Finite Group Databasebeta

← #17949935

Subgroups — #17949935

Order 4352.

646 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544639C212yesmaximal
2#14543605C212yesmaximal, complemented
3#14543604C212yesmaximal, complemented
4D17×#391C211yesmaximal, characteristic
5C2×C4⋊#3624—22no—
6C2×#10587543—22no—
7#11763675C2214yes—
8C22×Q8×D17C411yescharacteristic
9C2×C4⋊C4×D17C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11#11762716C2211yescharacteristic
12#11762715C411yescharacteristic
13#11762714C2211yescharacteristic
14#11762713—22no—
15#95×D17—22no—
16C17×#391C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C2×Q8×D17—22no—
19C2×Q8×D17—22no—
20C2×Q8×D17—42no—
21C2×Q8×D17—42no—
22C22×C4×D17—21no—
23C22×C4×D17—21no—
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D412yes—
26C22×Dic34—21no—
27C22×Dic34C2×C411yescharacteristic
28C22×#3624—24no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×#63225—21no—
33C2×C4×Dic17D412yes—
34C2×C4×Dic17C2311yescharacteristic
35C4⋊C4×D17—42no—
36C42×D17—42no—
37#10587543—44no—
38C4⋊#3624—44no—
39C17×#95—22no—
40#10587502—22no—
41#10587502—22no—
42#10587502—22no—
43C23×D17C22⋊C411yescharacteristic
44C2×Q8×C17—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17M1612yes—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22×C411yescharacteristic
52C22×Dic17—21no—
53C22×Dic17—21no—
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D412yes—
56Q8×D17—48no—
57Q8×D17—48no—
58C2×C4×D17—22no—
59C2×C4×D17—21no—
60C2×C4×D17—21no—
61C2×C4×D17—22no—
62C2×C4×D17—24no—
63C2×C4×D17—42no—
64C2×C4×D17—42no—
65C2×C4×D17—42no—
66C2×C4×D17—42no—
67C2×C4×D17—44no—
68C2×C4×D17—42no—
69C2×C4×D17SD1614yes—
70C2×C4×D17Q1614yes—
71C2×Dic34—24no—
72C2×Dic34—21no—
73C2×Dic34—42no—
74C2×Dic34—42no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34M1612yes—
78C2×#3624—44no—
79C2×#3624—44no—
80C2×#3624—44no—
81C4⋊C4×C17—41no—
82C42×C17—41no—
83C4⋊Dic17—41no—
84#63225—42no—
85C4×Dic17—24no—
86C4×Dic17—41no—
87#7643—171nomaximal, Sylow
88C23×C17C2×C22⋊C411yescharacteristic
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—22no—
92C22×D17#10012yes—
93C22×D17#10314yes—
94Q8×C17—42no—
95Q8×C17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—42no—
105C2×C4×C17#10314yes—
106C2×C4×C17C2×M1611yescharacteristic
107C2×C4×C17C2×M1611yescharacteristic
108C2×Dic17—21no—
109C2×Dic17—22no—
110C2×Dic17—21no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117C2×Dic17#9912yes—
118C2×Dic17C2×SD1612yes—
119C2×Dic17C2×Q1612yes—
120C4×D17—24no—
121C4×D17—24no—
122C4×D17—44no—
123C4×D17—44no—
124C4×D17—48no—
125C4×D17—84no—
126Dic34—48no—
127Dic34—42no—
128Dic34—42no—
129#3624—84no—
130C2×#391—171no—
131#1797—172no—
132#1272—172no—
133#1272—172no—
134C22×C17#40711yescharacteristic
135C22×C17C2×#9911yescharacteristic
136C22×C17C2×#10011yescharacteristic
137C22×C17C2×#10312yes—
138C22×C17#42112yes—
139C2×D17—22no—
140C2×D17—24no—
141C2×D17—24no—
142C2×D17—22no—
143C2×D17#33314yes—
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—24no—
147C4×C17—42no—
148C4×C17—81no—
149Dic17—22no—
150Dic17—22no—
151Dic17—22no—
152Dic17—42no—
153Dic17—81no—
