Finite Group Databasebeta

← #17960838

Subgroups — #17960838

Order 4352.

456 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 96 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#14545035C212yesmaximal
2#14544540C212yesmaximal, complemented
3#14543876C211yesmaximal, characteristic, complemented
4#14543873C211yesmaximal, characteristic, complemented
5#14537483C211yesmaximal, characteristic
6#11763795C2212yes—
7#11763788C2212yes—
8#11763772—22no—
9#11763711C414yescomplemented
10#11763709C414yescomplemented
11#11763691—22no—
12C2×#10587593C2211yescharacteristic
13C22×#63223C411yescharacteristic, complemented
14#11763320C2211yescharacteristic
15#11762884—22no—
16#11762883—22no—
17#11762837C411yescharacteristic, complemented
18#11762777C2211yescharacteristic
19#11762767C411yescharacteristic, complemented
20C17×#407C411yescharacteristic, complemented
21C23×C4×C17C2×C411yescharacteristic
22C2×#63252C2311yescharacteristic
23#10587677—22no—
24D17⋊M16—22no—
25C2×M16×C17C2×C411yescharacteristic
26C2×M16×C17C2×C411yescharacteristic
27#10587639C2×C412yes—
28#10587616—21no—
29#10587615C2×C412yes—
30C4×C22⋊D17—22no—
31#10587612C2×C411yescharacteristic
32#10587610—21no—
33C2×#63223—21no—
34C2×#63223—21no—
35C2×#63223—42no—
36C22×#3622C2×C411yescharacteristic
37C22×#3622C2×C411yescharacteristic
38#10587593—42no—
39C2×#63218D412yes—
40C2×C8×D17D412yes—
41#10587553C2×C414yes—
42#10587551C2×C414yes—
43#10587545—22no—
44#10587539—22no—
45#10587532—42no—
46C8×#595—42no—
47C17×C22⋊C8—22no—
48#10587491—22no—
49#10587486—21no—
50#10587485—22no—
51C8×Dic17—21no—
52C24×C17C4211yescharacteristic
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—41no—
56C22×C4×C17—41no—
57C22×C4×C17C4211yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C2×C22⋊D17—22no—
60C2×C22⋊D17C22×C412yes—
61#63252—41no—
62#63252—41no—
63C2×D68#4111yescharacteristic
64C2×C4×D17C2×D412yes—
65C2×Dic34#4111yescharacteristic
66C22⋊#595—24no—
67#63247—24no—
68C4⋊#595—42no—
69C4×#595—42no—
70#63243—42no—
71#63242—42no—
72M16×C17—41no—
73M16×C17—41no—
74C2×C8×C17—21no—
75C2×C8×C17C22⋊C412yes—
76C22⋊Dic17—22no—
77C22⋊Dic17C4212yes—
78#63227—22no—
79C4⋊Dic17—21no—
80#63225—22no—
81C4×Dic17—21no—
82#63223—42no—
83#63223—42no—
84C2×#3622—22no—
85C2×#3622—21no—
86C2×#3622—41no—
87C2×#3622C22⋊C411yescharacteristic
88C2×#3622C22⋊C411yescharacteristic
89#63218—42no—
90C8×D17—22no—
91C8×D17—22no—
92#12624—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—41no—
95C23×C17—81no—
96C23×C17C2×C4211yescharacteristic
97C22×D17C4×D412yes—
98C2×#595—44no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17C2×C22⋊C411yescharacteristic
107C2×C4×C17#11711yescharacteristic
108C22⋊D17—42no—
109C22⋊D17—82no—
110C2×Dic17—22no—
111C2×Dic17C4×D412yes—
112D68—41no—
113C4×D17—22no—
114C4×D17—22no—
115Dic34—41no—
116#3624—44no—
117C8×C17—22no—
118C8×C17—41no—
119#3622—21no—
120#3622—41no—
121#3622#9912yes—
122#2930—171no—
123#2120—172no—
124#1757—172no—
125#1392—171no—
126#1391—171no—
127C22×C17—21no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17—41no—
132C22×C17—82no—
133C22×C17C4×C22⋊C411yescharacteristic
134C2×D17—22no—
135C2×D17—42no—
136#595—84no—
137C4×C17—21no—
138C4×C17—41no—
