Finite Group Databasebeta

← #31783920

Subgroups — #31783920

Order 17536.

552 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984071C211yesmaximal, characteristic
2#22577303C211yesmaximal, characteristic
3#22577297C211yesmaximal, characteristic, complemented
4#22577294C211yesmaximal, characteristic, complemented
5#22577292C211yesmaximal, characteristic, complemented
6#22577291C211yesmaximal, characteristic, complemented
7#22577287C211yesmaximal, characteristic, complemented
8#22577285C211yesmaximal, characteristic, complemented
9#22577279C211yesmaximal, characteristic
10C4×#14555489C211yesmaximal, characteristic, complemented
11#22577254C211yesmaximal, characteristic, complemented
12C4×#14555487C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic137C211yesmaximal, characteristic, complemented
14#22577250C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18#17984071C418yescomplemented
19C2×#14555489C2211yescharacteristic
20C2×#14555489C2211yescharacteristic
21C2×#14555489C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×C22⋊Dic137C2211yescharacteristic
25C2×C22⋊Dic137C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×#14555487C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
29C2×#14555487C2211yescharacteristic
30C2×#14555487C2211yescharacteristic
31C2×#14555487C2211yescharacteristic
32C2×C4×Dic137C2211yescharacteristic
33C2×C4×Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36C2×C4×Dic137C2211yescharacteristic
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
52C23×D137C2311yescharacteristic
53C2×C22⋊D137—41no—
54C2×C22⋊D137—41no—
55C2×C22⋊D137C2×C414yes—
56C22×Dic137C2311yescharacteristic
57C22×Dic137C2311yescharacteristic
58C22×Dic137C2311yescharacteristic
59C22×Dic137C2311yescharacteristic
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C22×Dic137C2311yescharacteristic
63C2×C4×D137—22no—
64C2×C4×D137—22no—
65C2×C4×D137—41no—
66#14555489—22no—
67#14555489—22no—
68#14555489—41no—
69#14555489C2×C414yescomplemented
70#14555489C2×C414yescomplemented
71C22⋊Dic137—22no—
72C22⋊Dic137—22no—
73C22⋊Dic137C2×C414yescomplemented
74#14555487—22no—
75#14555487—22no—
76#14555487—41no—
77#14555487C2×C414yes—
78#14555487C2×C414yes—
79C4×Dic137—22no—
80C4×Dic137—22no—
81C4×Dic137—41no—
82C4×Dic137—41no—
83C4×Dic137—41no—
84C24×C137C2311yescharacteristic
85C22×C4×C137—41no—
86C22×C4×C137—41no—
87C22×C4×C137C2×C414yescomplemented
88C22×C4×C137C2311yescharacteristic
89C22×C4×C137C2311yescharacteristic
90C22×C4×C137C2311yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C22×C4×C137C2311yescharacteristic
94C22⋊C4×C137—22no—
95C22⋊C4×C137—22no—
96C42×C137—41no—
97C42×C137—41no—
98C22×D137—21no—
99C22×D137—21no—
100C22×D137—21no—
101C22×D137—21no—
102C22×D137—21no—
103C22×D137—21no—
104C22×D137C22×C412yes—
105C22⋊D137—44no—
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—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137C22×C412yes—
123C2×Dic137C22×C412yes—
124C2×Dic137C22×C412yes—
125C2×Dic137#4112yes—
126C2×Dic137#4112yes—
127C4×D137—44no—
128C23×C137—21no—
129C23×C137—21no—
130C23×C137—21no—
131C23×C137—21no—
132C23×C137—21no—
133C23×C137—21no—
134C23×C137C22×C412yes—
135C23×C137C2411yescharacteristic
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—44no—
151C2×C4×C137C22×C412yes—
152C2×C4×C137C22×C412yes—
153C2×C4×C137#4112yes—
154C2×C4×C137#4112yes—
155C22×C137—22no—
156C22×C137—22no—
157C22×C137—22no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137C23×C411yescharacteristic
163C22×C137C2×#4111yescharacteristic
164C22×C137C2×#4111yescharacteristic
165C22×C137C2×#4111yescharacteristic
166C22×C137C2×#4111yescharacteristic
167C22×C137H(5,𝔽2)11yescharacteristic
168C22×C137H(5,𝔽2)11yescharacteristic
169C2×D137—22no—
170C2×D137—22no—
171C2×D137—22no—
172C2×D137—41no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—41no—
176C4×C137—22no—
177C4×C137—22no—
178C4×C137—22no—
179C4×C137—22no—
180C4×C137—41no—
181C4×C137—41no—
182Dic137—22no—
183Dic137—22no—
184Dic137—22no—
185Dic137—22no—
186Dic137—22no—
187Dic137—41no—
188Dic137—41no—
189C2×C137—42no—
190C2×C137C4×#4111yescharacteristic
191C2×C137C4×#4111yescharacteristic
192C2×C137#51811yescharacteristic
193C2×C137#55111yescharacteristic
194C2×C137#55311yescharacteristic
195C2×C137#55311yescharacteristic
196C2×C137#55911yescharacteristic
197D137—42no—
198C137#237411yescharacteristic, complemented, Sylow
199#2374—1371nomaximal, Sylow
