Finite Group Databasebeta

← #30670555

Subgroups — #30670555

Order 16768.

440 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#22553240C211yesmaximal, characteristic, complemented
2#22553239C211yesmaximal, characteristic, complemented
3#22553239C211yesmaximal, characteristic, complemented
4#22553237C211yesmaximal, characteristic, complemented
5#22553212C211yesmaximal, characteristic
6#22553212C211yesmaximal, characteristic
7#22553212C211yesmaximal, characteristic
8#22553209C211yesmaximal, characteristic, complemented
9#22553205C211yesmaximal, characteristic, complemented
10#22553203C211yesmaximal, characteristic, complemented
11#22553202C211yesmaximal, characteristic, complemented
12#22553176C211yesmaximal, characteristic, complemented
13C4⋊C4×Dic131C211yesmaximal, characteristic, complemented
14C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
15#388×C131C211yesmaximal, characteristic, complemented
16C22×C22⋊D131C2211yescharacteristic
17C2×C4⋊Dic131C2211yescharacteristic
18C2×C4⋊Dic131C2211yescharacteristic
19C2×Dic131⋊C4C2211yescharacteristic
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×C22⋊Dic131C2211yescharacteristic
24C2×C22⋊Dic131C2211yescharacteristic
25C2×C22⋊Dic131C2211yescharacteristic
26C2×(C2×D131)⋊C4C2211yescharacteristic
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×C4×Dic131C2211yescharacteristic
33C2×C4×Dic131C2211yescharacteristic
34C2×C4×Dic131C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C131⋊#95C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C2×C4⋊C4×C131C2211yescharacteristic
45C2×C22⋊C4×C131C2211yescharacteristic
46C2×C22⋊C4×C131C2211yescharacteristic
47C2×C22⋊C4×C131C2211yescharacteristic
48C2×C42×C131C2211yescharacteristic
49#95×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51C23×D131C2311yescharacteristic
52C2×C22⋊D131—22no—
53C2×C22⋊D131—22no—
54C2×C22⋊D131—22no—
55C22×Dic131C2311yescharacteristic
56C22×Dic131C2311yescharacteristic
57C22×Dic131C2311yescharacteristic
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2311yescharacteristic
60C22×Dic131C2311yescharacteristic
61C22×Dic131C2311yescharacteristic
62C4⋊Dic131—22no—
63C4⋊Dic131—41no—
64Dic131⋊C4—22no—
65Dic131⋊C4—22no—
66Dic131⋊C4—22no—
67Dic131⋊C4—41no—
68C22⋊Dic131—41no—
69C22⋊Dic131—41no—
70C22⋊Dic131—41no—
71(C2×D131)⋊C4—22no—
72(C2×D131)⋊C4—22no—
73(C2×D131)⋊C4—41no—
74(C2×D131)⋊C4—41no—
75(C2×D131)⋊C4—41no—
76(C2×D131)⋊C4—41no—
77C4×Dic131—22no—
78C4×Dic131—22no—
79C4×Dic131—22no—
80C4×Dic131—41no—
81C24×C131C2311yescharacteristic
82C22×C4×C131C2311yescharacteristic
83C22×C4×C131C2311yescharacteristic
84C22×C4×C131C2311yescharacteristic
85C22×C4×C131C2311yescharacteristic
86C22×C4×C131C2311yescharacteristic
87C22×C4×C131C2311yescharacteristic
88C4⋊C4×C131—22no—
89C22⋊C4×C131—22no—
90C22⋊C4×C131—22no—
91C22⋊C4×C131—41no—
92C42×C131—41no—
93C22×D131—21no—
94C22×D131—21no—
95C22×D131—21no—
96C22×D131—21no—
97C22×D131—21no—
98C22×D131—21no—
99C22×D131—21no—
100C22⋊D131—44no—
101C2×Dic131—21no—
102C2×Dic131—21no—
103C2×Dic131—21no—
