Finite Group Databasebeta

← #16938734

Subgroups — #16938734

Order 4352.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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#14528971C212yesmaximal
2#14527040C211yesmaximal, characteristic
3#14526662C212yesmaximal
4#14526627C211yesmaximal, characteristic
5C17×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7C2×#10587478—22no—
8C2×C22⋊#3622C2212yes—
9C2×#10587502—21no—
10C2×#10587502—21no—
11C2×#10587502C2211yescharacteristic
12#11762571—22no—
13C2×C17×C22⋊Q8—21no—
14C17×#398—22no—
15C17×#395—22no—
16C17×#380—21no—
17C17×#380C2212yes—
18C2×C17×#95C2211yescharacteristic
19C22×Q8×C17—21no—
20C23×C4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17C2311yescharacteristic
23C23×Dic17—21no—
24C23×Dic17—21no—
25C17×C22⋊Q8—42no—
26C17×C22⋊Q8—42no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—22no—
34C2×C4⋊Dic17—22no—
35C2×C4⋊Dic17—42no—
36C22×#3622—22no—
37C17×#95—22no—
38C17×#95—22no—
39C17×#95—22no—
40#10587502—21no—
41#10587502—42no—
42#10587502—42no—
43#10587502Q812yescomplemented
44C22⋊#3622—22no—
45C22⋊#3622Q814yescomplemented
46#10587478—44no—
47C24×C17C2×D411yescharacteristic
48C2×Q8×C17—42no—
49C2×Q8×C17—81no—
50C2×Q8×C17—81no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17—41no—
60C22×C4×C17—41no—
61C22×C4×C17—41no—
62C22×C4×C17—41no—
63C22×C4×C17D812yes—
64C22×C4×C17Q1612yes—
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×Q812yes—
67C22×C4×C17C2×Q811yescharacteristic
68C22×C4×C17#4111yescharacteristic
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C22×Dic17—41no—
82C22×Dic17—41no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—42no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—44no—
87C4⋊C4×C17—81no—
88C4⋊C4×C17—81no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—42no—
91C4⋊Dic17—24no—
92C4⋊Dic17—42no—
93C2×#3622—24no—
94C2×#3622—42no—
95C2×#3622—42no—
96#10569—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17#12011yescharacteristic
102C23×C17C22⋊Q811yescharacteristic
103C23×C17C4⋊Q811yescharacteristic
104C23×C17C2×D811yescharacteristic
105C23×C17C2×Q1611yescharacteristic
106C23×C17AGL(1,ℤ/8)11yescharacteristic
107C23×C17#13711yescharacteristic
108Q8×C17—84no—
109C2×C4×C17—22no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—42no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17C22⋊Q812yes—
131C2×C4×C17C4⋊Q812yes—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—81no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17—81no—
142C2×Dic17—81no—
143C2×Dic17—81no—
144C2×Dic17—81no—
145C2×Dic17—81no—
146#3622—44no—
147#2659D1711yescharacteristic
148#1852—172no—
149#1442—171no—
150#1294—172no—
151#1280—171no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17#39511yescharacteristic
165C22×C17#44711yescharacteristic
166C22×C17#45111yescharacteristic
167C22×C17#45411yescharacteristic
168C22×C17#45511yescharacteristic
169C22×C17B(4,𝔽2)11yescharacteristic
170C22×C17#45811yescharacteristic
171C22×C17#47411yescharacteristic
172C22×C17#47711yescharacteristic
173C22×C17#50011yescharacteristic
174C22×C17#50111yescharacteristic
175C4×C17—22no—
176C4×C17—22no—
177C4×C17—42no—
178C4×C17—42no—
179C4×C17—82no—
180C4×C17—82no—
181Dic17—82no—
182Dic17—82no—
183C2×C22⋊Q8—21no—
184C22×C4⋊C4—171no—
185C2×#107—342no—
186C2×C22⋊C8—172no—
187#398—22no—
188#395—22no—
189#380—21no—
190#380C2×D1712yes—
191C2×#95—171no—
192C2×#95—341no—
193C2×#95—341no—
194C2×#95C2×D1711yescharacteristic
195#340—342no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—21no—
200C2×C17#159811yescharacteristic
201C2×C17#160211yescharacteristic
202C2×C17#160711yescharacteristic
203C2×C17#161711yescharacteristic
204C2×C17#205711yescharacteristic
205C2×C17#206111yescharacteristic
206C2×C17#206211yescharacteristic
207C22×Q8—21no—
208C23×C4—341no—
209C23×C4—341no—
210C23×C4C22⋊D1711yescharacteristic
211C23×C4C22⋊D1711yescharacteristic
212C23×C4C22×D1711yescharacteristic
213C22×C8—342no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—342no—
221C2×C4⋊C4—682no—
222C2×C4⋊C4C22⋊D1712yes—
223C2×C4⋊C4C22⋊D1712yes—
224C2×C22⋊C4—22no—
225#107—684no—
226C22⋊C8—174no—
227C22⋊C8—342no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—172no—
232#95—341no—
233#95—682no—
234#95—682no—
235C17#1056911yescharacteristic, complemented, Sylow
236C24C2×C22⋊D1711yescharacteristic
237C2×Q8—42no—
238C2×Q8—81no—
239C2×Q8—81no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—342no—
253C22×C4—342no—
254C22×C4—681no—
255C22×C4—681no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4—681no—
259C22×C4—681no—
260C22×C4—681no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4#6322812yes—
267C22×C4#6323112yes—
268C22×C4Q8×D1711yescharacteristic
269C22×C4Q8×D1712yes—
270C22×C4#6325611yescharacteristic
271C22×C4C2×C22⋊D1712yes—
272C2×C8—344no—
273C2×C8—682no—
274C2×C8—682no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—344no—
282C4⋊C4—682no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C2×#6322811yescharacteristic
290C23#1058761811yescharacteristic
291C23#1058762811yescharacteristic
292C23C2×#6323111yescharacteristic
293C23#1058763011yescharacteristic
294C23#1058763211yescharacteristic
295C23#1058763911yescharacteristic
296Q8—84no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—682no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—1361no—
325C2×C4—1361no—
326C2×C4—1361no—
327C2×C4—1361no—
328C2×C4—1361no—
329C2×C4—1361no—
330C2×C4—1361no—
331C2×C4—1361no—
332C2×C4#1058763012yes—
333C2×C4#1058763212yes—
334C8—684no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1176272811yescharacteristic
348C22#1176294311yescharacteristic
349C22#1176296511yescharacteristic
350C22#1176298211yescharacteristic
351C22#1176298711yescharacteristic
352C22#1176299611yescharacteristic
353C22#1176299911yescharacteristic
354C22#1176308611yescharacteristic
355C22#1176310111yescharacteristic
356C22#1176321211yescharacteristic
357C22#1176321511yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—1362no—
365C4—1362no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#1452771911yescharacteristic
371C2#1452774111yescharacteristic
372C2#1452777011yescharacteristic
373C2#1452782611yescharacteristic
374C2#1453010911yescharacteristic
375C2#1453013111yescharacteristic
376C2#1453013611yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#2659C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenter#2659D17
hypocenterC17#10569
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
43. #10587502Q8semidirect
45. C22⋊#3622Q8semidirect
235. C17#10569semidirect

Maximal subgroups

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

subgroupindex
#145289712
#145270402
#145266622
#145266272
C17×#26592
#1056917