Finite Group Databasebeta

← #17953591

Subgroups — #17953591

Order 4352.

526 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544511C211yesmaximal, characteristic
2#14544511C211yesmaximal, characteristic
3#14543660C214yesmaximal, complemented
4#14531189C211yesmaximal, characteristic
5C2×#10587543—21no—
6C2×#10587543—21no—
7C2×#10587542C2212yes—
8C2×#10587542C2212yes—
9#11763675—21no—
10#11763675—21no—
11C2×#10587578C412yes—
12C2×C42×D17C2211yescharacteristic
13#11762679C411yescharacteristic
14#11762674C2212yes—
15#11762666—21no—
16#11762666—21no—
17C4⋊C4×Dic17—21no—
18C4⋊C4×Dic17—21no—
19C17×#384C411yescharacteristic
20C22×C4×D17C2×C411yescharacteristic
21C22×C4×D17D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×#3624—22no—
24C22×#3624—22no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17C2×C412yes—
28C2×C42×C17C2×C411yescharacteristic
29C2×#63227—22no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C412yes—
33C2×C4×Dic17—22no—
34C2×C4×Dic17D411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36C2×C4×Dic17C2311yescharacteristic
37#10587578—44no—
38C42×D17—24no—
39#10587543—24no—
40#10587543—24no—
41#10587542—24no—
42#10587542—24no—
43#10587502—22no—
44#10587502—22no—
45C23×D17C22⋊C411yescharacteristic
46C22×C4×C17—22no—
47C22×C4×C17C22⋊C411yescharacteristic
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—22no—
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C2×D411yescharacteristic
53C22×Dic17C2×D411yescharacteristic
54C2×C4×D17—22no—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—41no—
62C2×C4×D17—41no—
63C2×#3624—24no—
64C2×#3624—24no—
65C2×#3624—42no—
66C2×#3624—42no—
67C2×#3624—42no—
68C2×#3624—42no—
69C4⋊C4×C17—42no—
70C4⋊C4×C17—42no—
71C4⋊C4×C17—42no—
72C42×C17—22no—
73#63227—44no—
74C4⋊Dic17—42no—
75C4⋊Dic17—42no—
76C4⋊Dic17—42no—
77C4×Dic17—22no—
78C4×Dic17—22no—
79C4×Dic17—44no—
80C4×Dic17#4114yes—
81#8722—171nomaximal, Sylow
82C23×C17C2×C22⋊C411yescharacteristic
83C22×D17—21no—
84C22×D17—21no—
85C22×D17—21no—
86C22×D17—21no—
87C22×D17—21no—
88C22×D17—21no—
89C22×D17#10012yes—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17#10412yes—
100C2×C4×C17#10412yes—
101C2×C4×C17#11712yes—
102C2×Dic17—21no—
103C2×Dic17—21no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—42no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17#10112yes—
112C2×Dic17#12312yes—
113C2×Dic17#12312yes—
114C4×D17—28no—
115C4×D17—44no—
116C4×D17—44no—
117#3624—44no—
118#3624—44no—
119#2285—171no—
120#1749—171no—
121#1749—171no—
122#1295—174no—
123C22×C17#38111yescharacteristic
124C22×C17C2×#10011yescharacteristic
125C22×C17C2×#10111yescharacteristic
126C22×C17C2×#10411yescharacteristic
127C22×C17C2×#10411yescharacteristic
128C22×C17#42111yescharacteristic
129C22×C17#42111yescharacteristic
130C2×D17—22no—
131C2×D17—22no—
132C2×D17—22no—
133C2×D17—41no—
134C2×D17—41no—
135C2×D17—41no—
136C2×D17—41no—
137C4×C17—22no—
138C4×C17—22no—
139C4×C17—22no—
140C4×C17—82no—
141Dic17—22no—
142Dic17—22no—
143Dic17—22no—
144Dic17—82no—
145C2×#117—172no—
146C22×C42—171no—
147C2×C22⋊C8—341no—
148C2×C22⋊C8—341no—
149C2×#97—172no—
