Finite Group Databasebeta

← #28348136

Subgroups — #28348136

Order 13696.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×D8×D107C212yesmaximal, complemented
2#20293262C212yesmaximal, complemented
3#20292860C212yesmaximal, complemented
4#20292819C212yesmaximal, complemented
5#20292759C214yesmaximal, complemented
6C4×D856C211yesmaximal, characteristic, complemented
7C8×D428C211yesmaximal, characteristic, complemented
8#493×C107C211yesmaximal, characteristic, complemented
9C2×D4×D107—22no—
10C2×D4×D107C2214yescomplemented
11C2×D856C2211yescharacteristic
12C2×#13133803C2214yescomplemented
13D8×D107—28no—
14C2×C8×D107C2212yescomplemented
15#16812080—22no—
16#16812080C2214yescomplemented
17#16812077—24no—
18C4×D428C2211yescharacteristic
19C4×D428C2212yescomplemented
20#16812043C2211yescharacteristic, complemented
21C4⋊(C107⋊C8)C2211yescharacteristic, complemented
22#16812031C2212yescomplemented
23#16812030C2214yescomplemented
24(C2×D107)⋊C8C2212yescomplemented
25C2×D8×C107C2212yescomplemented
26C2×D8×C107C2212yescomplemented
27#127×C107C2212yescomplemented
28C4×C8×C107C2211yescharacteristic, complemented
29C23×D107—24no—
30C2×D428D411yescharacteristic
31C2×D428C2312yes—
32C2×C22⋊D107—24no—
33C2×C22⋊D107—24no—
34D4×D107—28no—
35D4×D107—44no—
36D4×D107—48no—
37C2×C4×D107—22no—
38C2×C4×D107C2312yes—
39D856D414yescomplemented
40#13133803—28no—
41C2×C107⋊C8C2312yes—
42C8×D107—24no—
43C4⋊Dic107D411yescharacteristic
44C4⋊Dic107C2312yes—
45C22⋊Dic107—24no—
46(C2×D107)⋊C4—22no—
47(C2×D107)⋊C4D412yes—
48C2×D4×C107—22no—
49C2×D4×C107—22no—
50C2×D4×C107C2314yes—
51D8×C107—24no—
52D8×C107—42no—
53C2×C8×C107C2311yescharacteristic
54C2×C8×C107C2311yescharacteristic
55C42×C107C2311yescharacteristic
56C23×C107—24no—
57C22×D107—22no—
58C22×D107—48no—
59C22×D107—48no—
60C22×D107—84no—
61C22×D107C2×D412yes—
62D4×C107—24no—
63D4×C107—24no—
64D4×C107—44no—
65D4×C107—82no—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2411yescharacteristic
69C22⋊D107—48no—
70C22⋊D107—48no—
71C2×Dic107—22no—
72C2×Dic107C2×D412yes—
73D428—22no—
74D428D814yescomplemented
75D428C2×D414yescomplemented
76C4×D107—22no—
77C4×D107—22no—
78C4×D107—44no—
79C8×C107—21no—
80C8×C107C2×D412yescomplemented
81C107⋊C8—22no—
82C22×C107—44no—
83C22×C107—44no—
84C22×C107—44no—
85C22×C107C22×D411yescharacteristic
86C2×D107—22no—
87C2×D107—22no—
88C2×D107—24no—
89C2×D107—42no—
90C2×D107—42no—
91C2×D107—816no—
92C2×D107C2×D814yescomplemented
93C4×C107—21no—
94C4×C107C2×D812yescomplemented
95C4×C107C22×D411yescharacteristic
96C4×C107H(5,𝔽2)11yescharacteristic
97Dic107—22no—
98Dic107—42no—
99C2×C107—84no—
100C2×C107D4211yescharacteristic
101C2×C107C22×D811yescharacteristic
102C2×C107#57611yescharacteristic
103D107—24no—
104D107—44no—
105D4×D8—1071nomaximal, Sylow
106C107D4×D811yescharacteristic, complemented, Sylow
107C22×D8—1072no—
108D42—1072no—
109#493D10711yescharacteristic, complemented
110#466—1072no—
111#459—1072no—
112#447—1074no—
113C4×D8—1071no—
114C8×D4—1071no—
115C22×D4—1074no—
116C22×D4—2142no—
117C2×D8—1074no—
