Finite Group Databasebeta

← #16934883

Subgroups — #16934883

Order 4352.

804 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11762964C212yesmaximal
2C2×#11762546C212yesmaximal
3#14527054C211yesmaximal, characteristic
4#14526627C211yesmaximal, characteristic
5C17×#2659C211yesmaximal, characteristic
6C23×Dic34C2211yescharacteristic
7C22×#63231—22no—
8#11762964—24no—
9#11762964—24no—
10C2×#10587481—22no—
11C2×C22⋊#3622C2212yes—
12C2×#10587502—22no—
13C2×#10587502C2211yescharacteristic
14#11762546—24no—
15C2×C17×C22⋊Q8C2212yes—
16C17×#398—22no—
17C17×#395—22no—
18C17×#380—22no—
19C2×C17×#95C2211yescharacteristic
20C22×Q8×C17D412yes—
21C23×C4×C17D412yes—
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C22×Dic34—22no—
26C22×Dic34—22no—
27C22×Dic34—21no—
28C22×Dic34—44no—
29C22×Dic34D412yes—
30C17×C22⋊Q8—22no—
31C17×C22⋊Q8—22no—
32C17×C22⋊Q8—22no—
33C17×C22⋊Q8—22no—
34C2×C4⋊C4×C17—22no—
35C2×C4⋊C4×C17—22no—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17D412yes—
38C2×C22⋊C4×C17—22no—
39C2×#63231—28no—
40C2×#63231—48no—
41C22×#3622—22no—
42C17×#95—24no—
43C17×#95—22no—
44#10587502—22no—
45#10587502—44no—
46C22⋊#3622—22no—
47C22⋊#3622D414yes—
48#10587481—28no—
49C24×C17C2×D411yescharacteristic
50C2×Q8×C17—22no—
51C2×Q8×C17—22no—
52C2×Q8×C17—44no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17Q1614yescomplemented
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D412yes—
63C22×C4×C17C2×D412yes—
64C22×C4×C17#4111yescharacteristic
65C22×Dic17—24no—
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—42no—
69C22×Dic17—41no—
70C22×Dic17—41no—
71C22×Dic17—41no—
72C22×Dic17—42no—
73C22×Dic17—41no—
74C22×Dic17—42no—
75C2×Dic34—28no—
76C2×Dic34—24no—
77C2×Dic34—44no—
78C2×Dic34—48no—
79C2×Dic34—44no—
80C2×Dic34—44no—
81C2×Dic34—42no—
82C4⋊C4×C17—22no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—42no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—82no—
87C4⋊C4×C17—82no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90#63231—416no—
91C2×#3622—24no—
92C2×#3622—44no—
93#9967—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—22no—
97C23×C17#12011yescharacteristic
98C23×C17#12111yescharacteristic
99C23×C17#12711yescharacteristic
100C23×C17C2×Q1612yes—
101C23×C17#13712yes—
102Q8×C17—44no—
103Q8×C17—82no—
104C2×C4×C17—24no—
105C2×C4×C17—24no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—41no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—44no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—44no—
117C2×C4×C17—82no—
118C2×C4×C17#12012yes—
119C2×C4×C17#12112yes—
120C2×Dic17—24no—
121C2×Dic17—42no—
122C2×Dic17—44no—
123C2×Dic17—42no—
124C2×Dic17—44no—
125C2×Dic17—44no—
126C2×Dic17—42no—
127C2×Dic17—82no—
128C2×Dic17—81no—
129C2×Dic17—81no—
130Dic34—28no—
131Dic34—44no—
132Dic34—48no—
133#3622—44no—
134C2×#451—172no—
135#2659D1711yescharacteristic
136C2×#328—172no—
137#1446—171no—
138#1280—171no—
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17#39211yescharacteristic
148C22×C17#45112yes—
149C22×C17#45511yescharacteristic
150C22×C17#45611yescharacteristic
151C22×C17B(4,𝔽2)12yes—
152C22×C17#46212yes—
153C22×C17#49411yescharacteristic
154C22×C17#49711yescharacteristic
