Finite Group Databasebeta

← #31792293

Subgroups — #31792293

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D137C211yesmaximal, characteristic, complemented
2Q8×C22⋊D137C211yesmaximal, characteristic, complemented
3#22578041C211yesmaximal, characteristic
4#22577796C211yesmaximal, characteristic, complemented
5#22577649C211yesmaximal, characteristic, complemented
6#22577641C211yesmaximal, characteristic, complemented
7#22577627C211yesmaximal, characteristic, complemented
8#22577606C211yesmaximal, characteristic, complemented
9#22577565C211yesmaximal, characteristic, complemented
10#22577557C211yesmaximal, characteristic, complemented
11#22577552C211yesmaximal, characteristic, complemented
12#22577542C211yesmaximal, characteristic, complemented
13#22577504C211yesmaximal, characteristic, complemented
14#22577472C211yesmaximal, characteristic, complemented
15C2×#137×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×#14555500C2211yescharacteristic, complemented
19C2×#14555499C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C2×#14555497C2211yescharacteristic
22C2×#14555497C2211yescharacteristic
23SD16×D137—24no—
24C2×C8×D137C2211yescharacteristic
25#17984086—21no—
26#17984086C2211yescharacteristic, complemented
27#17984077C2211yescharacteristic
28#17984071—21no—
29#17984067—21no—
30#17984067C2211yescharacteristic, complemented
31#17984064—21no—
32#17984064C2211yescharacteristic
33#17984064C2211yescharacteristic
34#17984062C2211yescharacteristic
35#17984060—21no—
36#17984056—21no—
37Q8×Dic137C2211yescharacteristic, complemented
38D4×Dic137C2211yescharacteristic
39C4×C22⋊D137—21no—
40C4×C22⋊D137C2211yescharacteristic
41C2×#14555487—21no—
42#17984022C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984012C2211yescharacteristic, complemented
46#17984012C2211yescharacteristic, complemented
47#17984012C2211yescharacteristic, complemented
48#17984010C2211yescharacteristic
49#17984009C2211yescharacteristic
50C22⋊#11764040C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic
52#17983998C2211yescharacteristic, complemented
53C2×#41×C137C2211yescharacteristic
54C22×Q8×C137C2211yescharacteristic, complemented
55#137×C137—24no—
56C2×Q16×C137C2211yescharacteristic, complemented
57C2×Q16×C137C2211yescharacteristic, complemented
58C2×SD16×C137C2211yescharacteristic
59C2×SD16×C137C2211yescharacteristic
60C2×M16×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64Q8×D137—22no—
65Q8×D137—41no—
66C2×D548C2311yescharacteristic
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137D411yescharacteristic
70D4×D137—22no—
71D4×D137—41no—
72C22×Dic137—21no—
73C2×C4×D137—21no—
74C2×C4×D137C2311yescharacteristic
75#14555500—41no—
76#14555499—22no—
77#14555498—22no—
78#14555497—22no—
79#14555497—41no—
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C8×D137—22no—
83C4⋊Dic137—21no—
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489—21no—
91#14555489D411yescharacteristic
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487—21no—
99#14555487—21no—
100#14555487D411yescharacteristic
101C4×Dic137—21no—
102C4×Dic137C2311yescharacteristic
103#41×C137—22no—
104#41×C137—41no—
105C2×Q8×C137—21no—
106C2×Q8×C137—22no—
107C2×Q8×C137—41no—
108C2×Q8×C137C2311yescharacteristic
109C2×Q8×C137C2311yescharacteristic
110C2×Q8×C137C2311yescharacteristic
111C2×D4×C137—21no—
112C2×D4×C137C2311yescharacteristic
113C22×C4×C137—21no—
114C22×C4×C137—21no—
