Finite Group Databasebeta

← #31783716

Subgroups — #31783716

Order 17536.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic137C211yesmaximal, characteristic
2C2×#17984044C211yesmaximal, characteristic, complemented
3C22×#14555489C211yesmaximal, characteristic, complemented
4#22577297C211yesmaximal, characteristic, complemented
5#22577291C211yesmaximal, characteristic, complemented
6#22577288C211yesmaximal, characteristic, complemented
7#22577286C211yesmaximal, characteristic, complemented
8#22577283C211yesmaximal, characteristic
9#22577267C211yesmaximal, characteristic, complemented
10#22577261C211yesmaximal, characteristic, complemented
11#22577261C211yesmaximal, characteristic, complemented
12#22577254C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic137C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic137C211yesmaximal, characteristic, complemented
15#386×C137C211yesmaximal, characteristic
16C23×Dic137C2211yescharacteristic
17C23×Dic137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19D4×Dic137C418yescomplemented
20#17984044—24no—
21C2×C4⋊Dic137C2211yescharacteristic
22C2×#14555489—24no—
23C2×#14555489C2211yescharacteristic
24C2×#14555489C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×#14555489C2211yescharacteristic
27C2×#14555489C2211yescharacteristic
28C2×#14555489C2211yescharacteristic
29C2×C22⋊Dic137C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic
31C2×C22⋊Dic137C2211yescharacteristic
32C2×C22⋊Dic137C2211yescharacteristic
33C2×C22⋊Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36C2×C4×Dic137C2211yescharacteristic
37C2×C4×Dic137C2211yescharacteristic
38C2×C4×Dic137C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47C22×D4×C137C2211yescharacteristic
48C23×C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C22⋊C4×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53#95×C137C2211yescharacteristic
54C22×Dic137—22no—
55C22×Dic137—22no—
56C22×Dic137—22no—
57C22×Dic137—22no—
58C22×Dic137—22no—
59C22×Dic137—22no—
60C22×Dic137—41no—
61C22×Dic137C2×C414yes—
62C22×Dic137C2×C414yescomplemented
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C4⋊Dic137C2×C414yescomplemented
72#14555489—22no—
73#14555489—22no—
74#14555489—22no—
75#14555489—22no—
76#14555489—22no—
77#14555489D414yescomplemented
78C22⋊Dic137—22no—
79C22⋊Dic137—22no—
80C22⋊Dic137—41no—
81C22⋊Dic137C2×C414yes—
82C22⋊Dic137C2×C414yescomplemented
83C4×Dic137—22no—
84C4×Dic137—22no—
85C4×Dic137—41no—
86C4×Dic137—41no—
87C4×Dic137C2×C414yescomplemented
88C24×C137C2311yescharacteristic
89C24×C137C2311yescharacteristic
90C2×D4×C137—22no—
91C2×D4×C137—22no—
92C2×D4×C137C2×C414yes—
93C22×C4×C137—22no—
94C22×C4×C137—22no—
95C22×C4×C137—41no—
96C22×C4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C22⋊C4×C137—22no—
102C22⋊C4×C137—22no—
103C22⋊C4×C137—41no—
104C2×Dic137—21no—
105C2×Dic137—21no—
106C2×Dic137—21no—
107C2×Dic137—21no—
108C2×Dic137—21no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—28no—
113C2×Dic137—28no—
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—44no—
124C2×Dic137C22×C412yes—
