Finite Group Databasebeta

← #31792603

Subgroups — #31792603

Order 17536.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578057C211yesmaximal, characteristic, complemented
2D4×D548C211yesmaximal, characteristic, complemented
3#22577750C211yesmaximal, characteristic, complemented
4#22577717C212yesmaximal, complemented
5#22577684C211yesmaximal, characteristic, complemented
6#22577639C212yesmaximal, complemented
7#22577599C211yesmaximal, characteristic, complemented
8#22577539C212yesmaximal, complemented
9C4×D1096C211yesmaximal, characteristic, complemented
10C2×#17984010C212yesmaximal, complemented
11C8×D4×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—21no—
13C22×D548C2212yes—
14C2×D4×D137—21no—
15C2×D1096C2211yescharacteristic
16C2×D1096C2212yescomplemented
17#17984088C2211yescharacteristic, complemented
18#17984085C2211yescharacteristic
19#17984072—22no—
20#17984068—21no—
21#17984063—21no—
22#17984062C2212yes—
23#17984062C2212yescomplemented
24#17984058—22no—
25C4×D548C2211yescharacteristic
26C4×D548C2211yescharacteristic, complemented
27C2×C4⋊Dic137C2212yescomplemented
28#17984022C2212yescomplemented
29#17984022C2211yescharacteristic, complemented
30#17984010C2212yes—
31#17984010C2211yescharacteristic
32#17984010C2212yescomplemented
33#17984010C2212yescomplemented
34#17984010C2211yescharacteristic, complemented
35#17984010C2212yescomplemented
36C22×C8×C137C2212yescomplemented
37C4×D4×C137C2211yescharacteristic
38C4⋊C8×C137C2211yescharacteristic, complemented
39C22⋊C8×C137C2212yescomplemented
40C4×C8×C137C2211yescharacteristic, complemented
41C23×D137—22no—
42#14555509—44no—
43C2×Dic274—21no—
44C2×D548—24no—
45C2×D548—22no—
46C2×D548—21no—
47C2×D548—21no—
48C2×D548—42no—
49C2×D548C2312yes—
50C2×D548C2311yescharacteristic
51C2×D548C2311yescharacteristic
52C2×C22⋊D137—22no—
53C2×C22⋊D137—22no—
54D4×D137—44no—
55C22×Dic137—22no—
56C2×C4×D137—21no—
57C2×C4×D137—21no—
58D1096—22no—
59D1096—42no—
60C4⋊Dic137—22no—
61C4⋊Dic137—21no—
62C4⋊Dic137C2311yescharacteristic
63C4⋊Dic137C2312yes—
64C4⋊Dic137C2311yescharacteristic
65#14555487—21no—
66#14555487—21no—
67#14555487—22no—
68#14555487—22no—
69C2×D4×C137D411yescharacteristic
70C22×C4×C137C2312yes—
71C2×C8×C137—42no—
72C2×C8×C137C2311yescharacteristic
73C2×C8×C137C2312yes—
74C2×C8×C137C2311yescharacteristic
75C4⋊C4×C137D411yescharacteristic
76C22⋊C4×C137D412yes—
77C42×C137C2311yescharacteristic
78C22×D137—21no—
79C22×D137—21no—
80C22×D137—22no—
81C22×D137—44no—
82C22×D137—44no—
83C22×D137—82no—
84Dic274—42no—
85D548—21no—
86D548—22no—
87D548—21no—
88D548—21no—
89D548—22no—
90D548—44no—
91D548—42no—
92D548—81no—
93D548#4112yescomplemented
94C22⋊D137—44no—
95C22⋊D137—44no—
96C2×Dic137—21no—
97C2×Dic137—21no—
98C2×Dic137—22no—
99C2×Dic137—44no—
100C4×D137—42no—
101C4×D137—42no—
102C23×C137C2×D412yes—
103D4×C137D814yescomplemented
104C2×C4×C137—22no—
105C2×C4×C137—22no—
106C2×C4×C137C2×D411yescharacteristic
107C2×C4×C137C2×D412yes—
108C2×C4×C137C2×D411yescharacteristic
109C2×C4×C137C2411yescharacteristic
110C8×C137—21no—
111C8×C137—21no—
112C8×C137—22no—
113C22×C137—22no—
114C22×C137—22no—
115C22×C137C2×D814yes—
116C22×C137C22×D411yescharacteristic
117C2×D137—22no—
118C2×D137—41no—
119C2×D137—41no—
120C2×D137—42no—
121C2×D137—41no—
122C2×D137—41no—
123C2×D137—42no—
124C2×D137—42no—
125C2×D137—41no—
126C2×D137—88no—
127C4×C137—22no—
128C4×C137—21no—
129C4×C137C2×D812yes—
130C4×C137C2×#4111yescharacteristic
131C4×C137H(5,𝔽2)11yescharacteristic
132Dic137—41no—
133Dic137—41no—
134Dic137—42no—
135C2×C137—24no—
136C2×C137#54611yescharacteristic
137C2×C137C22×D811yescharacteristic
138C2×C137#57711yescharacteristic
139D137—42no—
140D137—81no—
141D137—82no—
142C137#329411yescharacteristic, complemented, Sylow
143#3294—1371nomaximal, Sylow
144#547—1371no—
145D42—1371no—
146#486—1371no—
147#480—1372no—
148#474—1371no—
