Finite Group Databasebeta

← #16929381

Subgroups — #16929381

Order 4352.

660 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 143 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×#11762739C211yesmaximal, characteristic
2#14531402C211yesmaximal, characteristic
3#14528871C212yesmaximal
4#14526664C212yesmaximal
5C2×C17×#324C211yesmaximal, characteristic
6C2×#10587612—21no—
7C22×C4⋊Dic17C2211yescharacteristic
8C22×#63225—21no—
9#11762739—22no—
10#11762739C414yescomplemented
11#11762708—24no—
12#11762669—21no—
13#11762669C2212yes—
14C2×#10587502—21no—
15C2×#10587502C2211yescharacteristic
16#11762572—24no—
17C22×C4⋊C4×C17C2211yescharacteristic
18C2×C17×C22⋊C8C2212yes—
19C17×#324—22no—
20C23×C4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
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—
31#10587612—42no—
32#10587612—42no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17—22no—
38C2×C4⋊Dic17C2×C412yes—
39C2×C4⋊Dic17D414yes—
40C2×#63225—24no—
41C2×#63225—42no—
42C2×#63225—42no—
43C2×C4×Dic17—22no—
44C17×C22⋊C8—22no—
45C17×C22⋊C8—22no—
46#10587502—22no—
47#10587502—24no—
48#10587502—22no—
49#10587502—21no—
50#10587502C2×C412yes—
51C24×C17C2×D411yescharacteristic
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—42no—
59C22×C4×C17—42no—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17D812yes—
63C22×C4×C17SD1612yes—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4112yes—
68C22×Dic17—24no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—24no—
72C22×Dic17—42no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C2×C8×C17—44no—
82C2×C8×C17—42no—
83C4⋊C4×C17—24no—
84C4⋊C4×C17—24no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87C4⋊Dic17—24no—
88C4⋊Dic17—44no—
89C4⋊Dic17—42no—
90C4⋊Dic17—44no—
91C4⋊Dic17—82no—
92#63225—48no—
93C4×Dic17—44no—
94#9027—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17—22no—
97C23×C17C2×C22⋊C411yescharacteristic
98C23×C17#12011yescharacteristic
99C23×C17#12311yescharacteristic
100C23×C17C2×D811yescharacteristic
101C23×C17C2×SD1611yescharacteristic
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—24no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—81no—
118C2×C4×C17#10214yes—
119C2×C4×C17C4×D412yes—
120C2×C4×C17#12112yes—
121C2×Dic17—24no—
122C2×Dic17—44no—
123C2×Dic17—44no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—44no—
130C2×Dic17—44no—
131C2×Dic17—44no—
132C2×Dic17—81no—
133C2×Dic17—81no—
134C2×Dic17—81no—
135C2×Dic17—81no—
136C8×C17—82no—
137C2×#398—171no—
138#2316—171no—
139#1823—172no—
140C2×#324D1711yescharacteristic
141#1294—172no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17#38611yescharacteristic
151C22×C17C2×#10211yescharacteristic
152C22×C17#41711yescharacteristic
153C22×C17#44711yescharacteristic
154C22×C17#44811yescharacteristic
155C22×C17#45612yes—
156C22×C17#45812yes—
157C22×C17#48011yescharacteristic
158C22×C17#48111yescharacteristic
159C4×C17—24no—
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—42no—
163Dic17—44no—
164Dic17—84no—
165Dic17—82no—
166C2×#117—341no—
167C22×C4⋊C4—171no—
168C22×C4⋊C4—341no—
169C22×C4⋊C4C2×D1711yescharacteristic
170C2×C22⋊C8C2×D1712yes—
171#398—174no—
172#398—342no—
173#389—344no—
174#380—172no—
175#380—341no—
176C2×#95—171no—
177C2×#95—341no—
178#340—344no—
179#324—22no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17#161012yes—
185C2×C17#161112yes—
186C2×C17#187411yescharacteristic
187C2×C17#188611yescharacteristic
188C2×C17#189011yescharacteristic
189C23×C4—341no—
190C23×C4—341no—
191C23×C4D6811yescharacteristic
192C23×C4D6811yescharacteristic
193C23×C4C22×D1711yescharacteristic
194C22×C8—22no—
195#117—682no—
196#117—682no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—172no—
200C2×C4⋊C4—174no—
201C2×C4⋊C4—342no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4—342no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—344no—
206C2×C4⋊C4—342no—
207C2×C4⋊C4—682no—
208C2×C4⋊C4—682no—
209C2×C4⋊C4C4×D1712yes—
210C2×C4⋊C4C22⋊D1712yes—
211C2×C22⋊C4—342no—
212C2×C42—342no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—172no—
216#95—342no—
217#95—342no—
218#95—341no—
219#95—344no—
220C17#902711yescharacteristic, complemented, Sylow
221C24C2×D6811yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—344no—
231C22×C4—342no—
232C22×C4—344no—
233C22×C4—342no—
234C22×C4—682no—
235C22×C4—681no—
236C22×C4—681no—
237C22×C4—681no—
238C22×C4—681no—
239C22×C4—681no—
240C22×C4—681no—
241C22×C4—681no—
242C22×C4—681no—
243C22×C4#6321912yes—
244C22×C4D13612yes—
245C22×C4#6322712yes—
246C22×C4#6322712yes—
247C22×C4C2×C4×D1711yescharacteristic
248C22×C4D4×D1712yes—
249C22×C4#6325412yes—
250C22×C4C2×C22⋊D1711yescharacteristic
251C2×C8—44no—
252C2×C8—42no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—344no—
256C4⋊C4—684no—
257C4⋊C4—684no—
258C4⋊C4—688no—
259C4⋊C4—682no—
260C4⋊C4—1362no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C42—684no—
264C23—22no—
265C23—22no—
266C23#1058756811yescharacteristic
267C23#1058757211yescharacteristic
268C23C2×#6321911yescharacteristic
269C23C2×D13611yescharacteristic
270C23#1058759411yescharacteristic
271C23#1058759511yescharacteristic
272C23C2×#6322711yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—81no—
287C2×C4—344no—
288C2×C4—684no—
289C2×C4—684no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—684no—
293C2×C4—684no—
294C2×C4—684no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—1361no—
299C2×C4—1361no—
300C2×C4—1361no—
301C2×C4—1361no—
302C2×C4#1058749214yes—
303C2×C4#1058756712yes—
304C2×C4#1058762512yes—
305C8—82no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22#1176268611yescharacteristic
315C22C2×#1058749211yescharacteristic
316C22#1176281111yescharacteristic
317C22#1176294511yescharacteristic
318C22#1176295011yescharacteristic
319C22#1176299412yes—
320C22#1176300012yes—
321C22#1176312311yescharacteristic
322C22#1176312811yescharacteristic
323C4—24no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—684no—
328C4—1364no—
329C4—1362no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#1452778912yes—
335C2#1452779412yes—
336C2#1452909911yescharacteristic
337C2#1452917511yescharacteristic
338C2#1452919511yescharacteristic
subgroupG/H
centerC23#10587568
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#324C2
socleC23×C17#120
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenterC2×#324D17
hypocenterC17#9027
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
10. #11762739C4semidirect
220. C17#9027semidirect

Maximal subgroups

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

subgroupindex
C2×#117627392
#145314022
#145288712
#145266642
C2×C17×#3242
#902717