154C23×Q8—171no—
155C22×C4⋊C4—171no—
156C22×C42—171no—
157C2×C4⋊C8—342no—
158C2×C22⋊C8—342no—
159#391—171no—
160#391—171no—
161#391—171no—
162#391—342no—
163#391#59511yescharacteristic
164C2×#95—342no—
165#336—174no—
166C2×C17#146211yescharacteristic
167C2×C17#148712yes—
168C2×C17#152012yes—
169C2×C17#153311yescharacteristic
170C2×C17C2×#33311yescharacteristic
171D17—24no—
172C22×Q8—171no—
173C22×Q8—341no—
174C22×Q8—342no—
175C22×Q8—342no—
176C22×Q8—682no—
177C22×Q8—682no—
178C22×Q8C2×#59511yescharacteristic
179C23×C4—172no—
180C23×C4—171no—
181C23×C4—341no—
182C23×C4—341no—
183C22×C8—344no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4—682no—
187C2×C4⋊C4C2×#59511yescharacteristic
188C2×C42—172no—
189C2×C42—171no—
190C2×C42—682no—
191C2×C42C2×#59511yescharacteristic
192C4⋊C8—684no—
193C22⋊C8—684no—
194#95—22no—
195#95—342no—
196#95—342no—
197#95—342no—
198C17#764311yescharacteristic, complemented, Sylow
199C24—171no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—172no—
204C2×Q8—344no—
205C2×Q8—341no—
206C2×Q8—681no—
207C2×Q8—681no—
208C2×Q8—682no—
209C2×Q8—682no—
210C2×Q8—688no—
211C2×Q8—688no—
212C2×Q8#6324312yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—171no—
216C22×C4—172no—
217C22×C4—174no—
218C22×C4—174no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—344no—
222C22×C4—341no—
223C22×C4—341no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4—682no—
227C22×C4—682no—
228C22×C4—682no—
229C22×C4—682no—
230C22×C4—682no—
231C22×C4—684no—
232C22×C4C22⋊#59512yes—
233C22×C4C22×#59511yescharacteristic
234C2×C8—684no—
235C2×C8—684no—
236C2×C8—684no—
237C4⋊C4—41no—
238C4⋊C4—681no—
239C4⋊C4—682no—
240C42—41no—
241C42—344no—
242C42—681no—
243C23—172no—
244C23—174no—
245C23—341no—
246C23—341no—
247C23—342no—
248C23C2×C22⋊#59511yescharacteristic
249Q8—42no—
250Q8—42no—
251Q8—682no—
252Q8—682no—
253Q8—688no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—172no—
264C2×C4—172no—
265C2×C4—172no—
266C2×C4—344no—
267C2×C4—344no—
268C2×C4—341no—
269C2×C4—342no—
270C2×C4—341no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—684no—
274C2×C4—684no—
275C2×C4—688no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—681no—
279C2×C4—682no—
280C2×C4—1364no—
281C2×C4#1058754914yes—
282C2×C4C2×#6324311yescharacteristic
283C2×C4D17⋊M1611yescharacteristic
284C8—1364no—
285C22—174no—
286C22—344no—
287C22—342no—
288C22—342no—
289C22—344no—
290C22#1176378011yescharacteristic
291C22D17⋊#9911yescharacteristic
292C22D17⋊#10011yescharacteristic
293C22C2×#1058754912yes—
294C22#1176380612yes—
295C4—24no—
296C4—21no—
297C4—21no—
298C4—42no—
299C4—81no—
300C4—342no—
301C4—342no—
302C4—342no—
303C4—682no—
304C4—1361no—
305C2—344no—
306C2#1454403011yescharacteristic
307C2#1454411312yes—
308C2#1454422212yes—
309C2#1454426411yescharacteristic
310C2D17⋊#33311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7643
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #14543605C2semidirect
3. #14543604C2semidirect
198. C17#7643semidirect

Maximal subgroups

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

subgroupindex
#145446392
#145436052
#145436042
D17×#3912
#764317