Finite Group Databasebeta

← #31797287

Subgroups — #31797287

Order 17536.

476 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578354C211yesmaximal, characteristic, complemented
2#22578354C211yesmaximal, characteristic, complemented
3#22578353C211yesmaximal, characteristic
4#22578353C211yesmaximal, characteristic
5#22578353C211yesmaximal, characteristic
6#22578352C211yesmaximal, characteristic, complemented
7#22578352C211yesmaximal, characteristic, complemented
8#22578352C211yesmaximal, characteristic, complemented
9#22578350C211yesmaximal, characteristic
10#22578350C211yesmaximal, characteristic
11#22578347C211yesmaximal, characteristic, complemented
12C4⋊C4×#10587739C211yesmaximal, characteristic, complemented
13C4×C4⋊#10587739C211yesmaximal, characteristic, complemented
14C22⋊C4×#10587739C211yesmaximal, characteristic
15#126×D137C211yesmaximal, characteristic
16C23×#10587739C2211yescharacteristic
17C2×C4⋊#10587739C2211yescharacteristic
18C2×C4⋊#10587739C2211yescharacteristic
19C2×C4⋊#10587739C2211yescharacteristic
20C2×C4⋊#10587739C2211yescharacteristic
21C2×C4⋊#10587739C2211yescharacteristic
22C2×C4⋊#10587739C2211yescharacteristic
23C2×C4⋊#10587739C2211yescharacteristic
24C2×C22⋊#10587739C2211yescharacteristic
25C2×C22⋊#10587739C2211yescharacteristic
26C2×C22⋊#10587739C2211yescharacteristic
27C2×C4×#10587739C2211yescharacteristic
28C2×C4×#10587739C2211yescharacteristic
29C2×C4×#10587739C2211yescharacteristic
30C2×C4×#10587739C2211yescharacteristic
31C2×C4×#10587739C2211yescharacteristic
32(C2×#10587739)⋊C4C2211yescharacteristic
33(C2×#10587739)⋊C4C2211yescharacteristic
34(C2×#10587739)⋊C4C2211yescharacteristic
35(C2×#10587739)⋊C4C2211yescharacteristic
36(C2×#10587739)⋊C4C2211yescharacteristic
37(C2×#10587739)⋊C4C2211yescharacteristic
38(C2×#10587739)⋊C4C2211yescharacteristic
39(C2×#10587739)⋊C4C2211yescharacteristic
40(C2×#10587739)⋊C4C2211yescharacteristic
41(C2×#10587739)⋊C4C2211yescharacteristic
42(C2×#10587739)⋊C4C2211yescharacteristic
43(C2×#10587739)⋊C4C2211yescharacteristic
44#17984083C411yescharacteristic, complemented
45#17984083C411yescharacteristic, complemented
46#17984083C411yescharacteristic, complemented
47#17984082C411yescharacteristic, complemented
48#17984081C411yescharacteristic, complemented
49#17984081C411yescharacteristic, complemented
50#17984081C411yescharacteristic, complemented
51C4⋊C4×D137C2211yescharacteristic
52C4⋊C4×D137C2211yescharacteristic
53C4⋊C4×D137C2211yescharacteristic
54C22⋊C4×D137C2211yescharacteristic
55C22⋊C4×D137C2211yescharacteristic
56C22⋊C4×D137C2211yescharacteristic
57C42×D137C2211yescharacteristic
58#126×C137C411yescharacteristic, complemented
59C22×#10587739—22no—
60C22×#10587739—41no—
61C22×#10587739—41no—
62C22×#10587739C2311yescharacteristic
63C22×#10587739C2311yescharacteristic
64C22×#10587739C2311yescharacteristic
65C22×#10587739C2311yescharacteristic
66C22×#10587739C2311yescharacteristic
67C22×#10587739C2311yescharacteristic
68C22×#10587739C2311yescharacteristic
69C22×#10587739C2311yescharacteristic
70C4⋊#10587739—22no—
71C4⋊#10587739—22no—
72C4⋊#10587739—22no—
73C4⋊#10587739—41no—
74C4⋊#10587739—41no—
75C4⋊#10587739—41no—
76C4⋊#10587739—41no—
77C22⋊#10587739—22no—
78C22⋊#10587739—41no—
79C22⋊#10587739—41no—