104C2×Dic131—21no—
105C2×Dic131—21no—
106C2×Dic131—21no—
107C2×Dic131—21no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—21no—
111C2×Dic131—21no—
112C2×Dic131—21no—
113C2×Dic131—21no—
114C2×Dic131—21no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C2×Dic131—21no—
119C2×Dic131#4112yes—
120C2×Dic131#4112yes—
121C2×Dic131#4112yes—
122C23×C131—21no—
123C23×C131—21no—
124C23×C131—21no—
125C23×C131—21no—
126C23×C131—21no—
127C23×C131—21no—
128C23×C131—21no—
129C23×C131C2411yescharacteristic
130C2×C4×C131—21no—
131C2×C4×C131—21no—
132C2×C4×C131—21no—
133C2×C4×C131—21no—
134C2×C4×C131—21no—
135C2×C4×C131—21no—
136C2×C4×C131—21no—
137C2×C4×C131—21no—
138C2×C4×C131—21no—
139C2×C4×C131—21no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131#4112yes—
147C2×C4×C131#4112yes—
148C22×C131—41no—
149C22×C131—41no—
150C22×C131—41no—
151C22×C131—41no—
152C22×C131—41no—
153C22×C131—41no—
154C22×C131—41no—
155C22×C131C2×#4111yescharacteristic
156C22×C131C2×#4111yescharacteristic
157C22×C131C2×#4111yescharacteristic
158C22×C131C2×#4111yescharacteristic
159C22×C131C2×#4111yescharacteristic
160C22×C131H(5,𝔽2)11yescharacteristic
161C22×C131H(5,𝔽2)11yescharacteristic
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—41no—
165C2×D131—41no—
166C2×D131—41no—
167C2×D131—41no—
168C2×D131—41no—
169C4×C131—22no—
170C4×C131—22no—
171C4×C131—41no—
172C4×C131—41no—
173C4×C131—41no—
174C4×C131—41no—
175Dic131—41no—
176Dic131—41no—
177Dic131—41no—
178Dic131—41no—
179Dic131—41no—
180Dic131—41no—
181Dic131#12614yes—
182C2×C131—81no—
183C2×C131C2×#12611yescharacteristic
184C2×C131#52911yescharacteristic
185C2×C131#54011yescharacteristic
186C2×C131#55311yescharacteristic
187C2×C131#55511yescharacteristic
188C2×C131#55911yescharacteristic
189C2×C131#55911yescharacteristic
190D131—81no—
191C131#251311yescharacteristic, complemented, Sylow
192#2513—1311nomaximal, Sylow
193#397—1311no—
194#397—1311no—
195#397—1311no—
196#397—1311no—
197#390—1311no—
198#390—1311no—
199#390—1311no—
200#389—1311no—
201#388—1311no—
202#388—1311no—
203#388—1311no—
204#388D13111yescharacteristic, complemented
205#382—1311no—
206C4×C4⋊C4—1311no—
207C4×C4⋊C4—1311no—
208C22×D4—1311no—
209C2×C4⋊C4—1311no—
210C2×C4⋊C4—1311no—
211C2×C4⋊C4—1311no—
212C2×C4⋊C4—1311no—
213C2×C4⋊C4—1311no—
214C2×C4⋊C4—1311no—
215C2×C4⋊C4C2×D13111yescharacteristic
216C2×C22⋊C4—1311no—
217C2×C22⋊C4—1311no—
218C2×C22⋊C4—1311no—
219C2×C22⋊C4—1311no—
220C2×C22⋊C4—1311no—
221C2×C22⋊C4—1311no—
222C2×C22⋊C4—1311no—
223C2×C22⋊C4—1311no—
224C2×C22⋊C4—1311no—
225C2×C22⋊C4C2×D13111yescharacteristic
226C2×C22⋊C4C2×D13111yescharacteristic
227C2×C22⋊C4C2×D13111yescharacteristic
228C2×C42—1311no—
229C2×C42—1311no—
230C2×C42—1311no—
231C2×C42—1311no—
232C2×C42C2×D13111yescharacteristic
233#95—1311no—
234#95—1311no—
235#95—1311no—
236#95—1311no—
237#95—1311no—
238#95—1311no—
239#95—1311no—
240#95—1311no—
241#95C2×D13111yescharacteristic
242#95C2×D13111yescharacteristic
243C24—1311no—
244C24C22×D13111yescharacteristic
245C2×D4—2622no—
246C2×D4—2622no—
