Finite Group Databasebeta

← #31783896

Subgroups — #31783896

Order 17536.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984074C211yesmaximal, characteristic
2#22577308C211yesmaximal, characteristic, complemented
3#22577307C211yesmaximal, characteristic, complemented
4#22577297C211yesmaximal, characteristic, complemented
5#22577293C211yesmaximal, characteristic, complemented
6#22577292C211yesmaximal, characteristic, complemented
7#22577280C211yesmaximal, characteristic
8#22577279C211yesmaximal, characteristic
9C22⋊C4×Dic137C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic137C211yesmaximal, characteristic, complemented
11C4×#14555487C211yesmaximal, characteristic, complemented
12#22577250C211yesmaximal, characteristic, complemented
13#22577249C211yesmaximal, characteristic, complemented
14#22577248C211yesmaximal, characteristic, complemented
15#388×C137C211yesmaximal, characteristic, complemented
16C22×Dic274C2211yescharacteristic
17C22×C22⋊D137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C22×C4×D137C2211yescharacteristic
20#17984074C418yescomplemented
21C2×#14555489C2211yescharacteristic
22C2×C22⋊Dic137C2211yescharacteristic
23C2×C22⋊Dic137C2211yescharacteristic
24C2×#14555487C2211yescharacteristic
25C2×#14555487C2211yescharacteristic
26C2×#14555487C2211yescharacteristic
27C2×#14555487C2211yescharacteristic
28C2×#14555487C2211yescharacteristic
29C2×C4×Dic137C2211yescharacteristic
30C2×C4×Dic137C2211yescharacteristic
31C2×C4×Dic137C2211yescharacteristic
32C2×C4×Dic137C2211yescharacteristic
33C2×C4×Dic137C2211yescharacteristic
34C2×C4×Dic137C2211yescharacteristic
35#17983995C2211yescharacteristic
36#17983995C2211yescharacteristic
37#17983995C2211yescharacteristic
38#17983995C2211yescharacteristic
39#17983995C2211yescharacteristic
40#17983995C2211yescharacteristic
41#17983995C2211yescharacteristic
42#17983995C2211yescharacteristic
43#17983995C2211yescharacteristic
44#17983995C2211yescharacteristic
45C2×C4⋊C4×C137C2211yescharacteristic
46C2×C22⋊C4×C137C2211yescharacteristic
47C2×C22⋊C4×C137C2211yescharacteristic
48C2×C22⋊C4×C137C2211yescharacteristic
49C2×C42×C137C2211yescharacteristic
50#95×C137C2211yescharacteristic
51#95×C137C2211yescharacteristic
52C23×D137C2311yescharacteristic
53C2×Dic274—41no—
54C2×Dic274—41no—
55C2×Dic274C2×C414yescomplemented
56C2×C22⋊D137—41no—
57C2×C22⋊D137—41no—
58C2×C22⋊D137C2×C414yes—
59C22×Dic137—22no—
60C22×Dic137—22no—
61C22×Dic137—41no—
62C22×Dic137C2311yescharacteristic
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2311yescharacteristic
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C22×Dic137C2311yescharacteristic
68C22×Dic137C2311yescharacteristic
69C2×C4×D137—22no—
70C2×C4×D137—22no—
71C2×C4×D137—41no—
72#14555489—41no—
73C22⋊Dic137—22no—
74C22⋊Dic137C2×C414yescomplemented
75#14555487—22no—
76#14555487—22no—
77#14555487—41no—
78#14555487C2×C414yescomplemented
79#14555487C2×C414yescomplemented
80C4×Dic137—22no—
81C4×Dic137—22no—
82C4×Dic137—22no—
83C4×Dic137—41no—
84C4×Dic137—41no—
85C4×Dic137C2×C414yescomplemented
86C24×C137C2311yescharacteristic
87C22×C4×C137C2311yescharacteristic
88C22×C4×C137C2311yescharacteristic
89C22×C4×C137C2311yescharacteristic
