Finite Group Databasebeta

← #16930370

Subgroups — #16930370

Order 4352.

560 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 127 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#14534597C211yesmaximal, characteristic
2#14531402C211yesmaximal, characteristic
3#14527357C212yesmaximal
4#14526652C212yesmaximal
5C2×C17×#324C211yesmaximal, characteristic
6C2×#10587610C2211yescharacteristic
7C2×#10587612—21no—
8#11762748C412yescomplemented
9#11762742—22no—
10#11762729C412yescomplemented
11#11762708—24no—
12#11762682—21no—
13#11762672—21no—
14#11762669—21no—
15#11762669C2211yescharacteristic
16#11762669C2212yes—
17#11762547—24no—
18C22×C4⋊C4×C17C2211yescharacteristic
19C2×C17×C22⋊C8C2212yes—
20C17×#324—22no—
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C22×C8×C17—22no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C4⋊C4×C17D412yes—
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—21no—
33#10587612—42no—
34#10587612—42no—
35C2×C4⋊Dic17—22no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40#10587610—22no—
41#10587610—22no—
42C2×#63225—22no—
43C2×#63225—21no—
44C2×C4×Dic17—22no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—22no—
50#10587502—22no—
51#10587502C2×C412yes—
52#10587502D414yes—
53C24×C17C2×D411yescharacteristic
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—81no—
61C22×C4×C17—81no—
62C22×C4×C17C22⋊C412yes—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17SD1612yes—
65C22×C4×C17Q1612yes—
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D412yes—
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17#4112yes—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—22no—
73C22×Dic17—24no—
74C2×Dic34—22no—
75C2×Dic34—81no—
76C2×C8×C17—42no—
77C2×C8×C17—82no—
78C4⋊C4×C17—24no—
79C4⋊C4×C17—24no—
80C22⋊Dic17—41no—
81C22⋊Dic17—41no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C22⋊Dic17—41no—
85C22⋊Dic17—41no—
86C4⋊Dic17—22no—
87C4⋊Dic17—44no—
88C4⋊Dic17—81no—
89C4⋊Dic17—82no—
90C4⋊Dic17—82no—
91#63225—42no—
92#63225—82no—
93C4×Dic17—44no—
94#9200—171nomaximal, Sylow
95C23×C17—41no—
96C23×C17—41no—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17#12011yescharacteristic
99C23×C17#12311yescharacteristic
100C23×C17C2×SD1611yescharacteristic
101C23×C17C2×Q1611yescharacteristic
102C23×C17AGL(1,ℤ/8)11yescharacteristic
103C23×C17#13711yescharacteristic
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—81no—
118C2×C4×C17#10314yes—
119C2×C4×C17C4×D412yes—
120C2×C4×C17#12112yes—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—44no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—44no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—42no—
130C2×Dic17—44no—
131Dic34—82no—
132C8×C17—82no—
133#2627—171no—
134#2316—171no—
135#1520—172no—
136C2×#324D1711yescharacteristic
137#1289—172no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17#38611yescharacteristic
149C22×C17C2×#10311yescharacteristic
150C22×C17#41711yescharacteristic
151C22×C17#45011yescharacteristic
152C22×C17#45111yescharacteristic
153C22×C17#48311yescharacteristic
154C22×C17#48411yescharacteristic
155C4×C17—24no—
156C4×C17—22no—
157C4×C17—22no—
158C4×C17—42no—
159Dic17—84no—
160Dic17—82no—
161Dic17—81no—
162Dic17—81no—
163C2×C22⋊Q8—171no—
164C2×#117—341no—
165C22×C4⋊C4C2×D1711yescharacteristic
166C2×C22⋊C8C2×D1712yes—
167#400—172no—
168#398—342no—
169#395—172no—
170#389—344no—
171#385—341no—
172#381—341no—
173#380—171no—
174#380—172no—
175#380—341no—
176#328—344no—
177#324—22no—
178C2×C17—21no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17#187211yescharacteristic
185C2×C17#191211yescharacteristic
186C2×C17#191311yescharacteristic
187C22×Q8—171no—
188C23×C4D6811yescharacteristic
189C23×C4D6811yescharacteristic
190C23×C4C22×D1711yescharacteristic
191C22×C8—22no—
192C22⋊Q8—342no—
193C22⋊Q8—342no—
194#117—682no—
195#117—682no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—342no—
205C2×C4⋊C4C4×D1712yes—
206C2×C4⋊C4C22⋊D1712yes—
207C2×C22⋊C4—342no—
208C2×C22⋊C4—341no—
209C2×C22⋊C4—341no—
210C2×C42—342no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213#95—172no—
214#95—174no—
215#95—342no—
216#95—342no—
217#95—342no—
218#95—342no—
219C17#920011yescharacteristic, complemented, Sylow
220C24C2×D6811yescharacteristic
221C2×Q8—342no—
222C2×Q8—1361no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—344no—
232C22×C4—342no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4#6321912yes—
236C22×C4Dic6812yes—
237C22×C4#6322712yes—
238C22×C4#6322712yes—
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4D4×D1712yes—
241C22×C4#6325412yes—
242C22×C4C2×C22⋊D1711yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—342no—
248C4⋊C4—684no—
249C4⋊C4—682no—
250C4⋊C4—1362no—
251C4⋊C4—1362no—
252C4⋊C4—1361no—
253C4⋊C4—1362no—
254C22⋊C4—681no—
255C22⋊C4—681no—
256C22⋊C4—682no—
257C22⋊C4—681no—
258C22⋊C4—682no—
259C22⋊C4—681no—
260C42—684no—
261C23—41no—
262C23—41no—
263C23#1058756811yescharacteristic
264C23#1058757211yescharacteristic
265C23C2×#6321911yescharacteristic
266C23C2×Dic6811yescharacteristic
267C23#1058759411yescharacteristic
268C23#1058759511yescharacteristic
269C23C2×#6322711yescharacteristic
270Q8—1362no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—684no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—684no—
289C2×C4—682no—
290C2×C4—682no—
291C2×C4—684no—
292C2×C4—682no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4#1058748714yes—
296C2×C4#1058756712yes—
297C2×C4#1058762512yes—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1176268611yescharacteristic
310C22C2×#1058748711yescharacteristic
311C22#1176281111yescharacteristic
312C22#1176296211yescharacteristic
313C22#1176296711yescharacteristic
314C22#1176314011yescharacteristic
315C22#1176314511yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—1364no—
321C4—1362no—
322C4—1361no—
323C4—1361no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#1452908911yescharacteristic
331C2#1452931911yescharacteristic
332C2#1452932411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#324C2
socleC22×C17#386
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenterC2×#324D17
hypocenterC17#9200
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
8. #11762748C4semidirect
10. #11762729C4semidirect
219. C17#9200semidirect

Maximal subgroups

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

subgroupindex
#145345972
#145314022
#145273572
#145266522
C2×C17×#3242
#920017