Finite Group Databasebeta

← #17952109

Subgroups — #17952109

Order 4352.

708 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C4×#10587544C212yesmaximal, complemented
2C2×#11763667C212yesmaximal
3#14543662C212yesmaximal, complemented
4#14531348C211yesmaximal, characteristic
5C2×#10587544C2214yes—
6C2×C4×#3624—22no—
7#11763675—22no—
8#11763667C418yescomplemented
9C2×#10587626C2211yescharacteristic
10C2×D4×Dic17C411yescharacteristic
11C2×#10587579C411yescharacteristic
12C2×#10587567—22no—
13#11762711C411yescharacteristic
14#11762709—22no—
15#11762678C2211yescharacteristic
16C42×Dic17C2211yescharacteristic
17C17×#390C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C22×C22⋊D17—22no—
20C23×Dic17—21no—
21C22×D68C2×C411yescharacteristic
22C22×C4×D17—21no—
23C22×#3624—24no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C22⋊C4×C17—22no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C22⋊Dic17—21no—
28#10587626—22no—
29#10587626C2×C414yes—
30D4×Dic17—42no—
31C2×#63227—22no—
32C2×#63227—21no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×#63225—22no—
35C2×C4×Dic17C2×C411yescharacteristic
36C2×C4×Dic17D412yes—
37C2×C4×Dic17D412yes—
38C2×C4×Dic17C2311yescharacteristic
39#10587579—42no—
40#10587567—44no—
41#10587567—44no—
42#10587544—24no—
43#10587544C2×C418yescomplemented
44C4×#3624—44no—
45C24×C17—21no—
46C23×D17—21no—
47C2×D4×C17—21no—
48C2×D4×C17—41no—
49C2×D4×C17—41no—
50C2×D4×C17C4212yes—
51C22×C4×C17—21no—
52C22×C4×C17C22⋊C412yes—
53C22×C4×C17C22×C411yescharacteristic
54C2×C22⋊D17—28no—
55C2×C22⋊D17—42no—
56C2×C22⋊D17—42no—
57C2×C22⋊D17—44no—
58C22×Dic17—21no—
59C22×Dic17—42no—
60C22×Dic17—44no—
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C2×D412yes—
64C2×D68—21no—
65C2×D68—41no—
66C2×D68—41no—
67C2×D68C4212yes—
68C2×C4×D17—42no—
69C2×C4×D17—44no—
70C2×#3624—28no—
71C2×#3624—44no—
72C2×#3624—44no—
73C4⋊C4×C17—41no—
74C22⋊C4×C17—44no—
75C42×C17—41no—
76C22⋊Dic17—42no—
77#63227—42no—
78#63227—44no—
79C4⋊Dic17—41no—
80#63225—42no—
81#63225—42no—
82C4×Dic17—22no—
83C4×Dic17—24no—
84C4×Dic17D814yes—
85C4×Dic17SD1614yes—
86C4×Dic17C22×C412yes—
87C4×Dic17#4112yes—
88C4×#331—171nomaximal, Sylow
89C23×C17—22no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17—42no—
93C23×C17—42no—
94C23×C17C2×C22⋊C411yescharacteristic
95C22×D17—22no—
96C22×D17—41no—
97C22×D17—41no—
98C22×D17—42no—
99C22×D17—42no—
100D4×C17—42no—
101D4×C17—42no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17#10214yes—
109C2×C4×C17C2×C4211yescharacteristic
110C2×C4×C17#11711yescharacteristic
111C22⋊D17—48no—
112C22⋊D17—48no—
113C2×Dic17—21no—
114C2×Dic17—22no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117C2×Dic17—84no—
118C2×Dic17#10012yes—
119C2×Dic17#10214yes—
120C2×Dic17C4×D414yes—
121C2×Dic17C2×D812yes—
122C2×Dic17C2×SD1612yes—
123C2×Dic17#13514yes—
124D68—42no—
125D68—42no—
126C4×D17—84no—
127#3624—48no—
128C4×#127—171no—
129C4×C4⋊C8—172no—
130C2×#331—172no—
131#1296—172no—
132C22×C17—42no—
133C22×C17—44no—
134C22×C17—81no—
135C22×C17—82no—
136C22×C17—81no—
137C22×C17C4×C22⋊C411yescharacteristic
138C22×C17C2×#10011yescharacteristic
139C22×C17#41311yescharacteristic
140C22×C17C2×#10212yes—
141C22×C17#41612yes—
142C2×D17—42no—
143C2×D17—44no—
144C2×D17—81no—
145C2×D17—82no—
146C2×D17—81no—
147C4×C17—24no—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—81no—
151Dic17—41no—
152Dic17—81no—
153Dic17#33114yescomplemented
