Finite Group Databasebeta

← #31789605

Subgroups — #31789605

Order 17536.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984120C211yesmaximal, characteristic
2C4×#14555511C211yesmaximal, characteristic
3C4×Q8×D137C211yesmaximal, characteristic, complemented
4#22577508C211yesmaximal, characteristic, complemented
5#22577505C211yesmaximal, characteristic, complemented
6#22577501C211yesmaximal, characteristic, complemented
7Q16×Dic137C211yesmaximal, characteristic, complemented
8#22577495C211yesmaximal, characteristic, complemented
9#22577490C211yesmaximal, characteristic, complemented
10#22577488C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577410C211yesmaximal, characteristic, complemented
13#22577403C211yesmaximal, characteristic
14#22577355C211yesmaximal, characteristic, complemented
15#441×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18#17984120C418yescomplemented
19C2×#14555500C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21C2×#14555497C2211yescharacteristic
22C2×#14555493C2211yescharacteristic
23C2×#14555493C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25Q8×Dic137C2211yescharacteristic
26C4×Dic274—21no—
27C4×Dic274C2211yescharacteristic
28#17984053—21no—
29#17984053C2211yescharacteristic
30C4×D548—21no—
31C4×D548C2211yescharacteristic
32#17984050—21no—
33#17984050C2211yescharacteristic
34#17984046—21no—
35#17984046C2211yescharacteristic
36C4⋊C4×D137—21no—
37C4⋊C4×D137C2211yescharacteristic
38C42×D137—21no—
39C42×D137—21no—
40C42×D137C2211yescharacteristic
41#17984022C2211yescharacteristic
42#17984021C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17983999C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C8×Dic137C2211yescharacteristic
53C2×Q16×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×Q8×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic
59#97×C137C2211yescharacteristic
60#14555511—22no—
61#14555511C2×C414yes—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64Q8×D137—22no—
65Q8×D137C2×C414yescomplemented
66C2×D548—21no—
67C2×D548C2311yescharacteristic
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137D411yescharacteristic
74C2×C4×D137D411yescharacteristic
75C2×C4×D137C2311yescharacteristic
76#14555500C2×C414yescomplemented
77#14555499C2×C414yescomplemented
78#14555497C2×C414yescomplemented
79#14555493—22no—
80#14555493C2×C414yescomplemented
81C2×#11764040C2311yescharacteristic
82C2×#11764040C2311yescharacteristic
83C4⋊Dic137—21no—
84C4⋊Dic137—21no—
85C4⋊Dic137C2311yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C4×Dic137D411yescharacteristic
99C4×Dic137D411yescharacteristic
100C4×Dic137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C2×Q8×C137C2311yescharacteristic
103Q16×C137C2×C414yescomplemented
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137—21no—
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C42×C137—21no—
111C42×C137—21no—
112C42×C137C2311yescharacteristic
113C22×D137—21no—
114C22×D137C2×D411yescharacteristic
115Dic274—21no—
116Dic274—24no—
117Dic274C22×C412yes—
118D548—24no—
119D548—21no—
120D548C22×C412yes—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—24no—
128C4×D137—24no—
129C4×D137—22no—
130C4×D137—41no—
131C4×D137—42no—
132C4×D137—42no—
133C4×D137—42no—
134C4×D137C22×C412yes—
135C4×D137#4112yes—
136#11764040—21no—
137#11764040C22×C412yes—
138Q8×C137—21no—
139Q8×C137—21no—
140Q8×C137C22×C412yes—
141Q8×C137C22×C412yes—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2411yescharacteristic
149C8×C137—21no—
150C8×C137C22×C412yes—
151C22×C137C22×D411yescharacteristic
152C2×D137—21no—
153C2×D137—21no—
154C2×D137—22no—
155C2×D137—41no—
156C2×D137—41no—
157C2×D137C4×D412yes—
158C4×C137—22no—
159C4×C137—22no—
160C4×C137—21no—
161C4×C137—21no—
162C4×C137—41no—
163C4×C137—41no—
164C4×C137C23×C411yescharacteristic
165C4×C137C2×#4111yescharacteristic
166Dic137—22no—
167Dic137—21no—
168Dic137—41no—
169Dic137—41no—
170Dic137C4×D412yes—
171Dic137#13712yescomplemented
172C2×C137C2×C4×D411yescharacteristic
173C2×C137C2×#13711yescharacteristic
174C2×C137#57511yescharacteristic
175D137—22no—
176D137—42no—
177C137C4×#13711yescharacteristic, complemented, Sylow
178C4×#137—1371nomaximal, Sylow
