Finite Group Databasebeta

← #17948888

Subgroups — #17948888

Order 4352.

402 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 84 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#14545009C211yesmaximal, characteristic
2#14544661C211yesmaximal, characteristic
3D17×#340C211yesmaximal, characteristic
4C2×#10587529—21no—
5C2×#10587534C412yescomplemented
6#11763751—21no—
7#11763704C412yescomplemented
8C22⋊#63216—21no—
9#11763683—21no—
10C22×C8×D17C2211yescharacteristic
11C2×C4⋊C4×D17—21no—
12#11762572C411yescharacteristic, complemented
13#11762571—21no—
14C17×#340C411yescharacteristic, complemented
15C22×C4×D17—41no—
16C22×C4×D17D411yescharacteristic
17C2×C22⋊#595—41no—
18C2×C4⋊#595—22no—
19C22×C8×C17C2×C411yescharacteristic
20C2×C4⋊C4×C17—21no—
21C2×C4⋊Dic17—21no—
22C2×#63225—41no—
23C22×#3622D411yescharacteristic
24C2×C8×D17—42no—
25C2×C8×D17C2×C411yescharacteristic
26C2×C8×D17C2×C411yescharacteristic
27C2×C8×D17D411yescharacteristic
28C2×C8×D17Q811yescharacteristic
29C4⋊C4×D17—24no—
30C4⋊C4×D17—42no—
31(C2×#595)⋊C4—41no—
32C2×#63216—22no—
33#10587534—42no—
34#10587529—42no—
35C23×D17D811yescharacteristic
36C22×#595—42no—
37C22×C4×C17—41no—
38C22×C4×C17C22⋊C411yescharacteristic
39C22×Dic17—41no—
40C22×Dic17SD1611yescharacteristic
41C2×C4×D17—22no—
42C2×C4×D17—22no—
43C2×C4×D17—44no—
44C2×C4×D17—82no—
45C2×C4×D17—81no—
46C2×C4×D17—81no—
47C2×C4×D17C22⋊C411yescharacteristic
48C2×C4×D17C4⋊C411yescharacteristic
49C2×C4×D17SD1611yescharacteristic
50C2×C4×D17Q1611yescharacteristic
51C22⋊#595—81no—
52C22⋊#595—81no—
53C4⋊#595—42no—
54C4⋊#595—42no—
55C2×C8×C17—41no—
56C2×C8×C17C4211yescharacteristic
57C2×C8×C17C4⋊C411yescharacteristic
58C4⋊C4×C17—22no—
59C4⋊C4×C17—41no—
60C4⋊Dic17—22no—
61C4⋊Dic17—41no—
62#63225—44no—
63C2×#3622—41no—
64C2×#3622C22⋊C411yescharacteristic
65C2×#3622C4⋊C411yescharacteristic
66C8×D17—22no—
67C8×D17—22no—
68C8×D17—41no—
69C8×D17—41no—
70#63216—42no—
71#7258—171nomaximal, Sylow
72C23×C17#10211yescharacteristic
73C22×D17—22no—
74C22×D17—22no—
75C22×D17#10211yescharacteristic
76C22×D17#10711yescharacteristic
77C22×D17D1611yescharacteristic
78C22×D17SD3211yescharacteristic
79C22×D17SD3211yescharacteristic
80C22×D17Q3211yescharacteristic
81C2×#595—82no—
82C2×#595—82no—
83C2×#595—82no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—42no—
87C2×C4×C17—81no—
88C2×C4×C17—81no—
89C2×C4×C17#9511yescharacteristic
90C2×C4×C17#10311yescharacteristic
91C2×Dic17—22no—
92C2×Dic17—42no—
93C2×Dic17—81no—
94C2×Dic17—81no—
95C2×Dic17#10311yescharacteristic
96C2×Dic17#10611yescharacteristic
97C4×D17—22no—
98C4×D17—21no—
99C4×D17—21no—
100C4×D17—22no—
101C4×D17—84no—
102C4×D17—84no—
103C4×D17#9912yes—
104C4×D17#10412yes—
105C8×C17—21no—
106C8×C17—21no—
107#3622—21no—
108#3622—21no—
109C2×C22⋊C16—171no—
110#1808—171no—
111C2×#340—171no—
112C22×C17—21no—
113C22×C17—21no—
114C22×C17#34011yescharacteristic
115C22×C17#35711yescharacteristic
116C22×C17#35811yescharacteristic
117C2×D17—21no—
118C2×D17—21no—
119C2×D17—22no—
120C2×D17—22no—
121C2×D17—22no—
122C2×D17—22no—
123C2×D17#32712yes—
124C2×D17#35711yescharacteristic
125C2×D17#35811yescharacteristic
126C2×D17#36012yes—
127C2×D17#36611yescharacteristic
128C2×D17#36711yescharacteristic
129#595—162no—
130C4×C17—21no—
131C4×C17—82no—
132C4×C17#33911yescharacteristic
133C4×C17#34211yescharacteristic
134Dic17—21no—
