Finite Group Databasebeta

← #31783475

Subgroups — #31783475

Order 17536.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984060C211yesmaximal, characteristic
2C2×#17984048C211yesmaximal, characteristic, complemented
3C2×#17984044C211yesmaximal, characteristic, complemented
4#22577303C211yesmaximal, characteristic, complemented
5#22577303C211yesmaximal, characteristic, complemented
6#22577298C211yesmaximal, characteristic, complemented
7#22577297C211yesmaximal, characteristic, complemented
8#22577294C211yesmaximal, characteristic, complemented
9#22577283C211yesmaximal, characteristic, complemented
10#22577279C211yesmaximal, characteristic, complemented
11#22577275C211yesmaximal, characteristic, complemented
12#22577274C211yesmaximal, characteristic, complemented
13#22577261C211yesmaximal, characteristic, complemented
14C4×#14555489C211yesmaximal, characteristic, complemented
15#390×C137C211yesmaximal, characteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C23×Dic137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21#17984060C418yescomplemented
22#17984048—24no—
23#17984044—24no—
24C2×#14555489C2211yescharacteristic
25C2×#14555489C2211yescharacteristic
26C2×#14555489C2211yescharacteristic
27C2×#14555489C2211yescharacteristic
28C2×#14555489C2211yescharacteristic
29C2×#14555489C2211yescharacteristic
30C2×C22⋊Dic137C2211yescharacteristic
31C2×C22⋊Dic137C2211yescharacteristic
32C2×C22⋊Dic137C2211yescharacteristic
33C2×C22⋊Dic137C2211yescharacteristic
34C2×#14555487C2211yescharacteristic
35C2×#14555487C2211yescharacteristic
36C2×#14555487C2211yescharacteristic
37C2×#14555487C2211yescharacteristic
38C2×C4×Dic137C2211yescharacteristic
39C2×C4×Dic137C2211yescharacteristic
40C2×C4×Dic137C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45#17983995C2211yescharacteristic
46#17983995C2211yescharacteristic
47C22×D4×C137C2211yescharacteristic
48C2×C4⋊C4×C137C2211yescharacteristic
49C2×C22⋊C4×C137C2211yescharacteristic
50C2×C22⋊C4×C137C2211yescharacteristic
51C2×C22⋊C4×C137C2211yescharacteristic
52C2×C22⋊C4×C137C2211yescharacteristic
53C2×C42×C137C2211yescharacteristic
54C23×D137C2311yescharacteristic
55C2×C22⋊D137—22no—
56C2×C22⋊D137—22no—
57C2×C22⋊D137—41no—
58C2×C22⋊D137—41no—
59C2×C22⋊D137C2×C414yes—
60C2×C22⋊D137C2×C414yescomplemented
61C22×Dic137—22no—
62C22×Dic137—22no—
63C22×Dic137—22no—
64C22×Dic137—41no—
65C22×Dic137—41no—
66C22×Dic137C2×C414yescomplemented
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C22×Dic137C2311yescharacteristic
70C22×Dic137C2311yescharacteristic
71C22×Dic137C2311yescharacteristic
72C22×Dic137C2311yescharacteristic
73C22×Dic137C2311yescharacteristic
74C2×C4×D137—22no—
75C2×C4×D137—41no—
76C2×C4×D137—41no—
77#14555489—22no—
78#14555489—22no—
79#14555489—22no—
80#14555489—22no—
81#14555489—41no—
82#14555489C2×C414yescomplemented
83C22⋊Dic137—22no—
84C22⋊Dic137—41no—
85C22⋊Dic137—41no—
86C22⋊Dic137C2×C414yescomplemented
87#14555487—22no—
88#14555487—41no—
89#14555487—41no—
90#14555487C2×C414yescomplemented
91C4×Dic137—22no—
92C4×Dic137—41no—
93C4×Dic137D414yescomplemented
94C24×C137C2311yescharacteristic
95C24×C137C2311yescharacteristic
96C2×D4×C137—41no—
97C2×D4×C137—41no—
98C2×D4×C137C2×C414yescomplemented
99C22×C4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101C22×C4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104C4⋊C4×C137—41no—
105C22⋊C4×C137—22no—
106C22⋊C4×C137—22no—
107C22⋊C4×C137—41no—
