Finite Group Databasebeta

← #31784090

Subgroups — #31784090

Order 17536.

432 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C137⋊#401C211yesmaximal, characteristic, complemented
2#22577345C211yesmaximal, characteristic, complemented
3#22577342C211yesmaximal, characteristic, complemented
4#22577341C211yesmaximal, characteristic, complemented
5#22577332C211yesmaximal, characteristic
6#22577330C211yesmaximal, characteristic
7#22577329C211yesmaximal, characteristic
8#22577325C211yesmaximal, characteristic
9C4×#14555489C211yesmaximal, characteristic, complemented
10C4×C4⋊Dic137C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic
12C4×#14555487C211yesmaximal, characteristic
13C4×#14555487C211yesmaximal, characteristic
14#22577249C211yesmaximal, characteristic, complemented
15C4×C4⋊C4×C137C211yesmaximal, characteristic, complemented
16C22×C4×D137C2211yescharacteristic
17C2×C4⋊Dic137C2211yescharacteristic
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555489C2211yescharacteristic
20C2×#14555489C2211yescharacteristic
21C2×#14555487C2211yescharacteristic
22C2×#14555487C2211yescharacteristic
23C2×#14555487C2211yescharacteristic
24C2×#14555487C2211yescharacteristic
25C2×#14555487C2211yescharacteristic
26C2×#14555487C2211yescharacteristic
27C2×C4×Dic137C2211yescharacteristic
28C2×C4×Dic137C2211yescharacteristic
29C2×C4×Dic137C2211yescharacteristic
30#17983995C2211yescharacteristic
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
44C2×C4⋊C4×C137C2211yescharacteristic
45C2×C4⋊C4×C137C2211yescharacteristic
46C2×C42×C137C2211yescharacteristic
47C2×C42×C137C2211yescharacteristic
48C2×C42×C137C2211yescharacteristic
49#95×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51C23×D137C2311yescharacteristic
52C22×Dic137C2311yescharacteristic
53C22×Dic137C2311yescharacteristic
54C22×Dic137C2311yescharacteristic
55C22×Dic137C2311yescharacteristic
56C22×Dic137C2311yescharacteristic
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2311yescharacteristic
59C2×C4×D137—22no—
60C2×C4×D137—22no—
61C2×C4×D137—22no—
62C4⋊Dic137—22no—
63C4⋊Dic137—22no—
64#14555489—22no—
65#14555489—22no—
66#14555487—22no—
67#14555487—22no—
68#14555487—22no—
69#14555487—22no—
70#14555487—22no—
71#14555487—22no—
72C4×Dic137—22no—
73C4×Dic137—22no—
74C4×Dic137—22no—
75C22×C4×C137C2311yescharacteristic
76C22×C4×C137C2311yescharacteristic
77C22×C4×C137C2311yescharacteristic
78C22×C4×C137C2311yescharacteristic
79C22×C4×C137C2311yescharacteristic
80C22×C4×C137C2311yescharacteristic
81C22×C4×C137C2311yescharacteristic
82C4⋊C4×C137—22no—
83C4⋊C4×C137—22no—
84C42×C137—22no—
85C42×C137—22no—
86C42×C137—22no—
87C22×D137—21no—
88C22×D137—21no—
89C22×D137—21no—
90C22×D137—21no—
91C22×D137—21no—
92C22×D137—21no—
93C22×D137#4112yes—
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—21no—
113C2×Dic137—21no—
114C2×Dic137#4112yes—
115C4×D137—44no—
116C23×C137C2411yescharacteristic
117C2×C4×C137—21no—
118C2×C4×C137—21no—
119C2×C4×C137—21no—
120C2×C4×C137—21no—
121C2×C4×C137—21no—
122C2×C4×C137—21no—
123C2×C4×C137—21no—
124C2×C4×C137—21no—
125C2×C4×C137—21no—
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#4112yes—
134C2×C4×C137#4112yes—
135C2×C4×C137#4112yes—
136C2×C4×C137#4112yes—
137C2×C4×C137#4112yes—
138C22×C137C2×#4111yescharacteristic
139C22×C137C2×#4111yescharacteristic
140C22×C137C2×#4111yescharacteristic
141C22×C137C2×#4111yescharacteristic
142C22×C137C2×#4111yescharacteristic
143C22×C137C2×#4111yescharacteristic
144C22×C137C2×#4111yescharacteristic
145C2×D137—22no—
146C2×D137—22no—
147C2×D137—22no—
148C2×D137—41no—
149C2×D137—41no—
150C2×D137—41no—
151C2×D137—41no—
152C4×C137—22no—
153C4×C137—22no—
154C4×C137—41no—
155C4×C137—41no—
156C4×C137—41no—
157C4×C137—41no—
158C4×C137#12614yes—
159Dic137—22no—
160Dic137—41no—
161Dic137—41no—
162Dic137—41no—
163Dic137—41no—
164Dic137—41no—
165Dic137—41no—
166C2×C137C2×#12611yescharacteristic
167C2×C137#52911yescharacteristic
168C2×C137#52911yescharacteristic
169C2×C137#52911yescharacteristic
170C2×C137#52911yescharacteristic
171C2×C137#52911yescharacteristic
