Finite Group Databasebeta

← #16926083

Subgroups — #16926083

Order 4352.

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

normal subgroups only

This group has 738 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
1C2×#11763532C211yesmaximal, characteristic
2#14527466C212yesmaximal, complemented
3#14527208C212yesmaximal
4#14527108C211yesmaximal, characteristic, complemented
5C17×#1801C211yesmaximal, characteristic, complemented
6#11763532—24no—
7#11763532—24no—
8C2×#10587555C2211yescharacteristic
9C2×#10587584—22no—
10C2×#10587561C412yes—
11C2×#10587632—21no—
12C2×C4×Dic34C412yes—
13C2×C4×D68C2211yescharacteristic
14C2×#10587578—21no—
15#11762862C2212yes—
16#11762557—24no—
17#11762536—24no—
18#11762535—22no—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#423C2212yes—
21C2×C17×#101—21no—
22C17×#343—21no—
23C22×Q8×C17D411yescharacteristic
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C22×Dic34C2×C412yes—
27C2×M16×C17—22no—
28C4×Q8×C17—22no—
29C4×Q8×C17—22no—
30C2×C4⋊C4×C17D412yes—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C42×C17C2×C412yes—
33C2×C42×C17C2311yescharacteristic
34#10587632—28no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—22no—
42C2×C4×Dic17—22no—
43#10587584—216no—
44#10587578—28no—
45#10587561—24no—
46#10587561—24no—
47C4×D68—22no—
48C4×D68D414yes—
49#10587555—22no—
50#10587555D414yes—
51C4×Dic34—24no—
52C4×Dic34—24no—
53C17×C4⋊C8—22no—
54C17×C4⋊C8—22no—
55C17×#101—42no—
56C22⋊#3622—22no—
57C4⋊#3622—22no—
58C4×#3622—22no—
59C23×D17—21no—
60C2×Q8×C17—22no—
61C2×Q8×C17—22no—
62C2×Q8×C17—21no—
63C2×Q8×C17—21no—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22⋊C412yes—
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—22no—
70C22×Dic17—21no—
71C2×D68—21no—
72C2×D68—42no—
73C2×D68C22⋊C412yes—
74C2×C4×D17—24no—
75C2×C4×D17—42no—
76C2×C4×D17—41no—
77C2×C4×D17—41no—
78C2×Dic34—22no—
79C2×Dic34—44no—
80C2×Dic34C22⋊C414yes—
81M16×C17—42no—
82M16×C17—42no—
83C2×C8×C17—42no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—24no—
86C42×C17—22no—
87C42×C17—22no—
88C42×C17C2×D412yes—
89C42×C17C2×D412yes—
90#63227—28no—
91#63227—28no—
92#63227—24no—
93C4⋊Dic17—21no—
94C4⋊Dic17—24no—
95C4⋊Dic17—24no—
96C4⋊Dic17C22⋊C412yes—
97#63225—28no—
98C4×Dic17—28no—
99C2×#3622—42no—
100#8394—171nomaximal, Sylow
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—22no—
103C22×D17—42no—
104C22×D17—42no—
105C22×D17—41no—
106C22×D17—41no—
107Q8×C17—42no—
108Q8×C17—42no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17C2×C22⋊C411yescharacteristic
117C2×C4×C17C2×C22⋊C411yescharacteristic
118C2×C4×C17#12012yes—
119C2×C4×C17#12012yes—
120C2×C4×C17#13512yes—
121C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C17#13711yescharacteristic
123C2×Dic17—24no—
124C2×Dic17—22no—
125C2×Dic17—44no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128D68—24no—
129D68—42no—
130C4×D17—44no—
131C4×D17—44no—
132Dic34—44no—
133Dic34—44no—
134C8×C17—82no—
135#3622—82no—
136C2×#556—171no—
137#1801D1711yescharacteristic, complemented
138#1548—172no—
139#1486—172no—
140#1461—171no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17#37911yescharacteristic
144C22×C17#41611yescharacteristic
145C22×C17#41711yescharacteristic
146C2×D17—42no—
147C2×D17—42no—
148C2×D17—42no—
149C2×D17—82no—
150C2×D17—82no—
151C4×C17—22no—
152C4×C17—21no—
153C4×C17—42no—
154C4×C17—42no—
155C4×C17#41312yes—
156C4×C17#44912yes—
157C4×C17#45212yes—
158Dic17—44no—
159Dic17—42no—
160#556—344no—
161#556—344no—