149#466—1372no—
150#459—1371no—
151#447—1372no—
152C4×D8—1371no—
153C8×D4D13711yescharacteristic, complemented
154C2×#102—1372no—
155C2×#41—2741no—
156C22×D4—2741no—
157C22×D4—1372no—
158C2×D8—1371no—
159C2×D8—1372no—
160C22×C8C2×D13712yescomplemented
161C4⋊Q8—1371no—
162#127—1371no—
163#123—2742no—
164C22⋊Q8—2741no—
165#121—2741no—
166#121—1372no—
167#121—1372no—
168#120—2742no—
169C4×D4—1371no—
170C4×D4—1371no—
171C4×D4C2×D13711yescharacteristic
172C2×C4⋊C4—1372no—
173#107—1371no—
174#107—1372no—
175C4⋊C8C2×D13711yescharacteristic, complemented
176#102—1371no—
177#102—1372no—
178#102—1372no—
179#102—1372no—
180#102—1371no—
181#102—1372no—
182C22⋊C8C2×D13712yescomplemented
183C4×C8C2×D13711yescharacteristic, complemented
184C24—2742no—
185#41—5484no—
186C2×Q8—2741no—
187C2×D4—2744no—
188C2×D4—2742no—
189C2×D4—2742no—
190C2×D4—2742no—
191C2×D4—2741no—
192C2×D4—2741no—
193C2×D4—5482no—
194C2×D4—5484no—
195C2×D4—1372no—
196C2×D4—1371no—
197C2×D4—1371no—
198C2×D4D54811yescharacteristic
199C22×C4—2742no—
200C22×C4—2741no—
201C22×C4—2741no—
202C22×C4C22×D13712yes—
203D8—2742no—
204D8—5482no—
205C2×C8—42no—
206C2×C8C22×D13711yescharacteristic
207C2×C8C22×D13711yescharacteristic
208C2×C8C22×D13712yes—
209C4⋊C4—2742no—
210C4⋊C4—2741no—
211C4⋊C4—1371no—
212C4⋊C4—1372no—
213C4⋊C4—1371no—
214C4⋊C4D54811yescharacteristic
215C22⋊C4—2741no—
216C22⋊C4—2741no—
217C22⋊C4—2742no—
218C22⋊C4—2742no—
219C22⋊C4D54812yes—
220C42C22×D13711yescharacteristic
221C23—2741no—
222C23—2741no—
223C23—2742no—
224C23—5484no—
225C23—5484no—
226C23—10962no—
227C23C2×D54812yes—
228Q8—5482no—
229D4—2741no—
230D4—2742no—
231D4—2741no—
232D4—2741no—
233D4—2742no—
234D4—5484no—
235D4—5484no—
236D4—5484no—
237D4—5482no—
238D4—10961no—
239D4—1372no—
240D4D109614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—2741no—
244C2×C4—2741no—
245C2×C4—2742no—
246C2×C4—5484no—
247C2×C4—5482no—
248C2×C4—5482no—
249C2×C4C2×D54811yescharacteristic
250C2×C4C2×D54812yes—
251C2×C4C2×D54811yescharacteristic
252C2×C4C23×D13711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—2742no—
259C22—5481no—
260C22—5481no—
261C22—5482no—
262C22—5481no—
263C22—5482no—
264C22—5481no—
265C22—5481no—
266C22—5482no—
267C22—10968no—
268C22C2×D109614yes—
269C22C22×D54811yescharacteristic
270C4—21no—
271C4—22no—
272C4—5481no—
273C4—5481no—
274C4—5482no—
275C4C2×D109612yes—
276C4#41×D13711yescharacteristic
277C4#1798413311yescharacteristic
278C2—24no—
279C2—5482no—
280C2—10962no—
281C2—10961no—
282C2#2257804811yescharacteristic
283C2C22×D109611yescharacteristic
284C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC8×D4×C137C2
socleC22×C137C22×D4
hypercenterC8×D4D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22578057C2semidirect
2. D4×D548C2semidirect
3. #22577750C2semidirect
4. #22577717C2semidirect
5. #22577684C2semidirect
6. #22577639C2semidirect
7. #22577599C2semidirect
8. #22577539C2semidirect
9. C4×D1096C2semidirect
10. C2×#17984010C2semidirect
11. C8×D4×C137C2semidirect
16. C2×D1096C22semidirect
17. #17984088C22semidirect
23. #17984062C22semidirect
26. C4×D548C22semidirect
27. C2×C4⋊Dic137C22semidirect
28. #17984022C22semidirect
29. #17984022C22semidirect
32. #17984010C22semidirect
33. #17984010C22semidirect
34. #17984010C22semidirect
35. #17984010C22semidirect
36. C22×C8×C137C22semidirect
38. C4⋊C8×C137C22semidirect
39. C22⋊C8×C137C22semidirect
40. C4×C8×C137C22semidirect
93. D548#41semidirect
103. D4×C137D8semidirect
142. C137#3294semidirect
153. C8×D4D137semidirect
160. C22×C8C2×D137semidirect
175. C4⋊C8C2×D137semidirect
182. C22⋊C8C2×D137semidirect
183. C4×C8C2×D137semidirect
240. D4D1096semidirect

Maximal subgroups

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

subgroupindex
#225780572
D4×D5482
#225777502
#225777172
#225776842
#225776392
#225775992
#225775392
C4×D10962
C2×#179840102
C8×D4×C1372
#3294137