154Dic17C4×D814yes—
155Dic17C4×SD1614yes—
156C2×#127—171no—
157C2×C4×D4—171no—
158C2×C4×D4—171no—
159C2×C4×D4—342no—
160C2×C4⋊C8—174no—
161C2×C4×C8—342no—
162#390—171no—
163#390—342no—
164#390#59511yescharacteristic
165#384—171no—
166C43—171no—
167#336—342no—
168#331—178no—
169C2×C17—82no—
170C2×C17C2×#33111yescharacteristic
171C2×C17#154012yes—
172C2×C17#154311yescharacteristic
173C2×C17C4×#10011yescharacteristic
174C2×C17C4×#10212yes—
175D17—82no—
176C22×D4—171no—
177C22×D4—342no—
178C22×D4C2×#59511yescharacteristic
179C23×C4—341no—
180C23×C4—341no—
181C22×C8—344no—
182#127—174no—
183#127—342no—
184C4×D4—684no—
185C4×D4—684no—
186C4×D4—682no—
187C4×D4—682no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—342no—
190C2×C4⋊C4C2×#59511yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—341no—
193C2×C22⋊C4—341no—
194C2×C22⋊C4—342no—
195C2×C42—172no—
196C2×C42—171no—
197C2×C42—172no—
198C2×C42—171no—
199C2×C42C2×#59511yescharacteristic
200C4⋊C8—178no—
201C4⋊C8—344no—
202C4×C8—684no—
203C17C4×#33111yescharacteristic, complemented, Sylow
204C24—21no—
205C24—341no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—172no—
210C2×D4—348no—
211C2×D4—341no—
212C2×D4—681no—
213C2×D4—681no—
214C2×D4—682no—
215C2×D4—682no—
216C2×D4—684no—
217C2×D4C4×#59512yes—
218C22×C4—21no—
219C22×C4—171no—
220C22×C4—172no—
221C22×C4—171no—
222C22×C4—341no—
223C22×C4—684no—
224C22×C4—682no—
225C22×C4—684no—
226C22×C4—682no—
227C22×C4C22⋊#59512yes—
228C22×C4C22×#59511yescharacteristic
229C2×C8—348no—
230C2×C8—684no—
231C2×C8—684no—
232C4⋊C4—41no—
233C4⋊C4—681no—
234C4⋊C4—682no—
235C4⋊C4—682no—
236C22⋊C4—44no—
237C22⋊C4—682no—
238C22⋊C4—682no—
239C22⋊C4—684no—
240C42—41no—
241C42—172no—
242C42—172no—
243C42—174no—
244C42—174no—
245C42—342no—
246C42—344no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—342no—
253C23—681no—
254C23—681no—
255C23—682no—
256C23—682no—
257C23C2×C22⋊#59511yescharacteristic
258D4—42no—
259D4—42no—
260D4—682no—
261D4—682no—
262D4—688no—
263D4—688no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—174no—
271C2×C4—172no—
272C2×C4—172no—
273C2×C4—172no—
274C2×C4—174no—
275C2×C4—174no—
276C2×C4—341no—
277C2×C4—342no—
278C2×C4—681no—
279C2×C4—682no—
280C2×C4—1364no—
281C2×C4—1364no—
282C2×C4#1058754714yes—
283C2×C4C2×C4×#59511yescharacteristic
284C2×C4#1058767711yescharacteristic
285C8—688no—
286C22—42no—
287C22—44no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22—682no—
292C22—684no—
293C22—1361no—
294C22—1362no—
295C22—1361no—
296C22C4×C22⋊#59511yescharacteristic
297C22D17⋊#10011yescharacteristic
298C22#1176379611yescharacteristic
299C22C2×#1058754712yes—
300C22#1176379912yes—
301C4—21no—
302C4—21no—
303C4—24no—
304C4—81no—
305C4—174no—
306C4—174no—
307C4—174no—
308C4—681no—
309C4—1361no—
310C2—82no—
311C2—1362no—
312C2D17⋊#33111yescharacteristic
313C2#1454428212yes—
314C2#1454428811yescharacteristic
315C2#1454453411yescharacteristic
316C2C4×#1058754712yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587626C22
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17C4×#331
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C4×#10587544C2semidirect
3. #14543662C2semidirect
8. #11763667C4semidirect
43. #10587544C2×C4semidirect
153. Dic17#331semidirect
203. C17C4×#331semidirect

Maximal subgroups

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

subgroupindex
C4×#105875442
C2×#117636672
#145436622
#145313482
C4×#33117