Finite Group Databasebeta

← #31789618

Subgroups — #31789618

Order 17536.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984112C211yesmaximal, characteristic
2#22577882C211yesmaximal, characteristic, complemented
3#22577875C211yesmaximal, characteristic, complemented
4#22577512C211yesmaximal, characteristic, complemented
5#22577505C211yesmaximal, characteristic, complemented
6#22577500C211yesmaximal, characteristic, complemented
7#22577490C211yesmaximal, characteristic, complemented
8C4×#14555498C211yesmaximal, characteristic, complemented
9D8×Dic137C211yesmaximal, characteristic, complemented
10C4×#14555495C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577396C211yesmaximal, characteristic, complemented
13#102×D137C211yesmaximal, characteristic, complemented
14#22577360C211yesmaximal, characteristic, complemented
15#442×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18#17984112C418yescomplemented
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C2×#14555495C2211yescharacteristic
22C2×#14555493C2211yescharacteristic
23C2×C8×D137C2211yescharacteristic
24C4×Dic274C2211yescharacteristic, complemented
25#17984053C2211yescharacteristic, complemented
26C4×D548C2211yescharacteristic
27D4×Dic137C2211yescharacteristic
28D4×Dic137C2211yescharacteristic, complemented
29#17984050C2211yescharacteristic
30C4×C22⋊D137—21no—
31C4×C22⋊D137—21no—
32#17984048—21no—
33#17984048—21no—
34#17984046C2211yescharacteristic
35#17984045C2211yescharacteristic
36#17984044—21no—
37C2×#14555489—21no—
38C4⋊C4×D137C2211yescharacteristic
39C2×#14555487—21no—
40C22⋊C4×D137—21no—
41#17984022C2211yescharacteristic, complemented
42#17984021C2211yescharacteristic, complemented
43#17984019C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984011C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C4×#11764040C2211yescharacteristic, complemented
52C8×Dic137C2211yescharacteristic, complemented
53C2×D8×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic, complemented
56#106×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic, complemented
59#97×C137C2211yescharacteristic, complemented
60C23×D137—21no—
61#14555509—22no—
62#14555509C2×C414yescomplemented
63C2×Dic274C2311yescharacteristic
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67D4×D137—22no—
68D4×D137C2×C414yescomplemented
69C22×Dic137—21no—
70C22×Dic137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137C2311yescharacteristic
76#14555499C2×C414yescomplemented
77#14555498C2×C414yescomplemented
78#14555495C2×C414yescomplemented
79#14555493C2×C414yescomplemented
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C8×D137—22no—
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89#14555489D411yescharacteristic
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×D4×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104C22×C4×C137—21no—
105D8×C137C2×C414yescomplemented
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22⋊C4×C137—21no—
112C42×C137C2311yescharacteristic
113C22×D137—21no—
114C22×D137—24no—
115C22×D137—81no—
116C22×D137—81no—
117C22×D137C2×D411yescharacteristic
118Dic274—21no—
119Dic274C22×C412yes—
120D548—21no—
121D548C22×C412yes—
122C22⋊D137—24no—
123C22⋊D137—24no—
