Finite Group Databasebeta

← #31783947

Subgroups — #31783947

Order 17536.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984060C211yesmaximal, characteristic
2#22577306C211yesmaximal, characteristic, complemented
3#22577303C211yesmaximal, characteristic, complemented
4#22577292C211yesmaximal, characteristic, complemented
5#22577291C211yesmaximal, characteristic, complemented
6#22577290C211yesmaximal, characteristic, complemented
7#22577285C211yesmaximal, characteristic, complemented
8#22577282C211yesmaximal, characteristic, complemented
9#22577279C211yesmaximal, characteristic, complemented
10#22577279C211yesmaximal, characteristic, complemented
11#22577277C211yesmaximal, characteristic, complemented
12#22577276C211yesmaximal, characteristic, complemented
13#22577264C211yesmaximal, characteristic, complemented
14#22577261C211yesmaximal, characteristic, complemented
15#386×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984060C418yescomplemented
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489C2211yescharacteristic
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×C22⋊Dic137C2211yescharacteristic
27C2×C22⋊Dic137C2211yescharacteristic
28C2×C22⋊Dic137C2211yescharacteristic
29C2×C22⋊Dic137C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic
31C2×#14555487C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36C2×C4×Dic137C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45C22×D4×C137C2211yescharacteristic
46C23×C4×C137C2211yescharacteristic
47C2×C22⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C23×D137C2311yescharacteristic
53C2×C22⋊D137—41no—
54C2×C22⋊D137—41no—
55C2×C22⋊D137—41no—
56C2×C22⋊D137—41no—
57C2×C22⋊D137C2×C414yes—
58C2×C22⋊D137C2×C414yescomplemented
59C22×Dic137—41no—
60C22×Dic137—41no—
61C22×Dic137C2×C414yescomplemented
62C22×Dic137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C2×C4×D137—41no—
70C2×C4×D137—41no—
71C2×C4×D137—41no—
72C4⋊Dic137—41no—
73#14555489—41no—
74#14555489—41no—
75#14555489—41no—
76#14555489C2×C414yescomplemented
77C22⋊Dic137—41no—
78C22⋊Dic137—41no—
79C22⋊Dic137—41no—
80C22⋊Dic137—41no—
81C22⋊Dic137C2×C414yescomplemented
82#14555487—41no—
83#14555487—41no—
84#14555487—41no—
85#14555487C2×C414yescomplemented
86C4×Dic137—41no—
87C4×Dic137—41no—
88C24×C137C2311yescharacteristic
89C24×C137C2311yescharacteristic
90C2×D4×C137—41no—
91C2×D4×C137—41no—
92C2×D4×C137C2×C414yescomplemented
93C22×C4×C137—41no—
94C22×C4×C137—41no—
95C22×C4×C137—41no—
96C22×C4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C22⋊C4×C137—41no—
102C22⋊C4×C137—41no—
103C22⋊C4×C137—41no—
104C22×D137—21no—
105C22×D137—21no—
106C22×D137—21no—
107C22×D137—21no—
108C22×D137—21no—
109C22×D137—21no—
110C22×D137C22×C412yes—
111C22⋊D137—44no—
112C22⋊D137—44no—
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×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
131C2×Dic137—44no—
132C2×Dic137C22×C412yes—
133C2×Dic137C22×C412yes—
134C2×Dic137C22×C412yes—
135C4×D137—44no—
136C23×C137—21no—
137C23×C137—21no—
138C23×C137—21no—
139C23×C137—21no—
140C23×C137—21no—
141C23×C137—21no—
142C23×C137—21no—
143C23×C137—21no—
144C23×C137—21no—
145C23×C137—21no—
146C23×C137—21no—
147C23×C137—21no—
148C23×C137C22×C412yes—
149C23×C137C22×C412yes—
150C23×C137C2411yescharacteristic
151D4×C137—44no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—21no—
155C2×C4×C137—21no—
156C2×C4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—21no—
162C2×C4×C137—21no—
163C2×C4×C137—21no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—44no—
167C2×C4×C137C22×C412yes—
168C22×C137—22no—
169C22×C137—22no—
170C22×C137—22no—
171C22×C137—22no—
172C22×C137—22no—
173C22×C137—22no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137C23×C411yescharacteristic
183C22×C137H(5,𝔽2)11yescharacteristic
184C22×C137H(5,𝔽2)11yescharacteristic
185C22×C137H(5,𝔽2)11yescharacteristic
186C22×C137H(5,𝔽2)11yescharacteristic
187C22×C137H(5,𝔽2)11yescharacteristic
188C22×C1372−1+411yescharacteristic
189C2×D137—22no—
190C2×D137—22no—
191C2×D137—22no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—41no—
195C2×D137—41no—
196C4×C137—22no—
197C4×C137—41no—
198C4×C137—41no—
199C4×C137—41no—
200C4×C137—41no—
201Dic137—22no—
202Dic137—22no—
203Dic137—22no—
204Dic137—41no—
205Dic137—41no—
206Dic137—41no—
207Dic137—41no—
208C2×C137—42no—
209C2×C137—42no—
210C2×C137#51811yescharacteristic
211C2×C137#51811yescharacteristic
212C2×C137#52011yescharacteristic
213C2×C137#56011yescharacteristic
214C2×C137#56011yescharacteristic
215C2×C137#56211yescharacteristic
