Finite Group Databasebeta

← #16929321

Subgroups — #16929321

Order 4352.

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

normal subgroups only

This group has 800 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×#11762729C211yesmaximal, characteristic
2#14531402C211yesmaximal, characteristic
3#14528720C212yesmaximal
4#14526664C212yesmaximal
5C2×C17×#324C211yesmaximal, characteristic
6C23×Dic34C2211yescharacteristic
7C2×#10587612—21no—
8C22×#63225—21no—
9C2×#10587487—24no—
10#11762729—22no—
11#11762729C414yescomplemented
12#11762708—24no—
13#11762669—21no—
14#11762669C2212yes—
15C2×#10587502—21no—
16C2×#10587502C2211yescharacteristic
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×Dic34—22no—
26C22×Dic34—22no—
27C22×Dic34—21no—
28C22×Dic34—44no—
29C22×Dic34C2×C412yes—
30C22×C8×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17D412yes—
35C2×C22⋊Dic17—22no—
36#10587612—42no—
37#10587612—42no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17—22no—
40C2×C4⋊Dic17D414yes—
41C2×#63225—24no—
42C2×#63225—44no—
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—
51#10587487—48no—
52C24×C17C2×D411yescharacteristic
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22⋊C412yes—
63C22×C4×C17SD1612yes—
64C22×C4×C17Q1612yes—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D412yes—
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17#4112yes—
69C22×Dic17—24no—
70C22×Dic17—24no—
71C22×Dic17—22no—
72C22×Dic17—22no—
73C22×Dic17—24no—
74C22×Dic17—42no—
75C22×Dic17—42no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C2×Dic34—28no—
81C2×Dic34—24no—
82C2×Dic34—44no—
83C2×Dic34—48no—
84C2×Dic34—44no—
85C2×Dic34—44no—
86C2×Dic34—42no—
87C2×C8×C17—44no—
88C2×C8×C17—42no—
89C4⋊C4×C17—24no—
90C4⋊C4×C17—24no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C4⋊Dic17—44no—
94C4⋊Dic17—44no—
95C4⋊Dic17—82no—
96#63225—48no—
97C4×Dic17—44no—
98#9015—171nomaximal, Sylow
99C23×C17—22no—
100C23×C17—22no—
101C23×C17C2×C22⋊C411yescharacteristic
102C23×C17#12011yescharacteristic
103C23×C17#12311yescharacteristic
104C23×C17C2×SD1611yescharacteristic
105C23×C17C2×Q1611yescharacteristic
106C23×C17AGL(1,ℤ/8)11yescharacteristic
107C23×C17#13711yescharacteristic
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—24no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—81no—
122C2×C4×C17#10314yescomplemented
123C2×C4×C17C4×D412yes—
124C2×C4×C17#12112yes—
125C2×Dic17—24no—
126C2×Dic17—24no—
127C2×Dic17—42no—
128C2×Dic17—44no—
129C2×Dic17—44no—
130C2×Dic17—44no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—44no—
135C2×Dic17—44no—
136C2×Dic17—44no—
137C2×Dic17—44no—
138Dic34—28no—
139Dic34—48no—
140Dic34—44no—
141C8×C17—82no—
142C2×#395—171no—
143#2316—171no—
144#1787—172no—
145C2×#324D1711yescharacteristic
146#1294—172no—
147C22×C17—22no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17#38611yescharacteristic
156C22×C17C2×#10311yescharacteristic
157C22×C17#41711yescharacteristic
158C22×C17#45011yescharacteristic
159C22×C17#45111yescharacteristic
160C22×C17#45612yes—
161C22×C17B(4,𝔽2)12yes—
162C22×C17#48311yescharacteristic
163C22×C17#48411yescharacteristic
164C4×C17—24no—
165C4×C17—22no—
166C4×C17—22no—
167C4×C17—42no—
168Dic17—44no—
169Dic17—44no—
170Dic17—84no—
171C23×Q8—171no—
172C2×#117—341no—
173C22×C4⋊C4—341no—
174C22×C4⋊C4C2×D1711yescharacteristic
175C2×#103—344no—
176C2×C22⋊C8C2×D1712yes—
177#395—174no—
178#395—342no—
179#389—344no—
