Finite Group Databasebeta

← #31789611

Subgroups — #31789611

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×#17984117C211yesmaximal, characteristic
2C4×#14555509C211yesmaximal, characteristic
3C4×Q8×D137C211yesmaximal, characteristic, complemented
4#22577511C211yesmaximal, characteristic, complemented
5#22577508C211yesmaximal, characteristic, complemented
6#22577503C211yesmaximal, characteristic, complemented
7#22577497C211yesmaximal, characteristic, complemented
8#22577495C211yesmaximal, characteristic, complemented
9#22577494C211yesmaximal, characteristic, complemented
10SD16×Dic137C211yesmaximal, characteristic, complemented
11#22577444C211yesmaximal, characteristic, complemented
12#22577410C211yesmaximal, characteristic, complemented
13#22577401C211yesmaximal, characteristic
14#22577355C211yesmaximal, characteristic, complemented
15#440×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18#17984117C418yescomplemented
19C2×Dic548C2211yescharacteristic
20C2×#14555500C2211yescharacteristic
21C2×#14555498C2211yescharacteristic
22C2×#14555493C2211yescharacteristic
23C2×#14555493C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25C4×Dic274—21no—
26C4×Dic274C2211yescharacteristic
27C4×Dic274C2211yescharacteristic
28#17984053—21no—
29#17984053C2211yescharacteristic
30#17984053C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32C4×C22⋊D137—21no—
33#17984048—21no—
34#17984046C2211yescharacteristic
35#17984044—21no—
36C4⋊C4×D137—21no—
37C4⋊C4×D137C2211yescharacteristic
38C2×C4×Dic137—21no—
39C42×D137—21no—
40C42×D137C2211yescharacteristic
41#17984021C2211yescharacteristic
42#17984020C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984013C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49#17984009C2211yescharacteristic
50#17983999C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C8×Dic137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic
56#107×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#97×C137C2211yescharacteristic
60#14555509—22no—
61#14555509C2×C414yes—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64C2×Dic274C2311yescharacteristic
65Q8×D137—22no—
66Q8×D137C2×C414yescomplemented
67C2×C22⋊D137—21no—
68C22×Dic137—21no—
69C22×Dic137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137D411yescharacteristic
73C2×C4×D137D411yescharacteristic
74C2×C4×D137C2311yescharacteristic
75Dic548C2×C414yescomplemented
76#14555500C2×C414yescomplemented
77#14555498C2×C414yescomplemented
78#14555493—22no—
79#14555493C2×C414yescomplemented
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90C22⋊Dic137—21no—
91#14555487—21no—
92C4×Dic137—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×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104SD16×C137C2×C414yescomplemented
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C42×C137—21no—
112C42×C137C2311yescharacteristic
113C22×D137C2×D411yescharacteristic
114Dic274—21no—
115Dic274—21no—
116Dic274—24no—
117Dic274C22×C412yes—
118Dic274C22×C412yes—
119C22⋊D137—24no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—24no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—42no—
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C4×D137—24no—
130C4×D137—22no—
131C4×D137—41no—
132C4×D137—42no—
133C4×D137C22×C412yes—
134C4×D137#4112yes—
135#11764040—21no—
136#11764040C22×C412yes—
137C23×C137—21no—
138Q8×C137—21no—
139Q8×C137C22×C412yes—
140D4×C137—21no—
141D4×C137C22×C412yes—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—42no—
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2411yescharacteristic
149C8×C137—21no—
150C8×C137C22×C412yes—
151C22×C137—22no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137C22×D411yescharacteristic
155C2×D137—21no—
156C2×D137—21no—
157C2×D137C4×D412yes—
158C4×C137—22no—
159C4×C137—21no—
160C4×C137—21no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137C23×C411yescharacteristic
