Finite Group Databasebeta

← D4×D136 #14540461

Subgroups — D4×D136 #14540461

Order 2176.

620 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×D136C212yesmaximal, complemented
2D4×D68C212yesmaximal, complemented
3#11763186C211yesmaximal, characteristic, complemented
4#11763045C212yesmaximal, complemented
5#11763005C212yesmaximal, complemented
6#11762945C214yesmaximal, complemented
7C4×D136C211yesmaximal, characteristic, complemented
8C8×D4×C17C211yesmaximal, characteristic, complemented
9C2×D4×D17—22no—
10C22×D68C2214yescomplemented
11C22×C8×C17C2212yescomplemented
12C4×D4×C17C2211yescharacteristic
13#10587616C2214yescomplemented
14C2×D136—28no—
15C2×D136C2211yescharacteristic
16C2×D136C2214yescomplemented
17C2×D136C2212yescomplemented
18C2×D136C2212yescomplemented
19#10587581—22no—
20#10587568—24no—
21#10587560C2212yescomplemented
22C4×D68C2212yescomplemented
23C17×C4⋊C8C2211yescharacteristic, complemented
24C17×C22⋊C8C2212yescomplemented
25C4×C8×C17C2211yescharacteristic, complemented
26#10587492C2212yescomplemented
27#10587492C2214yescomplemented
28#10587489C2211yescharacteristic, complemented
29C23×D17—24no—
30C2×D4×C17D411yescharacteristic
31C22×C4×C17C2312yes—
32C2×C22⋊D17—24no—
33D4×D17—48no—
34C2×D68—28no—
35C2×D68—24no—
36C2×D68—22no—
37C2×D68—22no—
38C2×D68—44no—
39C2×D68C2314yes—
40C2×D68C2312yes—
41C2×C4×D17—22no—
42C2×C8×C17—24no—
43C2×C8×C17C2312yes—
44C2×C8×C17C2311yescharacteristic
45C2×C8×C17C2311yescharacteristic
46C4⋊C4×C17D411yescharacteristic
47C22⋊C4×C17D412yes—
48C42×C17C2311yescharacteristic
49#63227—22no—
50#63227—24no—
51C4⋊Dic17C2312yes—
52D136—28no—
53D136—24no—
54D136—42no—
55D136D414yesdirect product factor
56C23×C17C2×D412yes—
57C22×D17—22no—
58C22×D17—24no—
59C22×D17—48no—
60C22×D17—48no—
61C22×D17—84no—
62D4×C17D814yescomplemented
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17C2×D411yescharacteristic
66C2×C4×C17C2×D412yes—
67C2×C4×C17C2×D411yescharacteristic
68C2×C4×C17C2411yescharacteristic
69C22⋊D17—48no—
70C2×Dic17—22no—
71D68—24no—
72D68—22no—
73D68—24no—
74D68—48no—
75D68—44no—
76D68—82no—
77D68C2×D414yescomplemented
78C4×D17—44no—
79C8×C17—22no—
80C8×C17—21no—
81C8×C17C2×D412yescomplemented
82D4×D8—171nomaximal, Sylow
83C22×C17—22no—
84C22×C17—22no—
85C22×C17C2×D814yescomplemented
86C22×C17C22×D411yescharacteristic
87C2×D17—24no—
88C2×D17—42no—
89C2×D17—42no—
90C2×D17—44no—
91C2×D17—44no—
92C2×D17—44no—
93C2×D17—816no—
94C4×C17—22no—
95C4×C17—21no—
96C4×C17C2×D812yescomplemented
97C4×C17C22×D411yescharacteristic
98C4×C17H(5,𝔽2)11yescharacteristic
99Dic17—42no—
100C22×D8—172no—
101D42—172no—
102#493—171no—
103#466—172no—
104#459—172no—
105#447—174no—
106C4×D8—171no—
107C8×D4D1711yescharacteristic, complemented
108C2×C17—24no—
109C2×C17D4211yescharacteristic
110C2×C17C22×D811yescharacteristic
111C2×C17#57611yescharacteristic
112D17—44no—
113D17—84no—
114C22×D4—174no—
115C22×D4—342no—
116C2×D8—171no—
117C2×D8—174no—
118C2×D8—172no—
119C2×D8—172no—
120C2×D8—348no—
121C22×C8C2×D1712yescomplemented
122#127—172no—
123#121—174no—
124#121—342no—
125#120—344no—