125C2×Dic137C22×C412yes—
126C2×Dic137C22×C412yes—
127C2×Dic137C22×C412yes—
128C2×Dic137C2×D412yes—
129C2×Dic137C2×D412yes—
130C2×Dic137#4112yes—
131C23×C137—21no—
132C23×C137—21no—
133C23×C137—21no—
134C23×C137—21no—
135C23×C137—21no—
136C23×C137—21no—
137C23×C137—21no—
138C23×C137—21no—
139C23×C137—21no—
140C23×C137—21no—
141C23×C137C22×C412yes—
142C23×C137C22×C412yes—
143C23×C137#4112yes—
144C23×C137#4112yes—
145C23×C137C2411yescharacteristic
146D4×C137—28no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
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—44no—
161C2×C4×C137C22×C412yes—
162C2×C4×C137C2×D412yes—
163C22×C137—22no—
164C22×C137—22no—
165C22×C137—22no—
166C22×C137—22no—
167C22×C137—22no—
168C22×C137—22no—
169C22×C137—22no—
170C22×C137—22no—
171C22×C137—22no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137#11714yescomplemented
177C22×C137C23×C411yescharacteristic
178C22×C137C22×D411yescharacteristic
179C22×C137C2×#4111yescharacteristic
180C22×C137C2×#4111yescharacteristic
181C22×C137C2×#4111yescharacteristic
182C22×C137H(5,𝔽2)11yescharacteristic
183C22×C1372−1+411yescharacteristic
184C4×C137—22no—
185C4×C137—22no—
186C4×C137—41no—
187C4×C137—41no—
188C4×C137—41no—
189Dic137—22no—
190Dic137—22no—
191Dic137—22no—
192Dic137—22no—
193Dic137—41no—
194Dic137—41no—
195Dic137—41no—
196Dic137C4×D414yes—
197C2×C137—24no—
198C2×C137—42no—
199C2×C137C2×#11711yescharacteristic
200C2×C137C2×C4×D411yescharacteristic
201C2×C137#52011yescharacteristic
202C2×C137#54611yescharacteristic
203C2×C137#54811yescharacteristic
204C2×C137#55211yescharacteristic
205C2×C137#55311yescharacteristic
206C137#235211yescharacteristic, complemented, Sylow
207#2352—1371nomaximal, Sylow
208C2×C4×D4—1371no—
209C2×#117—1371no—
210C22×C4⋊C4—1371no—
211#388—1371no—
212#388—1371no—
213#387—1371no—
214#387—1371no—
215#386—1371no—
216#386D13711yescharacteristic
217#382—1371no—
218#380—1371no—
219#380—1371no—
220C4×C4⋊C4—1371no—
221C4×C22⋊C4—1371no—
222C4×C22⋊C4—1371no—
223C22×D4C2×D13711yescharacteristic
224C23×C4—1371no—
225C23×C4—1371no—
226C23×C4—1371no—
227C23×C4C2×D13711yescharacteristic
228C4×D4—1378no—
229#117—2744no—
230C2×C4⋊C4—2744no—
231C2×C4⋊C4—1371no—
232C2×C4⋊C4—1371no—
233C2×C4⋊C4—1371no—
234C2×C4⋊C4—1371no—
235C2×C4⋊C4—1371no—
236C2×C4⋊C4—1371no—
237C2×C4⋊C4—1371no—
238C2×C22⋊C4—1371no—
239C2×C22⋊C4—1371no—
240C2×C22⋊C4—1371no—
241C2×C22⋊C4—1371no—
242C2×C22⋊C4—1371no—
243C2×C22⋊C4C2×D13711yescharacteristic
244C2×C22⋊C4C2×D13711yescharacteristic
245C2×C22⋊C4C2×D13711yescharacteristic
246C2×C42—1371no—
247C2×C42—1371no—
248C2×C42—1371no—
249C2×C42—1371no—
250C2×C42—1371no—
251#95—1371no—
252#95—1371no—
253#95—1371no—
254#95—1371no—
255#95—1371no—
256#95—1371no—
257#95—1371no—
258#95—1371no—
259#95C2×D13711yescharacteristic
260#95C2×D13711yescharacteristic
261C24C22×D13711yescharacteristic
262C24C22×D13711yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C4×D13714yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—2742no—
270C22×C4—2742no—
271C22×C4—2742no—
272C22×C4—2742no—
273C22×C4—2742no—
274C22×C4—2742no—
275C22×C4—5481no—