135Dic17—82no—
136Dic17#32812yes—
137C23×C8—171no—
138C22×C4⋊C4—341no—
139C22×C16—341no—
140C2×#107—172no—
141#380—341no—
142C22⋊C16—172no—
143C22⋊C16—341no—
144#340—171no—
145#340—341no—
146#340—341no—
147#340#59511yescharacteristic, complemented
148C2×C17—21no—
149C2×C17—21no—
150C2×C17#129411yescharacteristic
151C2×C17#138011yescharacteristic
152C2×C17#138611yescharacteristic
153D17—21no—
154D17—21no—
155D17#134712yes—
156D17#134812yes—
157C23×C4—171no—
158C23×C4—681no—
159C22×C8—171no—
160C22×C8—171no—
161C22×C8—171no—
162C22×C8—171no—
163C22×C8—171no—
164C22×C8—682no—
165C22×C8C2×#59511yescharacteristic
166C2×C4⋊C4—21no—
167C2×C4⋊C4—344no—
168C2×C4⋊C4—341no—
169C2×C4⋊C4—342no—
170C2×C4⋊C4—682no—
171C2×C4⋊C4—681no—
172C2×C22⋊C4—681no—
173C2×C16—342no—
174C2×C16—682no—
175#107—682no—
176#95—681no—
177C17#725811yescharacteristic, complemented, Sylow
178C24—171no—
179C22×C4—41no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—171no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—342no—
186C22×C4—342no—
187C22×C4—684no—
188C22×C4—681no—
189C22×C4—682no—
190C22×C4—1361no—
191C22×C4—1361no—
192C22×C4—1362no—
193C22×C4C22⋊#59511yescharacteristic
194C2×C8—41no—
195C2×C8—171no—
196C2×C8—171no—
197C2×C8—342no—
198C2×C8—342no—
199C2×C8—681no—
200C2×C8—681no—
201C2×C8—681no—
202C2×C8C4×#59511yescharacteristic
203C2×C8C4⋊#59511yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—342no—
207C4⋊C4—681no—
208C4⋊C4—684no—
209C4⋊C4—682no—
210C4⋊C4—682no—
211C22⋊C4—1361no—
212C22⋊C4—1361no—
213C16—682no—
214C23—171no—
215C23—171no—
216C23—171no—
217C23—171no—
218C23—171no—
219C23—171no—
220C23—342no—
221C23—342no—
222C23#1058754711yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—171no—
229C2×C4—172no—
230C2×C4—172no—
231C2×C4—171no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—341no—
236C2×C4—341no—
237C2×C4—682no—
238C2×C4—1364no—
239C2×C4—1364no—
240C2×C4—1362no—
241C2×C4—1362no—
242C2×C4—1362no—
243C2×C4—1361no—
244C2×C4—1361no—
245C2×C4(C2×#595)⋊C411yescharacteristic
246C2×C4#1058754911yescharacteristic
247C8—21no—
248C8—21no—
249C8—341no—
250C8—341no—
251C22—21no—
252C22—21no—
253C22—172no—
254C22—171no—
255C22—171no—
256C22—172no—
257C22—171no—
258C22—171no—
259C22—342no—
260C22—342no—
261C22—342no—
262C22—342no—
263C22—341no—
264C22—341no—
265C22#1176368211yescharacteristic
266C22#1176372111yescharacteristic
267C22#1176372211yescharacteristic
268C4—21no—
269C4—82no—
270C4—2722no—
271C4—172no—
272C4—341no—
273C4—1362no—
274C4#1176368011yescharacteristic
275C4#1176368611yescharacteristic
276C2—21no—
277C2—21no—
278C2—172no—
279C2—172no—
280C2—341no—
281C2—341no—
282C2#1454365811yescharacteristic
283C2#1454384111yescharacteristic
284C2#1454385711yescharacteristic
subgroupG/H
centerC22#11763682
commutatorC2×C8×C17C42
FrattiniC22×C8C2×#595
FittingC17×#340C4
socleC22×C17#340
prime radicalC23×D17D8
cosocleC22×C8×D17C22
hypercenter#340#595
hypocenterC17#7258
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #340

Complements

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

NH ≅ G/Ntype
5. C2×#10587534C4semidirect
7. #11763704C4semidirect
12. #11762572C4semidirect
14. C17×#340C4semidirect
147. #340#595semidirect
177. C17#7258semidirect

Maximal subgroups

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

subgroupindex
#145450092
#145446612
D17×#3402
#725817