164C4×C137C2×#4111yescharacteristic
165Dic137—22no—
166Dic137—22no—
167Dic137—21no—
168Dic137—41no—
169Dic137—41no—
170Dic137C4×D412yes—
171Dic137#13712yescomplemented
172C2×C137—42no—
173C2×C137C2×C4×D411yescharacteristic
174C2×C137C2×#13711yescharacteristic
175C2×C137#57511yescharacteristic
176D137—22no—
177C137C4×#13711yescharacteristic, complemented, Sylow
178C4×#137—1371nomaximal, Sylow
179C2×#137—1371no—
180C4×#41—1371no—
181C2×C4×Q8—1371no—
182#441—1371no—
183#441—1371no—
184#440—1371no—
185#440D13711yescharacteristic, complemented
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×Q16—1371no—
199C2×SD16—1371no—
200C2×SD16C2×D13711yescharacteristic
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×Q8—1371no—
210C4×Q8C2×D13711yescharacteristic
211C4×D4—2741no—
212C4×D4—2741no—
213C4×D4—1371no—
214C4×D4C2×D13711yescharacteristic
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#107C2×D13711yescharacteristic
224#106—1371no—
225#106—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103C2×D13711yescharacteristic
232#102—1371no—
233#102C2×D13711yescharacteristic
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×Q8—1371no—
245C2×Q8C22×D13711yescharacteristic
246C2×D4—2741no—
247C2×D4C22×D13711yescharacteristic
248C22×C4—21no—
249C22×C4—2741no—
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—1371no—
254C22×C4—1371no—
255C22×C4—1371no—
256Q16—1374no—
257Q16—1374no—
258SD16—1374no—
259SD16C4×D13714yescomplemented
260M16—2742no—
261M16—1374no—
262C2×C8—1371no—
263C2×C8—1371no—
264C2×C8C22×D13711yescharacteristic
265C2×C8C22×D13711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—2741no—
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—21no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C42—21no—
281C42—2741no—
282C42—2741no—
283C42—2741no—
284C42—2741no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C42—1371no—
289C42—1371no—
290C42C22×D13711yescharacteristic
291C23—21no—
292C23—1371no—
293Q8—21no—
294Q8—2744no—
295Q8—2741no—
296Q8—2741no—
297Q8—1372no—
298Q8—1372no—
299Q8C2×C4×D13712yes—
300D4—21no—
301D4—2744no—
302D4C2×C4×D13712yes—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—2744no—
308C2×C4—2742no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2744no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—5481no—
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—22no—
330C22—41no—
331C22—41no—
332C22—2741no—
333C22—2741no—
334C22—1372no—
335C22C2×D4×D13711yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—2741no—
342C4—2742no—
343C4—2742no—
344C4—5481no—
345C4—5481no—
346C4—1372no—
347C4—1372no—
348C4C22×C4×D13711yescharacteristic
349C4#41×D13711yescharacteristic
350C2—42no—
351C2—2742no—
352C2C4×D4×D13711yescharacteristic
353C2#137×D13711yescharacteristic
354C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting#440×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555509C22
cosocleC2×C4×C137C24
hypercenter#440D137
hypocenterC137C4×#137
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. C4×Q8×D137C2semidirect
4. #22577511C2semidirect
5. #22577508C2semidirect
6. #22577503C2semidirect
7. #22577497C2semidirect
8. #22577495C2semidirect
9. #22577494C2semidirect
10. SD16×Dic137C2semidirect
11. #22577444C2semidirect
12. #22577410C2semidirect
14. #22577355C2semidirect
15. #440×C137C2semidirect
18. #17984117C4semidirect
66. Q8×D137C2×C4semidirect
75. Dic548C2×C4semidirect
76. #14555500C2×C4semidirect
77. #14555498C2×C4semidirect
79. #14555493C2×C4semidirect
104. SD16×C137C2×C4semidirect
171. Dic137#137semidirect
177. C137C4×#137semidirect
185. #440D137semidirect
259. SD16C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841172
C4×#145555092
C4×Q8×D1372
#225775112
#225775082
#225775032
#225774972
#225774952
#225774942
SD16×Dic1372
#225774442
#225774102
#225774012
#225773552
#440×C1372
C4×#137137