276C22×C4—1371no—
277C22×C4—1371no—
278C22×C4—1371no—
279C22×C4—1374no—
280C22×C4—1374no—
281C22×C4—1371no—
282C22×C4—1371no—
283C22×C4—1371no—
284C22×C4—1371no—
285C22×C4—1371no—
286C22×C4C22×D13711yescharacteristic
287C22×C4C22×D13711yescharacteristic
288C22×C4C22×D13711yescharacteristic
289C22×C4C22×D13711yescharacteristic
290C22×C4C22×D13711yescharacteristic
291C4⋊C4—2742no—
292C4⋊C4—2742no—
293C4⋊C4—2742no—
294C4⋊C4—2742no—
295C4⋊C4—2742no—
296C4⋊C4—1374no—
297C4⋊C4—1374no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—41no—
301C22⋊C4—2742no—
302C22⋊C4—2742no—
303C22⋊C4—5481no—
304C22⋊C4—1374no—
305C22⋊C4—1374no—
306C42—2742no—
307C42—2742no—
308C42—5481no—
309C42—5481no—
310C42—1374no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×D13712yes—
322C23C2×C4×D13712yes—
323C23#1455550912yes—
324C23#1455550912yes—
325C23C23×D13711yescharacteristic
326D4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—2741no—
346C2×C4—2741no—
347C2×C4—2741no—
348C2×C4—2741no—
349C2×C4—2748no—
350C2×C4—2748no—
351C2×C4—2741no—
352C2×C4—2741no—
353C2×C4—2741no—
354C2×C4—2741no—
355C2×C4—2741no—
356C2×C4—2741no—
357C2×C4—2741no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—5484no—
361C2×C4—1372no—
362C2×C4—1372no—
363C2×C4—1372no—
364C2×C4—1372no—
365C2×C4—1372no—
366C2×C4—1372no—
367C2×C4—1372no—
368C2×C4C2×C4×D13712yes—
369C2×C4D4×D13712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22#1798404414yescomplemented
384C22C22×C4×D13711yescharacteristic
385C22C2×D4×D13711yescharacteristic
386C22#41×D13711yescharacteristic
387C22C2×#1455550911yescharacteristic
388C22C2×#1455550911yescharacteristic
389C22#1798413311yescharacteristic
390C22#1798413511yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—2742no—
397C4—2742no—
398C4—2742no—
399C4—2742no—
400C4—5481no—
401C4—5481no—
402C4—5481no—
403C4—1374no—
404C2—24no—
405C2—42no—
406C2C2×#1798404411yescharacteristic
407C2C4×D4×D13711yescharacteristic
408C2#2257788211yescharacteristic
409C2#2257804011yescharacteristic
410C2#2257805911yescharacteristic
411C2#2257809011yescharacteristic
412C2#2257809611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#386×C137C2
socle, cosocleC23×C137C24
prime radicalC22×D4×C137C22
hypercenter#386D137
hypocenterC137#2352
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#17984044C2semidirect
3. C22×#14555489C2semidirect
4. #22577297C2semidirect
5. #22577291C2semidirect
6. #22577288C2semidirect
7. #22577286C2semidirect
9. #22577267C2semidirect
10. #22577261C2semidirect
11. #22577261C2semidirect
12. #22577254C2semidirect
13. C22⋊C4×Dic137C2semidirect
14. C4×C22⋊Dic137C2semidirect
19. D4×Dic137C4semidirect
62. C22×Dic137C2×C4semidirect
71. C4⋊Dic137C2×C4semidirect
77. #14555489D4semidirect
82. C22⋊Dic137C2×C4semidirect
87. C4×Dic137C2×C4semidirect
176. C22×C137#117semidirect
206. C137#2352semidirect
383. C22#17984044semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1372
C2×#179840442
C22×#145554892
#225772972
#225772912
#225772882
#225772862
#225772832
#225772672
#225772612
#225772542
C22⋊C4×Dic1372
C4×C22⋊Dic1372
#386×C1372
#2352137