Finite Group Databasebeta

← #17949775

Subgroups — #17949775

Order 4352.

654 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 124 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#14544638C212yesmaximal, complemented
2#14543605C212yesmaximal, complemented
3#14543601C212yesmaximal, complemented
4#14531272C211yesmaximal, characteristic
5C2×C4⋊#3624—22no—
6C2×C22⋊#3624—22no—
7#11763675C2214yes—
8C22×#63252C411yescharacteristic
9C22×C4×Dic17C2211yescharacteristic
10#11762714C411yescharacteristic
11#11762714C2211yescharacteristic
12#11762710C411yescharacteristic
13#11762710C2211yescharacteristic
14#11762686—22no—
15C4×#63227—22no—
16C2×C17×#117C411yescharacteristic
17C23×C4×C17C2×C411yescharacteristic
18C22×C22⋊D17—21no—
19C23×Dic17D412yes—
20C2×#63252—22no—
21C2×#63252—42no—
22C2×#63252—42no—
23C2×#63252C814yescomplemented
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C22×Dic34C2×C411yescharacteristic
27C22×#3624—24no—
28C17×#117—42no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C42×C17—21no—
33C2×#63227—22no—
34C2×#63227—22no—
35C2×C4×Dic17—42no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17D412yes—
38C2×C4×Dic17C2311yescharacteristic
39C22⋊#3624—44no—
40C4⋊#3624—44no—
41#10587502—22no—
42#10587502—22no—
43C24×C17C22⋊C411yescharacteristic
44C23×D17—21no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C22⋊C411yescharacteristic
50C22×C4×C17C2×C812yes—
51C22×C4×C17M1612yes—
52C22×C4×C17C22×C411yescharacteristic
53C2×C22⋊D17—24no—
54C2×C22⋊D17—42no—
55C2×C22⋊D17—42no—
56C22×Dic17—21no—
57C22×Dic17—22no—
58C22×Dic17—22no—
59C22×Dic17—42no—
60C22×Dic17—42no—
61C22×Dic17C22⋊C412yes—
62C22×Dic17C2×D412yes—
63#63252—48no—
64#63252—48no—
65C2×D68—21no—
66C2×D68—41no—
67C2×D68—41no—
68C2×D68C2×C812yes—
69C2×C4×D17—24no—
70C2×C4×D17—42no—
71C2×C4×D17—42no—
72C2×Dic34—21no—
73C2×Dic34—41no—
74C2×Dic34—41no—
75C2×Dic34C2×C812yes—
76C2×#3624—44no—
77C2×#3624—44no—
78C2×#3624—44no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—42no—
82C42×C17—42no—
83#63227—44no—
84#63227—42no—
85#63227—42no—
86C4×Dic17—24no—
87C4×Dic17—41no—
88C4×Dic17—41no—
89#7597—171nomaximal, Sylow
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17C22⋊C812yes—
95C23×C17C2×C22⋊C411yescharacteristic
96C22×D17—22no—
97C22×D17—41no—
98C22×D17—41no—
99C22×D17—42no—
100C22×D17—42no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—24no—
104C2×C4×C17—24no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17C22⋊C812yes—
110C2×C4×C17#9912yes—
111C2×C4×C17#10012yes—
112C2×C4×C17C22×C811yescharacteristic
113C2×C4×C17C2×M1611yescharacteristic
114C22⋊D17—48no—
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—44no—
122C2×Dic17—44no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17AGL(1,ℤ/8)12yes—
126C2×Dic17#13712yes—
127D68—42no—
128D68—42no—
129C4×D17—48no—
130Dic34—42no—
131Dic34—42no—
132#3624—84no—
133#2297—171no—
134#1797—172no—
135#1272—172no—
136#1270—172no—
137C22×C17—22no—
138C22×C17—24no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—42no—
142C22×C17C2×C22⋊C811yescharacteristic
143C22×C17C2×#9911yescharacteristic
144C22×C17C2×#10011yescharacteristic
145C22×C17#41712yes—
146C22×C17#42112yes—
147C2×D17—42no—
148C2×D17—44no—
149C2×D17—81no—
150C2×D17—81no—
151C2×D17—82no—
152C4×C17—21no—
153C4×C17—21no—
154C4×C17—81no—
155C4×C17—81no—
156C4×C17#32314yes—
157Dic17—22no—
158Dic17—22no—
159Dic17—42no—
160Dic17—42no—
161C22×#41—171no—
