Finite Group Databasebeta

← #16944816

Subgroups — #16944816

Order 4352.

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

normal subgroups only

This group has 436 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#14528517C211yesmaximal, characteristic
3#14526809C212yesmaximal
4#14526785C212yesmaximal
5C17×#1464C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic, complemented
7C2×#10587612—21no—
8#11762748C411yescharacteristic, complemented
9#11762745C2211yescharacteristic
10#11762739C411yescharacteristic, complemented
11#11762734C2211yescharacteristic
12#11762704—21no—
13#11762703—21no—
14#11762693—21no—
15#11762682—21no—
16#11762655C411yescharacteristic, complemented
17#11762576—21no—
18#11762576C2212yes—
19C2×C17×#117C2211yescharacteristic
20C17×#432—21no—
21C17×#324—21no—
22C17×#323C2212yes—
23C23×C4×C17D411yescharacteristic
24C17×#117—41no—
25C17×#117—41no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35#10587612—24no—
36#10587612—42no—
37C2×C4⋊Dic17—41no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17—41no—
46#10587556—41no—
47#10587556—41no—
48C17×C4⋊C8—41no—
49C17×C22⋊C8—22no—
50C17×#97—41no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic
57#10587502C2×C411yescharacteristic
58C24×C17C2×D411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—42no—
75C2×C8×C17—42no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C22⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C42×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C22⋊Dic17—24no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C4⋊Dic17—44no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92#63225—42no—
93#63225—42no—
94#63225—41no—
95#63225—41no—
96#63225—42no—
97C4×Dic17—42no—
98C4×Dic17—44no—
99#11760—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17AGL(1,ℤ/8)11yescharacteristic
106C23×C17#13711yescharacteristic
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
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—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#10412yes—
126C2×Dic17—22no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—44no—
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—82no—
140C2×Dic17—82no—
141C8×C17—82no—
142#2611—171no—
143#1740—171no—
144#1464D1711yescharacteristic
145#1352—172no—
146#1344—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—81no—
156C22×C17C2×#9911yescharacteristic
157C22×C17#41711yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166Dic17—42no—
167Dic17—81no—
168Dic17—84no—
169Dic17—81no—
170Dic17—81no—
171C2×#125—171no—
172C2×#117—341no—
173C2×#117C2×D1711yescharacteristic
174#432—21no—
175#400—171no—
176#399—171no—
177#398—171no—
178#397—171no—
179#389—341no—
180#388—341no—
181#387—341no—
182#385—341no—
183#376—171no—
184#342—172no—
185#342—341no—
186#324—21no—
187#323C2×D1712yes—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17—41no—
191C2×C17—41no—
192C2×C17#150411yescharacteristic
193C2×C17#212311yescharacteristic
194C2×C17#212611yescharacteristic
195C23×C4D6811yescharacteristic
196#125—681no—
197#125—681no—
198#117—41no—
199#117—41no—
200#117—344no—
201#117—682no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—681no—
209C2×C4⋊C4C4×D1711yescharacteristic
210C2×C4⋊C4C22⋊D1711yescharacteristic
211C2×C22⋊C4—342no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C22⋊C4D6811yescharacteristic
215C2×C22⋊C4C22×D1711yescharacteristic
216C2×C42—341no—
217C2×C42—681no—
218C2×C42C4×D1711yescharacteristic
219C2×C42C22⋊D1711yescharacteristic
220C4⋊C8—41no—
221C22⋊C8—22no—
222#97—41no—
223#95—171no—
224#95—171no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230C17#1176011yescharacteristic, complemented, Sylow
231C24C2×D6811yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—682no—
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4#6322711yescharacteristic
246C22×C4C2×C4×D1711yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—682no—
254C4⋊C4—684no—
255C4⋊C4—682no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—682no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—344no—
262C22⋊C4—682no—
263C22⋊C4—681no—
264C22⋊C4—682no—
265C22⋊C4—681no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—684no—
271C42—682no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1058759411yescharacteristic
277C23#1058759511yescharacteristic
278C23C2×#6322711yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
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—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—342no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—684no—
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—681no—
308C2×C4—1362no—
309C2×C4—1362no—
310C2×C4#1058747711yescharacteristic
311C2×C4#1058747711yescharacteristic
312C2×C4#1058749612yes—
313C8—82no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1058747711yescharacteristic
324C22#1176281111yescharacteristic
325C22C2×#1058749611yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—682no—
334C4—1361no—
335C4—1364no—
336C4—1361no—
337C4—1361no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#1452728911yescharacteristic
343C2#1453043911yescharacteristic
344C2#1453044711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1464C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1464D17
hypocenterC17#11760
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
6. C2×#10587556C4semidirect
8. #11762748C4semidirect
10. #11762739C4semidirect
16. #11762655C4semidirect
230. C17#11760semidirect

Maximal subgroups

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

subgroupindex
#145344822
#145285172
#145268092
#145267852
C17×#14642
#1176017