216C2×C137#56211yescharacteristic
217D137—42no—
218C137#237711yescharacteristic, complemented, Sylow
219#2377—1371nomaximal, Sylow
220C2×#121—1371no—
221#390—1371no—
222#390—1371no—
223#388—1371no—
224#388—1371no—
225#387—1371no—
226#387—1371no—
227#386—1371no—
228#386—1371no—
229#386—1371no—
230#386D13711yescharacteristic, complemented
231#385—1371no—
232#385—1371no—
233#381—1371no—
234#380—1371no—
235C22×D4—1371no—
236C22×D4—1371no—
237C22×D4C2×D13711yescharacteristic
238C23×C4—1371no—
239C23×C4—1371no—
240C23×C4C2×D13711yescharacteristic
241#121—1378no—
242C2×C4⋊C4—1371no—
243C2×C4⋊C4—1371no—
244C2×C4⋊C4—1371no—
245C2×C4⋊C4—1371no—
246C2×C4⋊C4—1371no—
247C2×C22⋊C4—1371no—
248C2×C22⋊C4—1371no—
249C2×C22⋊C4—1371no—
250C2×C22⋊C4—1371no—
251C2×C22⋊C4—1371no—
252C2×C22⋊C4—1371no—
253C2×C22⋊C4—1371no—
254C2×C22⋊C4—1371no—
255C2×C22⋊C4—1371no—
256C2×C22⋊C4C2×D13711yescharacteristic
257C2×C22⋊C4C2×D13711yescharacteristic
258C2×C22⋊C4C2×D13711yescharacteristic
259C2×C42—1371no—
260C2×C42—1371no—
261#95—1371no—
262#95—1371no—
263#95—1371no—
264#95—1371no—
265#95—1371no—
266#95—1371no—
267#95—1371no—
268#95—1371no—
269#95C2×D13711yescharacteristic
270#95C2×D13711yescharacteristic
271C24—1371no—
272C24C22×D13711yescharacteristic
273C24C22×D13711yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—5481no—
277C2×D4—5481no—
278C2×D4—5481no—
279C2×D4—5481no—
280C2×D4—1374no—
281C2×D4—1374no—
282C2×D4C4×D13714yescomplemented
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—5481no—
287C22×C4—5481no—
288C22×C4—5481no—
289C22×C4—5481no—
290C22×C4—5481no—
291C22×C4—1371no—
292C22×C4—1371no—
293C22×C4—1371no—
294C22×C4—1371no—
295C22×C4—1371no—
296C22×C4—1371no—
297C22×C4—1371no—
298C22×C4—1374no—
299C22×C4C22×D13711yescharacteristic
300C22×C4C22×D13711yescharacteristic
301C22×C4C22×D13711yescharacteristic
302C22×C4C22×D13711yescharacteristic
303C22×C4C22×D13711yescharacteristic
304C4⋊C4—5481no—
305C4⋊C4—5481no—
306C4⋊C4—5481no—
307C4⋊C4—5481no—
308C4⋊C4—1374no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—5481no—
313C22⋊C4—5481no—
314C22⋊C4—5481no—
315C22⋊C4—5481no—
316C22⋊C4—5481no—
317C22⋊C4—5481no—
318C22⋊C4—5481no—
319C22⋊C4—1374no—
320C22⋊C4—1374no—
321C42—5481no—
322C42—5481no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—2741no—
336C23—2741no—
337C23—2741no—
338C23—2741no—
339C23—2741no—
340C23—2741no—
341C23—1372no—
342C23C2×C4×D13712yes—
343C23C2×C4×D13712yes—
344C23C23×D13711yescharacteristic
345D4—44no—
346D4—5484no—
347D4—5484no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—44no—
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—5484no—
382C2×C4—5484no—
383C2×C4—1372no—
384C2×C4—1372no—
385C2×C4—1372no—
386C2×C4C2×C4×D13712yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—2742no—
402C22—2742no—
403C22—2742no—
404C22—5481no—
405C22—5481no—
406C22—5481no—
407C22—5481no—
408C22C22×C4×D13711yescharacteristic
409C22#1798413311yescharacteristic
410C22#1798413311yescharacteristic
411C22#1798413311yescharacteristic
412C22#1798413311yescharacteristic
413C22#1798413411yescharacteristic
414C22#1798413511yescharacteristic
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—2742no—
421C4—2742no—
422C4—2742no—
423C4—5481no—
424C4—5481no—
425C4—5481no—
426C4—5481no—
427C2—42no—
428C2—42no—
429C2—5482no—
430C2#2257787511yescharacteristic
431C2#2257787611yescharacteristic
432C2#2257788211yescharacteristic
433C2#2257814111yescharacteristic
434C2#2257814211yescharacteristic
435C2#2257814811yescharacteristic
436C2#2257815011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#386×C137C2
socle, cosocleC23×C137C24
prime radicalC2×#17984060C2
hypercenter#386D137
hypocenterC137#2377
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #22577306C2semidirect
3. #22577303C2semidirect
4. #22577292C2semidirect
5. #22577291C2semidirect
6. #22577290C2semidirect
7. #22577285C2semidirect
8. #22577282C2semidirect
9. #22577279C2semidirect
10. #22577279C2semidirect
11. #22577277C2semidirect
12. #22577276C2semidirect
13. #22577264C2semidirect
14. #22577261C2semidirect
15. #386×C137C2semidirect
20. #17984060C4semidirect
58. C2×C22⋊D137C2×C4semidirect
61. C22×Dic137C2×C4semidirect
76. #14555489C2×C4semidirect
81. C22⋊Dic137C2×C4semidirect
85. #14555487C2×C4semidirect
92. C2×D4×C137C2×C4semidirect
218. C137#2377semidirect
230. #386D137semidirect
282. C2×D4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840602
#225773062
#225773032
#225772922
#225772912
#225772902
#225772852
#225772822
#225772792
#225772772
#225772762
#225772642
#225772612
#386×C1372
#2377137