Finite Group Databasebeta

← #31783687

Subgroups — #31783687

Order 17536.

776 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×C4×C22⋊D137C211yesmaximal, characteristic
2C2×#17984048C211yesmaximal, characteristic
3C2×#17984048C211yesmaximal, characteristic
4C22×#14555489C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D137C211yesmaximal, characteristic, complemented
6#22577300C211yesmaximal, characteristic, complemented
7#22577298C211yesmaximal, characteristic, complemented
8#22577278C211yesmaximal, characteristic, complemented
9#22577277C211yesmaximal, characteristic, complemented
10#22577274C211yesmaximal, characteristic, complemented
11#22577272C211yesmaximal, characteristic, complemented
12#22577262C211yesmaximal, characteristic, complemented
13#22577261C211yesmaximal, characteristic, complemented
14#22577254C211yesmaximal, characteristic, complemented
15#385×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C23×Dic137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21C22×C4×D137C2211yescharacteristic
22C4×C22⋊D137—24no—
23#17984048—24no—
24#17984048C418yescomplemented
25C2×C4⋊Dic137C2211yescharacteristic
26C2×C4⋊Dic137C2211yescharacteristic
27C2×#14555489—24no—
28C2×#14555489C2211yescharacteristic
29C2×#14555489C2211yescharacteristic
30C2×#14555489C2211yescharacteristic
31C2×#14555489C2211yescharacteristic
32C2×#14555489C2211yescharacteristic
33C2×#14555489C2211yescharacteristic
34C2×#14555489C2211yescharacteristic
35C2×#14555489C2211yescharacteristic
36C2×#14555489C2211yescharacteristic
37C4⋊C4×D137—24no—
38C2×C22⋊Dic137C2211yescharacteristic
39C2×#14555487C2211yescharacteristic
40C2×#14555487C2211yescharacteristic
41C2×#14555487C2211yescharacteristic
42C2×C4×Dic137C2211yescharacteristic
43C2×C4×Dic137C2211yescharacteristic
44C2×C4×Dic137C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47#17983995C2211yescharacteristic
48#17983995C2211yescharacteristic
49C23×C4×C137C2211yescharacteristic
50C2×C4⋊C4×C137C2211yescharacteristic
51C2×C4⋊C4×C137C2211yescharacteristic
52C2×C22⋊C4×C137C2211yescharacteristic
53C2×C22⋊C4×C137C2211yescharacteristic
54#95×C137C2211yescharacteristic
55#95×C137C2211yescharacteristic
56C23×D137C2311yescharacteristic
57C2×C22⋊D137C2×C414yes—
58C2×C22⋊D137D414yes—
59C2×C22⋊D137Q814yes—
60C22×Dic137—22no—
61C22×Dic137—22no—
62C22×Dic137—22no—
63C22×Dic137—22no—
64C22×Dic137—41no—
65C22×Dic137C2×C414yescomplemented
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C22×Dic137C2311yescharacteristic
72C22×Dic137C2311yescharacteristic
73C2×C4×D137—22no—
74C2×C4×D137—22no—
75C2×C4×D137—22no—
76C2×C4×D137—22no—
77C2×C4×D137—22no—
78C2×C4×D137—22no—
79C2×C4×D137—41no—
80C2×C4×D137—41no—
81C2×C4×D137C2×C414yescomplemented
82C4⋊Dic137—22no—
83C4⋊Dic137—41no—
84#14555489—22no—
85#14555489—22no—
86#14555489—22no—
87#14555489—22no—
88#14555489—22no—
89#14555489—22no—
90#14555489—22no—
91#14555489C2×C414yescomplemented
92#14555489D414yescomplemented
93C22⋊Dic137—22no—
94#14555487—22no—
95#14555487—22no—
96#14555487C2×C414yescomplemented
97C4×Dic137—22no—
98C4×Dic137—22no—
99C4×Dic137C2×C414yescomplemented
100C24×C137C2311yescharacteristic
101C22×C4×C137—22no—
102C22×C4×C137—22no—
103C22×C4×C137—41no—
104C22×C4×C137C2311yescharacteristic
105C22×C4×C137C2311yescharacteristic
106C22×C4×C137C2311yescharacteristic
107C22×C4×C137C2311yescharacteristic
108C22×C4×C137C2311yescharacteristic
109C22×C4×C137C2311yescharacteristic
110C4⋊C4×C137—22no—
111C4⋊C4×C137—41no—
112C22⋊C4×C137—22no—
113C22⋊C4×C137C2×C414yescomplemented
114C22×D137—21no—
115C22×D137—21no—
116C22×D137—21no—
117C22×D137—21no—
118C22×D137C22×C412yes—
