Finite Group Databasebeta

← #16945101

Subgroups — #16945101

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#14532068C211yesmaximal, characteristic
2#14528772C211yesmaximal, characteristic
3#14526806C212yesmaximal, complemented
4#14526785C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6C2×C4×Dic34C411yescharacteristic, complemented
7C2×#10587505—21no—
8#11762747—21no—
9#11762734C411yescharacteristic, complemented
10#11762734C2211yescharacteristic
11#11762723—21no—
12#11762721C411yescharacteristic, complemented
13#11762721C2211yescharacteristic
14#11762669—21no—
15C4×C22⋊Dic17—21no—
16C4×C22⋊Dic17—21no—
17#11762655C411yescharacteristic, complemented
18#11762576C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#324—21no—
22C17×#323C2212yes—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—41no—
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17C2311yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—22no—
39C2×C4⋊Dic17—41no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C4×Dic34—41no—
45C4×Dic34—41no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—41no—
48#10587505—42no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—41no—
55#10587502C2×C411yescharacteristic
56#10587502C2×C411yescharacteristic
57C24×C17C22⋊C411yescharacteristic
58C24×C17C2×D411yescharacteristic
59C2×D4×C17—21no—
60C2×D4×C17—21no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—42no—
79C2×Dic34—41no—
80C2×Dic34—42no—
81M16×C17—82no—
82C2×C8×C17—42no—
83C4⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C42×C17—21no—
87C42×C17—21no—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C22⋊Dic17—42no—
91C22⋊Dic17—41no—
92C22⋊Dic17—41no—
93C22⋊Dic17—42no—
94C22⋊Dic17—42no—
95C4⋊Dic17—41no—
96C4⋊Dic17—82no—
97#63225—42no—
98C4×Dic17—42no—
99C4×Dic17—41no—
100C4×Dic17—41no—
101#11823—171nomaximal, Sylow
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112C23×C17AGL(1,ℤ/8)11yescharacteristic
113C23×C17#13711yescharacteristic
114D4×C17—42no—
115D4×C17—42no—
116C2×C4×C17—22no—
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—42no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#9911yescharacteristic
132C2×Dic17—22no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—82no—
145C2×Dic17—82no—
146C2×Dic17—82no—
147Dic34—42no—
148Dic34—42no—
149C8×C17—82no—
150#2398—171no—
151#1799—171no—
152#1465D1711yescharacteristic
153#1351—172no—
154#1344—172no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—42no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17C2×#9911yescharacteristic
170C22×C17#41711yescharacteristic
171C22×C17#42111yescharacteristic
172C4×C17—21no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17#35412yes—
178Dic17—42no—
179Dic17—81no—
180Dic17—81no—
181Dic17—81no—
182Dic17—162no—
183C2×C4×Q8—171no—
184C2×C4×D4C2×D1711yescharacteristic
185C2×#99—341no—
186#407—21no—
187#400—341no—
188#397—171no—
189#397—171no—
190#394—341no—
191#393—171no—
192#393—171no—
193#380—341no—
194C4×C22⋊C4—341no—
195C4×C22⋊C4—341no—
196#376—171no—
197#342—172no—
198#324—21no—
199#323C2×D1712yes—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—41no—
203C2×C17—41no—
204C2×C17—81no—
205C2×C17#151311yescharacteristic
206C2×C17#212011yescharacteristic
207C2×C17C2×#35411yescharacteristic
208C22×Q8—171no—
209C22×D4D6811yescharacteristic
210C23×C4D6811yescharacteristic
211C23×C4C22⋊D1711yescharacteristic
212C2×M16—41no—
213C4×Q8—681no—
214C4×Q8—681no—
215C4×D4—41no—
216C4×D4—41no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—681no—
220C2×C4⋊C4C22⋊D1711yescharacteristic
221C2×C22⋊C4—341no—
222C2×C22⋊C4—341no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4—342no—
226C2×C22⋊C4C4×D1711yescharacteristic
227C2×C22⋊C4C22×D1711yescharacteristic
228C2×C42—341no—
229C2×C42—341no—
230C2×C42C4×D1711yescharacteristic
231#99—682no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234#95—171no—
235#95—171no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240#95—341no—
241#95—681no—
242C17#1182311yescharacteristic, complemented, Sylow
243C24#6322711yescharacteristic
244C24C2×D6811yescharacteristic
245C2×Q8—681no—
246C2×Q8—682no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—682no—
262C22×C4#6322711yescharacteristic
263C22×C4#6322711yescharacteristic
264C22×C4#6322711yescharacteristic
265C22×C4C2×C4×D1711yescharacteristic
266C22×C4C2×C22⋊D1711yescharacteristic
267M16—82no—
268C2×C8—42no—
269C4⋊C4—41no—
270C4⋊C4—681no—
271C4⋊C4—682no—
272C4⋊C4—1362no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—682no—
276C22⋊C4—682no—
277C22⋊C4—682no—
278C22⋊C4—681no—
279C22⋊C4—681no—
280C22⋊C4—682no—
281C22⋊C4—682no—
282C42—21no—
283C42—21no—
284C42—682no—
285C42—681no—
286C42—681no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#1058759411yescharacteristic
297C23#1058759511yescharacteristic
298C23C2×#6322711yescharacteristic
299Q8—682no—
300Q8—682no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—342no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—1362no—
330C2×C4—1362no—
331C2×C4—1362no—
332C2×C4#1058747711yescharacteristic
333C2×C4#1058747711yescharacteristic
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#1058747711yescharacteristic
350C22#1176281111yescharacteristic
351C22#1176282511yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2722no—
358C4—682no—
359C4—1361no—
360C4—1361no—
361C4—1361no—
362C4#1176260312yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#1452732811yescharacteristic
369C2#1453042611yescharacteristic
370C2C2×#1176260311yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11823
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526806C2semidirect
4. #14526785C2semidirect
6. C2×C4×Dic34C4semidirect
9. #11762734C4semidirect
12. #11762721C4semidirect
17. #11762655C4semidirect
242. C17#11823semidirect

Maximal subgroups

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

subgroupindex
#145320682
#145287722
#145268062
#145267852
C17×#14652
#1182317