162C2×C4⋊Q8—171no—
163C2×#126—342no—
164C2×#124—172no—
165C2×C22⋊Q8—341no—
166C2×C4×Q8—172no—
167C2×C4×Q8C2×D1711yescharacteristic
168C2×C4×D4—171no—
169C2×#117—341no—
170#431—172no—
171#423C2×D1712yes—
172C2×#101—21no—
173#343—21no—
174#333—344no—
175D4⋊C8—344no—
176#324—342no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17#178511yescharacteristic
180C2×C17#178811yescharacteristic
181C2×C17#179611yescharacteristic
182D17—82no—
183C22×Q8—172no—
184C22×Q8C22⋊D1711yescharacteristic
185C22×D4—171no—
186C23×C4—341no—
187C2×M16—22no—
188C4⋊Q8—174no—
189C4⋊Q8—342no—
190#126—3416no—
191#124—344no—
192#124—344no—
193C22⋊Q8—348no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—344no—
197C4×Q8—344no—
198C4×D4—174no—
199C4×D4—342no—
200#117—348no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—342no—
205C2×C4⋊C4D6812yes—
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C22⋊C4—341no—
208C2×C22⋊C4—342no—
209C2×C22⋊C4—342no—
210C2×C42—342no—
211C2×C42C4×D1712yes—
212C2×C42C22×D1711yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—342no—
216#101—42no—
217C22⋊C8—342no—
218C4×C8—342no—
219C17#839411yescharacteristic, complemented, Sylow
220C24—341no—
221C2×Q8—21no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—174no—
226C2×Q8—342no—
227C2×Q8—684no—
228C2×D4—172no—
229C2×D4—341no—
230C2×D4—682no—
231C22×C4—341no—
232C22×C4—342no—
233C22×C4—344no—
234C22×C4—681no—
235C22×C4—681no—
236C22×C4—682no—
237C22×C4#6322712yes—
238C22×C4#6322712yes—
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×D6811yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—682no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—172no—
249C4⋊C4—348no—
250C4⋊C4—341no—
251C4⋊C4—344no—
252C4⋊C4—344no—
253C22⋊C4—348no—
254C22⋊C4—348no—
255C22⋊C4—344no—
256C42—22no—
257C42—22no—
258C42—348no—
259C42D4×D1712yes—
260C42D4×D1712yes—
261C23—342no—
262C23—682no—
263C23—682no—
264C23—681no—
265C23—681no—
266C23C2×#6322711yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—684no—
270Q8—684no—
271D4—344no—
272D4—682no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—342no—
281C2×C4—344no—
282C2×C4—684no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—684no—
286C2×C4—684no—
287C2×C4C22⋊C4×D1711yescharacteristic
288C2×C4C22⋊C4×D1711yescharacteristic
289C2×C4#1058756812yes—
290C2×C4#1058761811yescharacteristic
291C2×C4#1058762312yes—
292C2×C4#1058762811yescharacteristic
293C2×C4#1058763612yes—
294C8—82no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—682no—
299C22—682no—
300C22—682no—
301C22—1362no—
302C22—1362no—
303C22#1176266411yescharacteristic
304C22#1176280311yescharacteristic
305C22#1176280811yescharacteristic
306C4—21no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—684no—
311C4—682no—
312C4#1176279212yes—
313C4#1176295212yes—
314C4#1176296912yes—
315C2—21no—
316C2—21no—
317C2—1362no—
318C2#1452871111yescharacteristic
319C2#1452872211yescharacteristic
320C2#1452876011yescharacteristic
subgroupG/H
centerC22#11762664
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1801C2
socleC22×C17#379
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#1801D17
hypocenterC17#8394
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #14527466C2semidirect
4. #14527108C2semidirect
5. C17×#1801C2semidirect
137. #1801D17semidirect
219. C17#8394semidirect

Maximal subgroups

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

subgroupindex
C2×#117635322
#145274662
#145272082
#145271082
C17×#18012
#839417