124C2×Dic137—21no—
125C2×Dic137—24no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—81no—
129C2×Dic137—81no—
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C4×D137—81no—
134C4×D137—81no—
135C4×D137—81no—
136C4×D137C22×C412yes—
137C4×D137#4112yes—
138#11764040—21no—
139#11764040C22×C412yes—
140C23×C137—21no—
141C23×C137—21no—
142D4×C137—21no—
143D4×C137—21no—
144D4×C137C22×C412yes—
145D4×C137C22×C412yes—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—81no—
149C2×C4×C137—81no—
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2×D411yescharacteristic
152C2×C4×C137C2411yescharacteristic
153C8×C137—21no—
154C8×C137C22×C412yes—
155C22×C137—22no—
156C22×C137—22no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—21no—
164C2×D137—22no—
165C2×D137—41no—
166C2×D137—41no—
167C2×D137—84no—
168C2×D137C4×D412yes—
169C4×C137—21no—
170C4×C137—21no—
171C4×C137—41no—
172C4×C137—41no—
173C4×C137C23×C411yescharacteristic
174C4×C137C2×#4111yescharacteristic
175Dic137—22no—
176Dic137—21no—
177Dic137—21no—
178Dic137—41no—
179Dic137—41no—
180Dic137C4×D412yes—
181C2×C137—42no—
182C2×C137—42no—
183C2×C137C2×C4×D411yescharacteristic
184C2×C137#57611yescharacteristic
185C2×C137#57711yescharacteristic
186D137—22no—
187D137—42no—
188C137#294611yescharacteristic, complemented, Sylow
189#2946—1371nomaximal, Sylow
190C2×AGL(1,ℤ/8)—1371no—
191#520—1371no—
192#518—1371no—
193#442—1371no—
194#442D13711yescharacteristic, complemented
195#440—1371no—
196#440—1371no—
197C4×SD16—1371no—
198C4×SD16—1371no—
199C4×D8—1371no—
200C4×D8—1371no—
201#428—1371no—
202#416—1371no—
203C2×#102—1371no—
204#405—1371no—
205C2×#41—1371no—
206C22×D4—1371no—
207AGL(1,ℤ/8)—1378no—
208C2×SD16—1371no—
209C2×SD16—1371no—
210C2×D8—1371no—
211C2×D8C2×D13711yescharacteristic
212C2×M16—1371no—
213C22×C8—1371no—
214C4×Q8—1371no—
215C4×Q8—1371no—
216C4×D4—2741no—
217C4×D4—2741no—
218C4×D4—2741no—
219C4×D4—2741no—
220C4×D4—1371no—
221C4×D4—1371no—
222C4×D4—1371no—
223C4×D4—1371no—
224C4×D4C2×D13711yescharacteristic
225C4×D4C2×D13711yescharacteristic, complemented
226#117—2741no—
227#117—1371no—
228#117—1371no—
229C2×C4⋊C4—2741no—
230C2×C4⋊C4—1371no—
231C2×C22⋊C4—2741no—
232C2×C22⋊C4—2741no—
233#107—1371no—
234#107—1371no—
235#106—1371no—
236#106C2×D13711yescharacteristic, complemented
237#103—1371no—
238#103—1371no—
239#102—1371no—
240#102—1371no—
241#102—1371no—
242#102—1371no—
243#102C2×D13711yescharacteristic
244#102C2×D13711yescharacteristic, complemented
245#97—1371no—
246#97C2×D13711yescharacteristic, complemented
247C4×C8—1371no—
248C4×C8—1371no—
249C24—2741no—
250#41—2742no—
251#41—1374no—
252C2×Q8—1371no—
253C2×D4—2741no—
254C2×D4—2742no—
255C2×D4—2741no—
256C2×D4—1374no—
257C2×D4—1371no—
258C2×D4C22×D13711yescharacteristic
259C2×D4C22×D13711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—2741no—
263C22×C4—2741no—
264C22×C4—2741no—
265C22×C4—2741no—
266C22×C4—2741no—
267C22×C4—2741no—
268C22×C4—1371no—
269SD16—1374no—
270SD16—1374no—
271D8—1374no—
272D8C4×D13714yescomplemented
273M16—1374no—
274C2×C8—2742no—
275C2×C8—1371no—
276C2×C8—1371no—
277C2×C8C22×D13711yescharacteristic
278C2×C8C22×D13711yescharacteristic
279C4⋊C4—2741no—
280C4⋊C4—2741no—
281C4⋊C4—2741no—
