Finite Group Databasebeta

← #16944007

Subgroups — #16944007

Order 4352.

424 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 83 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#14534127C211yesmaximal, characteristic
2#14527105C211yesmaximal, characteristic
3#14526733C212yesmaximal, complemented
4#14526731C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6C22×#63225C411yescharacteristic
7#11762862—21no—
8#11762742—21no—
9#11762732—21no—
10#11762693—21no—
11#11762689C411yescharacteristic
12#11762689C411yescharacteristic
13#11762682—21no—
14#11762681C2211yescharacteristic
15#11762681C2211yescharacteristic
16#11762669C411yescharacteristic
17#11762533—21no—
18#11762533C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#324—21no—
22C17×#323C2212yes—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×Dic17C2×C411yescharacteristic
27C23×Dic17C2×C411yescharacteristic
28C2×M16×C17—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2311yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17C2×C411yescharacteristic
38C2×C22⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—41no—
45C2×#63225—41no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—41no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C22⋊#3622—22no—
54C4⋊#3622—41no—
55C4×#3622—41no—
56C24×C17C22⋊C411yescharacteristic
57C24×C17C22×C411yescharacteristic
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—41no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—81no—
66C22×C4×C17—81no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—42no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—81no—
80M16×C17—82no—
81C2×C8×C17—42no—
82C4⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C42×C17—21no—
86C42×C17—21no—
87C22⋊Dic17—22no—
88C22⋊Dic17—22no—
89C22⋊Dic17—42no—
90C22⋊Dic17—81no—
91C4⋊Dic17—81no—
92#63225—42no—
93#63225—42no—
94#63225—41no—
95#63225—41no—
96#63225—81no—
97C2×#3622—42no—
98#11577—171nomaximal, Sylow
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17C2×C22⋊C411yescharacteristic
109C23×C17#13112yes—
110D4×C17—44no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—81no—
122C2×C4×C17—81no—
123C2×C4×C17—81no—
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#10112yes—
127C2×Dic17—22no—
128C2×Dic17—41no—
129C2×Dic17—42no—
130C2×Dic17—41no—
131C2×Dic17—42no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—82no—
142C8×C17—82no—
143#3622—82no—
144#2579—171no—
145#1465D1711yescharacteristic
146#1460—171no—
147#1326—172no—
148#1326—172no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—22no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—81no—
161C22×C17#40811yescharacteristic
162C22×C17C2×#9911yescharacteristic
163C22×C17C2×#10111yescharacteristic
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—22no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170Dic17—42no—
171Dic17—81no—
172Dic17—81no—
173Dic17—81no—
174C2×C4×D4C2×D1711yescharacteristic
175C22×C4⋊C4—171no—
176#431—341no—
177#407—21no—
178#398—341no—
179#396—341no—
180#387—341no—
181#386—171no—
182#386—171no—
183#385—171no—
184#385—171no—
185#385—341no—
186#380—171no—
187#324—21no—
188#323—172no—
189#323—341no—
190#323C2×D1712yes—
191C2×C17—21no—
192C2×C17—21no—
193C2×C17—42no—
194C2×C17—81no—
195C2×C17#147211yescharacteristic
196C2×C17#212011yescharacteristic
197C2×C17#212111yescharacteristic
198C22×D4C4×D1711yescharacteristic
199C23×C4—171no—
200C23×C4—171no—
201C23×C4C4×D1711yescharacteristic
202C23×C4C22⋊D1711yescharacteristic
203C2×M16—41no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—681no—
212C2×C4⋊C4—681no—
213C2×C4⋊C4C22⋊D1711yescharacteristic
214C2×C22⋊C4—171no—
215C2×C22⋊C4—171no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4D6811yescharacteristic
219C2×C22⋊C4C22×D1711yescharacteristic
220C2×C42D6811yescharacteristic
221C4⋊C8—681no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224C22⋊C8—342no—
225C4×C8—681no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230#95—341no—
231C17#1157711yescharacteristic, complemented, Sylow
232C24#6322711yescharacteristic
233C24C2×C4×D1711yescharacteristic
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—682no—
248C22×C4—681no—
249C22×C4—681no—
250C22×C4—1361no—
251C22×C4#6322711yescharacteristic
252C22×C4#6322711yescharacteristic
253C22×C4#6322711yescharacteristic
254C22×C4C2×D6811yescharacteristic
255C22×C4C2×C22⋊D1711yescharacteristic
256M16—82no—
257C2×C8—42no—
258C2×C8—682no—
259C4⋊C4—41no—
260C4⋊C4—682no—
261C4⋊C4—682no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C4⋊C4—1361no—
265C4⋊C4—1361no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—342no—
269C22⋊C4—342no—
270C22⋊C4—682no—
271C22⋊C4—1361no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1058759312yes—
284C23C2×#6322711yescharacteristic
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—342no—
300C2×C4—681no—
301C2×C4—682no—
302C2×C4—681no—
303C2×C4—682no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—1362no—
314C2×C4#1058747711yescharacteristic
315C2×C4#1058747711yescharacteristic
316C2×C4#1058749512yes—
317C8—82no—
318C8—1362no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1176277711yescharacteristic
332C22C2×#1058747711yescharacteristic
333C22C2×#1058749511yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—682no—
341C4—1361no—
342C4—1361no—
343C4—1361no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#1452714911yescharacteristic
349C2#1453042611yescharacteristic
350C2#1453042811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11577
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526733C2semidirect
4. #14526731C2semidirect
231. C17#11577semidirect

Maximal subgroups

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

subgroupindex
#145341272
#145271052
#145267332
#145267312
C17×#14652
#1157717