Finite Group Databasebeta

← #17950156

Subgroups — #17950156

Order 4352.

500 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 92 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#14544674C211yesmaximal, characteristic
2#14544510C211yesmaximal, characteristic
3#14543603C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14537134C211yesmaximal, characteristic
6C2×C22⋊#3624—21no—
7C2×C22⋊#3624—21no—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587574C411yescharacteristic
13C2×#10587564—21no—
14C22×C4×Dic17C2211yescharacteristic
15#11762713—21no—
16#11762708C2211yescharacteristic
17#11762706—21no—
18#11762705C2211yescharacteristic
19#11762678C411yescharacteristic
20#11762669C411yescharacteristic
21#11762667—21no—
22C17×#380C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17D411yescharacteristic
25C23×Dic17D411yescharacteristic
26C22×Dic34—21no—
27C22×#3624—22no—
28C22×#3624—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17—42no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587574—42no—
46#10587564—44no—
47C22⋊#3624—24no—
48C22⋊#3624—42no—
49#10587542—42no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C24×C17C22⋊C411yescharacteristic
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—41no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—22no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C2×Dic34—42no—
75C2×Dic34—42no—
76C2×Dic34—42no—
77C2×#3624—24no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—41no—
85C22⋊C4×C17—42no—
86C22⋊Dic17—42no—
87C4⋊Dic17—22no—
88C4⋊Dic17—42no—
89C4⋊Dic17—41no—
90#63225—42no—
91#63225—42no—
92#63225—42no—
93#63225—42no—
94C4×Dic17—22no—
95C4×Dic17—22no—
96C4×Dic17—41no—
97C4×Dic17—41no—
98#7705—171nomaximal, Sylow
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17#10112yes—
106C23×C17C2×C22⋊C411yescharacteristic
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—44no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17#13112yes—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—44no—
129C2×Dic17—44no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17#9912yes—
137C2×Dic17#10412yes—
138C2×Dic17AGL(1,ℤ/8)12yes—
139C2×Dic17#13712yes—
140Dic34—84no—
141#3624—44no—
142#3624—82no—
143#2857—171no—
144#1817—171no—
145#1749—171no—
146#1271—172no—
147#1271—172no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17#40811yescharacteristic
156C22×C17C2×#9911yescharacteristic
157C22×C17C2×#10111yescharacteristic
158C22×C17#41811yescharacteristic
159C22×C17#41911yescharacteristic
160C22×C17C2×#10411yescharacteristic
161C22×C17#42111yescharacteristic
162C4×C17—22no—
163C4×C17—41no—
164C4×C17—81no—
165C4×C17—81no—
166Dic17—22no—
167Dic17—22no—
168Dic17—22no—
169Dic17—22no—
170Dic17—81no—
171Dic17—81no—
172C2×C4⋊Q8—171no—
173C2×C22⋊Q8—341no—
174C22×C42—171no—
175C2×C22⋊C8—341no—
176C2×C22⋊C8—341no—
177C2×#97—341no—
178#391—341no—
179#389—171no—
180#389—171no—
181#389—341no—
182#384—171no—
183#380—171no—
184#380—341no—
185#380#59511yescharacteristic
186#336—172no—
187#336—172no—
188#336—341no—
189C2×C17—42no—
190C2×C17#147211yescharacteristic
191C2×C17#148511yescharacteristic
192C2×C17#149611yescharacteristic
193C2×C17#150711yescharacteristic
194C2×C17#151111yescharacteristic
195C2×C17#153111yescharacteristic
196C2×C17#155611yescharacteristic
197C22×Q8—341no—
198C23×C4—171no—
199C23×C4—171no—
200C23×C4C2×#59511yescharacteristic
201C22×C8—342no—
202C22×C8—342no—
203C4⋊Q8—682no—
204C22⋊Q8—684no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4C2×#59511yescharacteristic
213C2×C4⋊C4C2×#59511yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—341no—
216C2×C42—171no—
217C2×C42—171no—
218C2×C42—171no—
219C2×C42—171no—
220C2×C42—682no—
221C22⋊C8—344no—
222C22⋊C8—682no—
223#97—682no—
224#95—21no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#770511yescharacteristic, complemented, Sylow
229C24C22⋊#59511yescharacteristic
230C2×Q8—682no—
231C2×Q8—682no—
232C2×Q8—682no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—681no—
249C22×C4—681no—
250C22×C4C22⋊#59511yescharacteristic
251C22×C4C22×#59511yescharacteristic
252C2×C8—344no—
253C2×C8—682no—
254C2×C8—682no—
255C2×C8—682no—
256C2×C8—682no—
257C2×C8—682no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—342no—
261C4⋊C4—681no—
262C4⋊C4—682no—
263C4⋊C4—682no—
264C4⋊C4—682no—
265C4⋊C4—682no—
266C4⋊C4—682no—
267C22⋊C4—42no—
268C22⋊C4—682no—
269C42—342no—
270C42—342no—
271C42—681no—
272C42—681no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1058754512yes—
280C23C2×C22⋊#59511yescharacteristic
281Q8—1364no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—172no—
295C2×C4—172no—
296C2×C4—172no—
297C2×C4—172no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—684no—
307C2×C4—684no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4#1058767812yes—
315C8—684no—
316C8—1362no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22#1176378211yescharacteristic
325C22D17⋊#9911yescharacteristic
326C22C2×#1058754511yescharacteristic
327C22#1176380111yescharacteristic
328C22#1176380211yescharacteristic
329C22D17⋊#10411yescharacteristic
330C22#1176380611yescharacteristic
331C4—22no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—342no—
336C4—342no—
337C4—342no—
338C4—342no—
339C4—1361no—
340C4—1361no—
341C2—42no—
342C2#1454406311yescharacteristic
343C2#1454410611yescharacteristic
344C2#1454414311yescharacteristic
345C2#1454417911yescharacteristic
346C2#1454419311yescharacteristic
347C2#1454425611yescharacteristic
348C2#1454431311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#380C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#380#595
hypocenterC17#7705
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14543603C2semidirect
4. #14543602C2semidirect
228. C17#7705semidirect

Maximal subgroups

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

subgroupindex
#145446742
#145445102
#145436032
#145436022
#145371342
#770517