119C22×D137C2×D412yes—
120C22×D137C2×Q812yes—
121C22⋊D137C4⋊C4116yescomplemented
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—28no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
131C2×Dic137—21no—
132C2×Dic137—21no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—44no—
137C2×Dic137C22×C412yes—
138C2×Dic137C22×C412yes—
139C2×Dic137C22×C412yes—
140C2×Dic137C2×D412yes—
141C2×Dic137C2×D412yes—
142C2×Dic137C2×D412yes—
143C2×Dic137C2×Q812yes—
144C2×Dic137#4112yes—
145C4×D137—28no—
146C4×D137—44no—
147C4×D137—44no—
148C23×C137—21no—
149C23×C137—21no—
150C23×C137—21no—
151C23×C137—21no—
152C23×C137C22×C412yes—
153C23×C137C2×D412yes—
154C23×C137C2×Q812yes—
155C23×C137C2411yescharacteristic
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—21no—
167C2×C4×C137—21no—
168C2×C4×C137—21no—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137—44no—
172C2×C4×C137C22×C412yes—
173C2×C4×C137C22×C412yes—
174C2×C4×C137C2×D412yes—
175C22×C137—22no—
176C22×C137—22no—
177C22×C137—22no—
178C22×C137—22no—
179C22×C137—22no—
180C22×C137—22no—
181C22×C137C2×C4⋊C414yescomplemented
182C22×C137C23×C411yescharacteristic
183C22×C137C22×D411yescharacteristic
184C22×C137C22×D411yescharacteristic
185C22×C137C22×Q811yescharacteristic
186C22×C137C2×#4111yescharacteristic
187C22×C137H(5,𝔽2)11yescharacteristic
188C22×C1372−1+411yescharacteristic
189C2×D137—22no—
190C2×D137—22no—
191C2×D137—22no—
192C2×D137—22no—
193C2×D137—22no—
194C2×D137—22no—
195C2×D137C2×C4⋊C414yescomplemented
196C4×C137—22no—
197C4×C137—22no—
198C4×C137—22no—
199C4×C137—41no—
200C4×C137—41no—
201C4×C137—41no—
202Dic137—22no—
203Dic137—22no—
204Dic137—41no—
205Dic137—41no—
206Dic137—41no—
207Dic137C2×C4⋊C414yescomplemented
208Dic137C4×D414yescomplemented
209C2×C137—24no—
210C2×C137C22×C4⋊C411yescharacteristic
211C2×C137C2×C4×D411yescharacteristic
212C2×C137#52011yescharacteristic
213C2×C137D4211yescharacteristic
214C2×C137#54711yescharacteristic
215C2×C137D4×Q811yescharacteristic
216C2×C137#55411yescharacteristic
217D137—24no—
218C137D4×C4⋊C411yescharacteristic, complemented, Sylow
219D4×C4⋊C4—1371nomaximal, Sylow
220C2×C4×D4—1371no—
221C2×C4×D4—1371no—
222C2×C4×D4—1371no—
223C22×C4⋊C4—1371no—
224C22×C4⋊C4—1371no—
225#389—1371no—
226#389—1371no—
227#385—1371no—
228#385—1371no—
229#385—1371no—
230#385D13711yescharacteristic, complemented
231#384—1371no—
232#380—1371no—
233#380—1371no—
234C4×C4⋊C4—1371no—
235C22×D4—1371no—
236C23×C4—1371no—
237C23×C4—1371no—
238C23×C4—1371no—
239C23×C4—1371no—
240C23×C4—1371no—
241C23×C4C2×D13711yescharacteristic
242C4×D4—2744no—
243C4×D4—2744no—
244C4×D4—1378no—
245C2×C4⋊C4—2744no—
246C2×C4⋊C4—2744no—
247C2×C4⋊C4—1371no—
248C2×C4⋊C4—1371no—
249C2×C4⋊C4—1371no—
250C2×C4⋊C4—1371no—
251C2×C4⋊C4—1371no—
252C2×C4⋊C4—1371no—
253C2×C4⋊C4—1371no—
254C2×C4⋊C4—1371no—
255C2×C4⋊C4—1371no—
256C2×C4⋊C4—1371no—
257C2×C4⋊C4—1371no—
258C2×C4⋊C4C2×D13711yescharacteristic
259C2×C4⋊C4C2×D13711yescharacteristic
260C2×C22⋊C4—1371no—
261C2×C22⋊C4—1371no—
262C2×C22⋊C4—1371no—
263C2×C22⋊C4—1371no—
264C2×C22⋊C4C2×D13711yescharacteristic
265C2×C22⋊C4C2×D13711yescharacteristic
266C2×C42—1371no—
267C2×C42—1371no—
268C2×C42—1371no—
269#95—1371no—
270#95—1371no—
271#95—1371no—
272#95—1371no—
273#95C2×D13711yescharacteristic
274#95C2×D13711yescharacteristic
275C24—1371no—
276C24C22×D13711yescharacteristic
277C2×D4—1374no—
278C2×D4—1374no—
279C2×D4—1374no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—2742no—
284C22×C4—2742no—
285C22×C4—2742no—
286C22×C4—2742no—
287C22×C4—2742no—
288C22×C4—2742no—
289C22×C4—2742no—
290C22×C4—2742no—
291C22×C4—2742no—
292C22×C4—2742no—
