Finite Group Databasebeta

← #31783863

Subgroups — #31783863

Order 17536.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984074C211yesmaximal, characteristic
2C2×C4×Dic274C211yesmaximal, characteristic, complemented
3C2×#17984048C211yesmaximal, characteristic
4#22577307C211yesmaximal, characteristic, complemented
5#22577303C211yesmaximal, characteristic
6#22577297C211yesmaximal, characteristic, complemented
7#22577295C211yesmaximal, characteristic, complemented
8#22577293C211yesmaximal, characteristic, complemented
9#22577278C211yesmaximal, characteristic, complemented
10#22577277C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic137C211yesmaximal, characteristic, complemented
12#22577254C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic137C211yesmaximal, characteristic, complemented
14C4×#14555487C211yesmaximal, characteristic, complemented
15#388×C137C211yesmaximal, characteristic, complemented
16C22×Dic274C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984074C418yescomplemented
21C4×Dic274—24no—
22#17984048—24no—
23C2×C4⋊Dic137C2211yescharacteristic
24C2×C4⋊Dic137C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×#14555489C2211yescharacteristic
27C2×#14555489C2211yescharacteristic
28C2×C22⋊Dic137C2211yescharacteristic
29C2×C22⋊Dic137C2211yescharacteristic
30C2×#14555487C2211yescharacteristic
31C2×#14555487C2211yescharacteristic
32C2×#14555487C2211yescharacteristic
33C2×#14555487C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36C2×C4×Dic137C2211yescharacteristic
37C2×C4×Dic137C2211yescharacteristic
38C2×C4×Dic137C2211yescharacteristic
39C2×C4×Dic137C2211yescharacteristic
40C2×C4×Dic137C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47C2×C4⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C42×C137C2211yescharacteristic
52#95×C137C2211yescharacteristic
53#95×C137C2211yescharacteristic
54C23×D137C2311yescharacteristic
55C2×Dic274—22no—
56C2×Dic274—22no—
57C2×Dic274C2×C414yescomplemented
58C2×C22⋊D137—22no—
59C2×C22⋊D137—22no—
60C2×C22⋊D137C2×C414yes—
61C22×Dic137—22no—
62C22×Dic137—22no—
63C22×Dic137—41no—
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C2×C4×D137—22no—
72C2×C4×D137—22no—
73C2×C4×D137—41no—
74C4⋊Dic137—22no—
75C4⋊Dic137—22no—
76#14555489—22no—
77#14555489—22no—
78#14555489D414yes—
79C22⋊Dic137—41no—
80C22⋊Dic137C2×C414yescomplemented
81#14555487—22no—
82#14555487—22no—
83#14555487—41no—
84#14555487C2×C414yescomplemented
85#14555487C2×C414yescomplemented
86C4×Dic137—22no—
87C4×Dic137—22no—
88C4×Dic137—22no—
89C4×Dic137—41no—
90C4×Dic137—41no—
91C4×Dic137C2×C414yescomplemented
92C24×C137C2311yescharacteristic
93C22×C4×C137C2311yescharacteristic
94C22×C4×C137C2311yescharacteristic
95C22×C4×C137C2311yescharacteristic
96C22×C4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C4⋊C4×C137—22no—
100C22⋊C4×C137—22no—
101C22⋊C4×C137—22no—
102C22⋊C4×C137C2×C414yescomplemented
103C42×C137—22no—
104C22×D137—21no—
105C22×D137—21no—
106C22×D137—21no—
107C22×D137—21no—
108C22×D137—21no—
109C22×D137—21no—
110C22×D137C22×C412yes—
111Dic274—28no—
112C22⋊D137—28no—
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—44no—