108C22⋊C4×C137—41no—
109C42×C137—41no—
110C22×D137—21no—
111C22×D137—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—21no—
115C22×D137—21no—
116C22×D137C22×C412yes—
117C22⋊D137—28no—
118C22⋊D137—44no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—28no—
126C2×Dic137—21no—
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—44no—
136C2×Dic137C22×C412yes—
137C2×Dic137C22×C412yes—
138C2×Dic137C22×C412yes—
139C2×Dic137C2×D412yes—
140C2×Dic137C2×D412yes—
141C2×Dic137#4112yes—
142C4×D137—44no—
143C23×C137—21no—
144C23×C137—21no—
145C23×C137—21no—
146C23×C137—21no—
147C23×C137—21no—
148C23×C137—21no—
149C23×C137—21no—
150C23×C137—21no—
151C23×C137—21no—
152C23×C137—21no—
153C23×C137—21no—
154C23×C137—21no—
155C23×C137C22×C412yes—
156C23×C137C22×C412yes—
157C23×C137C2411yescharacteristic
158D4×C137—44no—
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—21no—
172C2×C4×C137C22×C412yes—
173C2×C4×C137C2×D412yes—
174C22×C137—22no—
175C22×C137—22no—
176C22×C137—22no—
177C22×C137—22no—
178C22×C137—22no—
179C22×C137—22no—
180C22×C137—41no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137C23×C411yescharacteristic
189C22×C137C22×D411yescharacteristic
190C22×C137C2×#4111yescharacteristic
191C22×C137H(5,𝔽2)11yescharacteristic
192C22×C137H(5,𝔽2)11yescharacteristic
193C22×C137H(5,𝔽2)11yescharacteristic
194C22×C1372−1+411yescharacteristic
195C2×D137—22no—
196C2×D137—22no—
197C2×D137—22no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C4×C137—22no—
203C4×C137—22no—
204C4×C137—41no—
205C4×C137—41no—
206C4×C137—41no—
207Dic137—22no—
208Dic137—22no—
209Dic137—22no—
210Dic137—41no—
211Dic137—41no—
212Dic137—41no—
213Dic137C4×D414yes—
214C2×C137—42no—
215C2×C137—42no—
216C2×C137C2×C4×D411yescharacteristic
217C2×C137#51811yescharacteristic
218C2×C137#52011yescharacteristic
219C2×C137#53511yescharacteristic
220C2×C137#53711yescharacteristic
221C2×C137#54011yescharacteristic
222C2×C137#54211yescharacteristic
223D137—42no—
224C137#232411yescharacteristic, complemented, Sylow
225#2324—1371nomaximal, Sylow
226C2×#121—1371no—
227C2×C4×D4—1371no—
228C2×#117—1371no—
229#390—1371no—
230#390—1371no—
231#390D13711yescharacteristic, complemented
232#389—1371no—
233#388—1371no—
234#388—1371no—
235#386—1371no—
236#386—1371no—
237#385—1371no—
238#385—1371no—
239#380—1371no—
240C4×C4⋊C4—1371no—
241C22×D4—1371no—
242C22×D4—1371no—
243C22×D4C2×D13711yescharacteristic
244C23×C4—1371no—
245C23×C4—1371no—
246C23×C4—1371no—
247#121—1378no—
248C4×D4—2744no—
249#117—2744no—
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⋊C4C2×D13711yescharacteristic
257C2×C22⋊C4—1371no—
258C2×C22⋊C4—1371no—
259C2×C22⋊C4—1371no—
260C2×C22⋊C4—1371no—
261C2×C22⋊C4—1371no—
262C2×C22⋊C4—1371no—
263C2×C22⋊C4—1371no—
264C2×C22⋊C4—1371no—
265C2×C22⋊C4C2×D13711yescharacteristic
266C2×C22⋊C4C2×D13711yescharacteristic
267C2×C22⋊C4C2×D13711yescharacteristic
268C2×C22⋊C4C2×D13711yescharacteristic
269C2×C42—1371no—
270C2×C42—1371no—
271C2×C42—1371no—
272C2×C42C2×D13711yescharacteristic
273#95—1371no—
274#95—1371no—
275#95—1371no—
276#95—1371no—
277#95—1371no—
278#95—1371no—
279C24—1371no—
280C24C22×D13711yescharacteristic
281C24C22×D13711yescharacteristic
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—2742no—
285C2×D4—2742no—
286C2×D4—5481no—
287C2×D4—5481no—
288C2×D4—1374no—
289C2×D4—1374no—
290C2×D4C4×D13714yescomplemented
291C22×C4—2742no—
292C22×C4—2742no—
293C22×C4—2742no—