126C4×D4—172no—
127C4×D4C2×D1711yescharacteristic
128#107—171no—
129C4⋊C8C2×D1711yescharacteristic, complemented
130#102—172no—
131#102—174no—
132C22⋊C8C2×D1712yescomplemented
133C4×C8C2×D1711yescharacteristic, complemented
134C17D4×D811yescharacteristic, complemented, Sylow
135C24—344no—
136C2×D4—172no—
137C2×D4—174no—
138C2×D4—348no—
139C2×D4—344no—
140C2×D4—344no—
141C2×D4—342no—
142C2×D4—342no—
143C2×D4—684no—
144C2×D4—688no—
145C2×D4D6811yescharacteristic
146C22×C4—342no—
147C22×C4C22×D1712yes—
148D8—174no—
149D8—348no—
150D8—344no—
151D8—682no—
152C2×C8—24no—
153C2×C8C22×D1711yescharacteristic
154C2×C8C22×D1712yes—
155C2×C8C22×D1711yescharacteristic
156C4⋊C4—172no—
157C4⋊C4D6811yescharacteristic
158C22⋊C4—342no—
159C22⋊C4—344no—
160C22⋊C4D6812yes—
161C42C22×D1711yescharacteristic
162C23—342no—
163C23—344no—
164C23—688no—
165C23—688no—
166C23—1364no—
167C23C2×D6812yes—
168D4—174no—
169D4—344no—
170D4—342no—
171D4—344no—
172D4—688no—
173D4—688no—
174D4—684no—
175D4—1362no—
176D4D13614yesdirect product factor
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—342no—
180C2×C4—684no—
181C2×C4C2×D6811yescharacteristic
182C2×C4C2×D6812yes—
183C2×C4C2×D6811yescharacteristic
184C2×C4C23×D1711yescharacteristic
185C8—22no—
186C8—21no—
187C8D4×D1712yescomplemented
188C22—22no—
189C22—22no—
190C22—344no—
191C22—682no—
192C22—682no—
193C22—684no—
194C22—684no—
195C22—684no—
196C22—13616no—
197C22C2×D13614yescomplemented
198C22C22×D6811yescharacteristic
199C4—21no—
200C4—22no—
201C4—682no—
202C4C2×D13612yescomplemented
203C4C2×D4×D1711yescharacteristic
204C4#1058769611yescharacteristic
205C2—24no—
206C2—684no—
207C2—1364no—
208C2D4×D6811yescharacteristic
209C2C22×D13611yescharacteristic
210C2#1176361411yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC8×D4×C17C2
socleC22×C17C22×D4
hypercenterC8×D4D17
hypocenterC17D4×D8
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×D136C2semidirect
2. D4×D68C2semidirect
3. #11763186C2semidirect
4. #11763045C2semidirect
5. #11763005C2semidirect
6. #11762945C2semidirect
7. C4×D136C2semidirect
8. C8×D4×C17C2semidirect
10. C22×D68C22semidirect
11. C22×C8×C17C22semidirect
13. #10587616C22semidirect
16. C2×D136C22semidirect
17. C2×D136C22semidirect
18. C2×D136C22semidirect
21. #10587560C22semidirect
22. C4×D68C22semidirect
23. C17×C4⋊C8C22semidirect
24. C17×C22⋊C8C22semidirect
25. C4×C8×C17C22semidirect
26. #10587492C22semidirect
27. #10587492C22semidirect
28. #10587489C22semidirect
55. D136D4direct (N × H)
62. D4×C17D8semidirect
77. D68C2×D4semidirect
81. C8×C17C2×D4semidirect
85. C22×C17C2×D8semidirect
96. C4×C17C2×D8semidirect
107. C8×D4D17semidirect
121. C22×C8C2×D17semidirect
129. C4⋊C8C2×D17semidirect
132. C22⋊C8C2×D17semidirect
133. C4×C8C2×D17semidirect
134. C17D4×D8semidirect
176. D4D136direct (N × H)
187. C8D4×D17semidirect
197. C22C2×D136semidirect
202. C4C2×D136semidirect

Maximal subgroups

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

subgroupindex
C22×D1362
D4×D682
#117631862
#117630452
#117630052
#117629452
C4×D1362
C8×D4×C172
D4×D817