80C4×#10587739—22no—
81C4×#10587739—41no—
82C4×#10587739—41no—
83C4×#10587739—41no—
84C4×#10587739—41no—
85C23×D137C2311yescharacteristic
86C2×C4×D137C2311yescharacteristic
87C2×C4×D137C2311yescharacteristic
88C2×C4×D137C2311yescharacteristic
89C2×C4×D137C2311yescharacteristic
90C2×C4×D137C2311yescharacteristic
91C2×C4×D137C2311yescharacteristic
92C4⋊Dic137C2×C411yescharacteristic, complemented
93C4⋊Dic137C2×C411yescharacteristic, complemented
94C4⋊Dic137C2×C411yescharacteristic, complemented
95#14555489C2×C411yescharacteristic, complemented
96#14555489C2×C411yescharacteristic, complemented
97#14555489C2×C411yescharacteristic, complemented
98#14555489C2×C411yescharacteristic, complemented
99#14555489C2×C411yescharacteristic, complemented
100#14555489C2×C411yescharacteristic, complemented
101C22⋊Dic137C2×C411yescharacteristic
102C22⋊Dic137C2×C411yescharacteristic
103C22⋊Dic137C2×C411yescharacteristic
104#14555487C2×C411yescharacteristic
105#14555487C2×C411yescharacteristic
106#14555487C2×C411yescharacteristic
107#14555487C2×C411yescharacteristic
108#14555487C2×C411yescharacteristic
109#14555487C2×C411yescharacteristic
110C4×Dic137C2×C411yescharacteristic, complemented
111C4×Dic137C2×C411yescharacteristic, complemented
112C4×Dic137C2×C411yescharacteristic, complemented
113C4⋊C4×C137C2×C411yescharacteristic, complemented
114C4⋊C4×C137C2×C411yescharacteristic, complemented
115C4⋊C4×C137C2×C411yescharacteristic, complemented
116C22⋊C4×C137C2×C411yescharacteristic
117C22⋊C4×C137C2×C411yescharacteristic
118C22⋊C4×C137C2×C411yescharacteristic
119C42×C137C2×C411yescharacteristic, complemented
120C2×#10587739—21no—
121C2×#10587739—21no—
122C2×#10587739—21no—
123C2×#10587739—21no—
124C2×#10587739—21no—
125C2×#10587739—21no—
126C2×#10587739—21no—
127C2×#10587739—21no—
128C2×#10587739—21no—
129C2×#10587739—21no—
130C2×#10587739—21no—
131C2×#10587739—21no—
132C2×#10587739—21no—
133C2×#10587739—21no—
134C2×#10587739—21no—
135C2×#10587739—21no—
136C2×#10587739—21no—
137C2×#10587739—21no—
138C2×#10587739—21no—
139C2×#10587739—21no—
140C2×#10587739—21no—
141C2×#10587739—21no—
142C2×#10587739—44no—
143C2×#10587739#4112yes—
144C2×#10587739#4112yes—
145C22×D137—21no—
146C22×D137—21no—
147C22×D137—21no—
148C22×D137—21no—
149C22×D137—21no—
150C22×D137—21no—
151C22×D137C22×C411yescharacteristic
152C22×D137C2411yescharacteristic
153C2×Dic137C22×C411yescharacteristic
154C2×Dic137C22×C411yescharacteristic
155C2×Dic137C22×C411yescharacteristic
156C2×Dic137C22×C411yescharacteristic
157C2×Dic137C22×C411yescharacteristic
158C2×Dic137C22×C411yescharacteristic
159C4×D137—21no—
160C4×D137—21no—
161C4×D137—21no—
162C4×D137—21no—
163C4×D137—21no—
164C4×D137—21no—
165C4×D137—21no—
166C4×D137—21no—
167C4×D137—21no—
168C4×D137—21no—
169C4×D137—21no—
170C4×D137—21no—
171C23×C137C22×C411yescharacteristic
172C2×C4×C137C22×C411yescharacteristic
173C2×C4×C137C22×C411yescharacteristic
174C2×C4×C137C22×C411yescharacteristic
175C2×C4×C137C22×C411yescharacteristic
176C2×C4×C137C22×C411yescharacteristic
177C2×C4×C137C22×C411yescharacteristic
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—21no—
181C22×C137C23×C411yescharacteristic
182C2×D137—21no—
183C2×D137—21no—
184C2×D137—21no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—41no—