162C2×#117#59511yescharacteristic
163C22×C42—171no—
164C2×C4⋊C8—342no—
165C2×C22⋊C8—342no—
166#391—171no—
167#391—171no—
168#390—171no—
169#390—171no—
170#386—342no—
171C4×C22⋊C4—342no—
172#336—174no—
173C2×C17—42no—
174C2×C17C2×#32311yescharacteristic
175C2×C17#147712yes—
176C2×C17#151312yes—
177C2×C17#153611yescharacteristic
178C2×C17#156011yescharacteristic
179D17—82no—
180C2×#41—174no—
181C2×#41—342no—
182C2×#41—682no—
183C2×#41—682no—
184C22×Q8—171no—
185C22×D4—171no—
186C22×D4—341no—
187C23×C4—172no—
188C23×C4—341no—
189C23×C4C2×#59511yescharacteristic
190C22×C8—344no—
191#117—42no—
192C2×C4⋊C4C2×#59511yescharacteristic
193C2×C4⋊C4C2×#59511yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4—342no—
197C2×C42—21no—
198C2×C42—172no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—682no—
202C4⋊C8—684no—
203C22⋊C8—684no—
204#95—342no—
205#95—342no—
206C17#759711yescharacteristic, complemented, Sylow
207C24—341no—
208C24C22⋊#59511yescharacteristic
209#41—688no—
210#41—688no—
211C2×Q8—172no—
212C2×Q8—341no—
213C2×Q8—681no—
214C2×Q8—681no—
215C2×D4—172no—
216C2×D4—344no—
217C2×D4—341no—
218C2×D4—681no—
219C2×D4—681no—
220C2×D4—682no—
221C2×D4—682no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—172no—
227C22×C4—172no—
228C22×C4—342no—
229C22×C4—342no—
230C22×C4—341no—
231C22×C4—344no—
232C22×C4—682no—
233C22×C4—682no—
234C22×C4—682no—
235C22×C4—682no—
236C22×C4#6324212yes—
237C22×C4#6324312yes—
238C22×C4C22⋊#59511yescharacteristic
239C22×C4C22×#59511yescharacteristic
240C2×C8—684no—
241C2×C8—684no—
242C2×C8—684no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—682no—
247C22⋊C4—682no—
248C22⋊C4—684no—
249C42—42no—
250C42—344no—
251C42—681no—
252C42—681no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—342no—
258C23—681no—
259C23—681no—
260C23—682no—
261C23—682no—
262C23#1058754312yes—
263C23C2×C22⋊#59511yescharacteristic
264Q8—682no—
265Q8—682no—
266D4—682no—
267D4—682no—
268D4—688no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—172no—
278C2×C4—172no—
279C2×C4—342no—
280C2×C4—342no—
281C2×C4—342no—
282C2×C4—342no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—684no—
286C2×C4—684no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—688no—
290C2×C4#1058753912yes—
291C2×C4#1058754312yes—
292C2×C4#1058755312yes—
293C2×C4C2×#6324211yescharacteristic
294C2×C4C2×#6324311yescharacteristic
295C8—1364no—
296C22—22no—
297C22—24no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—682no—
302C22—684no—
303C22—1362no—
304C22—1361no—
305C22—1361no—
306C22C2×#1058754311yescharacteristic
307C22C2×#1058753911yescharacteristic
308C22C2×#1058755311yescharacteristic
309C22#1176380012yes—
310C22#1176380612yes—
311C4—21no—
312C4—21no—
313C4—81no—
314C4—81no—
315C4—342no—
316C4—342no—
317C4—682no—
318C4—682no—
319C4#1176365314yes—
320C2—42no—
321C2—1362no—
322C2C2×#1176365311yescharacteristic
323C2#1454408112yes—
324C2#1454420112yes—
325C2#1454427211yescharacteristic
326C2#1454432111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC23×C17C2×C22⋊C4
prime radicalC22×#63252C4
cosocleC2×C4×Dic17C23
hypercenterC2×#117#595
hypocenterC17#7597
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #14544638C2semidirect
2. #14543605C2semidirect
3. #14543601C2semidirect
23. C2×#63252C8semidirect
206. C17#7597semidirect

Maximal subgroups

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

subgroupindex
#145446382
#145436052
#145436012
#145312722
#759717