293C22×C4—5481no—
294C22×C4—5481no—
295C22×C4—5481no—
296C22×C4—1371no—
297C22×C4—1371no—
298C22×C4—1371no—
299C22×C4—1374no—
300C22×C4—1371no—
301C22×C4—1371no—
302C22×C4—1374no—
303C22×C4—1371no—
304C22×C4—1371no—
305C22×C4C22×D13711yescharacteristic
306C22×C4C22×D13711yescharacteristic
307C22×C4C22×D13711yescharacteristic
308C22×C4C22×D13711yescharacteristic
309C22×C4C22×D13711yescharacteristic
310C22×C4C22×D13711yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—2742no—
314C4⋊C4—2742no—
315C4⋊C4—2742no—
316C4⋊C4—2742no—
317C4⋊C4—2742no—
318C4⋊C4—2742no—
319C4⋊C4—2742no—
320C4⋊C4—2742no—
321C4⋊C4—5481no—
322C4⋊C4—1374no—
323C4⋊C4—1374no—
324C22⋊C4—22no—
325C22⋊C4—2742no—
326C22⋊C4—2742no—
327C22⋊C4—2742no—
328C22⋊C4—1374no—
329C22⋊C4C4×D13714yescomplemented
330C42—2742no—
331C42—2742no—
332C42—1374no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—2741no—
338C23—2741no—
339C23—2741no—
340C23—2741no—
341C23—1372no—
342C23—1372no—
343C23—1372no—
344C23C2×C4×D13712yes—
345C23D4×D13712yes—
346C23Q8×D13712yes—
347C23C23×D13711yescharacteristic
348D4—13716no—
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—21no—
363C2×C4—21no—
364C2×C4—44no—
365C2×C4—2748no—
366C2×C4—2741no—
367C2×C4—2741no—
368C2×C4—2741no—
369C2×C4—2741no—
370C2×C4—2748no—
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—5484no—
381C2×C4—5484no—
382C2×C4—5484no—
383C2×C4—1372no—
384C2×C4—1372no—
385C2×C4—1372no—
386C2×C4—1372no—
387C2×C4—1372no—
388C2×C4—1372no—
389C2×C4—1372no—
390C2×C4—1372no—
391C2×C4C2×C4×D13712yes—
392C2×C4C2×C4×D13712yes—
393C2×C4D4×D13712yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—2742no—
401C22—2742no—
402C22—2742no—
403C22—2742no—
404C22—2742no—
405C22—2742no—
406C22—1374no—
407C22C4⋊C4×D13714yescomplemented
408C22C22×C4×D13711yescharacteristic
409C22C2×D4×D13711yescharacteristic
410C22C2×D4×D13711yescharacteristic
411C22C2×Q8×D13711yescharacteristic
412C22#41×D13711yescharacteristic
413C22#1798413311yescharacteristic
414C22#1798413511yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—2742no—
422C4—2742no—
423C4—5481no—
424C4—5481no—
425C4—5481no—
426C4—1374no—
427C4—1374no—
428C2—24no—
429C2—2744no—
430C2C2×C4⋊C4×D13711yescharacteristic
431C2C4×D4×D13711yescharacteristic
432C2#2257788211yescharacteristic
433C2#2257803511yescharacteristic
434C2#2257805111yescharacteristic
435C2#2257806611yescharacteristic
436C2#2257810711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#385×C137C2
socle, cosocleC23×C137C24
prime radicalC22×C22⋊D137C22
hypercenter#385D137
hypocenterC137D4×C4⋊C4
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. C22×#14555489C2semidirect
5. C2×C4⋊C4×D137C2semidirect
6. #22577300C2semidirect
7. #22577298C2semidirect
8. #22577278C2semidirect
9. #22577277C2semidirect
10. #22577274C2semidirect
11. #22577272C2semidirect
12. #22577262C2semidirect
13. #22577261C2semidirect
14. #22577254C2semidirect
15. #385×C137C2semidirect
24. #17984048C4semidirect
65. C22×Dic137C2×C4semidirect
81. C2×C4×D137C2×C4semidirect
91. #14555489C2×C4semidirect
92. #14555489D4semidirect
96. #14555487C2×C4semidirect
99. C4×Dic137C2×C4semidirect
113. C22⋊C4×C137C2×C4semidirect
121. C22⋊D137C4⋊C4semidirect
181. C22×C137C2×C4⋊C4semidirect
195. C2×D137C2×C4⋊C4semidirect
207. Dic137C2×C4⋊C4semidirect
208. Dic137C4×D4semidirect
218. C137D4×C4⋊C4semidirect
230. #385D137semidirect
329. C22⋊C4C4×D137semidirect
407. C22C4⋊C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1372
C2×#179840482
C22×#145554892
C2×C4⋊C4×D1372
#225773002
#225772982
#225772782
#225772772
#225772742
#225772722
#225772622
#225772612
#225772542
#385×C1372
D4×C4⋊C4137