Finite Group Databasebeta

← #17949904

Subgroups — #17949904

Order 4352.

556 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544511C211yesmaximal, characteristic
2#14544448C211yesmaximal, characteristic
3#14543605C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5C4×#10587583C211yesmaximal, characteristic
6C22×#63243—21no—
7C2×#10587543—21no—
8C2×#10587542—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587583C411yescharacteristic
13C2×C4×Dic34C411yescharacteristic
14C2×C42×D17C2211yescharacteristic
15#11762714C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762694—21no—
18#11762683—21no—
19#11762666C411yescharacteristic
20C4×C4⋊Dic17—21no—
21C4×#63227—21no—
22C4×C4⋊C4×C17C411yescharacteristic
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C2×#63243—24no—
30C2×#63243—42no—
31C2×#63243—42no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C42×C17—21no—
35C2×C42×C17C2×C411yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17C2311yescharacteristic
46#10587583—22no—
47#10587583—22no—
48C42×D17—42no—
49C4×Dic34—42no—
50#10587543—24no—
51#10587542—42no—
52C17×#95—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C23×D17C22⋊C411yescharacteristic
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—21no—
71C2×C4×D17—21no—
72C2×C4×D17—22no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—22no—
76C2×C4×D17—41no—
77C2×C4×D17—41no—
78C2×C4×D17—41no—
79C2×Dic34—21no—
80C2×Dic34—41no—
81C2×Dic34—41no—
82C2×Dic34M1612yescomplemented
83C2×#3624—24no—
84C2×#3624—24no—
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—42no—
88C2×#3624—42no—
89#63243—48no—
90C4⋊C4×C17—21no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17M1612yescomplemented
93C42×C17—22no—
94C42×C17—42no—
95#63227—24no—
96#63227—42no—
97C4⋊Dic17—24no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101#63225—42no—
102C4×Dic17—22no—
103C4×Dic17—41no—
104C4×Dic17—41no—
105C4×Dic17—42no—
106#7634—171nomaximal, Sylow
107C23×C17C2×C22⋊C411yescharacteristic
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C22×D17#10412yes—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—22no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17#9912yes—
128C2×C4×C17#10412yes—
129C2×C4×C17C2×M1611yescharacteristic
130C2×C4×C17C2×M1611yescharacteristic
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—22no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17AGL(1,ℤ/8)11yescharacteristic
145C2×Dic17#13711yescharacteristic
146C2×Dic17#13712yes—
147C4×D17—24no—
148C4×D17—24no—
149C4×D17—44no—
150C4×D17—42no—
151C4×D17—42no—
152Dic34—42no—
153Dic34—42no—
154#3624—44no—
155#3624—44no—
156C4×C22⋊Q8—171no—
157#1749—171no—
158#1728—171no—
159#1272—172no—
160#1271—172no—
161C22×C17#40711yescharacteristic
162C22×C17C2×#9911yescharacteristic
163C22×C17#41311yescharacteristic
164C22×C17#41711yescharacteristic
165C22×C17#41911yescharacteristic
166C22×C17C2×#10411yescharacteristic
167C22×C17C2×#10411yescharacteristic
168C2×D17—22no—
169C2×D17—22no—
170C2×D17—22no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—22no—
178C4×C17—22no—
179C4×C17—42no—
180C4×C17—81no—
181Dic17—21no—
182Dic17—21no—
183Dic17—22no—
184Dic17—42no—
185Dic17—41no—
186Dic17—81no—
187C22×M16—341no—
188C2×C22⋊Q8—171no—
189C2×C4×Q8—171no—
190C22×C42—171no—
191C2×C22⋊C8—341no—
192C2×#97—341no—
193#391—171no—
194#389—171no—
195#387—341no—
196#385—341no—
197#380—171no—
198C4×C4⋊C4—341no—
199C4×C4⋊C4#59511yescharacteristic
200C4×C22⋊C4—341no—
201#336—172no—
202#336—172no—
203#336—341no—
204C2×C17#146511yescharacteristic
205C2×C17#148911yescharacteristic
