Finite Group Databasebeta

← #17953331

Subgroups — #17953331

Order 4352.

486 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 106 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#14544648C212yesmaximal
2#14544341C212yesmaximal, complemented
3#14543654C212yesmaximal, complemented
4C2×#11762872C211yesmaximal, characteristic
5#11763812C2214yes—
6C2×#10587553—22no—
7#11763689—22no—
8#11763673—22no—
9#11763672—22no—
10C23×#3622C411yescharacteristic
11C22×C8×D17C411yescharacteristic
12C2×C4×C22⋊D17C2211yescharacteristic
13#11762872—22no—
14#11762872—22no—
15C2×#10587485C2211yescharacteristic
16C2×#10587491C411yescharacteristic
17C2×C8×Dic17C2211yescharacteristic
18C2×C17×C22⋊C8C411yescharacteristic
19C23×C4×C17C2×C411yescharacteristic
20C22×C22⋊D17D411yescharacteristic
21C22×C4×D17C2×C411yescharacteristic
22C2×#63247—24no—
23C22×C8×C17C2×C411yescharacteristic
24C22×C8×C17C2×C411yescharacteristic
25C2×C22⋊Dic17Q811yescharacteristic
26C4×C22⋊D17—22no—
27C4×C22⋊D17—22no—
28C2×#63227Q811yescharacteristic
29C2×#63225D411yescharacteristic
30C2×C4×Dic17C2311yescharacteristic
31C22×#3622—21no—
32C22×#3622—21no—
33C22×#3622—42no—
34C22×#3622C2×C411yescharacteristic
35C22×#3622C2×C411yescharacteristic
36C2×C8×D17—21no—
37C2×C8×D17—21no—
38C2×C8×D17—42no—
39#10587553—44no—
40#10587544—24no—
41C4⋊#3624—24no—
42C17×C22⋊C8—22no—
43#10587491—22no—
44#10587485—22no—
45C8×Dic17D412yes—
46C8×Dic17D412yes—
47C24×C17C4⋊C411yescharacteristic
48C23×D17C4⋊C411yescharacteristic
49C22×C4×C17—21no—
50C22×C4×C17—21no—
51C22×C4×C17—41no—
52C22×C4×C17—41no—
53C22×C4×C17C4⋊C411yescharacteristic
54C22×C4×C17C22×C411yescharacteristic
55C2×C22⋊D17—22no—
56C2×C22⋊D17—22no—
57C2×C22⋊D17D812yes—
58C2×C22⋊D17Q1612yes—
59C22×Dic17C4⋊C411yescharacteristic
60C22×Dic17C2×D411yescharacteristic
61C22×Dic17C2×Q811yescharacteristic
62C2×C4×D17—21no—
63C2×C4×D17—21no—
64C2×C4×D17—41no—
65C2×C4×D17—41no—
66#63247—44no—
67#63247—44no—
68C2×#3624—44no—
69C2×C8×C17—21no—
70C2×C8×C17—41no—
71C2×C8×C17—41no—
72C2×C8×C17C22⋊C412yes—
73C22⋊Dic17—22no—
74#63227—22no—
75#63225—22no—
76C4×Dic17C2×D412yes—
77C4×Dic17#4112yes—
78C2×#3622—22no—
79C2×#3622—21no—
80C2×#3622—42no—
81C2×#3622—42no—
82C2×#3622—41no—
83C2×#3622C22⋊C411yescharacteristic
84C2×#3622C22⋊C411yescharacteristic
85C8×D17—42no—
86C8×D17—42no—
87#8616—171nomaximal, Sylow
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17#10712yes—
93C23×C17C2×C4⋊C411yescharacteristic
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—41no—
97C22×D17—41no—
98C22×D17#10712yes—
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—24no—
109C22⋊D17—24no—
110C2×Dic17—21no—
111C2×Dic17—41no—
112C2×Dic17—41no—
113C2×Dic17#12112yes—
114C2×Dic17C22⋊Q812yes—
115C2×Dic17C2×D811yescharacteristic
116C2×Dic17C2×Q1611yescharacteristic
117C2×Dic17#13512yes—
118C4×D17—42no—
119C4×D17—41no—
120C4×D17—41no—
121#3624—84no—
122C8×C17—22no—
123C8×C17—41no—
124#3622—21no—
125#3622—41no—
126#3622#10012yescomplemented
127C2×C8×D4—171no—
128#1800—172no—
129#1567—172no—
130#1292—172no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—22no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—41no—
137C22×C17—41no—
138C22×C17#38011yescharacteristic
139C22×C17C2×#10711yescharacteristic
140C22×C17#42711yescharacteristic
141C2×D17—22no—
142C2×D17—42no—
143C2×D17—41no—
144C2×D17—41no—
145C2×D17—42no—
146C4×C17—21no—
147C4×C17—41no—
