Finite Group Databasebeta

← #16922980

Subgroups — #16922980

Order 4352.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#14537136C211yesmaximal, characteristic
2#14528852C211yesmaximal, characteristic
3#14526606C214yesmaximal, complemented
4C17×#1797C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6C2×#10587610—21no—
7C2×C22⋊#3622—21no—
8#11762715—21no—
9#11762707—21no—
10#11762707C2212yes—
11#11762679C411yescharacteristic
12#11762669—21no—
13#11762669C412yes—
14#11762564—21no—
15#11762564C2212yes—
16C22×C42×C17C2211yescharacteristic
17C2×C17×C4⋊C8—21no—
18C2×C17×C22⋊C8—21no—
19C17×#336C2212yes—
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C22×C8×C17—22no—
25C2×C42×C17—42no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C42×C17D411yescharacteristic
28C2×C42×C17D411yescharacteristic
29C2×C42×C17C2311yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C412yes—
35C2×C4⋊Dic17C2×C412yes—
36#10587610—44no—
37C2×#63225—22no—
38C22×#3622—22no—
39#10587555—42no—
40C17×C4⋊C8—42no—
41C17×C22⋊C8—42no—
42#10587502—22no—
43#10587502—22no—
44C22⋊#3622—42no—
45C24×C17C22⋊C411yescharacteristic
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17—22no—
50C22×C4×C17—41no—
51C22×C4×C17—41no—
52C22×C4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17SD1612yes—
58C22×C4×C17Q1612yes—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×Dic17—22no—
63C22×Dic17—22no—
64C2×Dic34—42no—
65C2×Dic34—42no—
66C2×Dic34—42no—
67C2×C8×C17—42no—
68C2×C8×C17—42no—
69C2×C8×C17—42no—
70C42×C17—22no—
71C42×C17—22no—
72C42×C17—41no—
73C42×C17—41no—
74C22⋊Dic17—44no—
75C4⋊Dic17—24no—
76C4⋊Dic17—42no—
77C4⋊Dic17—42no—
78C4⋊Dic17—42no—
79C4⋊Dic17—42no—
80#63225—44no—
81C2×#3622—42no—
82C2×#3622—42no—
83C2×#3622—42no—
84#7725—171nomaximal, Sylow
85C23×C17—21no—
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17#10312yes—
92C23×C17C2×C22⋊C411yescharacteristic
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—44no—
108C2×C4×C17—44no—
109C2×C4×C17#9912yes—
110C2×C4×C17#10112yes—
111C2×C4×C17#10312yes—
112C2×C4×C17#13112yes—
113C2×C4×C17C2×SD1611yescharacteristic
114C2×C4×C17C2×Q1611yescharacteristic
115C2×C4×C17AGL(1,ℤ/8)12yes—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122Dic34—84no—
123C8×C17—82no—
124#3622—82no—
125#2857—171no—
126#1817—171no—
127#1797D1711yescharacteristic
128#1271—174no—
129C22×C17—22no—
130C22×C17—22no—
131C22×C17—22no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17#40811yescharacteristic
137C22×C17C2×#9911yescharacteristic
138C22×C17C2×#10111yescharacteristic
139C22×C17C2×#10311yescharacteristic
140C22×C17#41811yescharacteristic
141C22×C17#42111yescharacteristic
142C22×C17#42111yescharacteristic
143C4×C17—22no—
144C4×C17—22no—
145C4×C17—22no—
146C4×C17—22no—
147C4×C17—21no—
148C4×C17—22no—
149C4×C17—21no—
150Dic17—82no—
151Dic17—82no—
152C2×C4⋊Q8—171no—
153C2×C22⋊Q8—341no—
154C22×C42C2×D1711yescharacteristic
155C2×C4⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8—341no—
158#391—341no—
159#389—172no—
160#389—341no—
161#384—171no—
162#380—172no—
163#380—341no—
164#336—172no—
165#336—341no—
166#336C2×D1712yes—
167C2×C17—42no—
168C2×C17#147211yescharacteristic
169C2×C17#148111yescharacteristic
170C2×C17#149611yescharacteristic
171C2×C17#151011yescharacteristic
172C2×C17#152011yescharacteristic
173C2×C17#153711yescharacteristic
174C2×C17#154911yescharacteristic
175C22×Q8—341no—
176C23×C4C4×D1711yescharacteristic
177C23×C4D6811yescharacteristic
178C23×C4C22⋊D1711yescharacteristic
179C22×C8—22no—
180C22×C8—342no—
181C4⋊Q8—682no—
182C22⋊Q8—684no—
183C2×C4⋊C4—172no—
184C2×C4⋊C4—172no—
185C2×C4⋊C4—342no—
186C2×C4⋊C4—341no—
187C2×C4⋊C4—341no—
188C2×C4⋊C4—341no—
189C2×C22⋊C4—342no—
190C2×C42—42no—
191C2×C42C4×D1711yescharacteristic
192C2×C42D6811yescharacteristic
193C2×C42C22⋊D1711yescharacteristic
194C2×C42C22×D1711yescharacteristic
195C4⋊C8—42no—
196C22⋊C8—42no—
197C22⋊C8—682no—
198#95—342no—
199#95—342no—
200C17#772511yescharacteristic, complemented, Sylow
201C24#6322711yescharacteristic
202C2×Q8—682no—
203C2×Q8—682no—
204C2×Q8—682no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—342no—
214C22×C4—342no—
215C22×C4#6322711yescharacteristic
216C22×C4#6322711yescharacteristic
217C22×C4#6322711yescharacteristic
218C22×C4#6322912yes—
219C22×C4#6323112yes—
220C22×C4C2×C4×D1711yescharacteristic
221C22×C4C2×D6811yescharacteristic
222C22×C4C2×C22⋊D1711yescharacteristic
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—682no—
227C2×C8—682no—
228C2×C8—682no—
229C4⋊C4—344no—
230C4⋊C4—682no—
231C4⋊C4—682no—
232C4⋊C4—682no—
233C4⋊C4—682no—
234C4⋊C4—684no—
235C22⋊C4—684no—
236C42—22no—
237C42—22no—
238C42—41no—
239C42—41no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23#1058748112yes—
247C23C2×#6322711yescharacteristic
248Q8—1364no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4—682no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4#1058747712yes—
272C2×C4#1058748112yes—
273C2×C4#1058749512yes—
274C2×C4#1058758712yes—
275C2×C4#1058759412yes—
276C2×C4C2×#6322911yescharacteristic
277C2×C4C2×#6323111yescharacteristic
278C8—82no—
279C8—1362no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22#1176277511yescharacteristic
288C22C2×#1058747711yescharacteristic
289C22C2×#1058749511yescharacteristic
290C22C2×#1058748111yescharacteristic
291C22#1176281511yescharacteristic
292C22#1176282511yescharacteristic
293C22#1176282711yescharacteristic
294C4—22no—
295C4—22no—
296C4—21no—
297C4—22no—
298C4—21no—
299C4—22no—
300C4—22no—
301C4—1362no—
302C4—1362no—
303C2—42no—
304C2#1452714711yescharacteristic
305C2#1452718611yescharacteristic
306C2#1452725411yescharacteristic
307C2#1452731411yescharacteristic
308C2#1452735511yescharacteristic
309C2#1452742111yescharacteristic
310C2#1452747011yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1797C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1797D17
hypocenterC17#7725
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #14526606C2semidirect
200. C17#7725semidirect

Maximal subgroups

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

subgroupindex
#145371362
#145288522
#145266062
C17×#17972
#772517