Finite Group Databasebeta

← #22578080

Subgroups — #22578080

Order 8768.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#17984083C211yesmaximal, characteristic, complemented
2#17984083C211yesmaximal, characteristic, complemented
3#17984077C211yesmaximal, characteristic, complemented
4#17984073C211yesmaximal, characteristic, complemented
5#17984073C211yesmaximal, characteristic, complemented
6#17984072C211yesmaximal, characteristic, complemented
7#17984064C211yesmaximal, characteristic, complemented
8#17984058C211yesmaximal, characteristic
9#17984050C211yesmaximal, characteristic, complemented
10C4×C22⋊D137C211yesmaximal, characteristic, complemented
11#17984046C211yesmaximal, characteristic, complemented
12#17984044C211yesmaximal, characteristic, complemented
13C2×#14555487C211yesmaximal, characteristic, complemented
14C22⋊C4×D137C211yesmaximal, characteristic, complemented
15C22⋊Q8×C137C211yesmaximal, characteristic, complemented
16C23×D137C2211yescharacteristic
17C2×D548C2211yescharacteristic
18C2×D548C2211yescharacteristic
19C2×C22⋊D137C2211yescharacteristic
20C22×Dic137C2211yescharacteristic
21C2×C4×D137C2211yescharacteristic
22C2×C4×D137C2211yescharacteristic
23C2×C4×D137C2211yescharacteristic, complemented
24C4⋊Dic137C2211yescharacteristic, complemented
25C4⋊Dic137C2211yescharacteristic, complemented
26#14555489C2211yescharacteristic, complemented
27#14555489C2211yescharacteristic, complemented
28#14555489C2211yescharacteristic, complemented
29C22⋊Dic137C2211yescharacteristic
30#14555487C2211yescharacteristic
31#14555487C2211yescharacteristic
32#14555487C2211yescharacteristic
33#14555487C2211yescharacteristic
34#14555487C2211yescharacteristic
35#14555487C2211yescharacteristic
36#14555487C2211yescharacteristic
37#14555487C2211yescharacteristic
38#14555487C2211yescharacteristic, complemented
39#14555487C2211yescharacteristic, complemented
40#14555487C2211yescharacteristic, complemented
41C4×Dic137C2211yescharacteristic, complemented
42C4×Dic137C2211yescharacteristic, complemented
43C4×Dic137C2211yescharacteristic, complemented
44C2×Q8×C137C2211yescharacteristic, complemented
45C22×C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic, complemented
47C4⋊C4×C137C2211yescharacteristic, complemented
48C4⋊C4×C137C2211yescharacteristic, complemented
49C22⋊C4×C137C2211yescharacteristic
50C22⋊C4×C137C2211yescharacteristic
51C22×D137—22no—
52C22×D137—22no—
53C22×D137—41no—
54C22×D137C2311yescharacteristic
55C22×D137C2311yescharacteristic
56C22×D137C2311yescharacteristic
57D548—22no—
58D548—41no—
59C22⋊D137—22no—
60C2×Dic137—22no—
61C2×Dic137C2311yescharacteristic
62C2×Dic137C2311yescharacteristic
63C2×Dic137C2311yescharacteristic
64C2×Dic137C2311yescharacteristic
65C2×Dic137C2311yescharacteristic
66C4×D137—22no—
67C4×D137—41no—
68C4×D137—41no—
69C23×C137C2311yescharacteristic
70Q8×C137—41no—
71C2×C4×C137—41no—
72C2×C4×C137C2311yescharacteristic
73C2×C4×C137C2311yescharacteristic
74C2×C4×C137C2311yescharacteristic
75C2×C4×C137C2311yescharacteristic
76C2×C4×C137C2311yescharacteristic
77C2×C4×C137C2311yescharacteristic
78C22×C137—21no—
79C22×C137—21no—
80C22×C137#4112yes—
81C22×C137C2411yescharacteristic
82C2×D137—21no—
83C2×D137—21no—
84C2×D137—21no—
85C2×D137—21no—
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—21no—
89C2×D137—21no—
90C2×D137—44no—
91C2×D137#4112yes—
92C4×C137—21no—
93C4×C137—21no—
94C4×C137—21no—
95C4×C137—21no—
96C4×C137—21no—
97C4×C137—21no—
98Dic137—21no—
99Dic137—21no—
100Dic137—21no—
