Finite Group Databasebeta

← #14536726

Subgroups — #14536726

Order 2176.

480 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762750C211yesmaximal, characteristic, complemented
2#11762747C211yesmaximal, characteristic, complemented
3#11762722C211yesmaximal, characteristic, complemented
4#11762722C211yesmaximal, characteristic, complemented
5#11762721C211yesmaximal, characteristic, complemented
6#11762721C211yesmaximal, characteristic, complemented
7#11762714C211yesmaximal, characteristic, complemented
8#11762713C211yesmaximal, characteristic, complemented
9#11762709C211yesmaximal, characteristic
10#11762688C211yesmaximal, characteristic
11#11762685C211yesmaximal, characteristic
12#11762682C211yesmaximal, characteristic, complemented
13#11762680C211yesmaximal, characteristic, complemented
14#11762680C211yesmaximal, characteristic, complemented
15C17×#385C211yesmaximal, characteristic, complemented
16C23×C4×C17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C22×Dic34C2211yescharacteristic
20C22×Dic34C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×C4×Dic17C2211yescharacteristic
37C17×#95C2211yescharacteristic
38C17×#95C2211yescharacteristic
39#10587502C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C23×D17C2311yescharacteristic
53C22×C4×C17—41no—
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C2×C22⋊D17—41no—
63C2×C22⋊D17—41no—
64C2×C22⋊D17Q814yescomplemented
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C2×C4×D17—41no—
73C2×C4×D17—41no—
74C2×C4×D17—41no—
75C2×Dic34—22no—
76C2×Dic34—22no—
77C2×Dic34—41no—
78C2×Dic34—41no—
79C2×Dic34—41no—
80C2×Dic34—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—41no—
85C22⋊Dic17—22no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88#63227—22no—
89#63227—22no—
90#63227—41no—
91#63227—41no—
92#63227—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×C17C2×Q812yes—
104C23×C17C2411yescharacteristic
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17C2×Q812yes—
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—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—44no—
130C2×C4×C17#4112yes—
131C22⋊D17—44no—
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×Dic17C2×Q812yes—
152C2×Dic17#4112yes—
153C4×D17—44no—
154Dic34—44no—
155Dic34—44no—
156#2804—171nomaximal, Sylow
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17C22×Q811yescharacteristic
165C22×C17C2×#4111yescharacteristic
166C22×C17C2×#4111yescharacteristic
167C22×C17H(5,𝔽2)11yescharacteristic
168C22×C17H(5,𝔽2)11yescharacteristic
169C22×C17H(5,𝔽2)11yescharacteristic
170C22×C17H(5,𝔽2)11yescharacteristic
171C2×D17—22no—
172C2×D17—22no—
173C2×D17—22no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C4×C17—22no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184Dic17—22no—
185Dic17—22no—
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189Dic17—41no—
190Dic17—41no—
191#400—171no—
192#400—171no—
193#393—171no—
194#393—171no—
195#393—171no—
196#393—171no—
197#391—171no—
198#391—171no—
199#390—171no—
200#386—171no—
201#386—171no—
202#385—171no—
203#385—171no—
204#385—171no—
205#385D1711yescharacteristic, complemented
206C2×C17—42no—
207C2×C17#53811yescharacteristic
208C2×C17#53811yescharacteristic
209C2×C17#54311yescharacteristic
210C2×C17#55411yescharacteristic
211C2×C17#55411yescharacteristic
212C2×C17#55911yescharacteristic
213C2×C17H(3,𝔽4)11yescharacteristic
214D17—42no—
215C22×Q8—171no—
216C22×Q8—171no—
217C22×D4—171no—
218C23×C4—171no—
219C23×C4C2×D1711yescharacteristic
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4C2×D1711yescharacteristic
224C2×C4⋊C4C2×D1711yescharacteristic
225C2×C22⋊C4—171no—
226C2×C22⋊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⋊C4C2×D1711yescharacteristic
234C2×C22⋊C4C2×D1711yescharacteristic
235C2×C42—171no—
236#95—171no—
237#95—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#95—171no—
247#95—171no—
248#95C2×D1711yescharacteristic
249#95C2×D1711yescharacteristic
250C17#280411yescharacteristic, complemented, Sylow
251C24—171no—
252C24C22×D1711yescharacteristic
253C2×Q8—342no—
254C2×Q8—342no—
255C2×Q8—681no—
256C2×Q8—681no—
257C2×Q8—681no—
258C2×Q8—681no—
259C2×D4—174no—
260C2×D4—681no—
261C2×D4—681no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—681no—
273C22×C4—681no—
274C22×C4—681no—
275C22×C4C22×D1711yescharacteristic
276C22×C4C22×D1711yescharacteristic
277C22×C4C22×D1711yescharacteristic
278C22×C4C22×D1711yescharacteristic
279C22×C4C22×D1711yescharacteristic
280C22×C4C22×D1711yescharacteristic
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—681no—
286C22⋊C4—22no—
287C22⋊C4—41no—
288C22⋊C4—342no—
289C22⋊C4—342no—
290C22⋊C4—342no—
291C22⋊C4—681no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C42—681no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—172no—
304C23—341no—
305C23—341no—
306C23—341no—
307C23—341no—
308C23—341no—
309C23—341no—
310C23Q8×D1712yes—
311C23C23×D1711yescharacteristic
312Q8—684no—
313Q8—684no—
314D4—684no—
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—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—44no—
333C2×C4—172no—
334C2×C4—172no—
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—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—684no—
355C2×C4#6325212yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—342no—
364C22—342no—
365C22—342no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22C2×#6325211yescharacteristic
371C22#1058769111yescharacteristic
372C22#1058769111yescharacteristic
373C22#1058769111yescharacteristic
374C22C2×Q8×D1711yescharacteristic
375C22#41×D1711yescharacteristic
376C22#1058769611yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—342no—
384C4—342no—
385C4—681no—
386C4—681no—
387C4—681no—
388C4—681no—
389C4—681no—
390C2—42no—
391C2—682no—
392C2#1176339211yescharacteristic
393C2#1176339511yescharacteristic
394C2#1176343411yescharacteristic
395C2#1176351311yescharacteristic
396C2#1176351411yescharacteristic
397C2#1176354211yescharacteristic
398C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC22×C22⋊D17C22
hypercenter#385D17
hypocenterC17#2804
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #11762750C2semidirect
2. #11762747C2semidirect
3. #11762722C2semidirect
4. #11762722C2semidirect
5. #11762721C2semidirect
6. #11762721C2semidirect
7. #11762714C2semidirect
8. #11762713C2semidirect
12. #11762682C2semidirect
13. #11762680C2semidirect
14. #11762680C2semidirect
15. C17×#385C2semidirect
64. C2×C22⋊D17Q8semidirect
205. #385D17semidirect
250. C17#2804semidirect

Maximal subgroups

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

subgroupindex
#117627502
#117627472
#117627222
#117627212
#117627142
#117627132
#117627092
#117626882
#117626852
#117626822
#117626802
C17×#3852
#280417