115C22×C4×C137C2311yescharacteristic
116Q16×C137—22no—
117Q16×C137—22no—
118SD16×C137—22no—
119SD16×C137D414yescomplemented
120M16×C137—22no—
121C2×C8×C137C2311yescharacteristic
122C2×C8×C137C2311yescharacteristic
123C22×D137—21no—
124C22×D137—42no—
125C22×D137—42no—
126C22×D137—81no—
127C22×D137C2×D411yescharacteristic
128Dic274—21no—
129Dic274—21no—
130Dic274—42no—
131D548—21no—
132D548—21no—
133C22⋊D137—22no—
134C22⋊D137—42no—
135C22⋊D137—81no—
136C2×Dic137—21no—
137C2×Dic137—21no—
138C2×Dic137—21no—
139C2×Dic137—81no—
140C2×Dic137C2×D411yescharacteristic
141C2×Dic137C2×D411yescharacteristic
142C2×Dic137C2×D411yescharacteristic
143C4×D137—21no—
144C4×D137—21no—
145C4×D137—42no—
146#11764040—21no—
147#11764040—21no—
148C23×C137—21no—
149C23×C137C2×D411yescharacteristic
150Q8×C137—21no—
151Q8×C137—21no—
152Q8×C137—21no—
153Q8×C137—21no—
154Q8×C137—21no—
155Q8×C137—42no—
156Q8×C137C2×D412yes—
157D4×C137—21no—
158D4×C137—42no—
159D4×C137C2×D412yes—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—42no—
166C2×C4×C137—42no—
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137C2×D412yes—
171C22×C137—21no—
172C22×C137—22no—
173C22×C137—21no—
174C22×C137—21no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137C22×D411yescharacteristic
178C2×D137—21no—
179C2×D137—21no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—41no—
183C2×D137—84no—
184C2×D137#13712yescomplemented
185C4×C137—21no—
186C4×C137—22no—
187C4×C137—41no—
188C4×C137—41no—
189C4×C137C22×D411yescharacteristic
190C4×C137H(5,𝔽2)11yescharacteristic
191Dic137—21no—
192Dic137—21no—
193Dic137—21no—
194Dic137—41no—
195Dic137—41no—
196Dic137—41no—
197C2×C137—42no—
198C2×C137—41no—
199C2×C137D4211yescharacteristic
200C2×C137C2×#13711yescharacteristic
201C2×C137#57711yescharacteristic
202D137—22no—
203D137—81no—
204C137#326611yescharacteristic, complemented, Sylow
205#3266—1371nomaximal, Sylow
206C2×#137D13711yescharacteristic, complemented
207C22×SD16—1371no—
208D4×Q8—1371no—
209#546—1371no—
210#497—1371no—
211#468—1371no—
212#467—1371no—
213#464—1371no—
214#461—1371no—
215#452—1371no—
216#451—1371no—
217#450—1371no—
218#448—1371no—
219#440—1371no—
220#435—1371no—
221C2×#41C2×D13711yescharacteristic
222C22×Q8—1371no—
223C22×Q8C2×D13711yescharacteristic, complemented
224C22×D4—1371no—
225#137—24no—
226C2×Q16—1371no—
227C2×Q16C2×D13711yescharacteristic, complemented
228C2×Q16C2×D13711yescharacteristic, complemented
229C2×SD16—2744no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×SD16—1371no—
233C2×SD16—1371no—
234C2×SD16C2×D13711yescharacteristic
235C2×SD16C2×D13711yescharacteristic
236C2×M16C2×D13711yescharacteristic, complemented
237C22×C8—1371no—
238C4⋊Q8—2741no—
239C4⋊Q8—1371no—
240#124—1371no—
241#123—2741no—
242C22⋊Q8—2741no—
243C22⋊Q8—2741no—
244C22⋊Q8—1371no—
245C22⋊Q8—1371no—
246C22⋊Q8—1371no—
247#121—2741no—
248#121—1371no—
249#120—2741no—
250C4×Q8—1371no—
251C4×D4—2741no—
252C4×D4—1371no—
253C4×D4—1371no—
254C2×C22⋊C4—2741no—
255#107—1371no—
256C4⋊C8—1371no—
257#103—1371no—
258#103—1371no—
259#103—1371no—
260#103—1371no—
261#102—1371no—
262#102—1371no—
263C22⋊C8—1371no—
264C22⋊C8—1371no—
265#97—1371no—
266C24—2741no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2741no—
273C2×Q8—2742no—
274C2×Q8—5481no—
275C2×Q8—1371no—
276C2×Q8C22×D13711yescharacteristic
277C2×Q8C22×D13711yescharacteristic
278C2×Q8C22×D13711yescharacteristic
279C2×D4—21no—
280C2×D4—2742no—
281C2×D4—2741no—
