroups

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

subgroupindex
#117638762
#117637892
#117637882
D17⋊#992
C2×#105875512
C2×#105876912
#288317