Finite Group Databasebeta

← #17953549

Subgroups — #17953549

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—42no—
38#10587578—42no—
39C42×D17—24no—
40#10587543—24no—
41#10587543—42no—
42C4⋊#3624—24no—
43C4×#3624—24no—
44#10587502—22no—
45#10587502—22no—
46C23×D17C22⋊C411yescharacteristic
47C22×C4×C17—22no—
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22⋊C411yescharacteristic
50C22×C4×C17C22×C411yescharacteristic
51C22×Dic17—22no—
52C22×Dic17C22⋊C411yescharacteristic
53C22×Dic17C2×D411yescharacteristic
54C22×Dic17C2×D411yescharacteristic
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—22no—
62C2×C4×D17—41no—
63C2×C4×D17D812yes—
64C2×C4×D17SD1612yes—
65C2×#3624—24no—
66C2×#3624—42no—
67C2×#3624—42no—
68C2×#3624—42no—
69C2×#3624—42no—
70C2×#3624—42no—
71C4⋊C4×C17—24no—
72C4⋊C4×C17—42no—
73C4⋊C4×C17—42no—
74C42×C17—22no—
75#63227—44no—
76C4⋊Dic17—24no—
77C4⋊Dic17—42no—
78C4⋊Dic17—42no—
79C4×Dic17—22no—
80C4×Dic17—22no—
81C4×Dic17—42no—
82C4×Dic17—42no—
83C4×Dic17#4114yes—
84#8709—171nomaximal, Sylow
85C23×C17C2×C22⋊C411yescharacteristic
86C22×D17—21no—
87C22×D17—21no—
88C22×D17—21no—
89C22×D17—21no—
90C22×D17—21no—
91C22×D17—21no—
92C22×D17#10212yes—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17#10212yes—
101C2×C4×C17#10412yes—
102C2×C4×C17#11712yes—
103C2×C4×C17#13112yes—
104C2×C4×C17#13112yes—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—42no—
111C2×Dic17—42no—
112C2×Dic17—42no—
113C2×Dic17#12312yes—
114C2×Dic17#12312yes—
115C2×Dic17C2×D811yescharacteristic
116C2×Dic17C2×SD1611yescharacteristic
117C2×Dic17#13712yes—
118C4×D17—24no—
119C4×D17—24no—
120C4×D17—44no—
121C4×D17—44no—
122#3624—44no—
123#3624—82no—
124#2285—171no—
125#1797—171no—
126C4×C22⋊C8—171no—
127#1279—174no—
128C22×C17#38111yescharacteristic
129C22×C17#40811yescharacteristic
130C22×C17#40811yescharacteristic
131C22×C17C2×#10211yescharacteristic
132C22×C17#41911yescharacteristic
133C22×C17C2×#10411yescharacteristic
134C22×C17#42111yescharacteristic
135C2×D17—22no—
136C2×D17—22no—
137C2×D17—22no—
138C2×D17—41no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17—41no—
142C4×C17—22no—
143C4×C17—22no—
144C4×C17—82no—
145C4×C17#43314yes—
146Dic17—22no—
147Dic17—41no—
148Dic17—82no—
149Dic17#48012yes—
150Dic17#48312yes—
151C2×#117—341no—
152C22×C42—171no—
153C2×C4⋊C8—172no—
154C2×C22⋊C8—341no—
155C2×C22⋊C8—341no—
156C2×C4×C8—172no—
157#384—171no—
158#384#59511yescharacteristic
159#382—341no—
160#380—172no—
161#380—341no—
162C4×C4⋊C4—172no—
163C4×C4⋊C4—341no—
164#336—341no—
165#336—341no—
166C2×C17#148011yescharacteristic
167C2×C17#148511yescharacteristic
168C2×C17#149211yescharacteristic
169C2×C17#149711yescharacteristic
170C2×C17#181811yescharacteristic
171C2×C17#182311yescharacteristic
172C2×C17#182611yescharacteristic
173D17—42no—
174C23×C4—171no—
175C23×C4—171no—
176C23×C4—171no—
177C22×C8—342no—
178C22×C8—342no—
179#117—682no—
180#117—682no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—172no—
184C2×C4⋊C4—341no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4C2×#59512yes—
187C2×C22⋊C4—342no—
188C2×C42—171no—
189C2×C42—171no—
190C2×C42—171no—
191C2×C42—344no—
192C2×C42—342no—
193C2×C42C2×#59511yescharacteristic
194C4⋊C8—344no—
195C22⋊C8—344no—
196C22⋊C8—682no—
197C4×C8—344no—
198#95—342no—
199#95—342no—
200C17#870911yescharacteristic, complemented, Sylow
201C24—171no—
202C22×C4—22no—
203C22×C4—171no—
204C22×C4—171no—
205C22×C4—172no—
206C22×C4—172no—
207C22×C4—171no—
208C22×C4—342no—
209C22×C4—342no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—342no—
214C22×C4—342no—
215C22×C4—342no—
216C22×C4—681no—
217C22×C4C22⋊#59511yescharacteristic
218C22×C4C22⋊#59511yescharacteristic
219C22×C4C22×#59511yescharacteristic
220C2×C8—344no—
221C2×C8—682no—
222C2×C8—682no—
223C2×C8—682no—
224C2×C8—682no—
225C2×C8—682no—
226C4⋊C4—24no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—344no—
230C4⋊C4—682no—
231C4⋊C4—682no—
232C22⋊C4—684no—
233C42—22no—
234C42—174no—
235C42—342no—
236C42—342no—
237C42—682no—
238C42—682no—
239C23—172no—
240C23—341no—
241C23—341no—
242C23—341no—
243C23—341no—
244C23—341no—
245C23—341no—
246C23C2×C22⋊#59511yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—172no—
255C2×C4—171no—
256C2×C4—171no—
257C2×C4—172no—
258C2×C4—172no—
259C2×C4—344no—
260C2×C4—344no—
261C2×C4—341no—
262C2×C4—341no—
263C2×C4—341no—
264C2×C4—341no—
265C2×C4—341no—
266C2×C4—684no—
267C2×C4—684no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4#1058754712yes—
272C2×C4#1058755012yes—
273C2×C4#1058767712yes—
274C2×C4#1058767812yes—
275C2×C4#1058768012yes—
276C8—684no—
277C8—1362no—
278C22—342no—
279C22—342no—
280C22—342no—
281C22—681no—
282C22—681no—
283C22—681no—
284C22—681no—
285C22#1176375211yescharacteristic
286C22#1176378311yescharacteristic
287C22#1176378411yescharacteristic
288C22C2×#1058754711yescharacteristic
289C22#1176380211yescharacteristic
290C22C2×#1058755011yescharacteristic
291C22#1176380711yescharacteristic
292C4—22no—
293C4—22no—
294C4—82no—
295C4—172no—
296C4—172no—
297C4—342no—
298C4—681no—
299C4—1362no—
300C4#1176383414yes—
301C2—682no—
302C2#1454409011yescharacteristic
303C2#1454410711yescharacteristic
304C2#1454413111yescharacteristic
305C2#1454414811yescharacteristic
306C2#1454467811yescharacteristic
307C2#1454468711yescharacteristic
308C2#1454469111yescharacteristic
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#8709
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
200. C17#8709semidirect

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
#870917