Finite Group Databasebeta

← #31783643

Subgroups — #31783643

Order 17536.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984044C211yesmaximal, characteristic
2#22577287C211yesmaximal, characteristic, complemented
3#22577287C211yesmaximal, characteristic, complemented
4#22577285C211yesmaximal, characteristic, complemented
5#22577285C211yesmaximal, characteristic, complemented
6#22577276C211yesmaximal, characteristic
7#22577275C211yesmaximal, characteristic
8#22577270C211yesmaximal, characteristic, complemented
9#22577267C211yesmaximal, characteristic, complemented
10#22577264C211yesmaximal, characteristic
11C4×#14555489C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic137C211yesmaximal, characteristic
13#22577249C211yesmaximal, characteristic, complemented
14C2×#17983995C211yesmaximal, characteristic
15C4×C22⋊C4×C137C211yesmaximal, characteristic
16C23×Dic137C2211yescharacteristic
17C23×Dic137C2211yescharacteristic
18#17984044C418yescomplemented
19C2×#14555489C2211yescharacteristic
20C2×#14555489C2211yescharacteristic
21C2×#14555489C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×C22⋊Dic137C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C4×Dic137C2211yescharacteristic
28C2×C4×Dic137C2211yescharacteristic
29C2×C4×Dic137C2211yescharacteristic
30C2×C4×Dic137C2211yescharacteristic
31#17983995C2211yescharacteristic
32#17983995C2211yescharacteristic
33#17983995C2211yescharacteristic
34#17983995C2211yescharacteristic
35#17983995C2211yescharacteristic
36#17983995C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45C23×C4×C137C2211yescharacteristic
46C2×C22⋊C4×C137C2211yescharacteristic
47C2×C22⋊C4×C137C2211yescharacteristic
48C2×C42×C137C2211yescharacteristic
49C2×C42×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C22×Dic137—22no—
53C22×Dic137—22no—
54C22×Dic137—22no—
55C22×Dic137—22no—
56C22×Dic137—41no—
57C22×Dic137C2×C414yes—
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2311yescharacteristic
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66#14555489—22no—
67#14555489—22no—
68#14555489—41no—
69#14555489—41no—
70#14555489C2×C414yescomplemented
71#14555489C2×C414yescomplemented
72C22⋊Dic137—22no—
73C22⋊Dic137C2×C414yes—
74C4×Dic137—22no—
75C4×Dic137—41no—
76C4×Dic137C2×C414yescomplemented
77C4×Dic137C2×C414yescomplemented
78C24×C137C2311yescharacteristic
79C22×C4×C137—22no—
80C22×C4×C137—22no—
81C22×C4×C137—41no—
82C22×C4×C137C2311yescharacteristic
83C22×C4×C137C2311yescharacteristic
84C22×C4×C137C2311yescharacteristic
85C22×C4×C137C2311yescharacteristic
86C22×C4×C137C2311yescharacteristic
87C22×C4×C137C2311yescharacteristic
88C22⋊C4×C137—22no—
89C22⋊C4×C137C2×C414yes—
90C42×C137—22no—
91C42×C137—41no—
92C2×Dic137—28no—
93C2×Dic137—21no—
94C2×Dic137—21no—
95C2×Dic137—21no—
96C2×Dic137—21no—
97C2×Dic137—21no—
98C2×Dic137—21no—
99C2×Dic137—21no—
100C2×Dic137—21no—
101C2×Dic137—21no—
102C2×Dic137—21no—
103C2×Dic137—21no—
104C2×Dic137—21no—
105C2×Dic137—21no—
106C2×Dic137—21no—
107C2×Dic137—21no—
108C2×Dic137—21no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—44no—
113C2×Dic137C22×C412yes—
114C2×Dic137C22×C412yes—
115C2×Dic137C22×C412yes—
116C2×Dic137C22×C412yes—
117C2×Dic137#4112yes—
118C23×C137—21no—
119C23×C137—21no—
120C23×C137—21no—
121C23×C137—21no—
122C23×C137C22×C412yes—
123C23×C137#4112yes—
124C23×C137#4112yes—
125C23×C137C2411yescharacteristic
126C2×C4×C137—21no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—44no—
142C2×C4×C137C22×C412yes—
143C2×C4×C137C22×C412yes—
144C2×C4×C137#4112yes—
145C22×C137—22no—
146C22×C137—22no—
147C22×C137—22no—
148C22×C137—22no—
149C22×C137—22no—
150C22×C137—22no—
151C22×C137#11714yes—
152C22×C137C23×C411yescharacteristic
153C22×C137C2×#4111yescharacteristic
154C22×C137C2×#4111yescharacteristic
155C22×C137C2×#4111yescharacteristic
156C22×C137C2×#4111yescharacteristic
157C22×C137H(5,𝔽2)11yescharacteristic
158C22×C1372−1+411yescharacteristic
159C4×C137—22no—
160C4×C137—22no—
161C4×C137—22no—
162C4×C137—41no—
163C4×C137—41no—
164C4×C137—41no—
165Dic137—22no—
166Dic137—22no—
167Dic137—22no—
168Dic137—22no—
169Dic137—22no—
170Dic137—41no—
171Dic137—41no—
172Dic137—41no—
173C2×C137—24no—
174C2×C137C2×#11711yescharacteristic
