Finite Group Databasebeta

← #17958299

Subgroups — #17958299

Order 4352.

616 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 90 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#14544021C212yesmaximal, complemented
2#14543727C212yesmaximal, complemented
3#14543712C212yesmaximal, complemented
4C4×#10587626C211yesmaximal, characteristic
5#11763777—22no—
6#11763654C2214yes—
7#11763653—22no—
8C2×#10587626C411yescharacteristic
9C2×C4×D68C2211yescharacteristic
10C2×C4×C22⋊D17—22no—
11#11762712C2211yescharacteristic
12#11762710C411yescharacteristic
13#11762709—22no—
14#11762678C2211yescharacteristic
15C42×Dic17C411yescharacteristic
16C2×C4×D4×C17C411yescharacteristic
17C22×D4×C17C2×C411yescharacteristic
18C23×C4×C17—21no—
19C22×C22⋊D17—22no—
20C22×D68C2×C411yescharacteristic
21C22×C4×D17D412yes—
22C2×#63242—42no—
23C4×D4×C17—41no—
24C4×D4×C17—41no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C22⋊C4×C17—21no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C22⋊Dic17—22no—
29#10587626—22no—
30#10587626C814yescomplemented
31C4×C22⋊D17—48no—
32C2×#63227—22no—
33C2×#63227D412yes—
34C2×C4⋊Dic17C2311yescharacteristic
35C2×#63225—22no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C4×D68—41no—
41C4×D68—41no—
42C22⋊#3624—24no—
43#10587543—42no—
44C24×C17—21no—
45C23×D17C22⋊C412yes—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17C2×C812yescomplemented
50C22×C4×C17—21no—
51C22×C4×C17—44no—
52C22×C4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22×C411yescharacteristic
57C2×C22⋊D17—28no—
58C2×C22⋊D17—44no—
59C2×C22⋊D17—44no—
60C22×Dic17—21no—
61C22×Dic17—21no—
62C22×Dic17C2×D412yes—
63C2×D68—41no—
64C2×D68—41no—
65C2×D68C2×C812yes—
66C2×D68M1612yes—
67C2×C4×D17—22no—
68C2×C4×D17—22no—
69C2×C4×D17—44no—
70C2×#3624—44no—
71#63242—84no—
72C4⋊C4×C17—41no—
73C22⋊C4×C17—42no—
74C42×C17—21no—
75C42×C17—21no—
76C22⋊Dic17—44no—
77#63227—42no—
78#63227—44no—
79C4⋊Dic17—41no—
80#63225—44no—
81C4×Dic17—24no—
82C4×Dic17—21no—
83C4×Dic17—42no—
84C4×Dic17—41no—
85C4×Dic17C2×C812yescomplemented
86#11475—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—42no—
96C22×D17—42no—
97C22×D17C22⋊C814yes—
98D4×C17—42no—
99D4×C17—42no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—41no—
105C2×C4×C17—42no—
106C2×C4×C17—41no—
107C2×C4×C17—82no—
108C2×C4×C17—82no—
109C2×C4×C17#9911yescharacteristic
110C2×C4×C17#9911yescharacteristic
111C2×C4×C17C22×C811yescharacteristic
112C2×C4×C17C2×M1611yescharacteristic
113C22⋊D17—416no—
114C2×Dic17—22no—
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—42no—
122D68—42no—
123D68—42no—
124C4×D17—44no—
125C4×D17—82no—
126#3624—84no—
127C4×#127—171no—
128#1459—172no—
129#1325—172no—
130#1318—172no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—44no—
134C22×C17—42no—
135C22×C17—82no—
136C22×C17—81no—
137C22×C17—81no—
138C22×C17C2×C22⋊C811yescharacteristic
139C22×C17C2×#9911yescharacteristic
140C22×C17#41311yescharacteristic
141C2×D17—24no—
142C2×D17—42no—
143C2×D17—42no—
144C2×D17—44no—
145C2×D17—82no—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—41no—
150C4×C17—81no—
151C4×C17#35312yes—
152Dic17—24no—
153Dic17—42no—
154Dic17—42no—
155C2×#127—171no—
156C2×C4×D4—171no—
157C2×C4×D4—342no—
158C2×C4×D4#59511yescharacteristic
159C2×C22⋊C8—342no—
160#390—171no—