200C2×#123—1371no—
201#390—1371no—
202#388—1371no—
203#388—1371no—
204#388—1371no—
205#388—1371no—
206#387—1371no—
207#387—1371no—
208#386—1371no—
209C4×C4⋊C4—1371no—
210C4×C4⋊C4—1371no—
211C4×C22⋊C4—1371no—
212C4×C22⋊C4—1371no—
213C4×C22⋊C4—1371no—
214C4×C22⋊C4D13711yescharacteristic, complemented
215C22×D4—1371no—
216C23×C4—1371no—
217C23×C4C2×D13711yescharacteristic
218#123—1378no—
219C2×C4⋊C4—1371no—
220C2×C4⋊C4—1371no—
221C2×C4⋊C4—1371no—
222C2×C4⋊C4—1371no—
223C2×C4⋊C4—1371no—
224C2×C22⋊C4—1371no—
225C2×C22⋊C4—1371no—
226C2×C22⋊C4—1371no—
227C2×C22⋊C4—1371no—
228C2×C22⋊C4—1371no—
229C2×C22⋊C4—1371no—
230C2×C22⋊C4—1371no—
231C2×C22⋊C4—1371no—
232C2×C22⋊C4C2×D13711yescharacteristic
233C2×C22⋊C4C2×D13711yescharacteristic
234C2×C42—1371no—
235C2×C42—1371no—
236C2×C42—1371no—
237C2×C42—1371no—
238C2×C42—1371no—
239C2×C42C2×D13711yescharacteristic
240C2×C42C2×D13711yescharacteristic
241#95—1371no—
242#95—1371no—
243#95—1371no—
244#95—1371no—
245#95—1371no—
246#95—1371no—
247#95—1371no—
248#95—1371no—
249#95C2×D13711yescharacteristic
250#95C2×D13711yescharacteristic
251C24—1371no—
252C24C22×D13711yescharacteristic
253C2×D4—5481no—
254C2×D4—5481no—
255C2×D4—1374no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—2742no—
259C22×C4—2742no—
260C22×C4—5481no—
261C22×C4—1371no—
262C22×C4—1371no—
263C22×C4—1371no—
264C22×C4—1371no—
265C22×C4—1371no—
266C22×C4—1371no—
267C22×C4—1371no—
268C22×C4C4×D13714yescomplemented
269C22×C4C22×D13711yescharacteristic
270C22×C4C22×D13711yescharacteristic
271C22×C4C22×D13711yescharacteristic
272C22×C4C22×D13711yescharacteristic
273C22×C4C22×D13711yescharacteristic
274C22×C4C22×D13711yescharacteristic
275C4⋊C4—2742no—
276C4⋊C4—2742no—
277C4⋊C4—5481no—
278C4⋊C4—1374no—
279C4⋊C4—1374no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—2742no—
283C22⋊C4—2742no—
284C22⋊C4—2742no—
285C22⋊C4—2742no—
286C22⋊C4—5481no—
287C22⋊C4—1374no—
288C22⋊C4—1374no—
289C22⋊C4—1374no—
290C42—41no—
291C42—41no—
292C42—2742no—
293C42—2742no—
294C42—5481no—
295C42—5481no—
296C42—5481no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—2741no—
304C23—2741no—
305C23—2741no—
306C23—2741no—
307C23—2741no—
308C23—2741no—
309C23—1372no—
310C23C2×C4×D13712yes—
311C23C23×D13711yescharacteristic
312D4—5484no—
313C2×C4—21no—
314C2×C4—21no—
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—44no—
328C2×C4—2741no—
329C2×C4—2741no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—5484no—
345C2×C4—1372no—
346C2×C4—1372no—
347C2×C4—1372no—
348C2×C4—1372no—
349C2×C4—1372no—
350C2×C4C2×C4×D13712yes—
351C2×C4C2×C4×D13712yes—
352C2×C4#1455551012yes—
353C2×C4#1455551012yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—2742no—
362C22—2742no—
363C22—2742no—
364C22—5481no—
365C22—5481no—
366C22—5481no—
367C22—5481no—
368C22C22×C4×D13711yescharacteristic
369C22#41×D13711yescharacteristic
370C22#41×D13711yescharacteristic
371C22C2×#1455551011yescharacteristic
372C22C2×#1455551011yescharacteristic
373C22#1798413311yescharacteristic
374C22#1798413311yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—2742no—
382C4—2742no—
383C4—2742no—
384C4—2742no—
385C4—2742no—
386C4—5481no—
387C4—5481no—
388C2—42no—
389C2—5482no—
390C2#2257787111yescharacteristic
391C2C4×#1455551011yescharacteristic
392C2#2257787611yescharacteristic
393C2#2257808211yescharacteristic
394C2#2257810211yescharacteristic
395C2#2257810311yescharacteristic
396C2#2257813611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
FittingC4×C22⋊C4×C137C2
socle, cosocleC23×C137C24
prime radicalC22×C22⋊D137C22
hypercenterC4×C22⋊C4D137
hypocenterC137#2374
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #22577297C2semidirect
4. #22577294C2semidirect
5. #22577292C2semidirect
6. #22577291C2semidirect
7. #22577287C2semidirect
8. #22577285C2semidirect
10. C4×#14555489C2semidirect
11. #22577254C2semidirect
12. C4×#14555487C2semidirect
13. C4×C22⋊Dic137C2semidirect
14. #22577250C2semidirect
15. C4×C22⋊C4×C137C2semidirect
18. #17984071C4semidirect
69. #14555489C2×C4semidirect
70. #14555489C2×C4semidirect
73. C22⋊Dic137C2×C4semidirect
87. C22×C4×C137C2×C4semidirect
198. C137#2374semidirect
214. C4×C22⋊C4D137semidirect
268. C22×C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840712
#225773032
#225772972
#225772942
#225772922
#225772912
#225772872
#225772852
#225772792
C4×#145554892
#225772542
C4×#145554872
C4×C22⋊Dic1372
#225772502
C4×C22⋊C4×C1372
#2374137