Finite Group Databasebeta

← #17953595

Subgroups — #17953595

Order 4352.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#14544641C211yesmaximal, characteristic
2#14544641C211yesmaximal, characteristic
3#14543662C214yesmaximal, complemented
4D17×#384C211yesmaximal, characteristic
5C2×#10587544C2212yes—
6C2×#10587544C2212yes—
7C2×#10587543—21no—
8C2×#10587543—21no—
9#11763675—21no—
10#11763675—21no—
11C2×C4⋊C4×D17—21no—
12C2×C4⋊C4×D17—21no—
13C2×C4⋊C4×D17C412yes—
14C2×C42×D17C2211yescharacteristic
15#11762679C411yescharacteristic
16#11762678—21no—
17#11762678—21no—
18#11762678C2212yes—
19C17×#384C411yescharacteristic
20C22×C4×D17—22no—
21C22×C4×D17C2×C411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C42×C17C2×C411yescharacteristic
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C412yes—
33C2×#63225—22no—
34C2×#63225—22no—
35C2×#63225—22no—
36C2×C4×Dic17D411yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17C2311yescharacteristic
39C4⋊C4×D17—44no—
40C4⋊C4×D17—44no—
41C4⋊C4×D17—42no—
42C4⋊C4×D17—42no—
43C42×D17—24no—
44#10587544—24no—
45#10587544—24no—
46#10587543—42no—
47#10587543—42no—
48C23×D17C22⋊C411yescharacteristic
49C22×C4×C17—22no—
50C22×C4×C17C22⋊C411yescharacteristic
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—22no—
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D411yescharacteristic
56C22×Dic17C2×D411yescharacteristic
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—22no—
62C2×C4×D17—22no—
63C2×C4×D17—22no—
64C2×C4×D17—44no—
65C2×C4×D17—44no—
66C2×C4×D17—44no—
67C2×C4×D17D812yes—
68C2×C4×D17SD1612yes—
69C2×C4×D17SD1612yes—
70C2×C4×D17Q1612yes—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76C2×#3624—42no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C42×C17—22no—
81C4⋊Dic17—42no—
82C4⋊Dic17—42no—
83C4⋊Dic17—42no—
84#63225—44no—
85#63225—42no—
86#63225—42no—
87#63225—44no—
88C4×Dic17—22no—
89C4×Dic17—22no—
90C4×Dic17#4114yes—
91#8724—171nomaximal, Sylow
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17#10112yes—
98C22×D17#10212yes—
99C22×D17#10312yes—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17#10212yes—
110C2×C4×C17#10312yes—
111C2×C4×C17#11712yes—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17#10012yes—
118C2×Dic17#12312yes—
119C2×Dic17#12312yes—
120C2×Dic17C2×D811yescharacteristic
121C2×Dic17C2×SD1611yescharacteristic
122C2×Dic17C2×SD1611yescharacteristic
123C2×Dic17C2×Q1611yescharacteristic
124C2×Dic17AGL(1,ℤ/8)12yes—
125C2×Dic17#13712yes—
126C4×D17—28no—
127C4×D17—24no—
128C4×D17—24no—
129C4×D17—24no—
130C4×D17—24no—
131C4×D17—88no—
132#3624—82no—
133#3624—82no—
134C2×#384—171no—
135#1797—171no—
136#1797—171no—
137#1296—174no—
138C22×C17#38111yescharacteristic
139C22×C17C2×#10011yescharacteristic
140C22×C17C2×#10111yescharacteristic
141C22×C17C2×#10211yescharacteristic
142C22×C17C2×#10311yescharacteristic
143C22×C17#41811yescharacteristic
144C22×C17#41911yescharacteristic
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—22no—
150C2×D17—22no—
151C2×D17#33214yes—
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—22no—
155C4×C17—82no—
156Dic17—22no—
157Dic17—82no—
158Dic17#48012yes—
159Dic17#48112yes—
160Dic17#48312yes—
161Dic17#48412yes—
162C22×C4⋊C4—172no—
163C22×C4⋊C4—341no—
164C22×C4⋊C4—341no—
