Finite Group Databasebeta

← #14536610

Subgroups — #14536610

Order 2176.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762747C211yesmaximal, characteristic, complemented
2#11762731C211yesmaximal, characteristic, complemented
3#11762725C211yesmaximal, characteristic
4#11762724C211yesmaximal, characteristic
5#11762724C211yesmaximal, characteristic
6#11762703C211yesmaximal, characteristic, complemented
7#11762702C211yesmaximal, characteristic, complemented
8#11762699C211yesmaximal, characteristic, complemented
9#11762699C211yesmaximal, characteristic, complemented
10#11762698C211yesmaximal, characteristic, complemented
11#11762697C211yesmaximal, characteristic, complemented
12#11762695C211yesmaximal, characteristic, complemented
13#11762694C211yesmaximal, characteristic, complemented
14#11762678C211yesmaximal, characteristic, complemented
15C17×#396C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×C4×Dic17C2211yescharacteristic
39C2×C4×Dic17C2211yescharacteristic
40C2×C4×Dic17C2211yescharacteristic
41C17×#95C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C23×D17C2311yescharacteristic
53C22×C4×C17C2311yescharacteristic
54C22×C4×C17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C2×C22⋊D17—41no—
60C2×C22⋊D17—41no—
61C2×C22⋊D17—41no—
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C4⋊C4×C17—22no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—41no—
72C22⋊C4×C17—22no—
73C22⋊C4×C17—41no—
74C22⋊C4×C17—41no—
75C22⋊Dic17—22no—
76C22⋊Dic17—22no—
77C22⋊Dic17—41no—
78#63227—22no—
79#63227—22no—
80#63227—22no—
81#63227—41no—
82#63227—41no—
83#63227—41no—
84C4⋊Dic17—22no—
85C4⋊Dic17—41no—
86#63225—41no—
87#63225—41no—
88#63225—41no—
89#63225—41no—
90C4×Dic17—22no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17C2411yescharacteristic
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
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#4112yes—
126C22⋊D17—44no—
127C2×Dic17—21no—
128C2×Dic17—21no—
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#4112yes—
147C2×Dic17#4112yes—
148#2795—171nomaximal, Sylow
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17C2×#4111yescharacteristic
157C22×C17C2×#4111yescharacteristic
158C22×C17C2×#4111yescharacteristic
159C22×C17H(5,𝔽2)11yescharacteristic
160C22×C17H(5,𝔽2)11yescharacteristic
161C22×C17H(5,𝔽2)11yescharacteristic
162C22×C172−1+411yescharacteristic
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
167C2×D17—41no—
168C2×D17—41no—
169C2×D17—41no—
170C4×C17—22no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176Dic17—22no—
177Dic17—22no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183#400—171no—
184#396—171no—
185#396D1711yescharacteristic, complemented
186#394—171no—
187#394—171no—
188#394—171no—
189#388—171no—
190#388—171no—
191#388—171no—
192#388—171no—
193#388—171no—
194#388—171no—
195#387—171no—
196#387—171no—
197#384—171no—
198C2×C17—81no—
199C2×C17#53911yescharacteristic
200C2×C17#54211yescharacteristic
201C2×C17#54211yescharacteristic
202C2×C17#55311yescharacteristic
203C2×C17#55311yescharacteristic
204C2×C17#55511yescharacteristic
205C2×C17#56011yescharacteristic
206D17—81no—
207C22×D4—171no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—171no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4C2×D1711yescharacteristic
215C2×C4⋊C4C2×D1711yescharacteristic
216C2×C4⋊C4C2×D1711yescharacteristic
217C2×C22⋊C4—171no—
218C2×C22⋊C4—171no—
219C2×C22⋊C4—171no—
220C2×C22⋊C4—171no—
221C2×C22⋊C4—171no—
222C2×C22⋊C4—171no—
223C2×C22⋊C4—171no—
224C2×C22⋊C4—171no—
225C2×C22⋊C4—171no—
226C2×C22⋊C4C2×D1711yescharacteristic
227C2×C22⋊C4C2×D1711yescharacteristic
228C2×C22⋊C4C2×D1711yescharacteristic
229C2×C42—171no—
230C2×C42—171no—
231C2×C42—171no—
232#95—171no—
233#95—171no—
234#95—171no—
235#95—171no—
236#95—171no—
237#95—171no—
238#95—171no—
239#95—171no—
240#95—171no—
241#95C2×D1711yescharacteristic
242C17#279511yescharacteristic, complemented, Sylow
243C24—171no—
244C24C22×D1711yescharacteristic
245C2×D4—681no—
246C2×D4—681no—
247C2×D4—681no—
248C22×C4—171no—
249C22×C4—171no—
250C22×C4—171no—
251C22×C4—171no—
252C22×C4—171no—
253C22×C4—171no—
254C22×C4—171no—
255C22×C4C22×D1711yescharacteristic
256C22×C4C22×D1711yescharacteristic
257C22×C4C22×D1711yescharacteristic
258C22×C4C22×D1711yescharacteristic
259C22×C4C22×D1711yescharacteristic
260C22×C4C22×D1711yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—342no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C4⋊C4—681no—
269C4⋊C4—681no—
270C22⋊C4—22no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—342no—
274C22⋊C4—342no—
275C22⋊C4—342no—
276C22⋊C4—342no—
277C22⋊C4—342no—
278C22⋊C4—681no—
279C22⋊C4—681no—
280C22⋊C4—681no—
281C22⋊C4—681no—
282C42—342no—
283C42—681no—
284C42—681no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—341no—
293C23—341no—
294C23—341no—
295C23—341no—
296C23—341no—
297C23—341no—
298C23—341no—
299C23C23×D1711yescharacteristic
300D4—684no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—172no—
319C2×C4—172no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
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#6325412yes—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—681no—
348C22—681no—
349C22—681no—
350C22—681no—
351C22—681no—
352C22—681no—
353C22—681no—
354C22C2×#6325411yescharacteristic
355C22#1058769111yescharacteristic
356C22#1058769111yescharacteristic
357C22#1058769411yescharacteristic
358C22#41×D1711yescharacteristic
359C22#41×D1711yescharacteristic
360C22#1058769611yescharacteristic
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—342no—
368C4—342no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4—681no—
373C4—681no—
374C2—81no—
375C2—1361no—
376C2#1176340511yescharacteristic
377C2#1176342911yescharacteristic
378C2#1176343211yescharacteristic
379C2#1176350211yescharacteristic
380C2#1176351011yescharacteristic
381C2#1176352311yescharacteristic
382C2#1176354711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#396C2
prime radicalC22×C22⋊D17C22
hypercenter#396D17
hypocenterC17#2795
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #11762747C2semidirect
2. #11762731C2semidirect
6. #11762703C2semidirect
7. #11762702C2semidirect
8. #11762699C2semidirect
9. #11762699C2semidirect
10. #11762698C2semidirect
11. #11762697C2semidirect
12. #11762695C2semidirect
13. #11762694C2semidirect
14. #11762678C2semidirect
15. C17×#396C2semidirect
185. #396D17semidirect
242. C17#2795semidirect

Maximal subgroups

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

subgroupindex
#117627472
#117627312
#117627252
#117627242
#117627032
#117627022
#117626992
#117626982
#117626972
#117626952
#117626942
#117626782
C17×#3962
#279517