Finite Group Databasebeta

← #22578005

Subgroups — #22578005

Order 8768.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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#17984084C211yesmaximal, characteristic, complemented
2#17984078C211yesmaximal, characteristic, complemented
3#17984073C211yesmaximal, characteristic, complemented
4#17984072C211yesmaximal, characteristic, complemented
5#17984070C211yesmaximal, characteristic, complemented
6#17984063C211yesmaximal, characteristic, complemented
7#17984062C211yesmaximal, characteristic, complemented
8#17984061C211yesmaximal, characteristic, complemented
9#17984061C211yesmaximal, characteristic, complemented
10#17984061C211yesmaximal, characteristic, complemented
11#17984060C211yesmaximal, characteristic, complemented
12C4×C22⋊D137C211yesmaximal, characteristic, complemented
13#17984048C211yesmaximal, characteristic, complemented
14#17984046C211yesmaximal, characteristic, complemented
15#123×C137C211yesmaximal, characteristic, complemented
16C2×D548C2211yescharacteristic
17C2×D548C2211yescharacteristic
18C2×D548C2211yescharacteristic
19C2×C22⋊D137C2211yescharacteristic
20C2×C22⋊D137C2211yescharacteristic
21C2×C22⋊D137C2211yescharacteristic
22C2×C22⋊D137C2211yescharacteristic
23C2×C22⋊D137C2211yescharacteristic
24C2×C22⋊D137C2211yescharacteristic
25C22×Dic137C2211yescharacteristic
26C2×C4×D137C2211yescharacteristic
27C2×C4×D137C2211yescharacteristic
28C2×C4×D137C2211yescharacteristic
29C4⋊Dic137C2211yescharacteristic
30C4⋊Dic137C2211yescharacteristic
31#14555489C2211yescharacteristic
32#14555489C2211yescharacteristic
33#14555489C2211yescharacteristic
34#14555489C2211yescharacteristic
35C22⋊Dic137C2211yescharacteristic
36#14555487C2211yescharacteristic
37#14555487C2211yescharacteristic
38#14555487C2211yescharacteristic
39#14555487C2211yescharacteristic
40#14555487C2211yescharacteristic
41#14555487C2211yescharacteristic
42C4×Dic137C2211yescharacteristic
43C4×Dic137C2211yescharacteristic
44C2×D4×C137C2211yescharacteristic
45C22×C4×C137C2211yescharacteristic
46C4⋊C4×C137C2211yescharacteristic
47C4⋊C4×C137C2211yescharacteristic
48C22⋊C4×C137C2211yescharacteristic
49C22⋊C4×C137C2211yescharacteristic
50C22⋊C4×C137C2211yescharacteristic
51C22×D137C2311yescharacteristic
52C22×D137C2311yescharacteristic
53C22×D137C2311yescharacteristic
54D548—41no—
55D548—41no—
56D548—41no—
57C22⋊D137—22no—
58C22⋊D137—22no—
59C22⋊D137—41no—
60C22⋊D137—41no—
61C22⋊D137—41no—
62C22⋊D137—41no—
63C2×Dic137—41no—
64C2×Dic137C2311yescharacteristic
65C2×Dic137C2311yescharacteristic
66C2×Dic137C2311yescharacteristic
67C2×Dic137C2311yescharacteristic
68C2×Dic137C2311yescharacteristic
69C4×D137—22no—
70C4×D137—41no—
71C4×D137—41no—
72C23×C137C2311yescharacteristic
73C23×C137C2311yescharacteristic
74D4×C137—41no—
75C2×C4×C137—41no—
76C2×C4×C137C2311yescharacteristic
77C2×C4×C137C2311yescharacteristic
78C2×C4×C137C2311yescharacteristic
79C2×C4×C137C2311yescharacteristic
80C2×C4×C137C2311yescharacteristic
81C22×C137—21no—
82C22×C137—21no—
83C22×C137—21no—
84C22×C137—21no—
85C22×C137—21no—
86C22×C137—21no—
87C22×C137C2411yescharacteristic
88C2×D137—21no—
89C2×D137—21no—
90C2×D137—21no—
91C2×D137—21no—
92C2×D137—21no—
