Finite Group Databasebeta

← #17958153

Subgroups — #17958153

Order 4352.

594 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 104 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×#11763654C212yesmaximal, complemented
2#14543715C212yesmaximal, complemented
3#14543713C212yesmaximal, complemented
4#14531372C211yesmaximal, characteristic
5C2×C22⋊#3624—22no—
6#11763654—22no—
7#11763654C2214yes—
8C22×#63252C411yescharacteristic
9C22×C4⋊Dic17C2211yescharacteristic
10#11762715C411yescharacteristic
11#11762715C2211yescharacteristic
12#11762711C411yescharacteristic
13#11762711C2211yescharacteristic
14#11762686—22no—
15#11762666—22no—
16C2×C17×#117C411yescharacteristic
17C23×C4×C17C2×C411yescharacteristic
18C22×C22⋊D17—21no—
19C23×Dic17D412yes—
20C2×#63252—22no—
21C2×#63252—42no—
22C2×#63252—42no—
23C2×#63252C814yescomplemented
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C22×Dic34C2×C411yescharacteristic
27C22×#3624—42no—
28C17×#117—42no—
29C2×C4⋊C4×C17—21no—
30C2×C22⋊C4×C17—21no—
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33C2×#63227—22no—
34C2×#63227—22no—
35C2×C4⋊Dic17—42no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17D412yes—
38C2×C4⋊Dic17C2311yescharacteristic
39C22⋊#3624—24no—
40C22⋊#3624—42no—
41#10587502—22no—
42#10587502—22no—
43C24×C17C22⋊C411yescharacteristic
44C23×D17—21no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C22⋊C411yescharacteristic
50C22×C4×C17C2×C812yes—
51C22×C4×C17M1612yes—
52C22×C4×C17C22×C411yescharacteristic
53C2×C22⋊D17—24no—
54C2×C22⋊D17—42no—
55C2×C22⋊D17—42no—
56C22×Dic17—22no—
57C22×Dic17—21no—
58C22×Dic17—22no—
59C22×Dic17—82no—
60C22×Dic17C22⋊C412yes—
61C22×Dic17C2×D412yes—
62#63252—48no—
63#63252—48no—
64C2×D68—21no—
65C2×D68—41no—
66C2×D68—41no—
67C2×D68C2×C812yescomplemented
68C2×C4×D17—24no—
69C2×C4×D17—42no—
70C2×C4×D17—42no—
71C2×Dic34—21no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×Dic34C2×C812yescomplemented
75C2×#3624—44no—
76C2×#3624—84no—
77C4⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C42×C17—41no—
80C42×C17—41no—
81#63227—82no—
82#63227—82no—
83C4⋊Dic17—41no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86#11432—171nomaximal, Sylow
87C23×C17—41no—
88C23×C17—41no—
89C23×C17C22⋊C812yes—
90C23×C17#9912yes—
91C23×C17#10012yes—
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—22no—
94C22×D17—42no—
95C22×D17—42no—
96C22×D17—41no—
97C22×D17—41no—
98C2×C4×C17—24no—
99C2×C4×C17—24no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17C22⋊C812yes—
111C2×C4×C17C22×C811yescharacteristic
112C2×C4×C17C2×M1611yescharacteristic
113C22⋊D17—48no—
114C2×Dic17—22no—
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17—44no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—42no—
121C2×Dic17—42no—
122C2×Dic17—82no—
123D68—42no—
124D68—42no—
125C4×D17—48no—
126Dic34—42no—
127Dic34—42no—
128#3624—84no—
129#2311—171no—
130C2×#324—172no—
131#1319—172no—
132#1318—172no—
133C22×C17—21no—
134C22×C17—22no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—42no—
138C22×C17—42no—
139C22×C17#32314yes—
140C22×C17C2×C22⋊C811yescharacteristic
141C22×C17C2×#9911yescharacteristic
142C22×C17C2×#10011yescharacteristic
143C2×D17—44no—
144C2×D17—42no—
145C2×D17—82no—
146C2×D17—81no—
147C2×D17—81no—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—22no—
151C4×C17—81no—
152C4×C17—81no—
153Dic17—42no—