180#380—172no—
181#380—341no—
182C2×#95—171no—
183C2×#95—341no—
184#324—22no—
185C2×C17—21no—
186C2×C17—21no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17#160412yes—
190C2×C17#160512yes—
191C2×C17#187211yescharacteristic
192C2×C17#188611yescharacteristic
193C2×C17#188911yescharacteristic
194C22×Q8—172no—
195C22×Q8—341no—
196C22×Q8—342no—
197C22×Q8—342no—
198C22×Q8—684no—
199C23×C4—341no—
200C23×C4—341no—
201C23×C4D6811yescharacteristic
202C23×C4D6811yescharacteristic
203C23×C4C22×D1711yescharacteristic
204C22×C8—22no—
205#117—682no—
206#117—682no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—174no—
210C2×C4⋊C4—342no—
211C2×C4⋊C4—344no—
212C2×C4⋊C4—342no—
213C2×C4⋊C4—684no—
214C2×C4⋊C4C4×D1712yes—
215C2×C4⋊C4C22⋊D1712yes—
216C2×C22⋊C4—342no—
217C2×C42—342no—
218#103—688no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221#95—172no—
222#95—342no—
223#95—342no—
224#95—341no—
225#95—344no—
226C17#901511yescharacteristic, complemented, Sylow
227C24C2×D6811yescharacteristic
228C2×Q8—348no—
229C2×Q8—344no—
230C2×Q8—684no—
231C2×Q8—684no—
232C2×Q8—684no—
233C2×Q8—688no—
234C2×Q8—682no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—344no—
244C22×C4—344no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—344no—
248C22×C4—682no—
249C22×C4—682no—
250C22×C4—681no—
251C22×C4—681no—
252C22×C4—681no—
253C22×C4—681no—
254C22×C4#6321912yes—
255C22×C4Dic6812yes—
256C22×C4#6322712yes—
257C22×C4#6322712yes—
258C22×C4C2×C4×D1711yescharacteristic
259C22×C4D4×D1712yes—
260C22×C4#6325412yes—
261C22×C4C2×C22⋊D1711yescharacteristic
262C2×C8—44no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—684no—
267C4⋊C4—684no—
268C4⋊C4—688no—
269C4⋊C4—1362no—
270C22⋊C4—682no—
271C22⋊C4—682no—
272C42—684no—
273C23—22no—
274C23—22no—
275C23#1058756811yescharacteristic
276C23#1058757211yescharacteristic
277C23C2×#6321911yescharacteristic
278C23C2×Dic6811yescharacteristic
279C23#1058759411yescharacteristic
280C23#1058759511yescharacteristic
281C23C2×#6322711yescharacteristic
282Q8—348no—
283Q8—688no—
284Q8—684no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—344no—
300C2×C4—344no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4—684no—
304C2×C4—684no—
305C2×C4—684no—
306C2×C4—684no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4—684no—
310C2×C4—684no—
311C2×C4—684no—
312C2×C4#1058748714yescomplemented
313C2×C4#1058756712yes—
314C2×C4#1058762512yes—
315C8—82no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22#1176268611yescharacteristic
325C22C2×#1058748711yescharacteristic
326C22#1176281111yescharacteristic
327C22#1176296211yescharacteristic
328C22#1176296711yescharacteristic
329C22#1176299412yes—
330C22#1176299712yes—
331C22#1176314011yescharacteristic
332C22#1176314511yescharacteristic
333C4—24no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—684no—
338C4—684no—
339C4—1364no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#1452775512yes—
345C2#1452776012yes—
346C2#1452908911yescharacteristic
347C2#1452917511yescharacteristic
348C2#1452919011yescharacteristic
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#9015
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
11. #11762729C4semidirect
122. C2×C4×C17#103semidirect
226. C17#9015semidirect
312. C2×C4#10587487semidirect

Maximal subgroups

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

subgroupindex
C2×#117627292
#145314022
#145287202
#145266642
C2×C17×#3242
#901517