Finite Group Databasebeta

← #14536262

Subgroups — #14536262

Order 2176.

408 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762745C211yesmaximal, characteristic
2#11762743C211yesmaximal, characteristic, complemented
3#11762742C211yesmaximal, characteristic, complemented
4#11762740C211yesmaximal, characteristic, complemented
5#11762739C211yesmaximal, characteristic, complemented
6#11762732C211yesmaximal, characteristic
7#11762708C211yesmaximal, characteristic, complemented
8#11762706C211yesmaximal, characteristic, complemented
9#11762703C211yesmaximal, characteristic
10#11762697C211yesmaximal, characteristic
11#11762694C211yesmaximal, characteristic, complemented
12#11762682C211yesmaximal, characteristic
13#11762673C211yesmaximal, characteristic, complemented
14#11762669C211yesmaximal, characteristic
15C17×#385C211yesmaximal, characteristic
16C23×C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×C4⋊Dic17C2211yescharacteristic
28C2×C4⋊Dic17C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×C4×Dic17C2211yescharacteristic
38C2×C4×Dic17C2211yescharacteristic
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C22×C4×C17—22no—
53C22×C4×C17—41no—
54C22×C4×C17—41no—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C4⋊C4×C17—22no—
70C4⋊C4×C17—41no—
71C22⋊C4×C17—22no—
72C22⋊C4×C17—41no—
73C22⋊Dic17—41no—
74C22⋊Dic17—41no—
75C22⋊Dic17—41no—
76C22⋊Dic17Q814yes—
77C4⋊Dic17—22no—
78C4⋊Dic17—41no—
79C4⋊Dic17—41no—
80C4⋊Dic17—41no—
81#63225—22no—
82#63225—41no—
83#63225—41no—
84#63225—41no—
85#63225—41no—
86#63225—41no—
87#63225—41no—
88#63225—41no—
89C4×Dic17—22no—
90C4×Dic17—41no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17C2×Q812yes—
98C23×C17C2411yescharacteristic
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—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—44no—
116C2×C4×C17#4112yes—
117C2×C4×C17#4112yes—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
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×Dic17C2×Q812yes—
141C2×Dic17C2×Q812yes—
142#2768—171nomaximal, Sylow
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17C22×Q811yescharacteristic
151C22×C17C2×#4111yescharacteristic
152C22×C17C2×#4111yescharacteristic
153C22×C17H(5,𝔽2)11yescharacteristic
154C22×C17H(5,𝔽2)11yescharacteristic
155C22×C17H(5,𝔽2)11yescharacteristic
156C22×C172−1+411yescharacteristic
157C4×C17—22no—
158C4×C17—22no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163Dic17—22no—
164Dic17—22no—
165Dic17—41no—
166Dic17—41no—
167Dic17—41no—
168Dic17—41no—
169Dic17—41no—
170Dic17—41no—
171#399—171no—
172#398—171no—
173#398—171no—
174#398—171no—
175#398—171no—
176#396—171no—
177#389—171no—
178#389—171no—
179#388—171no—
180#388—171no—
181#387—171no—
182#385—171no—
183#385D1711yescharacteristic
184#382—171no—
185#380—171no—
186C2×C17—42no—
187C2×C17#53811yescharacteristic
188C2×C17#54011yescharacteristic
189C2×C17#54411yescharacteristic
190C2×C17#55211yescharacteristic
191C2×C17#55411yescharacteristic
192C2×C17#55411yescharacteristic
193C2×C17#56211yescharacteristic
194C23×C4C2×D1711yescharacteristic
195C2×C4⋊C4—171no—
196C2×C4⋊C4—171no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4C2×D1711yescharacteristic
208C2×C4⋊C4C2×D1711yescharacteristic
209C2×C22⋊C4—171no—
210C2×C22⋊C4—171no—
211C2×C22⋊C4—171no—
212C2×C22⋊C4—171no—
213C2×C22⋊C4C2×D1711yescharacteristic
214C2×C22⋊C4C2×D1711yescharacteristic
215C2×C42—171no—
216C2×C42—171no—
217#95—171no—
218#95—171no—
219#95—171no—
220#95—171no—
221#95—171no—
222#95—171no—
223#95—171no—
224#95—171no—
225#95—171no—
226#95—171no—
227#95C2×D1711yescharacteristic
228#95C2×D1711yescharacteristic
229C17#276811yescharacteristic, complemented, Sylow
230C24C22×D1711yescharacteristic
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4C22×D1711yescharacteristic
243C22×C4C22×D1711yescharacteristic
244C22×C4C22×D1711yescharacteristic
245C22×C4C22×D1711yescharacteristic
246C22×C4C22×D1711yescharacteristic
247C22×C4C22×D1711yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—342no—
251C4⋊C4—342no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C4⋊C4—681no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—174no—
265C22⋊C4—681no—
266C22⋊C4—681no—
267C22⋊C4—681no—
268C42—342no—
269C42—681no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23Q8×D1712yes—
277C23C23×D1711yescharacteristic
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—172no—
296C2×C4—172no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4#6325212yes—
320C2×C4#6325412yes—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C2×#6325211yescharacteristic
329C22C2×#6325411yescharacteristic
330C22#1058769111yescharacteristic
331C22#1058769111yescharacteristic
332C22C2×Q8×D1711yescharacteristic
333C22#1058769611yescharacteristic
334C22#1058769711yescharacteristic
335C4—22no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—342no—
342C4—342no—
343C4—681no—
344C4—681no—
345C4—681no—
346C4—681no—
347C4—681no—
348C4—681no—
349C2—42no—
350C2#1176339411yescharacteristic
351C2#1176341011yescharacteristic
352C2#1176343811yescharacteristic
353C2#1176349911yescharacteristic
354C2#1176351411yescharacteristic
355C2#1176351711yescharacteristic
356C2#1176355611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2768
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #11762743C2semidirect
3. #11762742C2semidirect
4. #11762740C2semidirect
5. #11762739C2semidirect
7. #11762708C2semidirect
8. #11762706C2semidirect
11. #11762694C2semidirect
13. #11762673C2semidirect
229. C17#2768semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627432
#117627422
#117627402
#117627392
#117627322
#117627082
#117627062
#117627032
#117626972
#117626942
#117626822
#117626732
#117626692
C17×#3852
#276817