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—44no—
158C4×C17—82no—
159C4×C17—82no—
160Dic17—44no—
161Dic17—82no—
162C23×Q8—171no—
163C22×Q16—342no—
164C2×C22⋊Q8C2×D1712yes—
165#451—344no—
166#451—344no—
167C2×#103—342no—
168C2×C22⋊C8—172no—
169#398—22no—
170#395—22no—
171#380—22no—
172C2×#95—171no—
173C2×#95—342no—
174C2×#95C2×D1711yescharacteristic
175#328—344no—
176C2×C17—21no—
177C2×C17—22no—
178C2×C17—21no—
179C2×C17#160212yes—
180C2×C17#160512yes—
181C2×C17#200711yescharacteristic
182C2×C17#201611yescharacteristic
183C2×C17#202111yescharacteristic
184C22×Q8—172no—
185C22×Q8—341no—
186C22×Q8—342no—
187C22×Q8—342no—
188C22×Q8—684no—
189C22×Q8C22⋊D1712yes—
190C23×C4—341no—
191C23×C4—341no—
192C23×C4C22⋊D1712yes—
193C23×C4C22×D1711yescharacteristic
194C2×Q16—348no—
195C2×Q16—688no—
196C22×C8—342no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22⋊D1712yes—
205C2×C22⋊C4—22no—
206#103—348no—
207C22⋊C8—174no—
208C22⋊C8—342no—
209#95—24no—
210#95—22no—
211#95—342no—
212#95—684no—
213C17#996711yescharacteristic, complemented, Sylow
214C24C2×C22⋊D1711yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—348no—
219C2×Q8—344no—
220C2×Q8—684no—
221C2×Q8—684no—
222C2×Q8—684no—
223C2×Q8—688no—
224C2×Q8—682no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—344no—
233C22×C4—342no—
234C22×C4—342no—
235C22×C4—682no—
236C22×C4—681no—
237C22×C4—681no—
238C22×C4—682no—
239C22×C4—681no—
240C22×C4—681no—
241C22×C4—682no—
242C22×C4#6323114yescomplemented
243C22×C4D4×D1711yescharacteristic
244C22×C4D4×D1712yes—
245C22×C4#6325411yescharacteristic
246C22×C4C2×C22⋊D1712yes—
247Q16—6816no—
248C2×C8—344no—
249C2×C8—684no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#1058762411yescharacteristic
262C23#1058762611yescharacteristic
263C23#1058762812yes—
264C23C2×#6323112yes—
265C23#1058763911yescharacteristic
266Q8—44no—
267Q8—82no—
268Q8—348no—
269Q8—684no—
270Q8—688no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—344no—
286C2×C4—682no—
287C2×C4—684no—
288C2×C4—682no—
289C2×C4—684no—
290C2×C4—684no—
291C2×C4—682no—
292C2×C4—1361no—
293C2×C4—1362no—
294C2×C4—1361no—
295C2×C4#1058762312yes—
296C2×C4#1058762512yes—
297C8—684no—
298C22—22no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22#1176271711yescharacteristic
307C22#1176296512yes—
308C22#1176298711yescharacteristic
309C22#1176299211yescharacteristic
310C22#1176299612yes—
311C22#1176301612yes—
312C22#1176318811yescharacteristic
313C22#1176320111yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—82no—
318C4—82no—
319C4—684no—
320C4—1362no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2#1452774112yes—
325C2#1452775812yes—
326C2#1452983111yescharacteristic
327C2#1452988411yescharacteristic
328C2#1452990811yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#2659C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenter#2659D17
hypocenterC17#9967
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
60. C22×C4×C17Q16semidirect
213. C17#9967semidirect
242. C22×C4#63231semidirect

Maximal subgroups

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

subgroupindex
C2×#117629642
C2×#117625462
#145270542
#145266272
C17×#26592
#996717