Finite Group Databasebeta

← #17956490

Subgroups — #17956490

Order 4352.

664 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 122 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#14544640C211yesmaximal, characteristic
2#14544510C211yesmaximal, characteristic
3#14543622C214yesmaximal
4C2×#11762714C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×C22⋊#3624C2212yes—
7C2×C22⋊#3624C2212yes—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675—21no—
11C23×Dic34C411yescharacteristic
12C22×C4×Dic17C2211yescharacteristic
13#11762714—21no—
14#11762714—21no—
15#11762714—21no—
16#11762714—21no—
17C2×#10587502C412yes—
18C2×#10587502C2212yes—
19C22×C4⋊C4×C17C411yescharacteristic
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C412yes—
22C23×Dic17C2×C411yescharacteristic
23C23×Dic17C2×C412yes—
24C23×Dic17C2311yescharacteristic
25C22×Dic34—21no—
26C22×Dic34—21no—
27C22×Dic34—24no—
28C22×Dic34—42no—
29C22×Dic34—42no—
30C22×#3624—22no—
31C22×#3624—22no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—42no—
35C2×C4×Dic17—42no—
36C2×C4×Dic17D411yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C22⋊#3624—22no—
41C22⋊#3624—24no—
42C22⋊#3624Q814yescomplemented
43#10587544—42no—
44#10587542—42no—
45#10587502—22no—
46#10587502—22no—
47#10587502—22no—
48#10587502—22no—
49C24×C17C22×C411yescharacteristic
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—24no—
55C22×C4×C17—42no—
56C22×C4×C17—42no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—41no—
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17—42no—
67C22×Dic17—42no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17M1614yes—
71C22×Dic17C2×D411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×Q812yes—
74C22×Dic17#4112yes—
75C2×Dic34—24no—
76C2×Dic34—24no—
77C2×Dic34—416no—
78C2×Dic34—42no—
79C2×Dic34—42no—
80C2×Dic34—42no—
81C2×Dic34—42no—
82C2×#3624—24no—
83C2×#3624—42no—
84C2×#3624—42no—
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—42no—
88C4⋊C4×C17—42no—
89C4⋊C4×C17—42no—
90C4×Dic17—22no—
91C4×Dic17—22no—
92C4×Dic17—41no—
93C4×Dic17—41no—
94#9882—171nomaximal, Sylow
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17C2×C22⋊C411yescharacteristic
100C23×C17C4×Q812yes—
101C23×C17C2×M1612yes—
102C23×C17#13112yes—
103C2×C4×C17—28no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—84no—
112C2×C4×C17#9912yes—
113C2×C4×C17#10012yes—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—24no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—44no—
127C2×Dic17—44no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—44no—
131C2×Dic17#10112yes—
132C2×Dic17C22⋊Q812yes—
133C2×Dic17#12411yescharacteristic
134C2×Dic17C4⋊Q811yescharacteristic
135Dic34—48no—
136Dic34—48no—
137#3624—44no—
138#3624—82no—
139C2×#391—171no—
140#1797—171no—
141#1749—171no—
142#1280—174no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—22no—
148C22×C17—22no—
149C22×C17#39111yescharacteristic
150C22×C17#40711yescharacteristic
151C22×C17#40811yescharacteristic
152C22×C17C2×#9911yescharacteristic
153C22×C17#41011yescharacteristic
154C22×C17C2×#10011yescharacteristic
155C22×C17C2×#10111yescharacteristic
156C22×C17#44614yes—
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—82no—
160Dic17—22no—
161Dic17—22no—
162Dic17—21no—
163Dic17—21no—
164Dic17—44no—
165Dic17—41no—
166C23×Q8—171no—
167C22×C4⋊C4#59511yescharacteristic
168C22×C42—171no—
169C2×C4⋊C8—341no—
170C2×C22⋊C8—172no—
171C2×C22⋊C8—172no—
172C2×#97—341no—
173#391—341no—
174#391—341no—
175#391—341no—
176#391—341no—
177C2×#95—172no—
178C2×#95—172no—
179#336—341no—
180#336—341no—
181C2×C17—24no—
182C2×C17#146211yescharacteristic
183C2×C17#146711yescharacteristic
184C2×C17#146911yescharacteristic
185C2×C17#147211yescharacteristic
186C2×C17#198511yescharacteristic
187C2×C17#199611yescharacteristic
188C2×C17#199711yescharacteristic
189C22×Q8—344no—
190C22×Q8—341no—
191C22×Q8—341no—
192C22×Q8—682no—
193C22×Q8—682no—
194C23×C4—171no—
195C23×C4—171no—
196C23×C4—172no—
197C23×C4C2×#59512yes—
198C23×C4C2×#59511yescharacteristic
199C22×C8—342no—
200C22×C8—342no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42—171no—
207C2×C42—171no—
208C2×C42—682no—
209C4⋊C8—682no—
210C22⋊C8—174no—
211C22⋊C8—342no—
212C22⋊C8—344no—
213#97—682no—
214#95—342no—
215#95—342no—
216#95—342no—
217#95—342no—
218C17#988211yescharacteristic, complemented, Sylow
219C24C22×#59511yescharacteristic
220C2×Q8—344no—
221C2×Q8—344no—
222C2×Q8—6816no—
223C2×Q8—682no—
224C2×Q8—682no—
225C2×Q8—682no—
226C2×Q8—682no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—172no—
238C22×C4—172no—
239C22×C4—171no—
240C22×C4—174no—
241C22×C4—342no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—681no—
246C22×C4—681no—
247C22×C4—681no—
248C22×C4—682no—
249C22×C4—682no—
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—42no—
259C4⋊C4—42no—
260C42—342no—
261C42—342no—
262C42—681no—
263C42—681no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23D17⋊M1612yes—
269C23#1058767812yes—
270C23Q8×#59512yes—
271C23C2×C22⋊#59511yescharacteristic
272Q8—688no—
273Q8—688no—
274C2×C4—28no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—172no—
284C2×C4—171no—
285C2×C4—171no—
286C2×C4—172no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—344no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—684no—
300C2×C4—684no—
301C2×C4—682no—
302C2×C4—682no—
303C2×C4—684no—
304C2×C4#1058753912yes—
305C2×C4#1058755312yes—
306C8—684no—
307C8—1362no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22(C2×#595)⋊Q811yescharacteristic
315C22#1176378111yescharacteristic
316C22#1176378311yescharacteristic
317C22C2×#1058753911yescharacteristic
318C22#1176378911yescharacteristic
319C22C2×#1058755311yescharacteristic
320C22D17⋊#10111yescharacteristic
321C22#1176386114yes—
322C4—22no—
323C4—22no—
324C4—82no—
325C4—342no—
326C4—341no—
327C4—341no—
328C4—342no—
329C4—684no—
330C4—681no—
331C2—24no—
332C2#1454403111yescharacteristic
333C2#1454404811yescharacteristic
334C2#1454405411yescharacteristic
335C2#1454406211yescharacteristic
336C2#1454495611yescharacteristic
337C2#1454497411yescharacteristic
338C2#1454497611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutator, prime radicalC24×C17C22×C4
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#9882
LCSG ▷ C24×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
42. C22⋊#3624Q8semidirect
218. C17#9882semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145445102
#145436222
C2×#117627142
#988217