154Dic17—42no—
155Dic17—42no—
156C22×#41—171no—
157C2×#117#59511yescharacteristic
158C22×C4⋊C4—171no—
159C2×C22⋊C8—342no—
160#391—171no—
161#391—171no—
162#390—171no—
163#390—171no—
164#386—342no—
165#380—342no—
166#324—174no—
167#324—342no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—24no—
171C2×C17C2×#32311yescharacteristic
172C2×C17#212711yescharacteristic
173C2×C17#213311yescharacteristic
174D17—82no—
175C2×#41—174no—
176C2×#41—342no—
177C2×#41—682no—
178C2×#41—682no—
179C22×Q8—171no—
180C22×D4—171no—
181C22×D4—341no—
182C23×C4—172no—
183C23×C4—341no—
184C23×C4C2×#59511yescharacteristic
185C22×C8—682no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—172no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—682no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—342no—
194C2×C22⋊C4—342no—
195C2×C42C2×#59511yescharacteristic
196C2×C42C2×#59511yescharacteristic
197C22⋊C8—344no—
198C22⋊C8—682no—
199#95—342no—
200#95—342no—
201C17#1143211yescharacteristic, complemented, Sylow
202C24—341no—
203C24C22⋊#59511yescharacteristic
204#41—688no—
205#41—688no—
206C2×Q8—172no—
207C2×Q8—341no—
208C2×Q8—681no—
209C2×Q8—681no—
210C2×D4—172no—
211C2×D4—344no—
212C2×D4—341no—
213C2×D4—681no—
214C2×D4—681no—
215C2×D4—682no—
216C2×D4—682no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—172no—
222C22×C4—172no—
223C22×C4—341no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—344no—
227C22×C4—682no—
228C22×C4—682no—
229C22×C4—1362no—
230C22×C4#6324212yes—
231C22×C4#6324312yes—
232C22×C4C22⋊#59511yescharacteristic
233C22×C4C22×#59511yescharacteristic
234C2×C8—684no—
235C2×C8—1364no—
236C4⋊C4—42no—
237C4⋊C4—682no—
238C4⋊C4—681no—
239C4⋊C4—681no—
240C22⋊C4—42no—
241C22⋊C4—1362no—
242C22⋊C4—1362no—
243C42—41no—
244C42—41no—
245C23—41no—
246C23—41no—
247C23—342no—
248C23—682no—
249C23—682no—
250C23—681no—
251C23—681no—
252C23#1058753912yes—
253C23#1058754312yes—
254C23#1058755312yes—
255C23C2×C22⋊#59511yescharacteristic
256Q8—682no—
257Q8—682no—
258D4—682no—
259D4—682no—
260D4—688no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4—688no—
277C2×C4—684no—
278C2×C4—681no—
279C2×C4—681no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—1362no—
283C2×C4#1058754312yes—
284C2×C4C2×#6324211yescharacteristic
285C2×C4C2×#6324311yescharacteristic
286C8—1364no—
287C22—21no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—684no—
294C22—682no—
295C22—1362no—
296C22—1361no—
297C22—1361no—
298C22#1176365314yes—
299C22C2×#1058754311yescharacteristic
300C22C2×#1058753911yescharacteristic
301C22C2×#1058755311yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—81no—
306C4—81no—
307C4—682no—
308C4—682no—
309C4—682no—
310C2—21no—
311C2—21no—
312C2—24no—
313C2—1362no—
314C2C2×#1176365311yescharacteristic
315C2#1454506111yescharacteristic
316C2#1454508111yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC22×C17C2×#99
prime radicalC22×#63252C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×#117#595
hypocenterC17#11432
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#11763654C2semidirect
2. #14543715C2semidirect
3. #14543713C2semidirect
23. C2×#63252C8semidirect
67. C2×D68C2×C8semidirect
74. C2×Dic34C2×C8semidirect
201. C17#11432semidirect

Maximal subgroups

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

subgroupindex
C2×#117636542
#145437152
#145437132
#145313722
#1143217