Finite Group Databasebeta

← #16923007

Subgroups — #16923007

Order 4352.

450 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 107 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#14537094C211yesmaximal, characteristic
2#14528764C211yesmaximal, characteristic
3#14526608C212yesmaximal, complemented
4#14526606C212yesmaximal, complemented
5C17×#1817C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic
7C2×C4⋊#3622—21no—
8C2×C22⋊#3622—21no—
9#11762707C2211yescharacteristic
10#11762707C2211yescharacteristic
11#11762693—21no—
12#11762693—21no—
13#11762682—21no—
14#11762682—21no—
15#11762675C411yescharacteristic
16#11762669C411yescharacteristic
17#11762669C411yescharacteristic
18#11762564C2212yes—
19C22×C42×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C17×#336—21no—
22C17×#336C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×C8×C17—22no—
27C2×C42×C17—42no—
28C2×C42×C17C2×C411yescharacteristic
29C2×C42×C17D411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C42×C17C2311yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C22×#3622—22no—
43#10587556—42no—
44C17×C22⋊C8—42no—
45#10587502—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51C22⋊#3622—42no—
52C4⋊#3622—42no—
53C24×C17C22⋊C411yescharacteristic
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17—41no—
62C22×C4×C17—41no—
63C22×C4×C17—41no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×C8×C17—42no—
75C2×C8×C17—42no—
76C2×C8×C17—42no—
77C42×C17—22no—
78C42×C17—22no—
79C42×C17—41no—
80C42×C17—41no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C4⋊Dic17—22no—
84C4⋊Dic17—22no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87#63225—42no—
88#63225—42no—
89#63225—42no—
90#63225—41no—
91#63225—41no—
92C2×#3622—42no—
93C2×#3622—42no—
94C2×#3622—42no—
95#7731—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17#10112yes—
103C23×C17C2×C22⋊C411yescharacteristic
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—44no—
119C2×C4×C17—44no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17#9912yes—
123C2×C4×C17#13112yes—
124C2×C4×C17#13512yes—
125C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C17#13711yescharacteristic
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C8×C17—82no—
140#3622—82no—
141#2851—171no—
142#1817D1711yescharacteristic
143#1797—171no—
144#1271—172no—
145#1271—172no—
146C22×C17—22no—
147C22×C17—22no—
148C22×C17—22no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17#40811yescharacteristic
154C22×C17C2×#9911yescharacteristic
155C22×C17C2×#10111yescharacteristic
156C22×C17#41611yescharacteristic
157C22×C17#41711yescharacteristic
158C22×C17#42111yescharacteristic
159C22×C17#42111yescharacteristic
160C4×C17—22no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—41no—
166C4×C17—41no—
167Dic17—81no—
168Dic17—81no—
169Dic17—81no—
170Dic17—81no—
171C2×#125—171no—
172C22×C42C2×D1711yescharacteristic
173C2×C4⋊C8—341no—
174C2×C22⋊C8—21no—
175C2×C22⋊C8—341no—
176#389—171no—
177#389—171no—
178#387—341no—
179#387—341no—
180#385—341no—
181#385—341no—
182#382—171no—
183#380—171no—
184#380—171no—
185#336—21no—
186#336—172no—
187#336C2×D1712yes—
188C2×C17—42no—
189C2×C17#147211yescharacteristic
190C2×C17#148211yescharacteristic
191C2×C17#149511yescharacteristic
192C2×C17#151311yescharacteristic
193C2×C17#151611yescharacteristic
194C2×C17#153411yescharacteristic
195C2×C17#155211yescharacteristic
196C23×C4C4×D1711yescharacteristic
197C23×C4D6811yescharacteristic
198C23×C4C22⋊D1711yescharacteristic
199C22×C8—22no—
200C22×C8—342no—
201#125—682no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C42—42no—
213C2×C42C4×D1711yescharacteristic
214C2×C42D6811yescharacteristic
215C2×C42C22⋊D1711yescharacteristic
216C2×C42C22×D1711yescharacteristic
217C4⋊C8—682no—
218C22⋊C8—42no—
219C22⋊C8—682no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225#95—341no—
226C17#773111yescharacteristic, complemented, Sylow
227C24#6322711yescharacteristic
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4C2×C4×D1711yescharacteristic
246C22×C4C2×D6811yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—682no—
252C2×C8—682no—
253C2×C8—682no—
254C4⋊C4—342no—
255C4⋊C4—342no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—682no—
259C4⋊C4—682no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C4⋊C4—682no—
263C22⋊C4—682no—
264C22⋊C4—682no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23#1058749512yes—
276C23C2×#6322711yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4#1058747712yes—
308C2×C4#1058758712yes—
309C2×C4#1058759411yescharacteristic
310C2×C4#1058759511yescharacteristic
311C2×C4#1058763612yes—
312C8—82no—
313C8—1362no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22#1176277511yescharacteristic
322C22C2×#1058747711yescharacteristic
323C22C2×#1058749511yescharacteristic
324C22#1176280311yescharacteristic
325C22#1176281111yescharacteristic
326C22#1176282511yescharacteristic
327C22#1176282711yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—1361no—
336C4—1361no—
337C4—1361no—
338C4—1361no—
339C2—42no—
340C2#1452714711yescharacteristic
341C2#1452719011yescharacteristic
342C2#1452725011yescharacteristic
343C2#1452732811yescharacteristic
344C2#1452733811yescharacteristic
345C2#1452741011yescharacteristic
346C2#1452748511yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1817C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1817D17
hypocenterC17#7731
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #14526608C2semidirect
4. #14526606C2semidirect
226. C17#7731semidirect

Maximal subgroups

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

subgroupindex
#145370942
#145287642
#145266082
#145266062
C17×#18172
#773117