118C2×D8—1071no—
119C2×D8—2148no—
120C2×D8C2×D10712yescomplemented
121C2×D8C2×D10712yescomplemented
122C22×C8—1072no—
123#127C2×D10712yescomplemented
124#121—1074no—
125#121—2142no—
126#120—2144no—
127C4×D4—1071no—
128C4×D4—1072no—
129#107—1071no—
130C4⋊C8—1071no—
131#102—1074no—
132#102—1072no—
133C22⋊C8—1072no—
134C4×C8C2×D10711yescharacteristic, complemented
135C24—2144no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—1071no—
139C2×D4—1072no—
140C2×D4—4288no—
141C2×D4—4284no—
142C2×D4—2144no—
143C2×D4—2148no—
144C2×D4—2144no—
145C2×D4C22×D10714yes—
146C22×C4—1072no—
147C22×C4—2142no—
148D8—24no—
149D8—42no—
150D8—1074no—
151D8—2148no—
152C2×C8—1072no—
153C2×C8—2144no—
154C2×C8C22×D10711yescharacteristic
155C2×C8C22×D10711yescharacteristic
156C4⋊C4—1071no—
157C4⋊C4—1072no—
158C22⋊C4—1072no—
159C22⋊C4—2144no—
160C22⋊C4—2142no—
161C42C22×D10711yescharacteristic
162C23—24no—
163C23—8564no—
164C23—1072no—
165C23—4288no—
166C23—4288no—
167C23—2142no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—82no—
172D4—1074no—
173D4—1074no—
174D4—4288no—
175D4—4288no—
176D4—2142no—
177C2×C4—1072no—
178C2×C4—4284no—
179C2×C4—2142no—
180C2×C4—2142no—
181C2×C4—2142no—
182C2×C4D4×D10711yescharacteristic
183C2×C4D4×D10711yescharacteristic
184C2×C4C23×D10711yescharacteristic
185C8—21no—
186C8—2142no—
187C8D4×D10712yescomplemented
188C22—44no—
189C22—44no—
190C22—44no—
191C22—85616no—
192C22—1074no—
193C22—4282no—
194C22—4282no—
195C22—2144no—
196C22—2142no—
197C22—2142no—
198C22C2×D4×D10711yescharacteristic
199C4—21no—
200C4—4282no—
201C4—2142no—
202C4D8×D10712yescomplemented
203C4C2×D4×D10711yescharacteristic
204C4#1681215411yescharacteristic
205C2—84no—
206C2—4284no—
207C2—2144no—
208C2#2029326211yescharacteristic
209C2C2×D8×D10711yescharacteristic
210C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#493×C107C2
socleC22×C107C22×D4
hypercenter#493D107
hypocenterC107D4×D8
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×D8×D107C2semidirect
2. #20293262C2semidirect
3. #20292860C2semidirect
4. #20292819C2semidirect
5. #20292759C2semidirect
6. C4×D856C2semidirect
7. C8×D428C2semidirect
8. #493×C107C2semidirect
10. C2×D4×D107C22semidirect
12. C2×#13133803C22semidirect
14. C2×C8×D107C22semidirect
16. #16812080C22semidirect
19. C4×D428C22semidirect
20. #16812043C22semidirect
21. C4⋊(C107⋊C8)C22semidirect
22. #16812031C22semidirect
23. #16812030C22semidirect
24. (C2×D107)⋊C8C22semidirect
25. C2×D8×C107C22semidirect
26. C2×D8×C107C22semidirect
27. #127×C107C22semidirect
28. C4×C8×C107C22semidirect
39. D856D4semidirect
74. D428D8semidirect
75. D428C2×D4semidirect
80. C8×C107C2×D4semidirect
92. C2×D107C2×D8semidirect
94. C4×C107C2×D8semidirect
106. C107D4×D8semidirect
109. #493D107semidirect
120. C2×D8C2×D107semidirect
121. C2×D8C2×D107semidirect
123. #127C2×D107semidirect
134. C4×C8C2×D107semidirect
187. C8D4×D107semidirect
202. C4D8×D107semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1072
#202932622
#202928602
#202928192
#202927592
C4×D8562
C8×D4282
#493×C1072
D4×D8107