139C4×C17C2×#9911yescharacteristic
140C4×C17#41011yescharacteristic
141Dic17—22no—
142Dic17—42no—
143C2×#131—171no—
144C22×M16—171no—
145#525—171no—
146#436—342no—
147#435—342no—
148#424—171no—
149#413—172no—
150#410—172no—
151#408—171no—
152#407#59511yescharacteristic, complemented
153#405—171no—
154#405—342no—
155#352—174no—
156AΣL(1,𝔽16)—174no—
157#344—342no—
158C2×C17—21no—
159C2×C17—41no—
160C2×C17—81no—
161C2×C17C4×#9911yescharacteristic
162D17—42no—
163C2×#41—171no—
164C23×C4C2×#59511yescharacteristic
165#131—682no—
166C2×M16—172no—
167C2×M16—341no—
168C2×M16—341no—
169C2×M16—342no—
170C2×M16—682no—
171C2×M16C2×#59511yescharacteristic
172C2×M16C2×#59511yescharacteristic
173C22×C8—171no—
174C22×C8—172no—
175C22×C8—171no—
176#123—172no—
177C22⋊Q8—341no—
178#121—341no—
179#120—172no—
180C4×D4—342no—
181#117—171no—
182#117—342no—
183C4⋊C8—342no—
184#101—342no—
185#100—174no—
186#99—174no—
187#99—342no—
188C22⋊C8—22no—
189C22⋊C8—342no—
190#97—341no—
191#97—682no—
192C4×C8—341no—
193C4×C8—682no—
194C17#1262411yescharacteristic, complemented, Sylow
195C24C4×#59511yescharacteristic
196#41—681no—
197#41—681no—
198C2×Q8—171no—
199C2×D4—172no—
200C2×D4—171no—
201C2×D4—342no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—172no—
207C22×C4C4×#59511yescharacteristic
208C22×C4C22×#59511yescharacteristic
209M16—41no—
210M16—41no—
211M16—344no—
212M16—682no—
213M16—682no—
214M16—682no—
215M16—682no—
216C2×C8—21no—
217C2×C8—171no—
218C2×C8—171no—
219C2×C8—342no—
220C2×C8—342no—
221C2×C8—342no—
222C2×C8—341no—
223C2×C8—682no—
224C2×C8—681no—
225C2×C8C22⋊#59512yes—
226C4⋊C4—342no—
227C4⋊C4—341no—
228C4⋊C4—682no—
229C22⋊C4—172no—
230C22⋊C4—342no—
231C22⋊C4—342no—
232C22⋊C4—344no—
233C42—341no—
234C42—682no—
235C23—21no—
236C23—41no—
237C23—81no—
238C23—172no—
239C23C2×C4×#59511yescharacteristic
240Q8—681no—
241D4—681no—
242D4—682no—
243D4—1362no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—172no—
252C2×C4—342no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—684no—
256C2×C4#1058767711yescharacteristic
257C2×C4C2×C22⋊#59511yescharacteristic
258C8—22no—
259C8—41no—
260C8—172no—
261C8—341no—
262C8—684no—
263C8—681no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—82no—
270C22—342no—
271C22—682no—
272C22C4×C22⋊#59511yescharacteristic
273C4—21no—
274C4—41no—
275C4—342no—
276C4—682no—
277C4—1364no—
278C4D17⋊#9911yescharacteristic
279C4#1176378911yescharacteristic
280C2—21no—
281C2—41no—
282C2—81no—
283C2—682no—
284C2#1454453111yescharacteristic
subgroupG/H
centerC4D17⋊#99
commutatorC23×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#407C4
socleC2×C17C4×#99
prime radical#11763320C22
cosocleC2×#63252C23
hypercenter#407#595
hypocenterC17#12624
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #14544540C2semidirect
3. #14543876C2semidirect
4. #14543873C2semidirect
9. #11763711C4semidirect
10. #11763709C4semidirect
13. C22×#63223C4semidirect
17. #11762837C4semidirect
19. #11762767C4semidirect
20. C17×#407C4semidirect
152. #407#595semidirect
194. C17#12624semidirect

Maximal subgroups

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

subgroupindex
#145450352
#145445402
#145438762
#145438732
#145374832
#1262417