90C22×C4×C137C2311yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C22×C4×C137C2311yescharacteristic
93C4⋊C4×C137—41no—
94C22⋊C4×C137—22no—
95C22⋊C4×C137—41no—
96C22⋊C4×C137C2×C414yescomplemented
97C42×C137—22no—
98C22×D137—21no—
99C22×D137—21no—
100C22×D137—21no—
101C22×D137—21no—
102C22×D137—21no—
103C22×D137—21no—
104C22×D137C22×C412yes—
105Dic274—44no—
106C22⋊D137—44no—
107C2×Dic137—21no—
108C2×Dic137—21no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—21no—
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—44no—
123C2×Dic137C22×C412yes—
124C2×Dic137C22×C412yes—
125C2×Dic137C22×C412yes—
126C2×Dic137#4112yes—
127C2×Dic137#4112yes—
128C2×Dic137#4112yes—
129C4×D137—44no—
130C23×C137—21no—
131C23×C137—21no—
132C23×C137—21no—
133C23×C137—21no—
134C23×C137—21no—
135C23×C137—21no—
136C23×C137C22×C412yes—
137C23×C137C2411yescharacteristic
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
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×C137C22×C412yes—
154C2×C4×C137C22×C412yes—
155C2×C4×C137#4112yes—
156C22×C137—22no—
157C22×C137—22no—
158C22×C137—22no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137C23×C411yescharacteristic
164C22×C137C2×#4111yescharacteristic
165C22×C137C2×#4111yescharacteristic
166C22×C137C2×#4111yescharacteristic
167C22×C137C2×#4111yescharacteristic
168C22×C137H(5,𝔽2)11yescharacteristic
169C22×C137H(5,𝔽2)11yescharacteristic
170C2×D137—22no—
171C2×D137—22no—
172C2×D137—22no—
173C2×D137—41no—
174C2×D137—41no—
175C2×D137—41no—
176C2×D137—41no—
177C4×C137—22no—
178C4×C137—22no—
179C4×C137—22no—
180C4×C137—41no—
181C4×C137—41no—
182C4×C137—41no—
183Dic137—22no—
184Dic137—22no—
185Dic137—22no—
186Dic137—22no—
187Dic137—22no—
188Dic137—22no—
189Dic137—41no—
190C2×C137—42no—
191C2×C137C4×#4111yescharacteristic
192C2×C137C4×#4111yescharacteristic
193C2×C137#51811yescharacteristic
194C2×C137#55111yescharacteristic
195C2×C137#55111yescharacteristic
196C2×C137#55511yescharacteristic
197C2×C137#55511yescharacteristic
198D137—42no—
199C137#237211yescharacteristic, complemented, Sylow
200#2372—1371nomaximal, Sylow
201C2×#124—1371no—
202#391—1371no—
203#391—1371no—
204#388—1371no—
205#388—1371no—
206#388—1371no—
207#388D13711yescharacteristic, complemented
208#386—1371no—
209#386—1371no—
210C4×C22⋊C4—1371no—
211C4×C22⋊C4—1371no—
212C4×C22⋊C4—1371no—
213C4×C22⋊C4—1371no—
214#376—1371no—
215#376—1371no—
216C22×Q8—1371no—
217C22×D4—1371no—
218C23×C4—1371no—
219C23×C4—1371no—
220#124—1378no—
221C2×C4⋊C4—1371no—
222C2×C4⋊C4C2×D13711yescharacteristic
223C2×C22⋊C4—1371no—
224C2×C22⋊C4—1371no—
225C2×C22⋊C4—1371no—
226C2×C22⋊C4—1371no—
227C2×C22⋊C4—1371no—
228C2×C22⋊C4—1371no—
229C2×C22⋊C4—1371no—
230C2×C22⋊C4C2×D13711yescharacteristic
231C2×C22⋊C4C2×D13711yescharacteristic
232C2×C22⋊C4C2×D13711yescharacteristic
233C2×C42—1371no—
234C2×C42—1371no—
235C2×C42—1371no—
236C2×C42—1371no—
237C2×C42—1371no—
238C2×C42—1371no—
239C2×C42C2×D13711yescharacteristic
240#95—1371no—
241#95—1371no—
242#95—1371no—
243#95—1371no—
244#95—1371no—
245#95—1371no—
246#95—1371no—
247#95—1371no—
248#95—1371no—
249#95—1371no—
250#95C2×D13711yescharacteristic
251#95C2×D13711yescharacteristic
252C24—1371no—
253C24C22×D13711yescharacteristic
254C2×Q8—5481no—
255C2×Q8—5481no—
256C2×Q8—1374no—
257C2×D4—5481no—
258C2×D4—5481no—
