Finite Group Databasebeta

← #17953234

Subgroups — #17953234

Order 4352.

684 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 146 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#14544717C212yesmaximal, complemented
2C2×#11763667C212yesmaximal
3#14543662C212yesmaximal, complemented
4C4×#10587626C211yesmaximal, characteristic
5C2×#10587544—22no—
6C2×#10587544C2214yes—
7C2×C4⋊#3624—22no—
8#11763667—24no—
9C2×#10587626C2211yescharacteristic
10C2×C4×D68C411yescharacteristic
11C2×C4×C22⋊D17—22no—
12#11762712C411yescharacteristic
13#11762710C411yescharacteristic
14#11762709—22no—
15#11762678C2211yescharacteristic
16C42×Dic17C2211yescharacteristic
17C2×C4×D4×C17C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×C4×C17—21no—
20C22×C22⋊D17—22no—
21C22×D68C2×C411yescharacteristic
22C22×C4×D17—21no—
23C22×#3624—24no—
24C4×D4×C17—42no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C22⋊C4×C17—21no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C22⋊Dic17—22no—
29#10587626—22no—
30#10587626D412yes—
31#10587626Q812yes—
32C4×C22⋊D17—44no—
33C4×C22⋊D17—44no—
34C2×#63227—21no—
35C2×#63227—22no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—22no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17D412yes—
40C2×C4×Dic17D412yes—
41C2×C4×Dic17C2311yescharacteristic
42C4×D68—42no—
43#10587544—24no—
44#10587544—24no—
45#10587544—44no—
46C4⋊#3624—44no—
47C24×C17—21no—
48C23×D17—21no—
49C2×D4×C17—21no—
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17C4⋊C412yes—
53C22×C4×C17—21no—
54C22×C4×C17—42no—
55C22×C4×C17—44no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C2×C22⋊D17—24no—
60C2×C22⋊D17—24no—
61C2×C22⋊D17—42no—
62C2×C22⋊D17—42no—
63C2×C22⋊D17—42no—
64C2×C22⋊D17—42no—
65C22×Dic17—21no—
66C22×Dic17C22⋊C412yes—
67C22×Dic17C2×D412yes—
68C2×D68—21no—
69C2×D68—41no—
70C2×D68—41no—
71C2×D68C4⋊C412yes—
72C2×C4×D17—42no—
73C2×C4×D17—44no—
74C2×#3624—44no—
75C2×#3624—44no—
76C2×#3624—44no—
77C4⋊C4×C17—41no—
78C22⋊C4×C17—42no—
79C42×C17—22no—
80C22⋊Dic17—44no—
81#63227—42no—
82#63227—44no—
83C4⋊Dic17—41no—
84#63225—42no—
85#63225—42no—
86C4×Dic17—22no—
87C4×Dic17—22no—
88C4×Dic17—41no—
89C4×Dic17D814yes—
90C4×Dic17SD1614yes—
91C4×Dic17C2×D411yescharacteristic
92C4×Dic17C2×Q811yescharacteristic
93C4×Dic17#4112yes—
94#8585—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17C2×C22⋊C411yescharacteristic
101C22×D17—22no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17—42no—
105C22×D17—42no—
106D4×C17—42no—
107D4×C17—42no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—84no—
115C2×C4×C17#10012yes—
116C2×C4×C17#10214yes—
117C2×C4×C17C2×C4⋊C411yescharacteristic
118C2×C4×C17#11711yescharacteristic
119C22⋊D17—44no—
120C22⋊D17—44no—
121C22⋊D17—44no—
122C22⋊D17—44no—
123C2×Dic17—22no—
124C2×Dic17—22no—
125C2×Dic17—41no—
126C2×Dic17—42no—
127C2×Dic17#10214yes—
128C2×Dic17#12112yes—
129C2×Dic17C22⋊Q812yes—
130C2×Dic17C2×D812yes—
131C2×Dic17C2×SD1612yes—
132C2×Dic17AGL(1,ℤ/8)12yes—
133C2×Dic17#13712yes—
134D68—42no—
135D68—42no—
136C4×D17—84no—
137#3624—84no—
138C4×#127—171no—
139#1842—172no—
140C2×#331—172no—
141#1296—172no—
142C22×C17—42no—
143C22×C17—44no—
144C22×C17—81no—
145C22×C17—82no—
146C22×C17—81no—
147C22×C17#38011yescharacteristic
148C22×C17C2×#10011yescharacteristic
149C22×C17#41311yescharacteristic
150C22×C17C2×#10212yes—
151C22×C17#41712yes—
152C2×D17—42no—
153C2×D17—44no—
154C2×D17—81no—
155C2×D17—82no—
156C2×D17—81no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—41no—
160C4×C17—81no—
161C4×C17#33114yescomplemented
162Dic17—24no—
163Dic17—81no—
164Dic17#45912yes—
165Dic17#46112yes—
166Dic17#47412yes—
167Dic17#47612yes—
168C2×#127—171no—
