Finite Group Databasebeta

← #16944969

Subgroups — #16944969

Order 4352.

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

normal subgroups only

This group has 444 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#14533960C211yesmaximal, characteristic
2C4×#10587504C211yesmaximal, characteristic
3#14526815C212yesmaximal, complemented
4#14526812C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic
7C2×#10587504—21no—
8C4×#63223—21no—
9#11762748C2211yescharacteristic
10#11762745C411yescharacteristic
11#11762729C2211yescharacteristic
12#11762721C411yescharacteristic
13#11762693—21no—
14#11762689—21no—
15#11762682—21no—
16#11762667—21no—
17C2×#10587502C411yescharacteristic
18#11762578C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×Dic17—21no—
27C22×Dic34C2×C411yescharacteristic
28C2×M16×C17—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17C2311yescharacteristic
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39#10587610—41no—
40#10587610—41no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63223—22no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—41no—
47#10587504—24no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55#10587474—41no—
56C4×#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22⋊C411yescharacteristic
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—81no—
67C22×C4×C17—81no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—42no—
78C22×Dic17—42no—
79C22×Dic17—81no—
80C2×Dic34—41no—
81C2×Dic34—81no—
82M16×C17—82no—
83C2×C8×C17—42no—
84C4⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C42×C17—21no—
88C42×C17—21no—
89C22⋊Dic17—42no—
90C22⋊Dic17—42no—
91C22⋊Dic17—81no—
92C4⋊Dic17—41no—
93#63225—42no—
94#63225—41no—
95#63225—41no—
96#63225—81no—
97#63223—24no—
98#63223—42no—
99C2×#3622—42no—
100#11793—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17#10412yes—
111C23×C17C2×C22⋊C411yescharacteristic
112D4×C17—44no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17#13712yes—
129C2×Dic17—22no—
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—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—84no—
142Dic34—82no—
143C8×C17—82no—
144#3622—44no—
145#2564—171no—
146C4×#100—171no—
147#1465D1711yescharacteristic
148#1354—172no—
149#1353—172no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—22no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—81no—
162C22×C17C2×#9911yescharacteristic
163C22×C17#41911yescharacteristic
164C22×C17C2×#10411yescharacteristic
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—22no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—41no—
171Dic17—42no—
172Dic17—81no—
173Dic17—81no—
174Dic17—81no—
175C2×C22⋊Q8—171no—
176C2×C4×D4C2×D1711yescharacteristic
177C2×#100—341no—
178#407—21no—
179C4×M16—341no—
180#400—171no—
181#399—171no—
182#395—171no—
183#393—171no—
184#387—341no—
185#386—341no—
186#385—341no—
187#380—341no—
188C2×#95—171no—
189#343—172no—
190#324C2×D1712yes—
191#323—21no—
192C2×C17—21no—
193C2×C17—21no—
194C2×C17—42no—
195C2×C17—81no—
196C2×C17#150811yescharacteristic
197C2×C17#212111yescharacteristic
198C2×C17#213211yescharacteristic
199C22×Q8—171no—
200C22×D4D6811yescharacteristic
201C23×C4—341no—
202C23×C4C4×D1711yescharacteristic
203C23×C4C22⋊D1711yescharacteristic
204C2×M16—41no—
205C2×M16—342no—
206C22⋊Q8—681no—
207C22⋊Q8—681no—
208C4×D4—41no—
209C4×D4—41no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4C22×D1711yescharacteristic
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4C4×D1711yescharacteristic
219C2×C22⋊C4C22⋊D1711yescharacteristic
220C2×C42D6811yescharacteristic
221#100—344no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224#97—681no—
225C4×C8—681no—
226#95—171no—
227#95—171no—
228#95—341no—
229#95—341no—
230#95—341no—
231#95—341no—
232#95—341no—
233C17#1179311yescharacteristic, complemented, Sylow
234C24#6322711yescharacteristic
235C24#6322711yescharacteristic
236C2×Q8—681no—
237C2×Q8—1361no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—341no—
251C22×C4—682no—
252C22×C4—682no—
253C22×C4—1361no—
254C22×C4#6322711yescharacteristic
255C22×C4#6322711yescharacteristic
256C22×C4C2×C4×D1711yescharacteristic
257C22×C4C2×D6811yescharacteristic
258C22×C4C2×C22⋊D1711yescharacteristic
259M16—82no—
260M16—344no—
261M16—682no—
262C2×C8—42no—
263C2×C8—682no—
264C4⋊C4—41no—
265C4⋊C4—681no—
266C4⋊C4—682no—
267C4⋊C4—681no—
268C4⋊C4—681no—
269C4⋊C4—1361no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—682no—
273C22⋊C4—682no—
274C22⋊C4—1361no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1058749612yes—
287C23C2×#6322711yescharacteristic
288Q8—1362no—
289D4—44no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—342no—
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—681no—
314C2×C4—681no—
315C2×C4—1364no—
316C2×C4#1058747711yescharacteristic
317C2×C4#1058747711yescharacteristic
318C2×C4#1058759512yes—
319C8—82no—
320C8—684no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22C2×#1058747711yescharacteristic
334C22#1176281911yescharacteristic
335C22C2×#1058749611yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—682no—
343C4—1361no—
344C4—1361no—
345C4—1361no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#1452730511yescharacteristic
351C2#1453042811yescharacteristic
352C2#1453046711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11793
LCSG ▷ C22×C4×C17 ▷ C23×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. #14526815C2semidirect
4. #14526812C2semidirect
233. C17#11793semidirect

Maximal subgroups

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

subgroupindex
#145339602
C4×#105875042
#145268152
#145268122
C17×#14652
#1179317