Finite Group Databasebeta

← #17958152

Subgroups — #17958152

Order 4352.

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

normal subgroups only

This group has 594 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#14544013C212yesmaximal, complemented
2#14543715C212yesmaximal, complemented
3#14543712C212yesmaximal, complemented
4#14531374C211yesmaximal, characteristic
5#11763777—22no—
6#11763655—22no—
7#11763654C2214yes—
8C22×#63254C411yescharacteristic
9C2×#10587558C411yescharacteristic
10C2×C4⋊C4×D17C2211yescharacteristic
11#11762715C2211yescharacteristic
12#11762714C411yescharacteristic
13#11762712C2211yescharacteristic
14#11762685—22no—
15#11762667—22no—
16C17×#390C411yescharacteristic
17C22×D4×C17C2×C411yescharacteristic
18C22×C22⋊D17—21no—
19C23×Dic17—21no—
20C2×#63254—22no—
21C2×#63254—42no—
22C2×#63254—42no—
23C2×#63254C814yescomplemented
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D412yes—
26C22×Dic34C2×C411yescharacteristic
27C2×#63242—42no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17—22no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×#63227—21no—
33C2×C4⋊Dic17C2311yescharacteristic
34C2×#63225—21no—
35C2×#63225D412yes—
36C2×C4×Dic17C2×C411yescharacteristic
37C4⋊C4×D17—42no—
38#10587558—42no—
39C22⋊#3624—24no—
40#10587543—42no—
41#10587502—22no—
42#10587502—22no—
43C24×C17—21no—
44C23×D17C22⋊C411yescharacteristic
45C2×D4×C17—21no—
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D4×C17C2×C812yescomplemented
49C22×C4×C17—21no—
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22×C411yescharacteristic
52C2×C22⋊D17—24no—
53C2×C22⋊D17—42no—
54C2×C22⋊D17—42no—
55C22×Dic17—21no—
56C22×Dic17—24no—
57C22×Dic17—21no—
58C22×Dic17—42no—
59C22×Dic17—42no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C2×D412yes—
62#63254—48no—
63#63254—48no—
64C2×C4×D17—22no—
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17—22no—
68C2×C4×D17—82no—
69C2×C4×D17C2×C812yes—
70C2×C4×D17M1612yes—
71C2×Dic34—21no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×Dic34C2×C812yescomplemented
75C2×#3624—44no—
76#63242—84no—
77C4⋊C4×C17—41no—
78C22⋊C4×C17—82no—
79C42×C17—41no—
80C22⋊Dic17—82no—
81#63227—42no—
82C4⋊Dic17—41no—
83#63225—42no—
84#63225—42no—
85C4×Dic17—41no—
86#11432—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—42no—
89C23×C17—42no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—41no—
94C22×D17—41no—
95C22×D17C22⋊C812yes—
96C22×D17#9912yes—
97C22×D17#10012yes—
98D4×C17—42no—
99D4×C17—42no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—42no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17C22×C811yescharacteristic
108C2×C4×C17C2×M1611yescharacteristic
109C22⋊D17—48no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—48no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—42no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17C22⋊C812yes—
122C4×D17—24no—
123C4×D17—24no—
124C4×D17—44no—
125C4×D17—82no—
126Dic34—42no—
127Dic34—42no—
128#3624—84no—
129#2311—171no—
130C2×#324—172no—
131#1319—172no—
132#1318—172no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—44no—
136C22×C17—42no—
137C22×C17—82no—
138C22×C17—81no—
139C22×C17—81no—
140C22×C17C2×C22⋊C811yescharacteristic
141C22×C17C2×#9911yescharacteristic
142C22×C17C2×#10011yescharacteristic
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—42no—
146C2×D17—42no—
147C2×D17#32314yes—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—42no—
