Finite Group Databasebeta

← #17953249

Subgroups — #17953249

Order 4352.

644 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 116 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#14544700C212yesmaximal, complemented
2C2×#11763667C212yesmaximal
3#14543662C212yesmaximal, complemented
4#14531348C211yesmaximal, characteristic
5C2×#10587544C2214yes—
6#11763675—22no—
7#11763675—22no—
8#11763667—24no—
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#10587626D412yes—
30#10587626Q812yes—
31D4×Dic17—42no—
32C2×#63227—22no—
33C2×#63227—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×#63225—22no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17D412yes—
38C2×C4×Dic17D412yes—
39C2×C4×Dic17C2311yescharacteristic
40#10587579—42no—
41#10587567—48no—
42#10587544—24no—
43#10587544—24no—
44C24×C17—21no—
45C23×D17—21no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17C4⋊C412yes—
50C22×C4×C17—21no—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C2×C22⋊D17—24no—
54C2×C22⋊D17—24no—
55C2×C22⋊D17—44no—
56C2×C22⋊D17—44no—
57C22×Dic17—21no—
58C22×Dic17—44no—
59C22×Dic17—42no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C2×D412yes—
63C2×D68—21no—
64C2×D68—41no—
65C2×D68—41no—
66C2×D68C4⋊C412yes—
67C2×C4×D17—44no—
68C2×C4×D17—42no—
69C2×#3624—44no—
70C2×#3624—44no—
71C2×#3624—44no—
72C4⋊C4×C17—41no—
73C22⋊C4×C17—44no—
74C42×C17—41no—
75C22⋊Dic17—42no—
76#63227—42no—
77#63227—44no—
78C4⋊Dic17—41no—
79#63225—44no—
80C4×Dic17—22no—
81C4×Dic17—24no—
82C4×Dic17—24no—
83C4×Dic17C2×D411yescharacteristic
84C4×Dic17C2×Q811yescharacteristic
85C4×Dic17#4112yes—
86#8592—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—41no—
89C23×C17—41no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—22no—
94C22×D17—41no—
95C22×D17—41no—
96C22×D17—42no—
97C22×D17—42no—
98D4×C17—42no—
99D4×C17—42no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—41no—
106C2×C4×C17—42no—
107C2×C4×C17C2×C4⋊C411yescharacteristic
108C2×C4×C17#11711yescharacteristic
109C22⋊D17—48no—
110C22⋊D17—48no—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—42no—
116C2×Dic17—84no—
117C2×Dic17#10012yes—
118C2×Dic17#12112yes—
119C2×Dic17C22⋊Q812yes—
120C2×Dic17#13514yes—
121C2×Dic17AGL(1,ℤ/8)14yes—
122D68—42no—
123D68—42no—
124C4×D17—84no—
125#3624—84no—
126C4×#127—171no—
127#1831—172no—
128C2×#331—172no—
129#1296—172no—
130C22×C17—42no—
131C22×C17—44no—
132C22×C17—81no—
133C22×C17—82no—
134C22×C17—81no—
135C22×C17#38011yescharacteristic
136C22×C17C2×#10011yescharacteristic
137C22×C17#41311yescharacteristic
138C22×C17#41612yes—
139C22×C17#41812yes—
140C2×D17—42no—
141C2×D17—44no—
142C2×D17—81no—
143C2×D17—82no—
144C2×D17—81no—
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—42no—
148C4×C17—81no—
149Dic17—22no—
150Dic17—41no—
151Dic17—81no—
152Dic17#46314yes—
153Dic17#47814yes—
154C2×#127—171no—
155C2×C4×D4—171no—
156C2×C4×D4—171no—
157C2×C4×D4—342no—
158C2×C4⋊C8—174no—
159#390—171no—
160#390—342no—
161#390#59511yescharacteristic
162#384—171no—
163C43—171no—
164#336—342no—
165#336—342no—
