Finite Group Databasebeta

← #17955973

Subgroups — #17955973

Order 4352.

640 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 126 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
3C2×#11763654C214yesmaximal
4C2×#11762708C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×C22⋊#3624C2212yes—
7C2×C22⋊#3624C2212yes—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675—21no—
11#11763654—28no—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C4⋊Dic17C2212yes—
14C22×C4×Dic17C2211yescharacteristic
15#11762708—21no—
16#11762708—21no—
17#11762708—21no—
18#11762708—21no—
19C2×#10587502C412yes—
20C22×C4⋊C4×C17C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17C2×C412yes—
23C23×Dic17C2×C411yescharacteristic
24C23×Dic17C2×C412yes—
25C23×Dic17C2311yescharacteristic
26C22×#3624—22no—
27C22×#3624—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—24no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—24no—
36C2×C4⋊Dic17—24no—
37C2×C4⋊Dic17—42no—
38C2×C4×Dic17—42no—
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C22⋊#3624—22no—
44C22⋊#3624—24no—
45C22⋊#3624D414yes—
46#10587544—42no—
47#10587542—42no—
48#10587502—22no—
49#10587502—22no—
50C24×C17C22×C411yescharacteristic
51C22×C4×C17—22no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—24no—
56C22×C4×C17—22no—
57C22×C4×C17—24no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17M1614yes—
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—24no—
65C22×Dic17—41no—
66C22×Dic17—41no—
67C22×Dic17—41no—
68C22×Dic17—42no—
69C22×Dic17—42no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D412yes—
75C22×Dic17#4112yes—
76C2×#3624—24no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C4⋊C4×C17—24no—
83C4⋊C4×C17—24no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86C4⋊Dic17—44no—
87C4⋊Dic17—42no—
88C4⋊Dic17—42no—
89C4×Dic17—22no—
90C4×Dic17—22no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93#9654—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17C4×D412yes—
100C23×C17C2×M1612yes—
101C23×C17#13112yes—
102C2×C4×C17—28no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—24no—
108C2×C4×C17—28no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—44no—
112C2×C4×C17#9912yes—
113C2×C4×C17#10112yes—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—44no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—84no—
130C2×Dic17#10112yes—
131C2×Dic17#12112yes—
132C2×Dic17#12411yescharacteristic
133C2×Dic17#12711yescharacteristic
134#3624—44no—
135#3624—82no—
136C2×#389—171no—
137#1797—171no—
138#1749—171no—
139C2×#324—174no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17#39011yescharacteristic
147C22×C17#40711yescharacteristic
148C22×C17#40811yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17#41011yescharacteristic
151C22×C17C2×#10111yescharacteristic
152C22×C17C2×#10111yescharacteristic
153C22×C17#43514yes—
154C4×C17—22no—
155C4×C17—22no—
156C4×C17—44no—
157Dic17—22no—
158Dic17—21no—
159Dic17—21no—
160Dic17—22no—
161Dic17—41no—
162Dic17—82no—
163C22×C4⋊C4—172no—
164C22×C4⋊C4—171no—
165C22×C4⋊C4#59511yescharacteristic
166C22×C42—171no—
167C2×C4⋊C8—341no—
168C2×C22⋊C8—172no—
169C2×C22⋊C8—172no—
170C2×#97—341no—
171#389—341no—
172#389—341no—
173#389—341no—
174#389—341no—
175C2×#95—172no—
176#336—341no—
177#336—341no—
178#324—348no—
179C2×C17—24no—
180C2×C17#146311yescharacteristic
181C2×C17#146711yescharacteristic
182C2×C17#147011yescharacteristic
183C2×C17#147211yescharacteristic
184C2×C17#195511yescharacteristic
185C2×C17#196411yescharacteristic
186C2×C17#196611yescharacteristic
187C23×C4—171no—
188C23×C4—171no—
189C23×C4—172no—
190C23×C4C2×#59512yes—
191C23×C4C2×#59511yescharacteristic
192C22×C8—342no—
193C22×C8—342no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—344no—
198C2×C4⋊C4—342no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—344no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—682no—
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—
216C17#965411yescharacteristic, complemented, Sylow
217C24C22×#59511yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—172no—
229C22×C4—172no—
230C22×C4—171no—
231C22×C4—342no—
232C22×C4—342no—
233C22×C4—342no—
234C22×C4—344no—
235C22×C4—681no—
236C22×C4—681no—
237C22×C4—681no—
238C22×C4—682no—
239C22×C4—682no—
240C22×C4#6324714yes—
241C22×C4C22⋊#59511yescharacteristic
242C22×C4C22⋊#59511yescharacteristic
243C2×C8—344no—
244C2×C8—682no—
245C2×C8—682no—
246C2×C8—682no—
247C2×C8—682no—
248C2×C8—682no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—682no—
252C4⋊C4—684no—
253C4⋊C4—682no—
254C4⋊C4—682no—
255C4⋊C4—682no—
256C42—342no—
257C42—342no—
258C42—681no—
259C42—681no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#1058767812yes—
265C23D4×#59512yes—
266C23C2×#6324712yes—
267C23C2×C22⋊#59511yescharacteristic
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—28no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—172no—
279C2×C4—171no—
280C2×C4—171no—
281C2×C4—172no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—341no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—684no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—1364no—
298C2×C4#1058753912yes—
299C2×C4#1058754512yes—
300C8—684no—
301C8—1362no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22#1176376111yescharacteristic
309C22#1176377911yescharacteristic
310C22#1176378311yescharacteristic
311C22C2×#1058753911yescharacteristic
312C22#1176379011yescharacteristic
313C22C2×#1058754511yescharacteristic
314C22D17⋊#10111yescharacteristic
315C22#1176384014yes—
316C4—22no—
317C4—22no—
318C4—44no—
319C4—342no—
320C4—341no—
321C4—341no—
322C4—342no—
323C4—681no—
324C4—1362no—
325C2—24no—
326C2#1454403211yescharacteristic
327C2#1454404711yescharacteristic
328C2#1454405711yescharacteristic
329C2#1454406211yescharacteristic
330C2#1454490811yescharacteristic
331C2#1454492411yescharacteristic
332C2#1454492711yescharacteristic
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#9654
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
216. C17#9654semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145445102
C2×#117636542
C2×#117627082
#965417