Finite Group Databasebeta

← #16944848

Subgroups — #16944848

Order 4352.

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

normal subgroups only

This group has 420 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#14534482C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526810C212yesmaximal, complemented
4#14526789C212yesmaximal, complemented
5C17×#1742C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic
7#11762866—21no—
8#11762748C2211yescharacteristic
9#11762745C411yescharacteristic
10#11762739C2211yescharacteristic
11#11762734C411yescharacteristic
12#11762704—21no—
13#11762703—21no—
14#11762693—21no—
15#11762682—21no—
16#11762655C411yescharacteristic
17#11762535C2212yes—
18#11762533—21no—
19C22×M16×C17—21no—
20C2×C17×#117C2211yescharacteristic
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17D411yescharacteristic
24C2×M16×C17—24no—
25C2×M16×C17—22no—
26C2×M16×C17—41no—
27C2×M16×C17—42no—
28C22×C8×C17—41no—
29C17×#117—41no—
30C17×#117—41no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46#10587556—41no—
47#10587556—41no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57#10587474—41no—
58C24×C17C22⋊C411yescharacteristic
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
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—
74M16×C17—24no—
75M16×C17—44no—
76M16×C17—42no—
77C2×C8×C17—44no—
78C2×C8×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C22⋊Dic17—42no—
88C22⋊Dic17—42no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91#63225—42no—
92#63225—42no—
93#63225—41no—
94#63225—41no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—42no—
98C2×#3622—42no—
99#11767—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17#10412yes—
105C23×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
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—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C17#13711yescharacteristic
126C2×Dic17—22no—
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—
138C8×C17—44no—
139#3622—82no—
140#2611—171no—
141#1742D1711yescharacteristic
142#1464—171no—
143#1353—172no—
144#1346—172no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—42no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—81no—
152C22×C17C2×#9911yescharacteristic
153C22×C17#41711yescharacteristic
154C22×C17C2×#10411yescharacteristic
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163Dic17—42no—
164Dic17—81no—
165Dic17—81no—
166Dic17—81no—
167C22×M16—21no—
168C2×#125—171no—
169C2×#117C2×D1711yescharacteristic
170#432—341no—
171#400—171no—
172#399—171no—
173#398—171no—
174#397—171no—
175#389—341no—
176#388—341no—
177#387—341no—
178#385—341no—
179#376—171no—
180#343—21no—
181#343C2×D1712yes—
182#324—172no—
183#323—341no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17—42no—
187C2×C17#150411yescharacteristic
188C2×C17#212311yescharacteristic
189C2×C17#213211yescharacteristic
190C23×C4C22⋊D1711yescharacteristic
191C2×M16—24no—
192C2×M16—22no—
193C2×M16—41no—
194C2×M16—42no—
195C22×C8—41no—
196#125—681no—
197#125—681no—
198#117—41no—
199#117—41no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C4⋊C4C22×D1711yescharacteristic
208C2×C22⋊C4—341no—
209C2×C22⋊C4—341no—
210C2×C22⋊C4C4×D1711yescharacteristic
211C2×C22⋊C4D6811yescharacteristic
212C2×C42—341no—
213C2×C42C4×D1711yescharacteristic
214C2×C42D6811yescharacteristic
215C4⋊C8—681no—
216C22⋊C8—342no—
217#97—681no—
218#95—171no—
219#95—171no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225C17#1176711yescharacteristic, complemented, Sylow
226C24#6322711yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—341no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—341no—
236C22×C4#6322711yescharacteristic
237C22×C4#6322711yescharacteristic
238C22×C4#6322711yescharacteristic
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×D6811yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—24no—
243M16—44no—
244M16—42no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—682no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C4⋊C4—682no—
253C4⋊C4—682no—
254C4⋊C4—681no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—682no—
261C22⋊C4—682no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—682no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1058747612yes—
272C23C2×#6322711yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
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—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—342no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4#1058747711yescharacteristic
302C2×C4#1058747711yescharacteristic
303C2×C4#1058761811yescharacteristic
304C2×C4#1058762811yescharacteristic
305C8—44no—
306C8—1362no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1058747711yescharacteristic
315C22#1176280811yescharacteristic
316C22C2×#1058747611yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—682no—
326C4—1361no—
327C4—1361no—
328C4—1361no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#1452728611yescharacteristic
333C2#1453043911yescharacteristic
334C2#1453046711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1742C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1742D17
hypocenterC17#11767
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #14526810C2semidirect
4. #14526789C2semidirect
225. C17#11767semidirect

Maximal subgroups

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

subgroupindex
#145344822
#145271182
#145268102
#145267892
C17×#17422
#1176717