166#331—344no—
167C2×C17—82no—
168C2×C17#154311yescharacteristic
169C2×C17#155312yes—
170C2×C17#155511yescharacteristic
171C2×C17#180011yescharacteristic
172C2×C17#180412yes—
173D17—82no—
174C22×D4—171no—
175C22×D4—342no—
176C22×D4C2×#59511yescharacteristic
177C23×C4—341no—
178C23×C4—341no—
179C22×C8—344no—
180#127—172no—
181#127—172no—
182#127—342no—
183C4×D4—688no—
184C4×D4—682no—
185C4×D4—682no—
186C2×C4⋊C4—171no—
187C2×C4⋊C4—342no—
188C2×C4⋊C4C2×#59511yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—341no—
191C2×C22⋊C4—341no—
192C2×C22⋊C4—342no—
193C2×C42—171no—
194C2×C42—172no—
195C2×C42—172no—
196C2×C42—171no—
197C2×C42C2×#59511yescharacteristic
198C4⋊C8—344no—
199C4⋊C8—344no—
200C17#859211yescharacteristic, complemented, Sylow
201C24—21no—
202C24—341no—
203C2×D4—21no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—172no—
207C2×D4—344no—
208C2×D4—344no—
209C2×D4—341no—
210C2×D4—681no—
211C2×D4—681no—
212C2×D4—684no—
213C2×D4—684no—
214C2×D4C4⋊#59512yes—
215C22×C4—21no—
216C22×C4—172no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—341no—
220C22×C4—682no—
221C22×C4—684no—
222C22×C4—682no—
223C22×C4—684no—
224C22×C4C22⋊#59512yes—
225C22×C4C22×#59511yescharacteristic
226C2×C8—684no—
227C2×C8—684no—
228C2×C8—684no—
229C4⋊C4—41no—
230C4⋊C4—681no—
231C4⋊C4—684no—
232C22⋊C4—44no—
233C22⋊C4—684no—
234C22⋊C4—682no—
235C22⋊C4—682no—
236C42—41no—
237C42—171no—
238C42—171no—
239C42—172no—
240C42—342no—
241C42—344no—
242C42—344no—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—42no—
247C23—42no—
248C23—342no—
249C23—681no—
250C23—681no—
251C23—682no—
252C23—682no—
253C23C2×C22⋊#59511yescharacteristic
254D4—42no—
255D4—42no—
256D4—682no—
257D4—682no—
258D4—688no—
259D4—688no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—172no—
268C2×C4—172no—
269C2×C4—174no—
270C2×C4—174no—
271C2×C4—172no—
272C2×C4—342no—
273C2×C4—341no—
274C2×C4—342no—
275C2×C4—681no—
276C2×C4—682no—
277C2×C4—1364no—
278C2×C4—1364no—
279C2×C4C2×C4⋊#59511yescharacteristic
280C2×C4#1058767711yescharacteristic
281C8—1364no—
282C22—42no—
283C22—44no—
284C22—81no—
285C22—82no—
286C22—81no—
287C22—682no—
288C22—684no—
289C22—1361no—
290C22—1362no—
291C22—1361no—
292C22#1176375111yescharacteristic
293C22D17⋊#10011yescharacteristic
294C22#1176379611yescharacteristic
295C22#1176379912yes—
296C22#1176380112yes—
297C4—21no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—174no—
302C4—174no—
303C4—342no—
304C4—681no—
305C4—1361no—
306C2—82no—
307C2—1362no—
308C2#1454428811yescharacteristic
309C2#1454430812yes—
310C2#1454431211yescharacteristic
311C2#1454464911yescharacteristic
312C2#1454465412yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587626C22
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#8592
LCSG ▷ C22×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. #14544700C2semidirect
3. #14543662C2semidirect
200. C17#8592semidirect

Maximal subgroups

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

subgroupindex
#145447002
C2×#117636672
#145436622
#145313482
#859217