Finite Group Databasebeta

← #16943887

Subgroups — #16943887

Order 4352.

398 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 71 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#14533268C211yesmaximal, characteristic
2#14527112C211yesmaximal, characteristic
3D17×#324C211yesmaximal, characteristic
4#14526727C211yesmaximal, characteristic, complemented
5#14526727C211yesmaximal, characteristic, complemented
6#14526718C211yesmaximal, characteristic, complemented
7C17×#1319C211yesmaximal, characteristic, complemented
8C2×#10587632C411yescharacteristic
9C2×C4⋊C4×D17C2211yescharacteristic
10#11762771—21no—
11C22⋊C8×D17—21no—
12#11762750—21no—
13#11762750—21no—
14#11762716C411yescharacteristic
15#11762675C411yescharacteristic
16#11762674C2211yescharacteristic
17#11762671—21no—
18#11762671—21no—
19C4×#63227C411yescharacteristic
20#11762535C2211yescharacteristic
21#11762535C2211yescharacteristic
22#11762534—21no—
23#11762534—21no—
24C17×#391C2211yescharacteristic
25C17×#324C2211yescharacteristic
26C17×#324C2211yescharacteristic
27C22×Q8×C17C2×C411yescharacteristic
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C4×D17D411yescharacteristic
31C2×C4⋊C4×C17C2311yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33#10587632—41no—
34#10587632—41no—
35C2×#63227—21no—
36C2×#63227—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×#63225—21no—
40C2×#63225D411yescharacteristic
41C2×#63225D411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43C2×#63218—41no—
44C2×C8×D17—41no—
45C4⋊C4×D17—41no—
46C4⋊C4×D17—41no—
47C17×C22⋊C8—21no—
48C17×C22⋊C8—21no—
49C17×#95—21no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C22⋊#3622—21no—
59C22⋊#3622—21no—
60#10587491—41no—
61#10587491—41no—
62C23×D17C22⋊C411yescharacteristic
63C2×Q8×C17—21no—
64C2×Q8×C17—21no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C22⋊C411yescharacteristic
76C22×Dic17C22⋊C411yescharacteristic
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82C2×C4×D17—21no—
83C2×C4×D17—21no—
84C2×C4×D17—21no—
85C2×C4×D17—41no—
86C2×C4×D17—42no—
87C2×C4×D17—42no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C4⋊C4×C17—41no—
91C42×C17—41no—
92#63227—42no—
93#63227—41no—
94#63227—41no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97#63225—41no—
98#63225—41no—
99#63225—42no—
100C4×Dic17—21no—
101C4×Dic17—21no—
102C2×#3622—41no—
103C2×#3622—41no—
104#63218—82no—
105C8×D17—82no—
106#11551—171nomaximal, Sylow
107C23×C17C2×C22⋊C411yescharacteristic
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—22no—
111C22×D17—41no—
112C22×D17#9911yescharacteristic
113C22×D17#9911yescharacteristic
114Q8×C17—81no—
115Q8×C17—81no—
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—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17#13111yescharacteristic
129C2×C4×C17#13111yescharacteristic
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—42no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C4×D17—41no—
143C4×D17—41no—
144C4×D17—41no—
145C4×D17—41no—
146C4×D17—42no—
147C4×D17—42no—
148C4×D17—81no—
149C4×D17—81no—
150C8×C17—81no—
151C8×C17—81no—
152#3622—81no—
153#3622—81no—
154#2503—171no—
155#1463—171no—
156C2×#324—171no—
157#1324—171no—
158#1324—171no—
159#1319—171no—
160#1319D1711yescharacteristic, complemented
161C22×C17—21no—
162C22×C17—21no—
163C22×C17#40811yescharacteristic
164C22×C17C2×#9911yescharacteristic
165C22×C17#41311yescharacteristic
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—42no—
172C2×D17—42no—
173C2×D17#35412yes—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—81no—
179C4×C17—81no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—81no—
184Dic17—81no—
185C2×C22⋊Q8—171no—
186C22×C4⋊C4—171no—
187#407—341no—
188C2×C22⋊C8—341no—
189#400—341no—
190#400—341no—
191#391—171no—
192#391C2×D1711yescharacteristic
193#382—171no—
194#382—171no—
195#381—341no—
196#381—341no—
197C4×C22⋊C4—171no—
198#324—171no—
199#324—171no—
200#324—341no—
201#324—341no—
202#324C2×D1711yescharacteristic
203#324C2×D1711yescharacteristic
204C2×C17—21no—
205C2×C17—21no—
206C2×C17#146811yescharacteristic
207C2×C17C2×#35411yescharacteristic
208C2×C17#213311yescharacteristic
209D17—22no—
210D17—41no—
211C22×Q8C4×D1711yescharacteristic
212C23×C4—171no—
213C23×C4—171no—
214C23×C4—171no—
215C2×M16—681no—
216C22×C8—681no—
217C22⋊Q8—681no—
218C22⋊Q8—681no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—681no—
225C2×C4⋊C4—681no—
226C2×C4⋊C4C22×D1711yescharacteristic
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C42—171no—
230C2×C42C4×D1711yescharacteristic
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—341no—
234C22⋊C8—341no—
235C22⋊C8—681no—
236C22⋊C8—681no—
237#95—21no—
238#95—21no—
239#95—341no—
240#95—341no—
241#95—341no—
242#95—341no—
243#95—341no—
244#95—341no—
245#95—341no—
246C17#1155111yescharacteristic, complemented, Sylow
247C24—171no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—171no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4—681no—
268C22×C4—682no—
269C22×C4—682no—
270C22×C4C2×C4×D1711yescharacteristic
271C22×C4D4×D1711yescharacteristic
272C22×C4D4×D1711yescharacteristic
273M16—1362no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—681no—
277C2×C8—681no—
278C2×C8—1362no—
279C4⋊C4—41no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—682no—
285C22⋊C4—681no—
286C22⋊C4—681no—
287C22⋊C4—682no—
288C42—41no—
289C42—341no—
290C42—341no—
291C23—171no—
292C23—171no—
293C23—342no—
294C23—341no—
295C23—341no—
296C23—681no—
297C23C22⋊C4×D1711yescharacteristic
298Q8—81no—
299Q8—81no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—682no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—1361no—
331C2×C4—1361no—
332C2×C4#1058758711yescharacteristic
333C2×C4#1058759311yescharacteristic
334C8—81no—
335C8—81no—
336C8—1361no—
337C8—1361no—
338C22—21no—
339C22—21no—
340C22—172no—
341C22—341no—
342C22—341no—
343C22—341no—
344C22—341no—
345C22—341no—
346C22—682no—
347C22—682no—
348C22#1176277411yescharacteristic
349C22#99×D1711yescharacteristic
350C22#1176279011yescharacteristic
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—681no—
358C4—681no—
359C4—681no—
360C4—1361no—
361C4—1361no—
362C2—21no—
363C2—21no—
364C2—342no—
365C2—681no—
366C2#1452712911yescharacteristic
367C2D17×#35411yescharacteristic
368C2#1453047211yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1319C2
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1319D17
hypocenterC17#11551
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
4. #14526727C2semidirect
5. #14526727C2semidirect
6. #14526718C2semidirect
7. C17×#1319C2semidirect
160. #1319D17semidirect
246. C17#11551semidirect

Maximal subgroups

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

subgroupindex
#145332682
#145271122
D17×#3242
#145267272
#145267182
C17×#13192
#1155117