259C2×D4—1374no—
260C22×C4—2742no—
261C22×C4—2742no—
262C22×C4—2742no—
263C22×C4—2742no—
264C22×C4—5481no—
265C22×C4—5481no—
266C22×C4—1371no—
267C22×C4—1371no—
268C22×C4—1371no—
269C22×C4—1371no—
270C22×C4—1371no—
271C22×C4—1371no—
272C22×C4—1371no—
273C22×C4C22×D13711yescharacteristic
274C22×C4C22×D13711yescharacteristic
275C22×C4C22×D13711yescharacteristic
276C22×C4C22×D13711yescharacteristic
277C22×C4C22×D13711yescharacteristic
278C22×C4C22×D13711yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—5481no—
281C22⋊C4—22no—
282C22⋊C4—41no—
283C22⋊C4—2742no—
284C22⋊C4—2742no—
285C22⋊C4—2742no—
286C22⋊C4—5481no—
287C22⋊C4—1374no—
288C22⋊C4—1374no—
289C22⋊C4—1374no—
290C22⋊C4C4×D13714yescomplemented
291C42—22no—
292C42—2742no—
293C42—2742no—
294C42—2742no—
295C42—5481no—
296C42—5481no—
297C42—1374no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—2741no—
305C23—2741no—
306C23—2741no—
307C23—2741no—
308C23—2741no—
309C23—2741no—
310C23—1372no—
311C23C2×C4×D13712yes—
312C23C23×D13711yescharacteristic
313Q8—5484no—
314D4—5484no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2741no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—2741no—
340C2×C4—2741no—
341C2×C4—2741no—
342C2×C4—2741no—
343C2×C4—2741no—
344C2×C4—2741no—
345C2×C4—5484no—
346C2×C4—5484no—
347C2×C4—1372no—
348C2×C4—1372no—
349C2×C4—1372no—
350C2×C4—1372no—
351C2×C4—1372no—
352C2×C4—1372no—
353C2×C4C2×C4×D13712yes—
354C2×C4C2×C4×D13712yes—
355C2×C4#1455550912yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2742no—
364C22—2742no—
365C22—2742no—
366C22—5481no—
367C22—5481no—
368C22—5481no—
369C22—5481no—
370C22C22×C4×D13711yescharacteristic
371C22#41×D13711yescharacteristic
372C22#41×D13711yescharacteristic
373C22#41×D13711yescharacteristic
374C22C2×#1455550911yescharacteristic
375C22#1798413311yescharacteristic
376C22#1798413411yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—2742no—
384C4—2742no—
385C4—2742no—
386C4—2742no—
387C4—2742no—
388C4—2742no—
389C4—5481no—
390C2—42no—
391C2—5482no—
392C2C4×#1455550911yescharacteristic
393C2#2257787111yescharacteristic
394C2#2257787511yescharacteristic
395C2#2257807611yescharacteristic
396C2#2257807811yescharacteristic
397C2#2257811511yescharacteristic
398C2#2257811711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#388×C137C2
socle, cosocleC23×C137C24
prime radicalC22×C22⋊D137C22
hypercenter#388D137
hypocenterC137#2372
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #22577308C2semidirect
3. #22577307C2semidirect
4. #22577297C2semidirect
5. #22577293C2semidirect
6. #22577292C2semidirect
9. C22⋊C4×Dic137C2semidirect
10. C22⋊C4×Dic137C2semidirect
11. C4×#14555487C2semidirect
12. #22577250C2semidirect
13. #22577249C2semidirect
14. #22577248C2semidirect
15. #388×C137C2semidirect
20. #17984074C4semidirect
55. C2×Dic274C2×C4semidirect
74. C22⋊Dic137C2×C4semidirect
78. #14555487C2×C4semidirect
79. #14555487C2×C4semidirect
85. C4×Dic137C2×C4semidirect
96. C22⋊C4×C137C2×C4semidirect
199. C137#2372semidirect
207. #388D137semidirect
290. 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
#225773082
#225773072
#225772972
#225772932
#225772922
#225772802
#225772792
C22⋊C4×Dic1372
C4×#145554872
#225772502
#225772492
#225772482
#388×C1372
#2372137