Finite Group Databasebeta

← #16944904

Subgroups — #16944904

Order 4352.

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

normal subgroups only

This group has 446 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#14533689C211yesmaximal, characteristic
2C4×#10587507C211yesmaximal, characteristic, complemented
3#14526791C212yesmaximal, complemented
4#14526788C212yesmaximal
5C17×#1467C211yesmaximal, characteristic, complemented
6C2×#10587632—21no—
7C2×C4×D68C411yescharacteristic
8C2×#10587507—21no—
9C4×#63223—21no—
10#11762739C411yescharacteristic
11#11762739C2211yescharacteristic
12#11762726C411yescharacteristic
13#11762726C2211yescharacteristic
14#11762700—21no—
15#11762700—21no—
16#11762675C411yescharacteristic
17#11762666—21no—
18#11762578C2212yes—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#432—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C22×Q8×C17D411yescharacteristic
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C4×Q8×C17—41no—
27C4×Q8×C17—41no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C4⋊C4×C17C2311yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34#10587632—44no—
35C2×#63227—21no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63223—22no—
45C4×D68—41no—
46C4×D68—41no—
47C17×C4⋊C8—41no—
48C17×C22⋊C8—22no—
49C17×#97—41no—
50#10587507—24no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56#10587474—41no—
57C4×#3622—41no—
58C23×D17—21no—
59C2×Q8×C17—22no—
60C2×Q8×C17—22no—
61C2×Q8×C17—41no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C2×D68—41no—
73C2×D68—42no—
74C2×C4×D17—42no—
75C2×C4×D17—42no—
76C2×C4×D17—42no—
77C2×C8×C17—42no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87#63227—42no—
88#63227—42no—
89#63227—42no—
90#63227—41no—
91#63227—41no—
92#63227—42no—
93C4⋊Dic17—42no—
94C4⋊Dic17—41no—
95#63225—42no—
96#63225—42no—
97#63223—24no—
98#63223—42no—
99C2×#3622—42no—
100#11778—171nomaximal, Sylow
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—22no—
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17—41no—
107C22×D17—41no—
108C22×D17—41no—
109Q8×C17—44no—
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—21no—
119C2×C4×C17—21no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#10412yes—
127C2×C4×C17AGL(1,ℤ/8)12yes—
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—
137D68—42no—
138D68—42no—
139C4×D17—84no—
140C8×C17—82no—
141#3622—44no—
142#2536—171no—
143C4×#101—171no—
144#1467D1711yescharacteristic, complemented
145#1346—172no—
146#1345—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17C2×#9911yescharacteristic
150C22×C17#41811yescharacteristic
151C22×C17C2×#10411yescharacteristic
152C2×D17—42no—
153C2×D17—42no—
154C2×D17—42no—
155C2×D17—81no—
156C2×D17—81no—
157C2×D17—81no—
158C2×D17—81no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—22no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—41no—
167Dic17—81no—
168Dic17—81no—
169Dic17—81no—
170C2×C22⋊Q8—341no—
171C2×C4×Q8C2×D1711yescharacteristic
172C2×C4×D4—171no—
173#432—21no—
174C2×#101—341no—
175C4×M16—341no—
176#398—171no—
177#398—171no—
178#394—171no—
179#394—171no—
180#388—341no—
181#388—341no—
182#382—171no—
183#380—341no—
184#343—172no—
185#324—21no—
186#324C2×D1712yes—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17#150711yescharacteristic
190C2×C17#212011yescharacteristic
191C2×C17#213311yescharacteristic
192D17—82no—
193C22×Q8D6811yescharacteristic
194C22×D4—171no—
195C23×C4—341no—
196C2×M16—342no—
197C22⋊Q8—684no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×D4—681no—
201C4×D4—681no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4C4×D1711yescharacteristic
207C2×C4⋊C4C22⋊D1711yescharacteristic
208C2×C4⋊C4C22×D1711yescharacteristic
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C42C4×D1711yescharacteristic
215C2×C42D6811yescharacteristic
216C2×C42C22⋊D1711yescharacteristic
217C4⋊C8—41no—
218#101—344no—
219C22⋊C8—22no—
220#97—41no—
221#97—681no—
222C4×C8—681no—
223#95—171no—
224#95—171no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#1177811yescharacteristic, complemented, Sylow
229C24—341no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×D4—681no—
234C2×D4—682no—
235C22×C4—341no—
236C22×C4—341no—
237C22×C4—341no—
238C22×C4—682no—
239C22×C4—682no—
240C22×C4—682no—
241C22×C4#6322711yescharacteristic
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4C2×C4×D1711yescharacteristic
246C22×C4C2×D6811yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248M16—344no—
249M16—682no—
250C2×C8—42no—
251C2×C8—682no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—681no—
256C4⋊C4—682no—
257C4⋊C4—682no—
258C4⋊C4—682no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C22⋊C4—682no—
262C22⋊C4—681no—
263C22⋊C4—681no—
264C22⋊C4—682no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—342no—
272C23—681no—
273C23—681no—
274C23—681no—
275C23—681no—
276C23—681no—
277C23—681no—
278C23C2×#6322711yescharacteristic
279Q8—44no—
280D4—682no—
281D4—682no—
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—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
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—1364no—
306C2×C4#1058747711yescharacteristic
307C2×C4#1058747711yescharacteristic
308C2×C4#1058749612yes—
309C2×C4#1058759412yes—
310C8—82no—
311C8—684no—
312C22—21no—
313C22—21no—
314C22—682no—
315C22—682no—
316C22—682no—
317C22—1361no—
318C22—1361no—
319C22—1361no—
320C22—1361no—
321C22C2×#1058747711yescharacteristic
322C22#1176281511yescharacteristic
323C22C2×#1058749611yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—1361no—
333C4—1361no—
334C4—1361no—
335C2—21no—
336C2—21no—
337C2—1362no—
338C2#1452730111yescharacteristic
339C2#1453042311yescharacteristic
340C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1467C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1467D17
hypocenterC17#11778
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C4×#10587507C2semidirect
3. #14526791C2semidirect
5. C17×#1467C2semidirect
144. #1467D17semidirect
228. C17#11778semidirect

Maximal subgroups

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

subgroupindex
#145336892
C4×#105875072
#145267912
#145267882
C17×#14672
#1177817