165C22×C42—171no—
166C2×C4⋊C8—172no—
167C2×C4⋊C8—172no—
168C2×C22⋊C8—341no—
169C2×C22⋊C8—341no—
170#384—172no—
171#384—171no—
172#384—341no—
173#384—341no—
174#384#59511yescharacteristic
175#336—341no—
176#336—341no—
177C2×C17C2×#33211yescharacteristic
178C2×C17#154511yescharacteristic
179C2×C17#154811yescharacteristic
180C2×C17#155611yescharacteristic
181C2×C17#182111yescharacteristic
182C2×C17#182311yescharacteristic
183C2×C17#182411yescharacteristic
184D17—24no—
185C23×C4—171no—
186C23×C4—171no—
187C23×C4—171no—
188C23×C4—342no—
189C22×C8—342no—
190C22×C8—342no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—172no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4—682no—
200C2×C4⋊C4—682no—
201C2×C4⋊C4—684no—
202C2×C4⋊C4—684no—
203C2×C4⋊C4C2×#59512yes—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42—171no—
207C2×C42—344no—
208C2×C42C2×#59511yescharacteristic
209C4⋊C8—344no—
210C4⋊C8—344no—
211C22⋊C8—682no—
212C22⋊C8—682no—
213C17#872411yescharacteristic, complemented, Sylow
214C24—171no—
215C22×C4—22no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—172no—
220C22×C4—172no—
221C22×C4—172no—
222C22×C4—172no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—342no—
228C22×C4—342no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4—684no—
232C22×C4—684no—
233C22×C4—684no—
234C22×C4C22⋊#59511yescharacteristic
235C22×C4C22⋊#59511yescharacteristic
236C22×C4C22×#59511yescharacteristic
237C2×C8—682no—
238C2×C8—682no—
239C2×C8—682no—
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—682no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—682no—
247C4⋊C4—682no—
248C4⋊C4—682no—
249C4⋊C4—682no—
250C4⋊C4—684no—
251C4⋊C4—684no—
252C4⋊C4—682no—
253C42—22no—
254C42—174no—
255C42—342no—
256C42—342no—
257C23—172no—
258C23—172no—
259C23—172no—
260C23—341no—
261C23—341no—
262C23—341no—
263C23—341no—
264C23C2×C22⋊#59511yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—172no—
275C2×C4—172no—
276C2×C4—171no—
277C2×C4—171no—
278C2×C4—171no—
279C2×C4—171no—
280C2×C4—172no—
281C2×C4—172no—
282C2×C4—172no—
283C2×C4—344no—
284C2×C4—344no—
285C2×C4—344no—
286C2×C4—344no—
287C2×C4—348no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4—1368no—
294C2×C4#1058754712yes—
295C2×C4#1058754912yes—
296C2×C4#1058767712yes—
297C8—1362no—
298C8—1362no—
299C22—174no—
300C22—342no—
301C22—342no—
302C22—342no—
303C22—342no—
304C22—342no—
305C22—342no—
306C22#1176375211yescharacteristic
307C22D17⋊#10011yescharacteristic
308C22D17⋊#10111yescharacteristic
309C22C2×#1058754711yescharacteristic
310C22C2×#1058754911yescharacteristic
311C22#1176380111yescharacteristic
312C22#1176380211yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—82no—
317C4—172no—
318C4—172no—
319C4—172no—
320C4—172no—
321C4—342no—
322C4—1362no—
323C2—344no—
324C2D17⋊#33211yescharacteristic
325C2#1454429211yescharacteristic
326C2#1454429811yescharacteristic
327C2#1454431411yescharacteristic
328C2#1454468411yescharacteristic
329C2#1454468711yescharacteristic
330C2#1454468811yescharacteristic
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#8724
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #14543662C2semidirect
213. C17#8724semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145436622
D17×#3842
#872417