282C4⋊C4—1371no—
283C4⋊C4—1371no—
284C4⋊C4—1371no—
285C4⋊C4—1371no—
286C4⋊C4C22×D13711yescharacteristic
287C4⋊C4C22×D13711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—2741no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—1371no—
296C22⋊C4—1371no—
297C42—2741no—
298C42—2741no—
299C42—2741no—
300C42—1371no—
301C42C22×D13711yescharacteristic
302C23—21no—
303C23—21no—
304C23—2744no—
305C23—2741no—
306C23—10961no—
307C23—10961no—
308C23—1371no—
309Q8—2741no—
310Q8—1372no—
311D4—21no—
312D4—21no—
313D4—2744no—
314D4—2744no—
315D4—2741no—
316D4—1372no—
317D4C2×C4×D13712yes—
318D4C2×C4×D13712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—2741no—
324C2×C4—2744no—
325C2×C4—2741no—
326C2×C4—2741no—
327C2×C4—10961no—
328C2×C4—10961no—
329C2×C4—10961no—
330C2×C4—10961no—
331C2×C4—10961no—
332C2×C4—1371no—
333C2×C4—1371no—
334C2×C4—1371no—
335C2×C4—1372no—
336C2×C4—1372no—
337C2×C4D4×D13711yescharacteristic
338C2×C4D4×D13711yescharacteristic
339C2×C4C23×D13711yescharacteristic
340C8—21no—
341C8—2741no—
342C8—1372no—
343C8C2×C4×D13712yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—2741no—
351C22—2741no—
352C22—2742no—
353C22—5481no—
354C22—5481no—
355C22—10964no—
356C22—1372no—
357C22C2×D4×D13711yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—2741no—
363C4—2741no—
364C4—2742no—
365C4—5481no—
366C4—5481no—
367C4—1372no—
368C4C22×C4×D13711yescharacteristic
369C4#41×D13711yescharacteristic
370C2—42no—
371C2—42no—
372C2—2742no—
373C2—5482no—
374C2C4×D4×D13711yescharacteristic
375C2#2257820711yescharacteristic
376C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting#442×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984112C2
cosocleC2×C4×C137C24
hypercenter#442D137
hypocenterC137#2946
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #22577882C2semidirect
3. #22577875C2semidirect
4. #22577512C2semidirect
5. #22577505C2semidirect
6. #22577500C2semidirect
7. #22577490C2semidirect
8. C4×#14555498C2semidirect
9. D8×Dic137C2semidirect
10. C4×#14555495C2semidirect
11. #22577442C2semidirect
12. #22577396C2semidirect
13. #102×D137C2semidirect
14. #22577360C2semidirect
15. #442×C137C2semidirect
18. #17984112C4semidirect
24. C4×Dic274C22semidirect
25. #17984053C22semidirect
28. D4×Dic137C22semidirect
41. #17984022C22semidirect
42. #17984021C22semidirect
43. #17984019C22semidirect
44. #17984013C22semidirect
45. #17984011C22semidirect
48. #17984009C22semidirect
50. #17983998C22semidirect
51. C4×#11764040C22semidirect
52. C8×Dic137C22semidirect
55. C4×D4×C137C22semidirect
56. #106×C137C22semidirect
58. #102×C137C22semidirect
59. #97×C137C22semidirect
62. #14555509C2×C4semidirect
68. D4×D137C2×C4semidirect
76. #14555499C2×C4semidirect
77. #14555498C2×C4semidirect
78. #14555495C2×C4semidirect
79. #14555493C2×C4semidirect
105. D8×C137C2×C4semidirect
188. C137#2946semidirect
194. #442D137semidirect
225. C4×D4C2×D137semidirect
236. #106C2×D137semidirect
244. #102C2×D137semidirect
246. #97C2×D137semidirect
272. D8C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
#225778822
#225778752
#225775122
#225775052
#225775002
#225774902
C4×#145554982
D8×Dic1372
C4×#145554952
#225774422
#225773962
#102×D1372
#225773602
#442×C1372
#2946137