282C2×D4—2741no—
283C2×D4—5481no—
284C2×D4—1371no—
285C2×D4—1371no—
286C2×D4C22×D13711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—2741no—
290C22×C4—2741no—
291C22×C4—1371no—
292C22×C4C22×D13711yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—5481no—
296SD16—22no—
297SD16—2742no—
298SD16—2742no—
299SD16—2742no—
300SD16—5481no—
301SD16C22⋊D13714yescomplemented
302M16—22no—
303C2×C8—2742no—
304C2×C8—1371no—
305C2×C8—1371no—
306C2×C8C22×D13711yescharacteristic
307C2×C8C22×D13711yescharacteristic
308C4⋊C4—2741no—
309C4⋊C4—2741no—
310C4⋊C4—2741no—
311C4⋊C4—2741no—
312C4⋊C4—2741no—
313C4⋊C4—2741no—
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4—1371no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—1371no—
325C22⋊C4—1371no—
326C42—2741no—
327C42—1371no—
328C23—21no—
329C23—2741no—
330C23—5482no—
331C23—5482no—
332C23—10961no—
333C23—1371no—
334C23D4×D13711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—2741no—
342Q8—2741no—
343Q8—5482no—
344Q8C2×C22⋊D13712yes—
345D4—21no—
346D4—42no—
347D4—2742no—
348D4—2741no—
349D4—2741no—
350D4—5482no—
351D4—10961no—
352D4C2×C22⋊D13712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—2741no—
361C2×C4—2741no—
362C2×C4—2741no—
363C2×C4—2741no—
364C2×C4—2741no—
365C2×C4—5482no—
366C2×C4—10961no—
367C2×C4—1371no—
368C2×C4—1371no—
369C2×C4—1371no—
370C2×C4D4×D13711yescharacteristic
371C2×C4C23×D13711yescharacteristic
372C8—21no—
373C8—2741no—
374C8—2741no—
375C8C2×C22⋊D13712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—2741no—
383C22—2741no—
384C22—5481no—
385C22—5481no—
386C22—5481no—
387C22—10964no—
388C22—1372no—
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—2741no—
395C4—2741no—
396C4—2741no—
397C4—5481no—
398C4—5481no—
399C4—5481no—
400C4C22×C22⋊D13711yescharacteristic
401C4#1798413311yescharacteristic
402C2—41no—
403C2—42no—
404C2—2742no—
405C2—10961no—
406C2D4×C22⋊D13711yescharacteristic
407C2#137×D13711yescharacteristic
408C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#137×C137C2
socleC22×C137C22×D4
prime radical#22578041C2
hypercenterC2×#137D137
hypocenterC137#3266
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D137C2semidirect
2. Q8×C22⋊D137C2semidirect
4. #22577796C2semidirect
5. #22577649C2semidirect
6. #22577641C2semidirect
7. #22577627C2semidirect
8. #22577606C2semidirect
9. #22577565C2semidirect
10. #22577557C2semidirect
11. #22577552C2semidirect
12. #22577542C2semidirect
13. #22577504C2semidirect
14. #22577472C2semidirect
15. C2×#137×C137C2semidirect
18. C2×#14555500C22semidirect
26. #17984086C22semidirect
30. #17984067C22semidirect
37. Q8×Dic137C22semidirect
42. #17984022C22semidirect
43. #17984017C22semidirect
44. #17984013C22semidirect
45. #17984012C22semidirect
46. #17984012C22semidirect
47. #17984012C22semidirect
50. C22⋊#11764040C22semidirect
52. #17983998C22semidirect
54. C22×Q8×C137C22semidirect
56. C2×Q16×C137C22semidirect
57. C2×Q16×C137C22semidirect
60. C2×M16×C137C22semidirect
119. SD16×C137D4semidirect
184. C2×D137#137semidirect
204. C137#3266semidirect
206. C2×#137D137semidirect
223. C22×Q8C2×D137semidirect
227. C2×Q16C2×D137semidirect
228. C2×Q16C2×D137semidirect
236. C2×M16C2×D137semidirect
301. SD16C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1372
Q8×C22⋊D1372
#225780412
#225777962
#225776492
#225776412
#225776272
#225776062
#225775652
#225775572
#225775522
#225775422
#225775042
#225774722
C2×#137×C1372
#3266137