Finite Group Databasebeta

← #17958151

Subgroups — #17958151

Order 4352.

594 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 112 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#14543714C212yesmaximal, complemented
3#14543712C212yesmaximal, complemented
4#14531376C211yesmaximal, characteristic
5C2×C22⋊#3624—22no—
6#11763654—22no—
7#11763654C2214yes—
8C2×#10587612C411yescharacteristic
9C22×C4⋊Dic17C2211yescharacteristic
10#11762716C411yescharacteristic
11#11762716C2211yescharacteristic
12#11762712C411yescharacteristic
13#11762712C2211yescharacteristic
14#11762689—22no—
15#11762669—22no—
16C22×#41×C17C411yescharacteristic
17C2×#41×C17—22no—
18C2×#41×C17—42no—
19C2×#41×C17—42no—
20C2×#41×C17C814yescomplemented
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17—21no—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17—21no—
25C23×C4×C17C2×C411yescharacteristic
26C23×Dic17D412yes—
27C22×#3624—42no—
28C2×C22⋊Dic17—21no—
29C2×C22⋊Dic17—22no—
30C2×C22⋊Dic17—22no—
31#10587612—42no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—42no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17D412yes—
36C2×C4⋊Dic17C2311yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17C2×C411yescharacteristic
39C22⋊#3624—24no—
40C22⋊#3624—42no—
41#10587502—22no—
42#10587502—22no—
43C24×C17—21no—
44C24×C17C22⋊C411yescharacteristic
45#41×C17—48no—
46#41×C17—48no—
47C2×Q8×C17—21no—
48C2×Q8×C17—41no—
49C2×Q8×C17—41no—
50C2×Q8×C17C2×C812yescomplemented
51C2×D4×C17—24no—
52C2×D4×C17—21no—
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17—42no—
56C2×D4×C17—42no—
57C2×D4×C17C2×C812yescomplemented
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—24no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C2×C812yes—
66C22×C4×C17M1612yes—
67C22×C4×C17C22×C411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—82no—
73C22×Dic17C22⋊C412yes—
74C22×Dic17C2×D412yes—
75C2×#3624—44no—
76C2×#3624—84no—
77C22⋊Dic17—42no—
78C22⋊Dic17—82no—
79C22⋊Dic17—82no—
80C4⋊Dic17—41no—
81C4⋊Dic17—42no—
82C4⋊Dic17—41no—
83C4⋊Dic17—42no—
84C4×Dic17—41no—
85C4×Dic17—41no—
86#11432—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—41no—
89C23×C17—41no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17C22⋊C812yes—
95C23×C17#9912yes—
96C23×C17#10012yes—
97C23×C17C2×C22⋊C411yescharacteristic
98Q8×C17—42no—
99Q8×C17—42no—
100D4×C17—42no—
101D4×C17—42no—
102D4×C17—48no—
103C2×C4×C17—24no—
104C2×C4×C17—24no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—48no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17C22⋊C812yes—
114C2×C4×C17C22×C811yescharacteristic
115C2×C4×C17C2×M1611yescharacteristic
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—44no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—82no—
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—44no—
140C22×C17—42no—
141C22×C17—82no—
142C22×C17—81no—
143C22×C17—81no—
144C22×C17#32314yes—
145C22×C17C2×C22⋊C811yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10011yescharacteristic
148C4×C17—22no—
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—42no—
152Dic17—42no—
153Dic17—42no—
154Dic17—81no—
155Dic17—81no—
156C22×#41#59511yescharacteristic
157C2×#117—171no—
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×C17—82no—
172C2×C17C2×#32311yescharacteristic
173C2×C17#212711yescharacteristic
174C2×C17#213311yescharacteristic
175C2×#41—22no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41#362414yescomplemented
179C22×Q8C2×#59511yescharacteristic
180C22×D4—21no—
181C22×D4C2×#59511yescharacteristic
182C23×C4—21no—
183C23×C4—172no—
184C23×C4C2×#59511yescharacteristic
185C22×C8—682no—
186#117—682no—
187C2×C4⋊C4—172no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—682no—
192C2×C22⋊C4—341no—
193C2×C22⋊C4—342no—
194C2×C22⋊C4—342no—
195C2×C42—171no—
196C2×C42—171no—
197C22⋊C8—344no—
198C22⋊C8—682no—
199#95—342no—
200#95—342no—
201C17#1143211yescharacteristic, complemented, Sylow
202C24—21no—
203C24C22⋊#59511yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C2×#362412yescomplemented
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4C2×#362412yescomplemented
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—172no—
224C22×C4—172no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—1362no—
230C22×C4C2×#362412yes—
231C22×C4#6324712yes—
232C22×C4C22⋊#59511yescharacteristic
233C22×C4C22×#59511yescharacteristic
234C2×C8—684no—
235C2×C8—1364no—
236C4⋊C4—682no—
237C4⋊C4—681no—
238C4⋊C4—681no—
239C4⋊C4—682no—
240C22⋊C4—682no—
241C22⋊C4—1362no—
242C22⋊C4—1362no—
243C42—681no—
244C42—681no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23#1058755112yes—
253C23C22⋊#362412yes—
254C23#1058755312yes—
255C23C2×C22⋊#59511yescharacteristic
256Q8—42no—
257Q8—42no—
258D4—42no—
259D4—42no—
260D4—48no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—684no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—681no—
279C2×C4—681no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—1362no—
283C2×C4C22⋊#362412yes—
284C2×C4C22×#362411yescharacteristic
285C2×C4C2×#6324711yescharacteristic
286C8—1364no—
287C22—21no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—42no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22#1176365214yes—
299C22C2×C22⋊#362411yescharacteristic
300C22C2×#1058755111yescharacteristic
301C22C2×#1058755311yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—42no—
306C4—682no—
307C4—682no—
308C4—1361no—
309C4—1361no—
310C2—21no—
311C2—21no—
312C2—24no—
313C2—82no—
314C2C2×#1176365211yescharacteristic
315C2#1454505911yescharacteristic
316C2#1454507911yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
Fitting, prime radicalC22×#41×C17C4
socleC22×C17C2×#99
cosocleC2×C4⋊Dic17C23
hypercenterC22×#41#595
hypocenterC17#11432
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#11763654C2semidirect
2. #14543714C2semidirect
3. #14543712C2semidirect
20. C2×#41×C17C8semidirect
50. C2×Q8×C17C2×C8semidirect
57. C2×D4×C17C2×C8semidirect
178. C2×#41#3624semidirect
201. C17#11432semidirect
209. C2×Q8C2×#3624semidirect
216. C2×D4C2×#3624semidirect

Maximal subgroups

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

subgroupindex
C2×#117636542
#145437142
#145437122
#145313762
#1143217