172C2×C137#52911yescharacteristic
173D137—42no—
174C137#240111yescharacteristic, complemented, Sylow
175#2401—1371nomaximal, Sylow
176#401—1371no—
177#400—1371no—
178#400—1371no—
179#400—1371no—
180#397—1371no—
181#397—1371no—
182#397—1371no—
183#396—1371no—
184C4×C4⋊C4—1371no—
185C4×C4⋊C4—1371no—
186C4×C4⋊C4D13711yescharacteristic, complemented
187C4×C22⋊C4—1371no—
188C4×C22⋊C4—1371no—
189C4×C22⋊C4—1371no—
190#376—1371no—
191C23×C4—1371no—
192C2×C4⋊C4—1371no—
193C2×C4⋊C4—1371no—
194C2×C4⋊C4—1371no—
195C2×C4⋊C4—1371no—
196C2×C4⋊C4C2×D13711yescharacteristic
197C2×C4⋊C4C2×D13711yescharacteristic
198C2×C22⋊C4—1371no—
199C2×C22⋊C4—1371no—
200C2×C22⋊C4—1371no—
201C2×C22⋊C4—1371no—
202C2×C22⋊C4—1371no—
203C2×C22⋊C4—1371no—
204C2×C42—1371no—
205C2×C42—1371no—
206C2×C42—1371no—
207C2×C42C2×D13711yescharacteristic
208C2×C42C2×D13711yescharacteristic
209C2×C42C2×D13711yescharacteristic
210#95—1371no—
211#95—1371no—
212#95—1371no—
213#95—1371no—
214#95—1371no—
215#95—1371no—
216#95—1371no—
217#95—1371no—
218#95—1371no—
219#95—1371no—
220#95—1371no—
221#95—1371no—
222#95—1371no—
223#95—1371no—
224#95C2×D13711yescharacteristic
225#95C2×D13711yescharacteristic
226C24—1371no—
227C22×C4—2742no—
228C22×C4—2742no—
229C22×C4—2742no—
230C22×C4—1371no—
231C22×C4—1371no—
232C22×C4—1371no—
233C22×C4—1371no—
234C22×C4—1371no—
235C22×C4—1371no—
236C22×C4—1371no—
237C22×C4C22×D13711yescharacteristic
238C22×C4C22×D13711yescharacteristic
239C22×C4C22×D13711yescharacteristic
240C22×C4C22×D13711yescharacteristic
241C22×C4C22×D13711yescharacteristic
242C22×C4C22×D13711yescharacteristic
243C22×C4C22×D13711yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—2742no—
247C4⋊C4—2742no—
248C4⋊C4—2742no—
249C4⋊C4—2742no—
250C22⋊C4—2742no—
251C22⋊C4—2742no—
252C22⋊C4—2742no—
253C22⋊C4—2742no—
254C22⋊C4—2742no—
255C22⋊C4—2742no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—2742no—
260C42—2742no—
261C42—2742no—
262C23—2741no—
263C23—2741no—
264C23—2741no—
265C23—2741no—
266C23—2741no—
267C23—2741no—
268C23—1372no—
269C23C23×D13711yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
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—2741no—
287C2×C4—2741no—
288C2×C4—2741no—
289C2×C4—2741no—
290C2×C4—2741no—
291C2×C4—2741no—
292C2×C4—2741no—
293C2×C4—2741no—
294C2×C4—2741no—
295C2×C4—2741no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—5484no—
307C2×C4—1372no—
308C2×C4#1455550912yes—
309C2×C4#1455551012yes—
310C2×C4#1455551012yes—
311C2×C4#1455551012yes—
312C2×C4#1455551112yes—
313C22—2742no—
314C22—2742no—
315C22—2742no—
316C22—5481no—
317C22—5481no—
318C22—5481no—
319C22—5481no—
320C22#41×D13711yescharacteristic
321C22#41×D13711yescharacteristic
322C22C2×#1455550911yescharacteristic
323C22C2×#1455551011yescharacteristic
324C22C2×#1455551011yescharacteristic
325C22C2×#1455551011yescharacteristic
326C22C2×#1455551111yescharacteristic
327C4—22no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—2742no—
334C4—5481no—
335C4—5481no—
336C4—5481no—
337C4—5481no—
338C4—5481no—
339C4—5481no—
340C4#1798408314yes—
341C2—5482no—
342C2C2×#1798408311yescharacteristic
343C2#2257793111yescharacteristic
344C2#2257793311yescharacteristic
345C2#2257793511yescharacteristic
346C2#2257793711yescharacteristic
347C2#2257794011yescharacteristic
348C2#2257794011yescharacteristic
subgroupG/H
centerC22×C4C22×D137
commutator, socle, cosocleC23×C137C24
FrattiniC23C23×D137
FittingC4×C4⋊C4×C137C2
prime radicalC23×D137C23
hypercenterC4×C4⋊C4D137
hypocenterC137#2401
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C137⋊#401C2semidirect
2. #22577345C2semidirect
3. #22577342C2semidirect
4. #22577341C2semidirect
9. C4×#14555489C2semidirect
10. C4×C4⋊Dic137C2semidirect
14. #22577249C2semidirect
15. C4×C4⋊C4×C137C2semidirect
174. C137#2401semidirect
186. C4×C4⋊C4D137semidirect

Maximal subgroups

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

subgroupindex
C137⋊#4012
#225773452
#225773422
#225773412
#225773322
#225773302
#225773292
#225773252
C4×#145554892
C4×C4⋊Dic1372
C4×#145554872
#225772492
C4×C4⋊C4×C1372
#2401137