189C2×D137C2×#4111yescharacteristic
190C2×D137C2×#4111yescharacteristic
191C2×D137H(5,𝔽2)11yescharacteristic
192C2×D137H(5,𝔽2)11yescharacteristic
193C2×D1372−1+411yescharacteristic
194C2×D1372−1+411yescharacteristic
195#10587739—22no—
196#10587739—22no—
197#10587739—41no—
198#10587739—41no—
199#10587739—41no—
200#10587739—41no—
201#10587739—41no—
202#10587739—41no—
203C4×C137—21no—
204C4×C137—21no—
205C4×C137—21no—
206C4×C137—21no—
207C4×C137—21no—
208C4×C137—21no—
209Dic137—21no—
210Dic137—21no—
211Dic137—21no—
212Dic137—21no—
213Dic137—21no—
214Dic137—21no—
215C2×C137—41no—
216C2×C137C4×#4111yescharacteristic
217C2×C137#52011yescharacteristic
218C2×C137#52011yescharacteristic
219D137—41no—
220D137#53811yescharacteristic
221D137#53911yescharacteristic
222D137#54111yescharacteristic
223D137#54211yescharacteristic
224C137#233411yescharacteristic, complemented, Sylow
225#2334—1371nomaximal, Sylow
226C2×#126—1371no—
227#389—1371no—
228#389—1371no—
229#388—1371no—
230#388—1371no—
231#388—1371no—
232#387—1371no—
233#387—1371no—
234#387—1371no—
235#385—1371no—
236#385—1371no—
237#382—1371no—
238C4×C4⋊C4—1371no—
239C4×C4⋊C4—1371no—
240C4×C22⋊C4—1371no—
241C23×C4—1371no—
242#126—1371no—
243#126—1371no—
244#126—1371no—
245#126—1371no—
246#126—1371no—
247#126—1371no—
248#126—1371no—
249#126#1058773911yescharacteristic, complemented
250C2×C4⋊C4—1371no—
251C2×C4⋊C4—1371no—
252C2×C4⋊C4—1371no—
253C2×C4⋊C4—1371no—
254C2×C4⋊C4—1371no—
255C2×C4⋊C4—1371no—
256C2×C4⋊C4—1371no—
257C2×C4⋊C4—1371no—
258C2×C4⋊C4—1371no—
259C2×C4⋊C4—1371no—
260C2×C22⋊C4—1371no—
261C2×C22⋊C4—1371no—
262C2×C22⋊C4—1371no—
263C2×C22⋊C4—1371no—
264C2×C22⋊C4—1371no—
265C2×C22⋊C4—1371no—
266C2×C42—1371no—
267C2×C42—1371no—
268C2×C42—1371no—
269C2×C42—1371no—
270C2×C42—1371no—
271C2×C42—1371no—
272#95—1371no—
273#95—1371no—
274#95—1371no—
275#95—1371no—
276#95—1371no—
277#95—1371no—
278#95—1371no—
279#95—1371no—
280#95—1371no—
281#95—1371no—
282#95—1371no—
283#95—1371no—
284C24—1371no—
285C22×C4—2742no—
286C22×C4—5481no—
287C22×C4—5481no—
288C22×C4—1371no—
289C22×C4—1371no—
290C22×C4—1371no—
291C22×C4—1371no—
292C22×C4—1371no—
293C22×C4—1371no—
294C22×C4—1371no—
295C22×C4—1371no—
296C22×C4—1371no—
297C22×C4—1371no—
298C22×C4—1371no—
299C22×C4—1371no—
300C22×C4—1371no—
301C22×C4—1371no—
302C4⋊C4—2742no—
303C4⋊C4—2742no—
304C4⋊C4—2742no—
305C4⋊C4—5481no—
306C4⋊C4—5481no—
307C4⋊C4—5481no—
308C4⋊C4—5481no—
309C4⋊C4—1371no—
310C4⋊C4—1371no—
311C4⋊C4—1371no—
312C4⋊C4—1371no—
313C4⋊C4—1371no—
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4—1371no—
317C4⋊C4—1371no—
318C4⋊C4C2×#1058773911yescharacteristic, complemented
319C4⋊C4C2×#1058773911yescharacteristic, complemented
320C4⋊C4C2×#1058773911yescharacteristic, complemented
321C22⋊C4—2742no—
322C22⋊C4—5481no—
323C22⋊C4—5481no—
324C22⋊C4—1371no—
325C22⋊C4—1371no—
326C22⋊C4—1371no—
327C22⋊C4—1371no—
328C22⋊C4—1371no—
329C22⋊C4—1371no—
330C22⋊C4—1371no—
331C22⋊C4—1371no—
332C22⋊C4—1371no—
333C22⋊C4C2×#1058773911yescharacteristic
334C22⋊C4C2×#1058773911yescharacteristic
335C22⋊C4C2×#1058773911yescharacteristic
336C42—2742no—
337C42—5481no—
338C42—5481no—
339C42—5481no—
340C42—5481no—
341C42—1371no—
342C42—1371no—
343C42—1371no—
344C42C2×#1058773911yescharacteristic, complemented