128C2×Dic137C22×C412yes—
129C2×Dic137C22×C412yes—
130C2×Dic137C22×C412yes—
131C2×Dic137C2×D412yes—
132C2×Dic137C2×D412yes—
133C2×Dic137#4112yes—
134C2×Dic137#4112yes—
135C4×D137—44no—
136C23×C137—21no—
137C23×C137—21no—
138C23×C137—21no—
139C23×C137—21no—
140C23×C137—21no—
141C23×C137—21no—
142C23×C137C22×C412yes—
143C23×C137C2411yescharacteristic
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
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×C137C22×C412yes—
159C2×C4×C137C22×C412yes—
160C2×C4×C137C2×D412yes—
161C2×C4×C137#4112yes—
162C22×C137—22no—
163C22×C137—22no—
164C22×C137—22no—
165C22×C137—41no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137C23×C411yescharacteristic
170C22×C137C22×D411yescharacteristic
171C22×C137C2×#4111yescharacteristic
172C22×C137C2×#4111yescharacteristic
173C22×C137C2×#4111yescharacteristic
174C22×C137H(5,𝔽2)11yescharacteristic
175C22×C1372−1+411yescharacteristic
176C2×D137—22no—
177C2×D137—22no—
178C2×D137—22no—
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—41no—
183C4×C137—22no—
184C4×C137—22no—
185C4×C137—22no—
186C4×C137—22no—
187C4×C137—41no—
188C4×C137—41no—
189Dic137—22no—
190Dic137—22no—
191Dic137—22no—
192Dic137—22no—
193Dic137—41no—
194Dic137—41no—
195Dic137C4×D414yes—
196C2×C137—42no—
197C2×C137C2×C4×D411yescharacteristic
198C2×C137C4×#4111yescharacteristic
199C2×C137#52011yescharacteristic
200C2×C137#54611yescharacteristic
201C2×C137#54811yescharacteristic
202C2×C137#55611yescharacteristic
203C2×C137#55911yescharacteristic
204D137—42no—
205C137#236711yescharacteristic, complemented, Sylow
206#2367—1371nomaximal, Sylow
207C2×#124—1371no—
208C2×C4×Q8—1371no—
209C2×C4×D4—1371no—
210#391—1371no—
211#390—1371no—
212#388—1371no—
213#388—1371no—
214#388—1371no—
215#388D13711yescharacteristic, complemented
216#385—1371no—
217#385—1371no—
218C4×C4⋊C4—1371no—
219C4×C4⋊C4—1371no—
220C4×C22⋊C4—1371no—
221C4×C22⋊C4—1371no—
222C22×Q8—1371no—
223C22×D4—1371no—
224C23×C4—1371no—
225C23×C4—1371no—
226#124—1378no—
227C4×Q8—2744no—
228C4×D4—2744no—
229C2×C4⋊C4—1371no—
230C2×C4⋊C4—1371no—
231C2×C4⋊C4—1371no—
232C2×C4⋊C4—1371no—
233C2×C4⋊C4—1371no—
234C2×C4⋊C4C2×D13711yescharacteristic
235C2×C22⋊C4—1371no—
236C2×C22⋊C4—1371no—
237C2×C22⋊C4—1371no—
238C2×C22⋊C4—1371no—
239C2×C22⋊C4—1371no—
240C2×C22⋊C4—1371no—
241C2×C22⋊C4—1371no—
242C2×C22⋊C4C2×D13711yescharacteristic
243C2×C22⋊C4C2×D13711yescharacteristic
244C2×C22⋊C4C2×D13711yescharacteristic
245C2×C42—1371no—
246C2×C42—1371no—
247C2×C42—1371no—
248C2×C42—1371no—
249C2×C42—1371no—
250C2×C42—1371no—
251C2×C42C2×D13711yescharacteristic
252#95—1371no—
253#95—1371no—
254#95—1371no—
255#95—1371no—
256#95—1371no—
257#95—1371no—
258#95C2×D13711yescharacteristic
259#95C2×D13711yescharacteristic
260C24—1371no—
261C24C22×D13711yescharacteristic
262C2×Q8—2742no—
263C2×Q8—2742no—
264C2×Q8—1374no—
265C2×D4—2742no—
266C2×D4—2742no—
267C2×D4—1374no—
268C22×C4—2742no—
269C22×C4—2742no—
270C22×C4—2742no—
271C22×C4—2742no—
272C22×C4—5481no—
273C22×C4—5481no—
274C22×C4—1371no—
275C22×C4—1371no—
276C22×C4—1371no—
277C22×C4—1371no—
278C22×C4—1371no—
279C22×C4—1371no—
280C22×C4—1371no—
