Finite Group Databasebeta

← #16945013

Subgroups — #16945013

Order 4352.

472 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 79 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#14532020C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic, complemented
3#14526789C212yesmaximal, complemented
4#14526786C212yesmaximal, complemented
5C17×#1822C211yesmaximal, characteristic, complemented
6C2×C4×D68C411yescharacteristic
7C2×C4×C22⋊D17—21no—
8#11762866—21no—
9#11762739C2211yescharacteristic
10#11762734C411yescharacteristic
11#11762731—21no—
12#11762726C2211yescharacteristic
13#11762725—21no—
14#11762718C411yescharacteristic
15C4×C4⋊Dic17C411yescharacteristic
16C4×#63227—21no—
17#11762535C2212yes—
18#11762533—21no—
19C2×C17×#117C2211yescharacteristic
20C17×#407—21no—
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17D411yescharacteristic
24C22×C22⋊D17—21no—
25C22×D68C2×C411yescharacteristic
26C22×C4×D17—21no—
27C2×M16×C17—22no—
28C17×#117—41no—
29C17×#117—41no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C4⋊C4×C17C2311yescharacteristic
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C22⋊Dic17—21no—
37C4×C22⋊D17—44no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×#63225—21no—
45C2×C4×Dic17—21no—
46C4×D68—41no—
47C4×D68—41no—
48C17×C22⋊C8—41no—
49C17×C22⋊C8—41no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—22no—
55C4⋊#3622—41no—
56#10587474—41no—
57C24×C17C22⋊C411yescharacteristic
58C23×D17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—41no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C2×C22⋊D17—42no—
70C2×C22⋊D17—42no—
71C2×C22⋊D17—42no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C2×D68—41no—
76C2×D68—42no—
77C2×C4×D17—42no—
78C2×C4×D17—42no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81M16×C17—42no—
82M16×C17—42no—
83C2×C8×C17—42no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—21no—
92C22⋊Dic17—42no—
93#63227—42no—
94#63227—42no—
95#63227—42no—
96#63227—42no—
97C4⋊Dic17—42no—
98C4⋊Dic17—41no—
99#63225—42no—
100C4×Dic17—42no—
101C2×#3622—42no—
102#11804—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17C2×C22⋊C411yescharacteristic
109C22×D17—22no—
110C22×D17—41no—
111C22×D17—41no—
112C22×D17—41no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—41no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17AGL(1,ℤ/8)12yes—
133C22⋊D17—48no—
134C2×Dic17—22no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143D68—42no—
144D68—42no—
145C4×D17—82no—
146C4×D17—82no—
147C8×C17—82no—
148#3622—82no—
149#2392—171no—
150#1822D1711yescharacteristic, complemented
151#1464—171no—
152#1346—172no—
153#1345—172no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—81no—
161C22×C17C2×#9911yescharacteristic
162C22×C17#41811yescharacteristic
163C22×C17#42111yescharacteristic
164C2×D17—42no—
165C2×D17—42no—
166C2×D17—42no—
167C2×D17—81no—
168C2×D17—81no—
169C2×D17—81no—
170C2×D17—81no—
171C4×C17—22no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178Dic17—42no—
179Dic17—81no—
180Dic17—81no—
181C2×C4×D4—171no—
182C2×C4×D4—341no—
183C2×#117C2×D1711yescharacteristic
184#432—341no—
185#407—21no—
186#398—171no—
187#397—171no—
188#396—341no—
189#394—171no—
190#394—341no—
191#392—171no—
192C4×C4⋊C4—171no—
193C4×C22⋊C4—341no—
194#343—21no—
195#343C2×D1712yes—
196#324—172no—
197#323—341no—
198C2×C17—21no—
199C2×C17—21no—
200C2×C17—81no—
201C2×C17#151011yescharacteristic
202C2×C17#212011yescharacteristic
203C2×C17#213211yescharacteristic
204D17—82no—
205C22×D4—171no—
206C22×D4—341no—
207C23×C4—341no—
208C23×C4C22⋊D1711yescharacteristic
209C2×M16—22no—
210C4×D4—681no—
211C4×D4—681no—
212C4×D4—684no—
213#117—41no—
214#117—41no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4C22⋊D1711yescharacteristic
219C2×C4⋊C4C22×D1711yescharacteristic
220C2×C22⋊C4—341no—
221C2×C22⋊C4—341no—
222C2×C22⋊C4—341no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4C4×D1711yescharacteristic
226C2×C22⋊C4D6811yescharacteristic
227C2×C42—341no—
228C2×C42C4×D1711yescharacteristic
229C2×C42D6811yescharacteristic
230C4⋊C8—681no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—342no—
234#97—681no—
235#95—171no—
236#95—171no—
237#95—341no—
238#95—341no—
239C17#1180411yescharacteristic, complemented, Sylow
240C24—341no—
241C24#6322711yescharacteristic
242C2×D4—681no—
243C2×D4—682no—
244C2×D4—682no—
245C2×D4—682no—
246C2×D4—682no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—341no—
252C22×C4—341no—
253C22×C4—341no—
254C22×C4—682no—
255C22×C4—682no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4#6322711yescharacteristic
259C22×C4#6322711yescharacteristic
260C22×C4#6322711yescharacteristic
261C22×C4C2×C4×D1711yescharacteristic
262C22×C4C2×D6811yescharacteristic
263C22×C4C2×C22⋊D1711yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C2×C8—682no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—681no—
271C4⋊C4—682no—
272C4⋊C4—682no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—682no—
276C22⋊C4—682no—
277C22⋊C4—682no—
278C22⋊C4—682no—
279C22⋊C4—682no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—682no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—342no—
291C23—681no—
292C23—681no—
293C23—681no—
294C23—681no—
295C23—681no—
296C23—681no—
297C23C2×#6322711yescharacteristic
298D4—682no—
299D4—682no—
300D4—688no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—342no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—1362no—
325C2×C4—1362no—
326C2×C4#1058747711yescharacteristic
327C2×C4#1058747711yescharacteristic
328C2×C4#1058763512yes—
329C8—82no—
330C8—1362no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—682no—
339C22—682no—
340C22—682no—
341C22—1361no—
342C22—1361no—
343C22—1361no—
344C22—1361no—
345C22C2×#1058747711yescharacteristic
346C22#1176281311yescharacteristic
347C22#1176282711yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—682no—
356C4—1361no—
357C4—1361no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—1362no—
362C2#1452731211yescharacteristic
363C2#1453042611yescharacteristic
364C2#1453046711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1822C2
socleC22×C17C2×#99
prime radical#11762718C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1822D17
hypocenterC17#11804
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #14527118C2semidirect
3. #14526789C2semidirect
4. #14526786C2semidirect
5. C17×#1822C2semidirect
150. #1822D17semidirect
239. C17#11804semidirect

Maximal subgroups

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

subgroupindex
#145320202
#145271182
#145267892
#145267862
C17×#18222
#1180417