Finite Group Databasebeta

← #16929363

Subgroups — #16929363

Order 4352.

616 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#11762748C211yesmaximal, characteristic
2#14528872C212yesmaximal
3C2×#11762535C211yesmaximal, characteristic
4#14526662C212yesmaximal
5C17×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7C2×C22⋊#3622C2212yes—
8#11762748—22no—
9#11762748C414yescomplemented
10C2×#10587502—21no—
11C2×#10587502—21no—
12C2×#10587502C2211yescharacteristic
13#11762571—24no—
14#11762535—22no—
15C2×C17×#117—21no—
16C22×C4⋊C4×C17C2211yescharacteristic
17C17×#389—24no—
18C17×#380—21no—
19C17×#380C2212yes—
20C23×C4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C17×#117—42no—
26C17×#117—42no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C4⋊C4×C17D414yes—
34C2×C22⋊C4×C17—22no—
35C2×C42×C17—22no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17C2×C412yes—
40C22×#3622—22no—
41C17×#95—22no—
42C17×#95—22no—
43#10587502—24no—
44#10587502—24no—
45#10587502—21no—
46#10587502C2×C412yes—
47C22⋊#3622—22no—
48C22⋊#3622—22no—
49C24×C17C2×D411yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—24no—
57C22×C4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17SD1612yes—
62C22×C4×C17Q1612yes—
63C22×C4×C17C22×C411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×C4×C17#4112yes—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—42no—
70C22×Dic17—42no—
71C22×Dic17—44no—
72C22×Dic17—41no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C4⋊C4×C17—24no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—44no—
79C4⋊C4×C17—44no—
80C4⋊C4×C17—82no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—42no—
83C42×C17—44no—
84C4⋊Dic17—24no—
85C4⋊Dic17—42no—
86C2×#3622—44no—
87C2×#3622—42no—
88#9023—171nomaximal, Sylow
89C23×C17—22no—
90C23×C17—22no—
91C23×C17C2×C22⋊C411yescharacteristic
92C23×C17#12011yescharacteristic
93C23×C17#12311yescharacteristic
94C23×C17C2×SD1611yescharacteristic
95C23×C17C2×Q1611yescharacteristic
96C23×C17AGL(1,ℤ/8)11yescharacteristic
97C23×C17#13711yescharacteristic
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—24no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—44no—
112C2×C4×C17—44no—
113C2×C4×C17—44no—
114C2×C4×C17—81no—
115C2×C4×C17#10314yes—
116C2×C4×C17C4×D412yes—
117C2×C4×C17#12112yes—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—44no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—82no—
124C2×Dic17—82no—
125C2×Dic17—82no—
126C2×Dic17—82no—
127#3622—82no—
128C2×#400—171no—
129#2316D1711yescharacteristic
130#1824—172no—
131C2×#324—171no—
132#1294—172no—
133C22×C17—22no—
134C22×C17—22no—
135C22×C17—22no—
136C22×C17—22no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17#38611yescharacteristic
142C22×C17C2×#10311yescharacteristic
143C22×C17#41711yescharacteristic
144C22×C17#45011yescharacteristic
145C22×C17#45111yescharacteristic
146C22×C17#45412yes—
147C22×C17#45812yes—
148C22×C17#48311yescharacteristic
149C22×C17#48411yescharacteristic
150C4×C17—24no—
151C4×C17—22no—
152C4×C17—22no—
153C4×C17—42no—
154C4×C17—84no—
155Dic17—82no—
156Dic17—82no—
157C2×#117—21no—
158C22×C4⋊C4—171no—
159C22×C4⋊C4C2×D1711yescharacteristic
160C2×C22⋊C8—172no—
161#400—174no—
162#400—342no—
163#389—24no—
164#380—21no—
165#380C2×D1712yes—
166C2×#95—171no—
167C2×#95—341no—
168C2×#95—341no—
169#340—344no—
170#324—342no—
171C2×C17—21no—
172C2×C17—21no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17#161612yes—
176C2×C17#161712yes—
177C2×C17#187211yescharacteristic
178C2×C17#188811yescharacteristic
179C2×C17#189011yescharacteristic
180C23×C4—341no—
181C23×C4—341no—
182C23×C4C22⋊D1711yescharacteristic
183C23×C4C22⋊D1711yescharacteristic
184C23×C4C22×D1711yescharacteristic
185C22×C8—342no—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—172no—
193C2×C4⋊C4—342no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4C4×D1712yes—
197C2×C4⋊C4D6812yes—
198C2×C4⋊C4C22⋊D1714yes—
199C2×C22⋊C4—22no—
200C2×C42—22no—
201C22⋊C8—342no—
202C22⋊C8—342no—
203#95—22no—
204#95—22no—
205#95—172no—
206#95—341no—
207#95—344no—
208#95—344no—
209C17#902311yescharacteristic, complemented, Sylow
210C24C2×C22⋊D1711yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—342no—
221C22×C4—342no—
222C22×C4—682no—
223C22×C4—682no—
224C22×C4—684no—
225C22×C4—681no—
226C22×C4—681no—
227C22×C4—681no—
228C22×C4—681no—
229C22×C4#6322712yes—
230C22×C4#6322712yes—
231C22×C4#6322912yes—
232C22×C4#6323112yes—
233C22×C4C2×C4×D1711yescharacteristic
234C22×C4C2×D6811yescharacteristic
235C22×C4#6325212yes—
236C22×C4C2×C22⋊D1712yes—
237C2×C8—684no—
238C2×C8—682no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—82no—
244C4⋊C4—344no—
245C4⋊C4—682no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—44no—
249C23—22no—
250C23—22no—
251C23C2×#6322711yescharacteristic
252C23#1058761511yescharacteristic
253C23#1058761811yescharacteristic
254C23C2×#6322911yescharacteristic
255C23#1058762811yescharacteristic
256C23C2×#6323111yescharacteristic
257C23#1058763911yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—81no—
275C2×C4—682no—
276C2×C4—682no—
277C2×C4—684no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—1362no—
281C2×C4—1362no—
282C2×C4—1362no—
283C2×C4—1362no—
284C2×C4#1058748114yes—
285C2×C4C4×C22⋊D1712yes—
286C2×C4#1058761612yes—
287C8—1362no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22#1176268811yescharacteristic
297C22C2×#1058748111yescharacteristic
298C22#1176280811yescharacteristic
299C22#1176296011yescharacteristic
300C22#1176296511yescharacteristic
301C22#1176298212yes—
302C22#1176299912yes—
303C22#1176313911yescharacteristic
304C22#1176314411yescharacteristic
305C4—24no—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—84no—
310C4—1362no—
311C4—1362no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#1452782112yes—
317C2#1452782612yes—
318C2#1452909111yescharacteristic
319C2#1452918711yescharacteristic
320C2#1452919711yescharacteristic
subgroupG/H
centerC23#10587639
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2316C2
socleC23×C17#120
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#2316D17
hypocenterC17#9023
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
9. #11762748C4semidirect
209. C17#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#117627482
#145288722
C2×#117625352
#145266622
C17×#23162
#902317