281C22×C4C22×D13711yescharacteristic
282C22×C4C22×D13711yescharacteristic
283C22×C4C22×D13711yescharacteristic
284C22×C4C22×D13711yescharacteristic
285C22×C4C22×D13711yescharacteristic
286C22×C4C22×D13711yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—2742no—
289C4⋊C4—2742no—
290C4⋊C4—2742no—
291C4⋊C4—2742no—
292C4⋊C4—1374no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—2742no—
296C22⋊C4—2742no—
297C22⋊C4—5481no—
298C22⋊C4—5481no—
299C22⋊C4—1374no—
300C22⋊C4—1374no—
301C22⋊C4—1374no—
302C22⋊C4C4×D13714yescomplemented
303C42—22no—
304C42—2742no—
305C42—2742no—
306C42—2742no—
307C42—5481no—
308C42—5481no—
309C42—1374no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—2741no—
317C23—2741no—
318C23—2741no—
319C23—2741no—
320C23—2741no—
321C23—2741no—
322C23—1372no—
323C23C2×C4×D13712yes—
324C23C23×D13711yescharacteristic
325Q8—2748no—
326D4—2748no—
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—21no—
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—2741no—
350C2×C4—2741no—
351C2×C4—2741no—
352C2×C4—2741no—
353C2×C4—2741no—
354C2×C4—2741no—
355C2×C4—5484no—
356C2×C4—5484no—
357C2×C4—1372no—
358C2×C4—1372no—
359C2×C4—1372no—
360C2×C4—1372no—
361C2×C4—1372no—
362C2×C4—1372no—
363C2×C4—1372no—
364C2×C4C2×C4×D13712yes—
365C2×C4C2×C4×D13712yes—
366C2×C4D4×D13712yes—
367C2×C4#1455550912yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—2742no—
376C22—2742no—
377C22—2742no—
378C22—5481no—
379C22—5481no—
380C22—5481no—
381C22—5481no—
382C22C22×C4×D13711yescharacteristic
383C22C2×D4×D13711yescharacteristic
384C22#41×D13711yescharacteristic
385C22#41×D13711yescharacteristic
386C22C2×#1455550911yescharacteristic
387C22#1798413411yescharacteristic
388C22#1798413511yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—2742no—
396C4—2742no—
397C4—2742no—
398C4—2742no—
399C4—5481no—
400C4—5481no—
401C4—1374no—
402C2—42no—
403C2—5482no—
404C2C4×D4×D13711yescharacteristic
405C2C4×#1455550911yescharacteristic
406C2#2257788311yescharacteristic
407C2#2257804511yescharacteristic
408C2#2257805911yescharacteristic
409C2#2257812011yescharacteristic
410C2#2257813511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#388×C137C2
socle, cosocleC23×C137C24
prime radicalC22×C22⋊D137C22
hypercenter#388D137
hypocenterC137#2367
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic274C2semidirect
4. #22577307C2semidirect
6. #22577297C2semidirect
7. #22577295C2semidirect
8. #22577293C2semidirect
9. #22577278C2semidirect
10. #22577277C2semidirect
11. C4⋊C4×Dic137C2semidirect
12. #22577254C2semidirect
13. C22⋊C4×Dic137C2semidirect
14. C4×#14555487C2semidirect
15. #388×C137C2semidirect
20. #17984074C4semidirect
57. C2×Dic274C2×C4semidirect
80. C22⋊Dic137C2×C4semidirect
84. #14555487C2×C4semidirect
85. #14555487C2×C4semidirect
91. C4×Dic137C2×C4semidirect
102. C22⋊C4×C137C2×C4semidirect
205. C137#2367semidirect
215. #388D137semidirect
302. C22⋊C4C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840742
C2×C4×Dic2742
C2×#179840482
#225773072
#225773032
#225772972
#225772952
#225772932
#225772782
#225772772
C4⋊C4×Dic1372
#225772542
C22⋊C4×Dic1372
C4×#145554872
#388×C1372
#2367137