151C4×C17—81no—
152Dic17—22no—
153Dic17—42no—
154Dic17—42no—
155Dic17—81no—
156C22×#41—171no—
157C2×#117—171no—
158C22×C4⋊C4—171no—
159C2×C22⋊C8—342no—
160#391—171no—
161#391—171no—
162#390—171no—
163#390#59511yescharacteristic
164#386—342no—
165#380—342no—
166#324—174no—
167#324—342no—
168C2×C17—21no—
169C2×C17—21no—
170C2×C17—82no—
171C2×C17C2×#32311yescharacteristic
172C2×C17#212711yescharacteristic
173C2×C17#213311yescharacteristic
174D17—24no—
175C2×#41—174no—
176C2×#41—342no—
177C2×#41—682no—
178C2×#41—682no—
179C22×Q8—171no—
180C22×D4—341no—
181C22×D4C2×#59511yescharacteristic
182C23×C4—171no—
183C23×C4—172no—
184C23×C4—341no—
185C22×C8—682no—
186#117—682no—
187C2×C4⋊C4—172no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—682no—
191C2×C4⋊C4C2×#59511yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—341no—
194C2×C22⋊C4—342no—
195C2×C42—171no—
196C2×C42C2×#59511yescharacteristic
197C22⋊C8—344no—
198C22⋊C8—682no—
199#95—342no—
200#95—342no—
201C17#1143211yescharacteristic, complemented, Sylow
202C24—21no—
203C24—171no—
204#41—688no—
205#41—688no—
206C2×Q8—172no—
207C2×Q8—341no—
208C2×Q8—681no—
209C2×Q8—681no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—344no—
214C2×D4—682no—
215C2×D4—682no—
216C2×D4#6324212yescomplemented
217C22×C4—21no—
218C22×C4—171no—
219C22×C4—172no—
220C22×C4—172no—
221C22×C4—172no—
222C22×C4—342no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—344no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—682no—
230C22×C4—682no—
231C22×C4—1362no—
232C22×C4C22⋊#59512yes—
233C22×C4C22×#59511yescharacteristic
234C2×C8—684no—
235C2×C8—1364no—
236C4⋊C4—41no—
237C4⋊C4—682no—
238C4⋊C4—681no—
239C4⋊C4—682no—
240C22⋊C4—82no—
241C22⋊C4—682no—
242C22⋊C4—1362no—
243C42—41no—
244C42—681no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—172no—
251C23—172no—
252C23—172no—
253C23—681no—
254C23—681no—
255C23C2×C22⋊#59511yescharacteristic
256Q8—682no—
257Q8—682no—
258D4—42no—
259D4—42no—
260D4—688no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—172no—
269C2×C4—344no—
270C2×C4—344no—
271C2×C4—341no—
272C2×C4—341no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—684no—
276C2×C4—688no—
277C2×C4—681no—
278C2×C4—681no—
279C2×C4—682no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—1362no—
284C2×C4C2×#6324211yescharacteristic
285C2×C4D17⋊M1611yescharacteristic
286C8—1364no—
287C22—21no—
288C22—21no—
289C22—44no—
290C22—42no—
291C22—82no—
292C22—81no—
293C22—81no—
294C22—174no—
295C22—342no—
296C22—342no—
297C22—682no—
298C22—682no—
299C22#1176377711yescharacteristic
300C22D17⋊#9911yescharacteristic
301C22D17⋊#10011yescharacteristic
302C4—21no—
303C4—21no—
304C4—42no—
305C4—81no—
306C4—342no—
307C4—682no—
308C4—682no—
309C4—1361no—
310C2—21no—
311C2—21no—
312C2—82no—
313C2—344no—
314C2#1454401011yescharacteristic
315C2#1454506011yescharacteristic
316C2#1454508011yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC22×C17C2×#99
prime radicalC22×#63254C4
cosocleC2×C4⋊Dic17C23
hypercenter#390#595
hypocenterC17#11432
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #14544013C2semidirect
2. #14543715C2semidirect
3. #14543712C2semidirect
23. C2×#63254C8semidirect
48. C2×D4×C17C2×C8semidirect
74. C2×Dic34C2×C8semidirect
201. C17#11432semidirect
216. C2×D4#63242semidirect

Maximal subgroups

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

subgroupindex
#145440132
#145437152
#145437122
#145313742
#1143217