Finite Group Databasebeta

← #16926077

Subgroups — #16926077

Order 4352.

998 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 159 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#14528776C211yesmaximal, characteristic, complemented
2#14527465C212yesmaximal, complemented
3#14527209C212yesmaximal, complemented
4#14527105C211yesmaximal, characteristic, complemented
5C2×C17×#547C211yesmaximal, characteristic
6#11762862C2212yes—
7#11762833C2212yes—
8C2×#10587504—21no—
9#11762578—21no—
10#11762558—24no—
11D4⋊#3622—24no—
12#11762533—22no—
13C22×#41×C17C412yes—
14C17×#547—24no—
15C17×#547—24no—
16C2×C17×C4⋊Q8C2211yescharacteristic
17C2×C17×#123—22no—
18C2×C17×C22⋊Q8C412yes—
19C2×C17×#121—21no—
20C2×C4×D4×C17C2211yescharacteristic
21C2×C4×D4×C17C2211yescharacteristic
22C22×C4⋊C4×C17—21no—
23C2×#41×C17—24no—
24C2×#41×C17—24no—
25C2×#41×C17—48no—
26C22×Q8×C17C2×C412yes—
27C22×D4×C17—22no—
28C22×D4×C17C2×C411yescharacteristic
29C22×D4×C17D411yescharacteristic
30C23×C4×C17—22no—
31C23×C4×C17—21no—
32C23×C4×C17C2×C412yes—
33C17×C4⋊Q8—22no—
34C17×C4⋊Q8D414yes—
35C17×#123—216no—
36C17×C22⋊Q8—24no—
37C17×C22⋊Q8—24no—
38C17×#121—28no—
39C4×D4×C17—22no—
40C4×D4×C17—22no—
41C4×D4×C17—22no—
42C4×D4×C17D414yes—
43C2×C4⋊C4×C17—28no—
44C2×C4⋊C4×C17—22no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17C2×C411yescharacteristic
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—22no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊C4×C17D412yes—
52C2×C42×C17C2311yescharacteristic
53C2×#63223—22no—
54#10587504—42no—
55C22⋊#3622—22no—
56C4⋊#3622—22no—
57C4⋊#3622—22no—
58C4⋊#3622—22no—
59C4×#3622—22no—
60C24×C17—21no—
61C24×C17C22⋊C412yes—
62#41×C17—416no—
63#41×C17—48no—
64#41×C17—48no—
65C2×Q8×C17—22no—
66C2×Q8×C17—44no—
67C2×Q8×C17C22⋊C414yes—
68C2×D4×C17—24no—
69C2×D4×C17—28no—
70C2×D4×C17—21no—
71C2×D4×C17—22no—
72C2×D4×C17—22no—
73C2×D4×C17—21no—
74C2×D4×C17—21no—
75C2×D4×C17—48no—
76C2×D4×C17C22⋊C412yes—
77C22×C4×C17—28no—
78C22×C4×C17—22no—
79C22×C4×C17—21no—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—24no—
83C22×C4×C17—44no—
84C22×C4×C17—48no—
85C22×C4×C17—42no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17C22⋊C412yes—
89C22×C4×C17C22×C411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C4⋊C4×C17—28no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—24no—
95C4⋊C4×C17—24no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17C22⋊C412yes—
98C22⋊C4×C17—28no—
99C22⋊C4×C17—24no—
100C22⋊C4×C17—24no—
101C42×C17C2×D412yes—
102C42×C17C2×D412yes—
103#63223—42no—
104#63223—42no—
105C2×#3622—42no—
106C2×#3622—42no—
107#8393—171nomaximal, Sylow
108C23×C17—24no—
109C23×C17—22no—
110C23×C17—22no—
111C23×C17—22no—
112C23×C17—22no—
113C23×C17—42no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17C2×C22⋊C411yescharacteristic
119Q8×C17—44no—
120Q8×C17—44no—
121D4×C17—24no—
122D4×C17—22no—
123D4×C17—416no—
124D4×C17—42no—
125D4×C17—42no—
126D4×C17#10014yescomplemented
127C2×C4×C17—22no—
128C2×C4×C17—22no—
129C2×C4×C17—22no—
130C2×C4×C17—24no—
131C2×C4×C17—41no—
132C2×C4×C17—44no—
133C2×C4×C17—44no—
134C2×C4×C17—416no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—44no—
138C2×C4×C17—44no—
139C2×C4×C17—44no—
140C2×C4×C17—42no—
141C2×C4×C17C2×C22⋊C411yescharacteristic
142C2×C4×C17C2×C22⋊C411yescharacteristic
143C2×C4×C17#12012yes—
144C2×C4×C17#12012yes—
145C2×C4×C17#13512yes—
146C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C17#13711yescharacteristic
148#3622—82no—
149#3622—82no—
150C2×#547D1711yescharacteristic
151#1800—171no—
152#1548—172no—
153#1486—172no—
154#1460—171no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—24no—
158C22×C17—44no—
159C22×C17—44no—
160C22×C17—42no—
161C22×C17—42no—
162C22×C17—42no—
163C22×C17—42no—
164C22×C17—42no—
165C22×C17—82no—