175C2×C137C4×#4111yescharacteristic
176C2×C137#52011yescharacteristic
177C2×C137#55111yescharacteristic
178C2×C137#55111yescharacteristic
179C2×C137#55211yescharacteristic
180C2×C137#55211yescharacteristic
181C137#234311yescharacteristic, complemented, Sylow
182#2343—1371nomaximal, Sylow
183C2×#117—1371no—
184#387—1371no—
185#387—1371no—
186#387—1371no—
187#387—1371no—
188#385—1371no—
189#385—1371no—
190#383—1371no—
191#382—1371no—
192#381—1371no—
193C4×C4⋊C4—1371no—
194C4×C22⋊C4—1371no—
195C4×C22⋊C4D13711yescharacteristic
196#376—1371no—
197C2×#95—1371no—
198C23×C4—1371no—
199C23×C4—1371no—
200C23×C4C2×D13711yescharacteristic
201#117—1378no—
202C2×C4⋊C4—1371no—
203C2×C4⋊C4—1371no—
204C2×C4⋊C4—1371no—
205C2×C4⋊C4—1371no—
206C2×C4⋊C4—1371no—
207C2×C4⋊C4—1371no—
208C2×C22⋊C4—1371no—
209C2×C22⋊C4—1371no—
210C2×C22⋊C4C2×D13711yescharacteristic
211C2×C22⋊C4C2×D13711yescharacteristic
212C2×C42—1371no—
213C2×C42—1371no—
214C2×C42—1371no—
215C2×C42—1371no—
216C2×C42C2×D13711yescharacteristic
217C2×C42C2×D13711yescharacteristic
218#95—1371no—
219#95—1371no—
220#95—1371no—
221#95—1371no—
222#95—1371no—
223#95—1371no—
224#95—1371no—
225#95—1371no—
226#95—1371no—
227#95—1371no—
228#95—1371no—
229#95—1371no—
230#95—1371no—
231#95—1371no—
232#95C2×D13711yescharacteristic
233#95C2×D13711yescharacteristic
234C24C22×D13711yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—2742no—
239C22×C4—2742no—
240C22×C4—2742no—
241C22×C4—2742no—
242C22×C4—5481no—
243C22×C4—1371no—
244C22×C4—1371no—
245C22×C4—1371no—
246C22×C4—1374no—
247C22×C4—1371no—
248C22×C4—1371no—
249C22×C4—1371no—
250C22×C4—1371no—
251C22×C4—1371no—
252C22×C4C22×D13711yescharacteristic
253C22×C4C22×D13711yescharacteristic
254C22×C4C22×D13711yescharacteristic
255C22×C4C22×D13711yescharacteristic
256C22×C4C22×D13711yescharacteristic
257C22×C4C22×D13711yescharacteristic
258C4⋊C4—2742no—
259C4⋊C4—2742no—
260C4⋊C4—5481no—
261C4⋊C4—5481no—
262C4⋊C4—1374no—
263C4⋊C4—1374no—
264C22⋊C4—22no—
265C22⋊C4—2742no—
266C22⋊C4—1374no—
267C22⋊C4C4×D13714yes—
268C42—22no—
269C42—41no—
270C42—2742no—
271C42—5481no—
272C42—1374no—
273C42—1374no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D13712yes—
279C23#1455551012yes—
280C23#1455551012yes—
281C23C23×D13711yescharacteristic
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—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—2748no—
299C2×C4—2741no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—2741no—
318C2×C4—5484no—
319C2×C4—1372no—
320C2×C4—1372no—
321C2×C4—1372no—
322C2×C4—1372no—
323C2×C4—1372no—
324C2×C4C2×C4×D13712yes—
325C2×C4C2×C4×D13712yes—
326C2×C4#1455550912yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#1798404514yes—
334C22C22×C4×D13711yescharacteristic
335C22#41×D13711yescharacteristic
336C22C2×#1455550911yescharacteristic
337C22C2×#1455551011yescharacteristic
338C22C2×#1455551011yescharacteristic
339C22#1798413311yescharacteristic
340C22#1798413511yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—2742no—
348C4—2742no—
349C4—2742no—
350C4—2742no—
351C4—2742no—
352C4—5481no—
353C4—5481no—
354C4—5481no—
355C2—24no—
356C2C2×#1798404511yescharacteristic
357C2C4×#1455550911yescharacteristic
358C2#2257788211yescharacteristic
359C2#2257807911yescharacteristic
360C2#2257808211yescharacteristic
361C2#2257808611yescharacteristic
362C2#2257809311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
FittingC4×C22⋊C4×C137C2
socle, cosocleC23×C137C24
prime radicalC24×C137C23
hypercenterC4×C22⋊C4D137
hypocenterC137#2343
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22577287C2semidirect
3. #22577287C2semidirect
4. #22577285C2semidirect
5. #22577285C2semidirect
8. #22577270C2semidirect
9. #22577267C2semidirect
11. C4×#14555489C2semidirect
13. #22577249C2semidirect
18. #17984044C4semidirect
70. #14555489C2×C4semidirect
71. #14555489C2×C4semidirect
76. C4×Dic137C2×C4semidirect
77. C4×Dic137C2×C4semidirect
181. C137#2343semidirect

Maximal subgroups

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

subgroupindex
C2×#179840442
#225772872
#225772852
#225772762
#225772752
#225772702
#225772672
#225772642
C4×#145554892
C22⋊C4×Dic1372
#225772492
C2×#179839952
C4×C22⋊C4×C1372
#2343137