93C2×D137—21no—
94C2×D137—21no—
95C2×D137—21no—
96C2×D137—21no—
97C4×C137—21no—
98C4×C137—21no—
99C4×C137—21no—
100C4×C137—21no—
101C4×C137—21no—
102Dic137—21no—
103Dic137—21no—
104Dic137—21no—
105Dic137—21no—
106Dic137#4112yes—
107C2×C137—41no—
108C2×C137—41no—
109C2×C137C2×#4111yescharacteristic
110C2×C137H(5,𝔽2)11yescharacteristic
111C2×C137H(5,𝔽2)11yescharacteristic
112D137—41no—
113D137—41no—
114D137—41no—
115C137#54011yescharacteristic, complemented, Sylow
116#540—1371nomaximal, Sylow
117#127—1371no—
118#125—1371no—
119#123—1371no—
120#123—1371no—
121#123—1371no—
122#123D13711yescharacteristic, complemented
123#121—1371no—
124#121—1371no—
125#121—1371no—
126#121—1371no—
127#121—1371no—
128#121—1371no—
129C4×D4—1371no—
130C4×D4—1371no—
131#117—1371no—
132C2×D4—1371no—
133C2×D4—1371no—
134C2×D4—1371no—
135C2×D4—1371no—
136C2×D4—1371no—
137C2×D4—1371no—
138C2×D4—1371no—
139C2×D4—1371no—
140C2×D4—1371no—
141C2×D4C2×D13711yescharacteristic
142C22×C4—1371no—
143C22×C4—1371no—
144C22×C4—1371no—
145C22×C4—1371no—
146C22×C4C2×D13711yescharacteristic
147C4⋊C4—1371no—
148C4⋊C4—1371no—
149C4⋊C4—1371no—
150C4⋊C4—1371no—
151C4⋊C4—1371no—
152C4⋊C4—1371no—
153C4⋊C4C2×D13711yescharacteristic
154C4⋊C4C2×D13711yescharacteristic
155C22⋊C4—1371no—
156C22⋊C4—1371no—
157C22⋊C4—1371no—
158C22⋊C4—1371no—
159C22⋊C4—1371no—
160C22⋊C4—1371no—
161C22⋊C4—1371no—
162C22⋊C4C2×D13711yescharacteristic
163C22⋊C4C2×D13711yescharacteristic
164C22⋊C4C2×D13711yescharacteristic
165C42—1371no—
166C42—1371no—
167C23—1371no—
168C23—1371no—
169C23—1371no—
170C23C22×D13711yescharacteristic
171C23C22×D13711yescharacteristic
172D4—41no—
173D4—2742no—
174D4—2742no—
175D4—5481no—
176D4—5481no—
177D4—5481no—
178D4—5481no—
179D4—5481no—
180D4—5481no—
181D4—5481no—
182C2×C4—41no—
183C2×C4—2742no—
184C2×C4—5481no—
185C2×C4—5481no—
186C2×C4—5481no—
187C2×C4—1371no—
188C2×C4—1371no—
189C2×C4—1371no—
190C2×C4—1371no—
191C2×C4—1371no—
192C2×C4C22×D13711yescharacteristic
193C2×C4C22×D13711yescharacteristic
194C2×C4C22×D13711yescharacteristic
195C2×C4C22×D13711yescharacteristic
196C2×C4C22×D13711yescharacteristic
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—2741no—
204C22—2741no—
205C22—2741no—
206C22—2741no—
207C22—2741no—
208C22—2741no—
209C22—2741no—
210C22—2741no—
211C22—2741no—
212C22C23×D13711yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—2741no—
219C4—2741no—
220C4—2741no—
221C4—2741no—
222C4—1372no—
223C2—41no—
224C2—41no—
225C2—5481no—
226C2—5481no—
227C2—5481no—
228C2#41×D13711yescharacteristic
229C2#1798413311yescharacteristic
230C2#1798413411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D137
commutator, socleC22×C137C24
Fitting#123×C137C2
hypercenter#123D137
hypocenterC137#540
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #17984084C2semidirect
2. #17984078C2semidirect
3. #17984073C2semidirect
4. #17984072C2semidirect
5. #17984070C2semidirect
6. #17984063C2semidirect
7. #17984062C2semidirect
8. #17984061C2semidirect
9. #17984061C2semidirect
10. #17984061C2semidirect
11. #17984060C2semidirect
12. C4×C22⋊D137C2semidirect
13. #17984048C2semidirect
14. #17984046C2semidirect
15. #123×C137C2semidirect
115. C137#540semidirect
122. #123D137semidirect