166C22×C17—82no—
167C22×C17#37911yescharacteristic
168C22×C17#41611yescharacteristic
169C22×C17#41711yescharacteristic
170C4×C17—22no—
171C4×C17—21no—
172C4×C17—42no—
173C4×C17—44no—
174C4×C17—42no—
175C4×C17C2×#10012yes—
176C4×C17#44912yes—
177C4×C17#45212yes—
178C22×#41Dic1712yes—
179#547—24no—
180#547—24no—
181C2×C4⋊Q8C2×D1711yescharacteristic
182C2×#123—22no—
183C2×C22⋊Q8Dic1712yes—
184C2×#121—21no—
185C2×C4×D4C2×D1711yescharacteristic
186C2×C4×D4C2×D1711yescharacteristic
187C22×C4⋊C4—21no—
188#431—172no—
189#423—172no—
190C2×#100—341no—
191#343—341no—
192#333—344no—
193D4⋊C8—344no—
194#323—342no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17—44no—
198C2×C17—82no—
199C2×C17#178311yescharacteristic
200C2×C17#178811yescharacteristic
201C2×C17#179611yescharacteristic
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—48no—
205C22×Q8C2×Dic1712yes—
206C22×D4—22no—
207C22×D4C2×Dic1711yescharacteristic
208C22×D4C22⋊D1711yescharacteristic
209C23×C4—22no—
210C23×C4—21no—
211C23×C4C2×Dic1712yes—
212C2×M16—342no—
213C4⋊Q8—22no—
214C4⋊Q8C22⋊D1714yes—
215#123—216no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218#121—28no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4C22⋊D1714yes—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C2×Dic1711yescharacteristic
228C2×C4⋊C4C22⋊D1711yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4C22⋊D1712yes—
232C2×C42C22×D1711yescharacteristic
233C4⋊C8—342no—
234C4⋊C8—342no—
235C4⋊C8—342no—
236#100—682no—
237C22⋊C8—342no—
238C4×C8—342no—
239C17#839311yescharacteristic, complemented, Sylow
240C24—21no—
241C24C22⋊Dic1712yes—
242#41—416no—
243#41—48no—
244#41—48no—
245C2×Q8—22no—
246C2×Q8—44no—
247C2×Q8C22⋊Dic1714yes—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—48no—
256C2×D4C22⋊Dic1712yes—
257C22×C4—28no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4C22⋊Dic1712yes—
269C22×C4C22×Dic1711yescharacteristic
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C2×C22⋊D1711yescharacteristic
272M16—682no—
273M16—682no—
274C2×C8—682no—
275C2×C8—682no—
276C4⋊C4—28no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4C22⋊Dic1712yes—
282C22⋊C4—28no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C42C2×C22⋊D1712yes—
286C42C2×C22⋊D1712yes—
287C23—24no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic1711yescharacteristic
298Q8—44no—
299Q8—44no—
300D4—24no—
301D4—22no—
302D4—416no—
303D4—42no—
304D4—42no—
305D4#1058750414yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4#1058763511yescharacteristic
321C2×C4#1058763612yes—
322C2×C4#1058763711yescharacteristic
323C2×C4C2×C22⋊Dic1711yescharacteristic
324C2×C4C2×C22⋊Dic1711yescharacteristic
325C2×C4#1058763912yes—
326C2×C4#1058763912yes—
327C8—1362no—
328C8—1362no—
329C22—21no—
330C22—21no—
331C22—24no—
332C22—44no—
333C22—44no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—82no—
341C22#1176266511yescharacteristic
342C22#1176280511yescharacteristic
343C22#1176281011yescharacteristic
344C4—21no—
345C4—22no—
346C4—42no—
347C4—44no—
348C4—42no—
349C4C2×#1058750412yes—
350C4#1176295412yes—
351C4#1176297112yes—
352C2—21no—
353C2—21no—
354C2—44no—
355C2—82no—
356C2#1452870511yescharacteristic
357C2#1452872411yescharacteristic
358C2#1452875911yescharacteristic
subgroupG/H
centerC22#11762665
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
Fitting, prime radicalC2×C17×#547C2
socleC22×C17#379
cosocleC2×C42×C17C23
hypercenterC2×#547D17
hypocenterC17#8393
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #14528776C2semidirect
2. #14527465C2semidirect
3. #14527209C2semidirect
4. #14527105C2semidirect
126. D4×C17#100semidirect
239. C17#8393semidirect
305. D4#10587504semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145274652
#145272092
#145271052
C2×C17×#5472
#839317