169C2×C4×D4—171no—
170C2×C4×D4—342no—
171C2×C4×D4#59511yescharacteristic
172C2×C4⋊C8—174no—
173C2×C4⋊C8—342no—
174C2×C4⋊C8—342no—
175#390—171no—
176#390—171no—
177#390—342no—
178#384—171no—
179C43—171no—
180#331—344no—
181C2×C17—82no—
182C2×C17C2×#33111yescharacteristic
183C2×C17#154712yes—
184C2×C17#156011yescharacteristic
185C2×C17#180011yescharacteristic
186C2×C17#180212yes—
187D17—82no—
188C22×D4—171no—
189C22×D4—342no—
190C22×D4C2×#59511yescharacteristic
191C23×C4—21no—
192C23×C4—341no—
193C22×C8—344no—
194#127—172no—
195#127—172no—
196#127—342no—
197C4×D4—42no—
198C4×D4—682no—
199C4×D4—684no—
200C4×D4—684no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—342no—
203C2×C4⋊C4C2×#59511yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—342no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4—341no—
208C2×C42—172no—
209C2×C42—171no—
210C2×C42—172no—
211C2×C42—171no—
212C2×C42C2×#59511yescharacteristic
213C4⋊C8—344no—
214C4⋊C8—344no—
215C4⋊C8—684no—
216C4⋊C8—684no—
217C17#858511yescharacteristic, complemented, Sylow
218C24—21no—
219C24—341no—
220C2×D4—21no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—172no—
224C2×D4—344no—
225C2×D4—344no—
226C2×D4—341no—
227C2×D4—681no—
228C2×D4—681no—
229C2×D4—682no—
230C2×D4—682no—
231C2×D4—682no—
232C2×D4—682no—
233C2×D4C4⋊#59512yes—
234C22×C4—21no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—172no—
238C22×C4—172no—
239C22×C4—341no—
240C22×C4—684no—
241C22×C4—682no—
242C22×C4C22⋊#59511yescharacteristic
243C22×C4C22⋊#59511yescharacteristic
244C22×C4C22×#59511yescharacteristic
245C2×C8—684no—
246C2×C8—684no—
247C2×C8—684no—
248C4⋊C4—41no—
249C4⋊C4—681no—
250C4⋊C4—682no—
251C4⋊C4—682no—
252C22⋊C4—42no—
253C22⋊C4—682no—
254C22⋊C4—684no—
255C22⋊C4—684no—
256C42—22no—
257C42—171no—
258C42—171no—
259C42—172no—
260C42—174no—
261C42—174no—
262C42—342no—
263C42—342no—
264C42—681no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—342no—
271C23—681no—
272C23—681no—
273C23—682no—
274C23—682no—
275C23C2×C22⋊#59511yescharacteristic
276D4—42no—
277D4—42no—
278D4—682no—
279D4—682no—
280D4—684no—
281D4—684no—
282D4—684no—
283D4—684no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4—172no—
292C2×C4—172no—
293C2×C4—172no—
294C2×C4—172no—
295C2×C4—172no—
296C2×C4—172no—
297C2×C4—174no—
298C2×C4—342no—
299C2×C4—342no—
300C2×C4—681no—
301C2×C4—682no—
302C2×C4—1364no—
303C2×C4#1058754714yes—
304C2×C4#1058755312yes—
305C2×C4C2×C4⋊#59511yescharacteristic
306C2×C4#1058767711yescharacteristic
307C8—1364no—
308C22—42no—
309C22—44no—
310C22—81no—
311C22—82no—
312C22—81no—
313C22—682no—
314C22—684no—
315C22—1361no—
316C22—1362no—
317C22—1361no—
318C22#1176375111yescharacteristic
319C22C2×#1058755311yescharacteristic
320C22#1176379511yescharacteristic
321C22C2×#1058754712yes—
322C22#1176380012yes—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—81no—
327C4—172no—
328C4—172no—
329C4—172no—
330C4—172no—
331C4—344no—
332C4—1361no—
333C4#1176366714yescomplemented
334C2—82no—
335C2—1362no—
336C2C2×#1176366711yescharacteristic
337C2#1454429512yes—
338C2#1454432111yescharacteristic
339C2#1454464811yescharacteristic
340C2#1454465212yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×D4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587626C22
cosocleC2×C4×Dic17C23
hypercenterC2×C4×D4#595
hypocenterC17#8585
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #14544717C2semidirect
3. #14543662C2semidirect
161. C4×C17#331semidirect
217. C17#8585semidirect
333. C4#11763667semidirect

Maximal subgroups

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

subgroupindex
#145447172
C2×#117636672
#145436622
C4×#105876262
#858517