150C2×#97—172no—
151#384—171no—
152#384#59511yescharacteristic
153#382—172no—
154#380—341no—
155#380—341no—
156C4×C4⋊C4—341no—
157C4×C4⋊C4—341no—
158#336—341no—
159#336—341no—
160C2×C17#152611yescharacteristic
161C2×C17#153011yescharacteristic
162C2×C17#153011yescharacteristic
163C2×C17#153711yescharacteristic
164C2×C17#182111yescharacteristic
165C2×C17#182611yescharacteristic
166C2×C17#182611yescharacteristic
167D17—42no—
168C23×C4—171no—
169C23×C4—171no—
170C23×C4—171no—
171C22×C8—342no—
172C22×C8—342no—
173#117—684no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—172no—
177C2×C4⋊C4—341no—
178C2×C4⋊C4—341no—
179C2×C4⋊C4C2×#59512yes—
180C2×C22⋊C4—342no—
181C2×C42—171no—
182C2×C42—171no—
183C2×C42—171no—
184C2×C42—344no—
185C2×C42—342no—
186C2×C42C2×#59511yescharacteristic
187C22⋊C8—344no—
188C22⋊C8—344no—
189#97—344no—
190#97—344no—
191#95—342no—
192#95—342no—
193C17#872211yescharacteristic, complemented, Sylow
194C24—171no—
195C22×C4—22no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—171no—
199C22×C4—342no—
200C22×C4—342no—
201C22×C4—342no—
202C22×C4—342no—
203C22×C4—342no—
204C22×C4—342no—
205C22×C4—342no—
206C22×C4—342no—
207C22×C4—681no—
208C22×C4—681no—
209C22×C4C22⋊#59511yescharacteristic
210C22×C4C22⋊#59511yescharacteristic
211C22×C4C22×#59511yescharacteristic
212C2×C8—344no—
213C2×C8—344no—
214C2×C8—682no—
215C2×C8—682no—
216C2×C8—682no—
217C2×C8—682no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—682no—
222C4⋊C4—682no—
223C4⋊C4—682no—
224C22⋊C4—684no—
225C42—22no—
226C42—174no—
227C42—342no—
228C42—342no—
229C42—684no—
230C23—172no—
231C23—341no—
232C23—341no—
233C23—341no—
234C23—341no—
235C23—341no—
236C23—341no—
237C23C2×C22⋊#59511yescharacteristic
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—172no—
248C2×C4—172no—
249C2×C4—172no—
250C2×C4—348no—
251C2×C4—341no—
252C2×C4—341no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—341no—
256C2×C4—341no—
257C2×C4—684no—
258C2×C4—684no—
259C2×C4—682no—
260C2×C4—682no—
261C2×C4—682no—
262C2×C4#1058754812yes—
263C2×C4#1058755012yes—
264C2×C4#1058767712yes—
265C8—684no—
266C8—684no—
267C22—342no—
268C22—342no—
269C22—342no—
270C22—681no—
271C22—681no—
272C22—681no—
273C22—681no—
274C22#1176375211yescharacteristic
275C22D17⋊#10011yescharacteristic
276C22D17⋊#10111yescharacteristic
277C22C2×#1058754811yescharacteristic
278C22C2×#1058755011yescharacteristic
279C22#1176380711yescharacteristic
280C22#1176380811yescharacteristic
281C4—22no—
282C4—22no—
283C4—22no—
284C4—82no—
285C4—342no—
286C4—342no—
287C4—342no—
288C4—1362no—
289C2—682no—
290C2#1454424211yescharacteristic
291C2#1454425411yescharacteristic
292C2#1454425511yescharacteristic
293C2#1454427611yescharacteristic
294C2#1454468411yescharacteristic
295C2#1454469011yescharacteristic
296C2#1454469111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#384#595
hypocenterC17#8722
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #14543660C2semidirect
193. C17#8722semidirect

Maximal subgroups

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

subgroupindex
#145445112
#145436602
#145311892
#872217