Finite Group Databasebeta

← #17953552

Subgroups — #17953552

Order 4352.

540 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544639C211yesmaximal, characteristic
2C4×#10587543C211yesmaximal, characteristic
3#14543620C214yesmaximal, complemented
4#14531189C211yesmaximal, characteristic
5C2×C4⋊#3624C2212yes—
6C2×#10587543—21no—
7C2×#10587543—21no—
8C2×C4×#3624C2212yes—
9#11763675—21no—
10#11763675—21no—
11C2×#10587578—21no—
12C2×C42×D17C2211yescharacteristic
13#11762679C411yescharacteristic
14#11762674—21no—
15#11762666—21no—
16#11762666C412yes—
17C4⋊C4×Dic17—21no—
18C4⋊C4×Dic17C2212yes—
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—42no—
41C4⋊#3624—24no—
42C4×#3624—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×D17SD1612yes—
63C2×C4×D17Q1612yes—
64C2×#3624—24no—
65C2×#3624—42no—
66C2×#3624—42no—
67C2×#3624—42no—
68C2×#3624—42no—
69C2×#3624—42no—
70C4⋊C4×C17—24no—
71C4⋊C4×C17—42no—
72C4⋊C4×C17—42no—
73C42×C17—22no—
74#63227—44no—
75C4⋊Dic17—24no—
76C4⋊Dic17—42no—
77C4⋊Dic17—42no—
78C4×Dic17—22no—
79C4×Dic17—22no—
80C4×Dic17—44no—
81C4×Dic17#4114yes—
82#8710—171nomaximal, Sylow
83C23×C17C2×C22⋊C411yescharacteristic
84C22×D17—21no—
85C22×D17—21no—
86C22×D17—21no—
87C22×D17—21no—
88C22×D17—21no—
89C22×D17—21no—
90C22×D17#10312yes—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17#10312yes—
99C2×C4×C17#10412yes—
100C2×C4×C17#11712yes—
101C2×C4×C17#13112yes—
102C2×C4×C17#13112yes—
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#12312yes—
112C2×Dic17#12312yes—
113C2×Dic17C2×SD1611yescharacteristic
114C2×Dic17C2×Q1611yescharacteristic
115C2×Dic17AGL(1,ℤ/8)12yes—
116C4×D17—24no—
117C4×D17—24no—
118C4×D17—44no—
119C4×D17—44no—
120#3624—44no—
121#3624—82no—
122#2285—171no—
123#1797—171no—
124C4×C22⋊C8—171no—
125#1279—174no—
126C22×C17#38111yescharacteristic
127C22×C17#40811yescharacteristic
128C22×C17#40811yescharacteristic
129C22×C17C2×#10311yescharacteristic
130C22×C17#41811yescharacteristic
131C22×C17C2×#10411yescharacteristic
132C22×C17#42111yescharacteristic
133C2×D17—22no—
134C2×D17—22no—
135C2×D17—22no—
136C2×D17—41no—
137C2×D17—41no—
138C2×D17—41no—
139C2×D17—41no—
140C4×C17—22no—
141C4×C17—22no—
142C4×C17—82no—
143C4×C17#43314yes—
144Dic17—22no—
145Dic17—41no—
146Dic17—82no—
147Dic17#48112yes—
148Dic17#48412yes—
149C2×#117—341no—
150C22×C42—171no—
151C2×C4⋊C8—172no—
152C2×C22⋊C8—341no—
153C2×C22⋊C8—341no—
154C2×C4×C8—172no—
155#384—171no—
156#384#59511yescharacteristic
157#382—341no—
158#380—172no—
159#380—341no—
160C4×C4⋊C4—172no—
161C4×C4⋊C4—341no—
162#336—341no—
163#336—341no—
164C2×C17#148111yescharacteristic
165C2×C17#148411yescharacteristic
166C2×C17#149311yescharacteristic
167C2×C17#149611yescharacteristic
168C2×C17#181811yescharacteristic
169C2×C17#182411yescharacteristic
170C2×C17#182611yescharacteristic
171D17—42no—
172C23×C4—171no—
173C23×C4—171no—
174C23×C4—171no—
175C22×C8—342no—
176C22×C8—342no—
177#117—684no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—172no—
181C2×C4⋊C4—341no—
182C2×C4⋊C4—341no—
183C2×C4⋊C4C2×#59512yes—
184C2×C22⋊C4—342no—
185C2×C42—171no—
186C2×C42—171no—
187C2×C42—171no—
188C2×C42—344no—
189C2×C42—342no—
190C2×C42C2×#59511yescharacteristic
191C4⋊C8—344no—
192C22⋊C8—344no—
193C22⋊C8—682no—
194C4×C8—344no—
195#95—342no—
196#95—342no—
197C17#871011yescharacteristic, complemented, Sylow
198C24—171no—
199C22×C4—22no—
200C22×C4—171no—
201C22×C4—171no—
202C22×C4—172no—
203C22×C4—172no—
204C22×C4—171no—
205C22×C4—342no—
206C22×C4—342no—
207C22×C4—342no—
208C22×C4—342no—
209C22×C4—342no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—681no—
214C22×C4C22⋊#59511yescharacteristic
215C22×C4C22⋊#59511yescharacteristic
216C22×C4C22×#59511yescharacteristic
217C2×C8—344no—
218C2×C8—682no—
219C2×C8—682no—
220C2×C8—682no—
221C2×C8—682no—
222C2×C8—682no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—344no—
227C4⋊C4—682no—
228C4⋊C4—682no—
229C22⋊C4—684no—
230C42—22no—
231C42—174no—
232C42—342no—
233C42—342no—
234C42—684no—
235C23—172no—
236C23—341no—
237C23—341no—
238C23—341no—
239C23—341no—
240C23—341no—
241C23—341no—
242C23C2×C22⋊#59511yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—172no—
251C2×C4—171no—
252C2×C4—171no—
253C2×C4—172no—
254C2×C4—172no—
255C2×C4—344no—
256C2×C4—344no—
257C2×C4—341no—
258C2×C4—341no—
259C2×C4—341no—
260C2×C4—341no—
261C2×C4—341no—
262C2×C4—684no—
263C2×C4—684no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4#1058754812yes—
268C2×C4#1058754912yes—
269C2×C4#1058767712yes—
270C2×C4#1058767812yes—
271C2×C4#1058768012yes—
272C8—684no—
273C8—1362no—
274C22—342no—
275C22—342no—
276C22—342no—
277C22—681no—
278C22—681no—
279C22—681no—
280C22—681no—
281C22#1176375211yescharacteristic
282C22#1176378311yescharacteristic
283C22#1176378411yescharacteristic
284C22C2×#1058754911yescharacteristic
285C22#1176380111yescharacteristic
286C22C2×#1058754811yescharacteristic
287C22#1176380811yescharacteristic
288C4—22no—
289C4—22no—
290C4—82no—
291C4—172no—
292C4—172no—
293C4—342no—
294C4—681no—
295C4—1362no—
296C4#1176383414yes—
297C2—682no—
298C2#1454409311yescharacteristic
299C2#1454410411yescharacteristic
300C2#1454413411yescharacteristic
301C2#1454414511yescharacteristic
302C2#1454467811yescharacteristic
303C2#1454468811yescharacteristic
304C2#1454469011yescharacteristic
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#8710
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #14543620C2semidirect
197. C17#8710semidirect

Maximal subgroups

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

subgroupindex
#145446392
C4×#105875432
#145436202
#145311892
#871017