161#390—171no—
162#390—342no—
163#384—171no—
164C43—171no—
165#324—174no—
166#323—342no—
167C2×C17—21no—
168C2×C17—21no—
169C2×C17—82no—
170C2×C17#145911yescharacteristic
171C2×C17C2×#35311yescharacteristic
172C2×C17#213211yescharacteristic
173D17—44no—
174C22×D4—171no—
175C22×D4—342no—
176C22×D4C2×#59511yescharacteristic
177C23×C4—21no—
178C23×C4—172no—
179C22×C8—682no—
180#127—174no—
181#127—342no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—681no—
185C4×D4—681no—
186C4×D4—688no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4—342no—
189C2×C4⋊C4C2×#59511yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—172no—
192C2×C22⋊C4—342no—
193C2×C22⋊C4—342no—
194C2×C42—171no—
195C2×C42—171no—
196C2×C42—341no—
197C2×C42—341no—
198C2×C42C2×#59511yescharacteristic
199C22⋊C8—344no—
200C22⋊C8—682no—
201C17#1147511yescharacteristic, complemented, Sylow
202C24—21no—
203C24—172no—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—172no—
208C2×D4—172no—
209C2×D4—348no—
210C2×D4—681no—
211C2×D4—681no—
212C2×D4—684no—
213C2×D4—684no—
214C2×D4#6324212yescomplemented
215C22×C4—21no—
216C22×C4—44no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—172no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4—342no—
223C22×C4—342no—
224C22×C4—684no—
225C22×C4C22⋊#59511yescharacteristic
226C22×C4C22⋊#59511yescharacteristic
227C22×C4C22×#59511yescharacteristic
228C2×C8—684no—
229C2×C8—1364no—
230C4⋊C4—41no—
231C4⋊C4—681no—
232C4⋊C4—684no—
233C22⋊C4—42no—
234C22⋊C4—682no—
235C22⋊C4—684no—
236C22⋊C4—684no—
237C42—21no—
238C42—21no—
239C42—172no—
240C42—344no—
241C42—341no—
242C42—681no—
243C42—682no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—174no—
250C23—342no—
251C23—342no—
252C23—682no—
253C23—682no—
254C23C2×C22⋊#59511yescharacteristic
255D4—42no—
256D4—42no—
257D4—682no—
258D4—682no—
259D4—6816no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4—684no—
277C2×C4—682no—
278C2×C4—1362no—
279C2×C4#1058755111yescharacteristic
280C2×C4#1058755111yescharacteristic
281C2×C4C2×#6324211yescharacteristic
282C2×C4D17⋊M1611yescharacteristic
283C8—1364no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—42no—
288C22—82no—
289C22—81no—
290C22—81no—
291C22—344no—
292C22—684no—
293C22—682no—
294C22—682no—
295C22—1362no—
296C22#1176377711yescharacteristic
297C22C2×#1058755111yescharacteristic
298C22#1176379511yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—41no—
303C4—81no—
304C4—344no—
305C4—682no—
306C4—682no—
307C4#1176371312yes—
308C2—21no—
309C2—21no—
310C2—82no—
311C2—684no—
312C2#1454401911yescharacteristic
313C2C2×#1176371311yescharacteristic
314C2#1454507511yescharacteristic
subgroupG/H
centerC2×C4#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C4×D4×C17C4
socleC22×C17C2×#99
prime radicalC2×#10587626C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×C4×D4#595
hypocenterC17#11475
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #14544021C2semidirect
2. #14543727C2semidirect
3. #14543712C2semidirect
30. #10587626C8semidirect
49. C2×D4×C17C2×C8semidirect
85. C4×Dic17C2×C8semidirect
201. C17#11475semidirect
214. C2×D4#63242semidirect

Maximal subgroups

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

subgroupindex
#145440212
#145437272
#145437122
C4×#105876262
#1147517