345C23—2741no—
346C23—2741no—
347C23—2741no—
348C23—2741no—
349C23—2741no—
350C23—2741no—
351C23—1371no—
352C23—1371no—
353C23C22×#1058773911yescharacteristic
354C2×C4—2741no—
355C2×C4—2741no—
356C2×C4—2741no—
357C2×C4—2741no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—2741no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—2741no—
364C2×C4—2741no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—2741no—
369C2×C4—2741no—
370C2×C4—2741no—
371C2×C4—2741no—
372C2×C4—2741no—
373C2×C4—2741no—
374C2×C4—2741no—
375C2×C4—2741no—
376C2×C4—2741no—
377C2×C4—2741no—
378C2×C4—2741no—
379C2×C4—2741no—
380C2×C4—2741no—
381C2×C4—2741no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—2741no—
386C2×C4—2741no—
387C2×C4—2741no—
388C2×C4—5484no—
389C2×C4—1372no—
390C2×C4—1372no—
391C2×C4—1371no—
392C2×C4—1371no—
393C2×C4—1371no—
394C2×C4—1371no—
395C2×C4—1371no—
396C2×C4—1371no—
397C2×C4C22×#1058773911yescharacteristic
398C2×C4C22×#1058773911yescharacteristic
399C2×C4C22×#1058773911yescharacteristic
400C2×C4C22×#1058773911yescharacteristic
401C2×C4C22×#1058773911yescharacteristic
402C2×C4C22×#1058773911yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—2741no—
407C22—2741no—
408C22—2741no—
409C22—5481no—
410C22—5481no—
411C22—5481no—
412C22—5481no—
413C22—1371no—
414C22—1371no—
415C22—1371no—
416C22—1371no—
417C22—1371no—
418C22—1371no—
419C22C23×#1058773911yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—2742no—
427C4—2742no—
428C4—2741no—
429C4—2741no—
430C4—2741no—
431C4—2741no—
432C4—2741no—
433C4—2741no—
434C4—5481no—
435C4—5481no—
436C4—5481no—
437C4—5481no—
438C4—5481no—
439C4—5481no—
440C2—41no—
441C2—5481no—
442C2—1371no—
443C2—1371no—
444C2—1371no—
445C2—1371no—
446C2#41×#1058773911yescharacteristic
447C2#2257847211yescharacteristic
448C2#2257847211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
Fitting#126×C137C4
prime radicalC23×D137C23
cosocleC22×D137C24
hypercenter#126#10587739
hypocenterC137#2334
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #22578354C2semidirect
2. #22578354C2semidirect
6. #22578352C2semidirect
7. #22578352C2semidirect
8. #22578352C2semidirect
11. #22578347C2semidirect
12. C4⋊C4×#10587739C2semidirect
13. C4×C4⋊#10587739C2semidirect
44. #17984083C4semidirect
45. #17984083C4semidirect
46. #17984083C4semidirect
47. #17984082C4semidirect
48. #17984081C4semidirect
49. #17984081C4semidirect
50. #17984081C4semidirect
58. #126×C137C4semidirect
92. C4⋊Dic137C2×C4semidirect
93. C4⋊Dic137C2×C4semidirect
94. C4⋊Dic137C2×C4semidirect
95. #14555489C2×C4semidirect
96. #14555489C2×C4semidirect
97. #14555489C2×C4semidirect
98. #14555489C2×C4semidirect
99. #14555489C2×C4semidirect
100. #14555489C2×C4semidirect
110. C4×Dic137C2×C4semidirect
111. C4×Dic137C2×C4semidirect
112. C4×Dic137C2×C4semidirect
113. C4⋊C4×C137C2×C4semidirect
114. C4⋊C4×C137C2×C4semidirect
115. C4⋊C4×C137C2×C4semidirect
119. C42×C137C2×C4semidirect
224. C137#2334semidirect
249. #126#10587739semidirect
318. C4⋊C4C2×#10587739semidirect
319. C4⋊C4C2×#10587739semidirect
320. C4⋊C4C2×#10587739semidirect
344. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783542
#225783532
#225783522
#225783502
#225783472
C4⋊C4×#105877392
C4×C4⋊#105877392
C22⋊C4×#105877392
#126×D1372
#2334137