179C2×#137—1371no—
180C4×#41—1371no—
181C2×C4×Q8—1371no—
182#441—1371no—
183#441D13711yescharacteristic, complemented
184#440—1371no—
185#440—1371no—
186C4×Q16—1371no—
187C4×Q16—1371no—
188C4×SD16—1371no—
189C4×SD16—1371no—
190#428—1371no—
191#419—1371no—
192#417—1371no—
193C4×M16—1371no—
194C2×#41—1371no—
195C22×Q8—1371no—
196#137—1378no—
197C2×Q16—1371no—
198C2×Q16C2×D13711yescharacteristic
199C2×SD16—1371no—
200C2×SD16—1371no—
201C2×M16—1371no—
202C2×M16—1371no—
203C4×Q8—2741no—
204C4×Q8—2741no—
205C4×Q8—1371no—
206C4×Q8—1371no—
207C4×Q8—1371no—
208C4×Q8—1371no—
209C4×Q8C2×D13711yescharacteristic
210C4×Q8C2×D13711yescharacteristic
211C4×D4—2741no—
212C4×D4—2741no—
213C4×D4—1371no—
214C4×D4—1371no—
215#117—2741no—
216#117—1371no—
217C2×C4⋊C4—2741no—
218C2×C4⋊C4—1371no—
219C2×C42—2741no—
220C2×C42—2741no—
221C2×C42—1371no—
222#107—1371no—
223#107—1371no—
224#106—1371no—
225#106C2×D13711yescharacteristic
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103C2×D13711yescharacteristic
231#103C2×D13711yescharacteristic
232#102—1371no—
233#102—1371no—
234#97—1371no—
235#97C2×D13711yescharacteristic
236C4×C8—1371no—
237C4×C8—1371no—
238#41—2742no—
239#41—1374no—
240C2×Q8—2741no—
241C2×Q8—2742no—
242C2×Q8—1374no—
243C2×Q8—1371no—
244C2×Q8C22×D13711yescharacteristic
245C2×Q8C22×D13711yescharacteristic
246C2×D4—2741no—
247C2×D4—1371no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—1371no—
254C22×C4—1371no—
255C22×C4—1371no—
256Q16—1374no—
257Q16C4×D13714yescomplemented
258SD16—1374no—
259SD16—1374no—
260M16—2742no—
261M16—1374no—
262C2×C8—1371no—
263C2×C8—1371no—
264C2×C8C22×D13711yescharacteristic
265C2×C8C22×D13711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—2741no—
273C4⋊C4—1371no—
274C4⋊C4—1371no—
275C4⋊C4C22×D13711yescharacteristic
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—2741no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C42—21no—
281C42—21no—
282C42—2741no—
283C42—2741no—
284C42—2741no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C42—1371no—
289C42—1371no—
290C42C22×D13711yescharacteristic
291C23—2741no—
292C23—1371no—
293Q8—21no—
294Q8—21no—
295Q8—2744no—
296Q8—2741no—
297Q8—1372no—
298Q8C2×C4×D13712yes—
299Q8C2×C4×D13712yes—
300D4—2744no—
301D4—2741no—
302D4—1372no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—2744no—
308C2×C4—2744no—
309C2×C4—2742no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—5481no—
314C2×C4—5482no—
315C2×C4—5482no—
316C2×C4—5482no—
317C2×C4—1371no—
318C2×C4—1371no—
319C2×C4—1371no—
320C2×C4—1372no—
321C2×C4—1372no—
322C2×C4D4×D13711yescharacteristic
323C2×C4D4×D13711yescharacteristic
324C2×C4C23×D13711yescharacteristic
325C8—21no—
326C8—2741no—
327C8—1372no—
328C8C2×C4×D13712yes—
329C22—2741no—
330C22—2741no—
331C22—2742no—
332C22—5481no—
333C22—5481no—
334C22—1372no—
335C22C2×D4×D13711yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—2741no—
343C4—2742no—
344C4—5481no—
345C4—5481no—
346C4—1372no—
347C4—1372no—
348C4C22×C4×D13711yescharacteristic
349C4#41×D13711yescharacteristic
350C2—2742no—
351C2—5482no—
352C2C4×D4×D13711yescharacteristic
353C2#137×D13711yescharacteristic
354C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting#441×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555511C22
cosocleC2×C4×C137C24
hypercenter#441D137
hypocenterC137C4×#137
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. C4×Q8×D137C2semidirect
4. #22577508C2semidirect
5. #22577505C2semidirect
6. #22577501C2semidirect
7. Q16×Dic137C2semidirect
8. #22577495C2semidirect
9. #22577490C2semidirect
10. #22577488C2semidirect
11. #22577442C2semidirect
12. #22577410C2semidirect
14. #22577355C2semidirect
15. #441×C137C2semidirect
18. #17984120C4semidirect
65. Q8×D137C2×C4semidirect
76. #14555500C2×C4semidirect
77. #14555499C2×C4semidirect
78. #14555497C2×C4semidirect
80. #14555493C2×C4semidirect
103. Q16×C137C2×C4semidirect
171. Dic137#137semidirect
177. C137C4×#137semidirect
183. #441D137semidirect
257. Q16C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841202
C4×#145555112
C4×Q8×D1372
#225775082
#225775052
#225775012
Q16×Dic1372
#225774952
#225774902
#225774882
#225774422
#225774102
#225774032
#225773552
#441×C1372
C4×#137137