Finite Group Databasebeta

← #14536436

Subgroups — #14536436

Order 2176.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762751C211yesmaximal, characteristic, complemented
2#11762745C211yesmaximal, characteristic, complemented
3#11762741C211yesmaximal, characteristic, complemented
4#11762732C211yesmaximal, characteristic, complemented
5#11762732C211yesmaximal, characteristic, complemented
6#11762732C211yesmaximal, characteristic, complemented
7#11762712C211yesmaximal, characteristic
8#11762706C211yesmaximal, characteristic, complemented
9#11762696C211yesmaximal, characteristic, complemented
10#11762689C211yesmaximal, characteristic
11#11762684C211yesmaximal, characteristic, complemented
12#11762682C211yesmaximal, characteristic, complemented
13#11762681C211yesmaximal, characteristic, complemented
14#11762669C211yesmaximal, characteristic, complemented
15C17×#386C211yesmaximal, characteristic
16C22×D4×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C23×Dic17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C4⋊Dic17C2211yescharacteristic
30C2×C4⋊Dic17C2211yescharacteristic
31C2×C4⋊Dic17C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×C4×Dic17C2211yescharacteristic
41C17×#95C2211yescharacteristic
42C17×#95C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C24×C17C2311yescharacteristic
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17Q814yes—
56C22×C4×C17—22no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17—41no—
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22⋊C4×C17—22no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊Dic17—22no—
79C22⋊Dic17—22no—
80C22⋊Dic17—41no—
81C22⋊Dic17—41no—
82C22⋊Dic17—41no—
83C22⋊Dic17—41no—
84C22⋊Dic17—41no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90#63225—22no—
91#63225—22no—
92#63225—41no—
93#63225—41no—
94#63225—41no—
95#63225—41no—
96C4×Dic17—41no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17C2×Q812yes—
110C23×C17C2×Q812yes—
111C23×C17C2411yescharacteristic
112D4×C17—44no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
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—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—44no—
127C2×C4×C17C2×Q812yes—
128C2×C4×C17#4112yes—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—44no—
153C2×Dic17#4112yes—
154#2780—171nomaximal, Sylow
155C22×C17—22no—
156C22×C17—22no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17C22×Q811yescharacteristic
170C22×C17C2×#4111yescharacteristic
171C22×C17C2×#4111yescharacteristic
172C22×C17H(5,𝔽2)11yescharacteristic
173C22×C17H(5,𝔽2)11yescharacteristic
174C22×C17H(5,𝔽2)11yescharacteristic
175C22×C17H(5,𝔽2)11yescharacteristic
176C4×C17—22no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181Dic17—22no—
182Dic17—41no—
183Dic17—41no—
184Dic17—41no—
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189#400—171no—
190#399—171no—
191#398—171no—
192#396—171no—
193#396—171no—
194#396—171no—
195#390—171no—
196#389—171no—
197#387—171no—
198#386—171no—
199#386D1711yescharacteristic
200#385—171no—
201#385—171no—
202#385—171no—
203#380—171no—
204C2×C17—42no—
205C2×C17—42no—
206C2×C17#53811yescharacteristic
207C2×C17#54011yescharacteristic
208C2×C17#54311yescharacteristic
209C2×C17#55311yescharacteristic
210C2×C17#55411yescharacteristic
211C2×C17#55411yescharacteristic
212C2×C17#56011yescharacteristic
213C22×D4C2×D1711yescharacteristic
214C23×C4—171no—
215C23×C4C2×D1711yescharacteristic
216C2×C4⋊C4—171no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C22⋊C4—171no—
228C2×C22⋊C4—171no—
229C2×C22⋊C4—171no—
230C2×C22⋊C4—171no—
231C2×C22⋊C4—171no—
232C2×C22⋊C4—171no—
233C2×C22⋊C4—171no—
234C2×C22⋊C4C2×D1711yescharacteristic
235C2×C22⋊C4C2×D1711yescharacteristic
236C2×C22⋊C4C2×D1711yescharacteristic
237C2×C42—171no—
238#95—171no—
239#95—171no—
240#95—171no—
241#95—171no—
242#95—171no—
243#95—171no—
244#95—171no—
245#95—171no—
246#95C2×D1711yescharacteristic
247#95C2×D1711yescharacteristic
248C17#278011yescharacteristic, complemented, Sylow
249C24C22×D1711yescharacteristic
250C24C22×D1711yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Dic3414yes—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4—171no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4C22×D1711yescharacteristic
269C22×C4C22×D1711yescharacteristic
270C22×C4C22×D1711yescharacteristic
271C22×C4C22×D1711yescharacteristic
272C22×C4C22×D1711yescharacteristic
273C4⋊C4—342no—
274C4⋊C4—342no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—342no—
288C22⋊C4—342no—
289C22⋊C4—681no—
290C22⋊C4—681no—
291C22⋊C4—681no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C42—681no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23C2×Dic3412yes—
308C23C2×Dic3412yes—
309C23C23×D1711yescharacteristic
310D4—44no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—172no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—684no—
350C2×C4C2×Dic3412yes—
351C2×C4#6325412yes—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22C22×Dic3411yescharacteristic
367C22C2×#6325411yescharacteristic
368C22#1058769111yescharacteristic
369C22#1058769111yescharacteristic
370C22#1058769111yescharacteristic
371C22#41×D1711yescharacteristic
372C22#1058769611yescharacteristic
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—342no—
379C4—681no—
380C4—681no—
381C4—681no—
382C4—681no—
383C4—681no—
384C4—681no—
385C4—681no—
386C2—42no—
387C2—42no—
388C2#1176339411yescharacteristic
389C2#1176341111yescharacteristic
390C2#1176343511yescharacteristic
391C2#1176350611yescharacteristic
392C2#1176351611yescharacteristic
393C2#1176351811yescharacteristic
394C2#1176354711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2780
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762745C2semidirect
3. #11762741C2semidirect
4. #11762732C2semidirect
5. #11762732C2semidirect
6. #11762732C2semidirect
8. #11762706C2semidirect
9. #11762696C2semidirect
11. #11762684C2semidirect
12. #11762682C2semidirect
13. #11762681C2semidirect
14. #11762669C2semidirect
248. C17#2780semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627452
#117627412
#117627322
#117627122
#117627062
#117626962
#117626892
#117626842
#117626822
#117626812
#117626692
C17×#3862
#278017