148C4×C17C2×#10011yescharacteristic
149C4×C17#41311yescharacteristic
150Dic17—22no—
151Dic17—41no—
152Dic17#46612yes—
153Dic17#46712yes—
154C23×C8—171no—
155C23×C8—171no—
156C2×C4×D4—171no—
157C8×D4—342no—
158C8×D4—342no—
159#423—174no—
160C2×C4⋊C8—171no—
161C2×#100—342no—
162C2×C22⋊C8—171no—
163C2×C22⋊C8#59511yescharacteristic
164C2×C4×C8—171no—
165#343—342no—
166#335—342no—
167#335—342no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—42no—
171C2×C17#180011yescharacteristic
172C2×C17#180811yescharacteristic
173C2×C17#181311yescharacteristic
174D17—42no—
175C22×D4—171no—
176C23×C4—171no—
177C23×C4C2×#59511yescharacteristic
178C2×M16—344no—
179C22×C8—171no—
180C22×C8—171no—
181C22×C8—341no—
182C22×C8—341no—
183C22×C8—341no—
184C22×C8—341no—
185C22×C8—682no—
186C22×C8—682no—
187C22×C8C2×#59511yescharacteristic
188C22×C8C2×#59511yescharacteristic
189C4×D4—342no—
190C4×D4—342no—
191C2×C4⋊C4—171no—
192C2×C22⋊C4—171no—
193C2×C22⋊C4—171no—
194C2×C42—171no—
195C4⋊C8—342no—
196C4⋊C8—344no—
197C4⋊C8—344no—
198#100—684no—
199C22⋊C8—22no—
200C22⋊C8—342no—
201C4×C8—172no—
202C4×C8—172no—
203C17#861611yescharacteristic, complemented, Sylow
204C24—171no—
205C24C4⋊#59511yescharacteristic
206C2×D4—172no—
207C2×D4—172no—
208C2×D4—342no—
209C2×D4—342no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—341no—
218C22×C4—341no—
219C22×C4—681no—
220C22×C4—681no—
221C22×C4C4⋊#59511yescharacteristic
222C22×C4C22×#59511yescharacteristic
223M16—684no—
224M16—684no—
225C2×C8—21no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—171no—
229C2×C8—171no—
230C2×C8—342no—
231C2×C8—341no—
232C2×C8—682no—
233C2×C8—682no—
234C2×C8—682no—
235C2×C8—682no—
236C2×C8—681no—
237C2×C8—684no—
238C2×C8C22⋊#59512yes—
239C4⋊C4—342no—
240C22⋊C4—342no—
241C22⋊C4—342no—
242C42—172no—
243C42—172no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—172no—
249C23—341no—
250C23—341no—
251C23—681no—
252C23—681no—
253C23#1058753412yes—
254C23C2×C4⋊#59511yescharacteristic
255D4—344no—
256D4—344no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—172no—
265C2×C4—171no—
266C2×C4—171no—
267C2×C4—172no—
268C2×C4—172no—
269C2×C4—341no—
270C2×C4—682no—
271C2×C4—681no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—681no—
275C2×C4#1058767711yescharacteristic
276C2×C4C2×C22⋊#59511yescharacteristic
277C8—22no—
278C8—41no—
279C8—172no—
280C8—341no—
281C8—681no—
282C8—1364no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—342no—
291C22—682no—
292C22—682no—
293C22—681no—
294C22—681no—
295C22#1176375111yescharacteristic
296C22C2×#1058753411yescharacteristic
297C22#1176381911yescharacteristic
298C4—21no—
299C4—41no—
300C4—172no—
301C4—172no—
302C4—342no—
303C4—681no—
304C4D17⋊#10011yescharacteristic
305C4#1176379611yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—682no—
310C2#1454464911yescharacteristic
311C2#1454466111yescharacteristic
312C2#1454466911yescharacteristic
subgroupG/H
centerC22#11763751
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C22⋊C8C4
socleC22×C17#380
prime radicalC22×C22⋊D17D4
cosocleC2×C4×Dic17C23
hypercenterC2×C22⋊C8#595
hypocenterC17#8616
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #14544341C2semidirect
3. #14543654C2semidirect
126. #3622#100semidirect
203. C17#8616semidirect

Maximal subgroups

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

subgroupindex
#145446482
#145443412
#145436542
C2×#117628722
#861617