294C22×C4—2742no—
295C22×C4—5481no—
296C22×C4—5481no—
297C22×C4—5481no—
298C22×C4—5481no—
299C22×C4—1371no—
300C22×C4—1371no—
301C22×C4—1371no—
302C22×C4—1374no—
303C22×C4—1371no—
304C22×C4—1371no—
305C22×C4—1371no—
306C22×C4—1371no—
307C22×C4C22×D13711yescharacteristic
308C22×C4C22×D13711yescharacteristic
309C22×C4C22×D13711yescharacteristic
310C22×C4C22×D13711yescharacteristic
311C22×C4C22×D13711yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—2742no—
314C4⋊C4—2742no—
315C4⋊C4—2742no—
316C4⋊C4—2742no—
317C4⋊C4—5481no—
318C4⋊C4—1374no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—2742no—
324C22⋊C4—2742no—
325C22⋊C4—5481no—
326C22⋊C4—5481no—
327C22⋊C4—5481no—
328C22⋊C4—5481no—
329C22⋊C4—1374no—
330C22⋊C4—1374no—
331C42—41no—
332C42—2742no—
333C42—5481no—
334C42—1374no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—2741no—
348C23—2741no—
349C23—2741no—
350C23—2741no—
351C23—2741no—
352C23—2741no—
353C23—1372no—
354C23C2×C4×D13712yes—
355C23C2×C4×D13712yes—
356C23C23×D13711yescharacteristic
357D4—44no—
358D4—2748no—
359D4—5484no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—2741no—
374C2×C4—2741no—
375C2×C4—2741no—
376C2×C4—2741no—
377C2×C4—2741no—
378C2×C4—2741no—
379C2×C4—2748no—
380C2×C4—2741no—
381C2×C4—2741no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—2741no—
386C2×C4—2741no—
387C2×C4—2741no—
388C2×C4—2741no—
389C2×C4—5484no—
390C2×C4—5484no—
391C2×C4—1372no—
392C2×C4—1372no—
393C2×C4—1372no—
394C2×C4—1372no—
395C2×C4—1372no—
396C2×C4—1372no—
397C2×C4C2×C4×D13712yes—
398C2×C4D4×D13712yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2742no—
414C22—2742no—
415C22—2742no—
416C22—5481no—
417C22—5481no—
418C22—5481no—
419C22—5481no—
420C22C22×C4×D13711yescharacteristic
421C22C2×D4×D13711yescharacteristic
422C22#41×D13711yescharacteristic
423C22#1798413311yescharacteristic
424C22#1798413311yescharacteristic
425C22#1798413311yescharacteristic
426C22#1798413511yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—2742no—
433C4—2742no—
434C4—2742no—
435C4—5481no—
436C4—5481no—
437C4—5481no—
438C4—1374no—
439C2—42no—
440C2—42no—
441C2—5482no—
442C2C4×D4×D13711yescharacteristic
443C2#2257787611yescharacteristic
444C2#2257788211yescharacteristic
445C2#2257796611yescharacteristic
446C2#2257798011yescharacteristic
447C2#2257800311yescharacteristic
448C2#2257801711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#390×C137C2
socle, cosocleC23×C137C24
prime radicalC2×#17984060C2
hypercenter#390D137
hypocenterC137#2324
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#17984048C2semidirect
3. C2×#17984044C2semidirect
4. #22577303C2semidirect
5. #22577303C2semidirect
6. #22577298C2semidirect
7. #22577297C2semidirect
8. #22577294C2semidirect
9. #22577283C2semidirect
10. #22577279C2semidirect
11. #22577275C2semidirect
12. #22577274C2semidirect
13. #22577261C2semidirect
14. C4×#14555489C2semidirect
15. #390×C137C2semidirect
21. #17984060C4semidirect
60. C2×C22⋊D137C2×C4semidirect
66. C22×Dic137C2×C4semidirect
82. #14555489C2×C4semidirect
86. C22⋊Dic137C2×C4semidirect
90. #14555487C2×C4semidirect
93. C4×Dic137D4semidirect
98. C2×D4×C137C2×C4semidirect
224. C137#2324semidirect
231. #390D137semidirect
290. 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
C2×#179840482
C2×#179840442
#225773032
#225772982
#225772972
#225772942
#225772832
#225772792
#225772752
#225772742
#225772612
C4×#145554892
#390×C1372
#2324137