247C2×D4—2622no—
248C22×C4—1311no—
249C22×C4—1311no—
250C22×C4—1311no—
251C22×C4—1311no—
252C22×C4—1311no—
253C22×C4—1311no—
254C22×C4—1311no—
255C22×C4C22×D13111yescharacteristic
256C22×C4C22×D13111yescharacteristic
257C22×C4C22×D13111yescharacteristic
258C22×C4C22×D13111yescharacteristic
259C22×C4C22×D13111yescharacteristic
260C22×C4C22×D13111yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—2622no—
263C4⋊C4—2622no—
264C4⋊C4—2622no—
265C4⋊C4—2622no—
266C4⋊C4—5241no—
267C4⋊C4—5241no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—41no—
271C22⋊C4—2622no—
272C22⋊C4—2622no—
273C22⋊C4—5241no—
274C22⋊C4—5241no—
275C22⋊C4—5241no—
276C22⋊C4—5241no—
277C22⋊C4—5241no—
278C22⋊C4—5241no—
279C22⋊C4—5241no—
280C42—41no—
281C42—2622no—
282C42—2622no—
283C42—2622no—
284C42—5241no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—2621no—
293C23—2621no—
294C23—2621no—
295C23—2621no—
296C23—2621no—
297C23—2621no—
298C23—2621no—
299C23C23×D13111yescharacteristic
300D4—5244no—
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—2621no—
318C2×C4—2621no—
319C2×C4—2621no—
320C2×C4—2621no—
321C2×C4—2621no—
322C2×C4—2621no—
323C2×C4—2621no—
324C2×C4—2621no—
325C2×C4—2621no—
326C2×C4—2621no—
327C2×C4—2621no—
328C2×C4—2621no—
329C2×C4—2621no—
330C2×C4—2621no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—2621no—
334C2×C4—2621no—
335C2×C4—1312no—
336C2×C4—1312no—
337C2×C4—1312no—
338C2×C4#1452351212yes—
339C2×C4#1452351312yes—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—5241no—
348C22—5241no—
349C22—5241no—
350C22—5241no—
351C22—5241no—
352C22—5241no—
353C22—5241no—
354C22#41×D13111yescharacteristic
355C22#41×D13111yescharacteristic
356C22#41×D13111yescharacteristic
357C22C2×#1452351211yescharacteristic
358C22C2×#1452351311yescharacteristic
359C22#1686229111yescharacteristic
360C22#1686229211yescharacteristic
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—5241no—
368C4—5241no—
369C4—5241no—
370C4—5241no—
371C4—5241no—
372C4—5241no—
373C4—1314no—
374C2—81no—
375C2—10481no—
376C2#126×D13111yescharacteristic
377C2#2255384211yescharacteristic
378C2#2255391411yescharacteristic
379C2#2255400511yescharacteristic
380C2#2255402511yescharacteristic
381C2#2255404411yescharacteristic
382C2#2255404511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#388×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#388D131
hypocenterC131#2513
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. #22553240C2semidirect
2. #22553239C2semidirect
3. #22553239C2semidirect
4. #22553237C2semidirect
8. #22553209C2semidirect
9. #22553205C2semidirect
10. #22553203C2semidirect
11. #22553202C2semidirect
12. #22553176C2semidirect
13. C4⋊C4×Dic131C2semidirect
14. C4×Dic131⋊C4C2semidirect
15. #388×C131C2semidirect
191. C131#2513semidirect
204. #388D131semidirect

Maximal subgroups

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

subgroupindex
#225532402
#225532392
#225532372
#225532122
#225532092
#225532052
#225532032
#225532022
#225531762
C4⋊C4×Dic1312
C4×Dic131⋊C42
#388×C1312
#2513131