101Dic137—21no—
102Dic137—21no—
103C2×C137—22no—
104C2×C137C2×#4111yescharacteristic
105C2×C137C2×#4111yescharacteristic
106C2×C137H(5,𝔽2)11yescharacteristic
107D137—22no—
108D137—41no—
109D137—41no—
110C137#55111yescharacteristic, complemented, Sylow
111#551—1371nomaximal, Sylow
112#126—1371no—
113#126—1371no—
114#124—1371no—
115#123—1371no—
116#123—1371no—
117#123—1371no—
118C22⋊Q8—1371no—
119C22⋊Q8D13711yescharacteristic, complemented
120#120—1371no—
121C4×D4—1371no—
122C4×D4—1371no—
123#117—1371no—
124#117—1371no—
125C2×C22⋊C4—1371no—
126C2×C22⋊C4—1371no—
127C24—1371no—
128C2×Q8C2×D13711yescharacteristic, complemented
129C2×D4—1371no—
130C2×D4—1371no—
131C2×D4—1371no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4C2×D13711yescharacteristic
137C4⋊C4—1371no—
138C4⋊C4—1371no—
139C4⋊C4—1371no—
140C4⋊C4—1371no—
141C4⋊C4—1371no—
142C4⋊C4C2×D13711yescharacteristic, complemented
143C4⋊C4C2×D13711yescharacteristic, complemented
144C4⋊C4C2×D13711yescharacteristic, complemented
145C22⋊C4—1371no—
146C22⋊C4—1371no—
147C22⋊C4—1371no—
148C22⋊C4—1371no—
149C22⋊C4—1371no—
150C22⋊C4—1371no—
151C22⋊C4—1371no—
152C22⋊C4—1371no—
153C22⋊C4—1371no—
154C22⋊C4—1371no—
155C22⋊C4—1371no—
156C22⋊C4—1371no—
157C22⋊C4C2×D13711yescharacteristic
158C22⋊C4C2×D13711yescharacteristic
159C42—1371no—
160C42—1371no—
161C42—1371no—
162C23—2742no—
163C23—2742no—
164C23—5481no—
165C23—1371no—
166C23—1371no—
167C23—1371no—
168C23C22×D13711yescharacteristic
169Q8—41no—
170D4—2742no—
171D4—2742no—
172D4—5481no—
173C2×C4—41no—
174C2×C4—2742no—
175C2×C4—2742no—
176C2×C4—5481no—
177C2×C4—5481no—
178C2×C4—1371no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1371no—
183C2×C4C22×D13711yescharacteristic
184C2×C4C22×D13711yescharacteristic
185C2×C4C22×D13711yescharacteristic
186C2×C4C22×D13711yescharacteristic
187C2×C4C22×D13711yescharacteristic
188C2×C4C22×D13711yescharacteristic
189C22—21no—
190C22—21no—
191C22—2741no—
192C22—2741no—
193C22—2741no—
194C22—2741no—
195C22—2741no—
196C22—2741no—
197C22—2741no—
198C22—2741no—
199C22—5484no—
200C22—1372no—
201C22#1455551112yes—
202C22C23×D13711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—2741no—
210C4—2741no—
211C4—2741no—
212C4—2741no—
213C4—2741no—
214C2—22no—
215C2—2742no—
216C2—5481no—
217C2—5481no—
218C2#41×D13711yescharacteristic
219C2C2×#1455551111yescharacteristic
220C2#1798413311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
FittingC22⋊Q8×C137C2
hypercenterC22⋊Q8D137
hypocenterC137#551
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #17984083C2semidirect
2. #17984083C2semidirect
3. #17984077C2semidirect
4. #17984073C2semidirect
5. #17984073C2semidirect
6. #17984072C2semidirect
7. #17984064C2semidirect
9. #17984050C2semidirect
10. C4×C22⋊D137C2semidirect
11. #17984046C2semidirect
12. #17984044C2semidirect
13. C2×#14555487C2semidirect
14. C22⋊C4×D137C2semidirect
15. C22⋊Q8×C137C2semidirect
23. C2×C4×D137C22semidirect
24. C4⋊Dic137C22semidirect
25. C4⋊Dic137C22semidirect
26. #14555489C22semidirect
27. #14555489C22semidirect
28. #14555489C22semidirect
38. #14555487C22semidirect
39. #14555487C22semidirect
40. #14555487C22semidirect
41. C4×Dic137C22semidirect
42. C4×Dic137C22semidirect
43. C4×Dic137C22semidirect
44. C2×Q8×C137C22semidirect
46. C4⋊C4×C137C22semidirect
47. C4⋊C4×C137C22semidirect
48. C4⋊C4×C137C22semidirect
110. C137#551semidirect
119. C22⋊Q8D137semidirect
128. C2×Q8C2×D137semidirect
142. C4⋊C4C2×D137semidirect
143. C4⋊C4C2×D137semidirect
144. C4⋊C4C2×D137semidirect