Finite Group Databasebeta

← #16944002

Subgroups — #16944002

Order 4352.

426 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533101C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526731C214yesmaximal, complemented
4C17×#1466C211yesmaximal, characteristic
5C22×C4⋊Dic17C411yescharacteristic
6C2×C22⋊#3622—21no—
7#11762748—21no—
8#11762745—21no—
9#11762707—21no—
10#11762689C412yes—
11#11762681C2212yes—
12#11762669—21no—
13#11762669C411yescharacteristic
14#11762535—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#432—21no—
18C17×#323—21no—
19C17×#323C2212yes—
20C22×D4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17C2×C412yes—
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C22⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17C2311yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17C2×C412yes—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—41no—
36C2×C4⋊Dic17—41no—
37C2×#63225—22no—
38C22×#3622—41no—
39C17×C4⋊C8—41no—
40C17×C22⋊C8—22no—
41C17×#97—41no—
42#10587502—22no—
43#10587502—21no—
44#10587502—22no—
45C22⋊#3622—22no—
46C22⋊#3622—41no—
47C24×C17C22⋊C411yescharacteristic
48C24×C17C22×C411yescharacteristic
49C2×D4×C17—22no—
50C2×D4×C17—22no—
51C2×D4×C17—41no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—42no—
57C22×C4×C17—42no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—42no—
66C22×Dic17—82no—
67C2×C8×C17—42no—
68C4⋊C4×C17—41no—
69C22⋊C4×C17—41no—
70C22⋊C4×C17—41no—
71C42×C17—21no—
72C42×C17—21no—
73C22⋊Dic17—24no—
74C22⋊Dic17—82no—
75C4⋊Dic17—42no—
76C4⋊Dic17—42no—
77C4⋊Dic17—41no—
78C4⋊Dic17—41no—
79#63225—82no—
80C2×#3622—42no—
81C2×#3622—82no—
82#11576—171nomaximal, Sylow
83C23×C17—21no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—21no—
87C23×C17—41no—
88C23×C17—41no—
89C23×C17—41no—
90C23×C17—41no—
91C23×C17#10112yes—
92C23×C17C2×C22⋊C411yescharacteristic
93C23×C17#13112yes—
94D4×C17—44no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—81no—
109C2×C4×C17#9911yescharacteristic
110C2×C4×C17#9911yescharacteristic
111C2×Dic17—42no—
112C2×Dic17—42no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—84no—
118C8×C17—82no—
119#3622—82no—
120#2487—171no—
121#1466D1711yescharacteristic
122#1459—171no—
123#1326—174no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—22no—
127C22×C17—22no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—81no—
136C22×C17#40811yescharacteristic
137C22×C17C2×#9911yescharacteristic
138C22×C17C2×#10111yescharacteristic
139C4×C17—21no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17—41no—
143C4×C17—41no—
144C4×C17#35212yes—
145Dic17—82no—
146Dic17—82no—
147C2×C4×D4C2×D1711yescharacteristic
148C22×C4⋊C4—171no—
149#432—21no—
150C2×C22⋊C8—341no—
151#400—341no—
152#399—341no—
153#389—341no—
154#386—172no—
155#385—172no—
156#380—171no—
157#380—341no—
158#324—341no—
159#323—21no—
160#323—172no—
161#323C2×D1712yes—
162C2×C17—21no—
163C2×C17—21no—
164C2×C17—42no—
165C2×C17—42no—
166C2×C17#147211yescharacteristic
167C2×C17C2×#35211yescharacteristic
168C2×C17#212211yescharacteristic
169C22×D4C4×D1711yescharacteristic
170C23×C4—172no—
171C23×C4C4×D1711yescharacteristic
172C23×C4D6811yescharacteristic
173C22×C8—681no—
174C4×D4—41no—
175C4×D4—41no—
176C2×C4⋊C4—341no—
177C2×C4⋊C4—341no—
178C2×C4⋊C4—341no—
179C2×C4⋊C4—342no—
180C2×C4⋊C4—681no—
181C2×C4⋊C4—681no—
182C2×C4⋊C4D6811yescharacteristic
183C2×C22⋊C4—172no—
184C2×C22⋊C4—342no—
185C2×C22⋊C4C22⋊D1711yescharacteristic
186C2×C22⋊C4C22×D1711yescharacteristic
187C2×C42C22⋊D1711yescharacteristic
188C4⋊C8—41no—
189C22⋊C8—22no—
190C22⋊C8—342no—
191C22⋊C8—681no—
192#97—41no—
193#95—342no—
194#95—341no—
195#95—342no—
196C17#1157611yescharacteristic, complemented, Sylow
197C24#6322711yescharacteristic
198C24C2×C4×D1711yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—342no—
209C22×C4—342no—
210C22×C4—682no—
211C22×C4—1362no—
212C22×C4#6322711yescharacteristic
213C22×C4#6322711yescharacteristic
214C22×C4#6322711yescharacteristic
215C22×C4C2×D6811yescharacteristic
216C22×C4C2×C22⋊D1711yescharacteristic
217C2×C8—42no—
218C2×C8—682no—
219C2×C8—1362no—
220C4⋊C4—41no—
221C4⋊C4—682no—
222C4⋊C4—682no—
223C4⋊C4—681no—
224C4⋊C4—681no—
225C4⋊C4—1362no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—344no—
229C22⋊C4—1362no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23#1058749512yes—
241C23#1058759312yes—
242C23C2×#6322711yescharacteristic
243D4—44no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4—682no—
261C2×C4—682no—
262C2×C4—682no—
263C2×C4—682no—
264C2×C4—1364no—
265C2×C4#1058747711yescharacteristic
266C2×C4#1058747711yescharacteristic
267C8—82no—
268C8—1362no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#1176277711yescharacteristic
282C22C2×#1058747711yescharacteristic
283C22C2×#1058749511yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—1362no—
290C4—1362no—
291C4#1176260012yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2#1452714911yescharacteristic
297C2C2×#1176260011yescharacteristic
298C2#1453043311yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1466C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#1466D17
hypocenterC17#11576
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #14526731C2semidirect
196. C17#11576semidirect

Maximal subgroups

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

subgroupindex
#145331012
#145271022
#145267312
C17×#14662
#1157617