206C2×C17#149111yescharacteristic
207C2×C17#150411yescharacteristic
208C2×C17#150811yescharacteristic
209C2×C17#152411yescharacteristic
210C2×C17#155911yescharacteristic
211D17—42no—
212C22×Q8—171no—
213C23×C4—171no—
214C23×C4—171no—
215C23×C4—171no—
216C2×M16—344no—
217C2×M16—682no—
218C2×M16—682no—
219C22×C8—342no—
220C22×C8—342no—
221C22⋊Q8—342no—
222C22⋊Q8—342no—
223C4×Q8—682no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4C2×#59511yescharacteristic
229C2×C4⋊C4C2×#59511yescharacteristic
230C2×C22⋊C4—341no—
231C2×C22⋊C4—341no—
232C2×C42—21no—
233C2×C42—171no—
234C2×C42—171no—
235C2×C42—171no—
236C2×C42—341no—
237C2×C42—682no—
238C2×C42C2×#59511yescharacteristic
239C22⋊C8—344no—
240#97—682no—
241#95—21no—
242#95—341no—
243#95—341no—
244#95—341no—
245#95—341no—
246C17#763411yescharacteristic, complemented, Sylow
247C24—171no—
248C2×Q8—172no—
249C2×Q8—341no—
250C2×Q8—681no—
251C2×Q8—681no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—171no—
255C22×C4—171no—
256C22×C4—171no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—342no—
260C22×C4—342no—
261C22×C4—342no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—342no—
266C22×C4—341no—
267C22×C4—681no—
268C22×C4—681no—
269C22×C4—681no—
270C22×C4C22⋊#59511yescharacteristic
271C22×C4C22⋊#59511yescharacteristic
272C22×C4C22×#59511yescharacteristic
273M16—688no—
274C2×C8—344no—
275C2×C8—344no—
276C2×C8—682no—
277C2×C8—682no—
278C2×C8—682no—
279C2×C8—682no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—344no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—681no—
286C4⋊C4—682no—
287C4⋊C4#6324312yescomplemented
288C22⋊C4—344no—
289C22⋊C4—682no—
290C42—22no—
291C42—42no—
292C42—342no—
293C42—681no—
294C42—681no—
295C42—682no—
296C23—172no—
297C23—341no—
298C23—341no—
299C23—341no—
300C23—341no—
301C23—341no—
302C23—341no—
303C23C2×C22⋊#59511yescharacteristic
304Q8—682no—
305Q8—682no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—171no—
319C2×C4—171no—
320C2×C4—172no—
321C2×C4—344no—
322C2×C4—344no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—342no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—684no—
332C2×C4—682no—
333C2×C4—682no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4—681no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4#1058753912yes—
340C2×C4#1058754812yes—
341C2×C4C2×#6324311yescharacteristic
342C2×C4D17⋊M1611yescharacteristic
343C8—684no—
344C8—684no—
345C22—342no—
346C22—342no—
347C22—342no—
348C22—681no—
349C22—681no—
350C22—681no—
351C22—681no—
352C22#1176378011yescharacteristic
353C22C2×#1058753911yescharacteristic
354C22#1176379511yescharacteristic
355C22#1176380011yescharacteristic
356C22#1176380211yescharacteristic
357C22C2×#1058754811yescharacteristic
358C22D17⋊#10411yescharacteristic
359C4—22no—
360C4—22no—
361C4—21no—
362C4—21no—
363C4—42no—
364C4—81no—
365C4—341no—
366C4—341no—
367C4—342no—
368C4—681no—
369C4—682no—
370C4—1361no—
371C2—682no—
372C2#1454404111yescharacteristic
373C2#1454412011yescharacteristic
374C2#1454412711yescharacteristic
375C2#1454417211yescharacteristic
376C2#1454418311yescharacteristic
377C2#1454423411yescharacteristic
378C2#1454431911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#7634
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543605C2semidirect
4. #14543602C2semidirect
82. C2×Dic34M16semidirect
92. C4⋊C4×C17M16semidirect
246. C17#7634semidirect
287. C4⋊C4#63243semidirect

Maximal subgroups

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

subgroupindex
#145445112
#145444482
#145436052
#145436022
C4×#105875832
#763417