Finite Group Databasebeta

← #16932114

Subgroups — #16932114

Order 4352.

556 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 123 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#14537813C211yesmaximal, characteristic
2#14528874C212yesmaximal, complemented
3#14528772C211yesmaximal, characteristic
4#14526651C212yesmaximal, complemented
5C17×#2487C211yesmaximal, characteristic
6#11762881—24no—
7#11762866—21no—
8C2×#10587505—21no—
9#11762773—21no—
10C2×C22⋊#3622C2212yes—
11#11762669C2212yes—
12#11762576—21no—
13#11762571—24no—
14C2×C4×D4×C17C2211yescharacteristic
15C22×C4⋊C4×C17C2211yescharacteristic
16C17×#400C412yescomplemented
17C17×#399C412yescomplemented
18C17×#389—21no—
19C17×#386—21no—
20C17×#385—21no—
21C17×#380—21no—
22C17×#380C2211yescharacteristic
23C22×D4×C17D411yescharacteristic
24C23×C4×C17—21no—
25C23×C4×C17D411yescharacteristic
26C23×C4×C17C2311yescharacteristic
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17C2×C412yes—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×C22⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×#63223—22no—
44C22×#3622D414yes—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95C2×C412yes—
48#10587505—42no—
49#10587502—22no—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C4⋊#3622—22no—
54#10587474—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C2×D411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—21no—
59C2×D4×C17—22no—
60C2×D4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—22no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17—82no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D412yes—
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17#4112yes—
76C22×Dic17—42no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—24no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—82no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—82no—
88C42×C17—21no—
89C42×C17—21no—
90C22⋊Dic17—42no—
91C22⋊Dic17—42no—
92C4⋊Dic17—44no—
93#63223—82no—
94C2×#3622—24no—
95C2×#3622—44no—
96#9473—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17#12112yes—
106C23×C17AGL(1,ℤ/8)11yescharacteristic
107C23×C17#13712yes—
108C23×C17#13711yescharacteristic
109D4×C17—42no—
110D4×C17—42no—
111C2×C4×C17—22no—
112C2×C4×C17—24no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—81no—
130C2×C4×C17—84no—
131C2×C4×C17C4×D412yes—
132C2×C4×C17#12011yescharacteristic
133C2×C4×C17#12311yescharacteristic
134C2×Dic17—82no—
135C2×Dic17—82no—
136C2×Dic17—82no—
137#3622—44no—
138#2973—171no—
139#2487D1711yescharacteristic
140#1824—172no—
141#1799—171no—
142#1289—172no—
143C22×C17—22no—
144C22×C17—21no—
145C22×C17—22no—
146C22×C17—21no—
147C22×C17—41no—
148C22×C17—42no—
149C22×C17—41no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—81no—
155C22×C17#38611yescharacteristic
156C22×C17#41711yescharacteristic
157C22×C17#41911yescharacteristic
158C22×C17#46812yes—
159C22×C17#47012yes—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—82no—
166C4×C17—82no—
167C4×C17#45812yes—
168Dic17—162no—
169C2×C4×D4C2×D1711yescharacteristic
170C22×C4⋊C4C2×D1711yescharacteristic
171#435—344no—
172#432—341no—
173C2×#99—341no—
174#407—341no—
175C2×C22⋊C8—172no—
176#400Dic1712yescomplemented
177#399Dic1712yescomplemented
178#389—21no—
179#386—21no—
180#385—21no—
181#380—21no—
182#380—172no—
183#380C2×D1711yescharacteristic
184#342—341no—
185#340—344no—
186C2×C17—22no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—22no—
190C2×C17—81no—
191C2×C17#189011yescharacteristic
192C2×C17#189411yescharacteristic
193C2×C17#191011yescharacteristic
194C22×D4C22⋊D1711yescharacteristic
195C23×C4—21no—
196C23×C4C22⋊D1711yescharacteristic
197C23×C4C22×D1711yescharacteristic
198C2×M16—342no—
199C22×C8—174no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—342no—
208C2×C4⋊C4C2×Dic1712yes—
209C2×C4⋊C4C2×Dic1711yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4C2×Dic1711yescharacteristic
214C2×C22⋊C4C22⋊D1711yescharacteristic
215C2×C42C22⋊D1711yescharacteristic
216C4⋊C8—342no—
217#99—682no—
218C22⋊C8—342no—
219C22⋊C8—342no—
220C22⋊C8—342no—
221#97—342no—
222#95—22no—
223#95—22no—
224#95—342no—
225#95C2×Dic1712yes—
226C17#947311yescharacteristic, complemented, Sylow
227C24C22⋊Dic1711yescharacteristic
228C24C2×C22⋊D1711yescharacteristic
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—21no—
232C2×D4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—682no—
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C22⋊Dic1711yescharacteristic
245C22×C4D4×D1712yes—
246C22×C4#6325412yes—
247C22×C4C22×Dic1711yescharacteristic
248C22×C4C2×C22⋊D1711yescharacteristic
249M16—1362no—
250C2×C8—344no—
251C2×C8—684no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—82no—
258C4⋊C4—684no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—82no—
264C22⋊C4—682no—
265C22⋊C4—682no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23#1058762512yes—
276C23#1058762812yes—
277C23#1058763511yescharacteristic
278C23#1058763711yescharacteristic
279C23C2×C22⋊Dic1711yescharacteristic
280D4—42no—
281D4—42no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—1362no—
303C2×C4—1362no—
304C2×C4—1362no—
305C2×C4D4×Dic1712yes—
306C2×C4#1058762111yescharacteristic
307C2×C4#1058762311yescharacteristic
308C8—684no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1176268911yescharacteristic
322C22#1176281011yescharacteristic
323C22#1176281811yescharacteristic
324C22#1176305712yes—
325C22#1176306512yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—2722no—
334C4#1176299812yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#1452919811yescharacteristic
341C2#1452921811yescharacteristic
342C2#1452931211yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2487C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#2487D17
hypocenterC17#9473
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #14528874C2semidirect
4. #14526651C2semidirect
16. C17×#400C4semidirect
17. C17×#399C4semidirect
176. #400Dic17semidirect
177. #399Dic17semidirect
226. C17#9473semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145288742
#145287722
#145266512
C17×#24872
#947317