Finite Group Databasebeta

← #28338444

Subgroups — #28338444

Order 13696.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#20293262C211yesmaximal, characteristic, complemented
2#20292814C212yesmaximal, complemented
3#20292420C211yesmaximal, characteristic, complemented
4#20292415C212yesmaximal, complemented
5#453×C107C211yesmaximal, characteristic, complemented
6C2×D4×D107—22no—
7#16812080—22no—
8#16812077—22no—
9#16812077—21no—
10#16812077C2212yes—
11C4×D428—21no—
12#16812024—22no—
13#16812024C2212yescomplemented
14H(5,𝔽2)×C107C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C107—21no—
16#127×C107C2211yescharacteristic
17#104×C107—21no—
18#100×C107C2211yescharacteristic, complemented
19C23×D107—21no—
20C23×D107D412yescomplemented
21C2×D428—21no—
22C2×C22⋊D107—22no—
23C2×C22⋊D107—22no—
24C2×C22⋊D107—42no—
25D4×D107—48no—
26C2×C4×D107—22no—
27C4⋊Dic107—21no—
28C22⋊Dic107—21no—
29C22⋊Dic107—22no—
30C22⋊Dic107D412yescomplemented
31(C2×D107)⋊C4—22no—
32(C2×D107)⋊C4—42no—
33#41×C107—21no—
34#41×C107—41no—
35C2×D4×C107—22no—
36C2×D4×C107—22no—
37C2×D4×C107—41no—
38C2×D4×C107D411yescharacteristic, complemented
39C2×D4×C107C2311yescharacteristic
40SD16×C107—41no—
41D8×C107—41no—
42M16×C107—21no—
43C42×C107D411yescharacteristic, complemented
44C23×C107—22no—
45C23×C107—21no—
46C23×C107C2×D412yes—
47C22×D107—22no—
48C22×D107—44no—
49C22×D107—44no—
50C22×D107—42no—
51C22×D107—81no—
52Q8×C107—21no—
53D4×C107—21no—
54D4×C107—41no—
55D4×C107—41no—
56D4×C107—44no—
57D4×C107—42no—
58D4×C107—41no—
59C2×C4×C107—21no—
60C2×C4×C107—22no—
61C2×C4×C107—41no—
62C2×C4×C107C2×D411yescharacteristic
63C22⋊D107—44no—
64C22⋊D107—44no—
65C22⋊D107—84no—
66C2×Dic107—22no—
67C2×Dic107—42no—
68D428—24no—
69C4×D107—44no—
70C8×C107—41no—
71C22×C107—22no—
72C22×C107—42no—
73C22×C107—42no—
74C22×C107—41no—
75C22×C107—41no—
76C22×C107—82no—
77C22×C107#12011yescharacteristic
78C2×D107—24no—
79C2×D107—42no—
80C2×D107—42no—
81C2×D107—44no—
82C2×D107—82no—
83C2×D107—84no—
84C4×C107—22no—
85C4×C107—21no—
86C4×C107—41no—
87Dic107—42no—
88Dic107—82no—
89C2×C107—21no—
90C2×C107—42no—
91C2×C107—81no—
92C2×C107—81no—
93C2×C107B(4,𝔽2)11yescharacteristic
94D107—44no—
95#2196—1071nomaximal, Sylow
96C107#219611yescharacteristic, complemented, Sylow
97D42—1071no—
98B(4,𝔽2)—1072no—
99#453D10711yescharacteristic, complemented
100#353—1071no—
101AΣL(1,𝔽16)—1072no—
102H(5,𝔽2)C2×D10711yescharacteristic, complemented
103C22×D4—2142no—
104AGL(1,ℤ/8)—21no—
105#127C2×D10711yescharacteristic
106#121—2142no—
107#120—1072no—
108#120—2142no—
109#120—2141no—
110C4×D4—2141no—
111#104—21no—
112#100C2×D10711yescharacteristic, complemented
113#99—1072no—
114#99—2142no—
115C24—1072no—
116C24—2141no—
117#41—21no—
118#41—41no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—41no—
122C2×D4—4288no—
123C2×D4—4282no—
124C2×D4—2141no—
125C2×D4—2142no—
126C2×D4—2142no—
127C2×D4C22⋊D10711yescharacteristic, complemented
128C2×D4C22×D10711yescharacteristic
129C22×C4—2142no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—2141no—
134C22⋊C4—1072no—
135C22⋊C4—4282no—
136C22⋊C4—2142no—
137C22⋊C4—2141no—
138C22⋊C4—2142no—
139C42C22⋊D10711yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—8561no—
143C23—4282no—
144C23—4284no—
145C23—4284no—
146C23—2142no—
147C23D4×D10712yes—
148Q8—21no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—42no—
154D4—41no—
155D4—8564no—
156D4—4284no—
157D4—4284no—
158D4—2144no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—4284no—
163C2×C4—4282no—
164C2×C4—2142no—
165C2×C4C2×C22⋊D10711yescharacteristic
166C8—41no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—82no—
173C22—8562no—
174C22—8564no—
175C22—4282no—
176C22—4284no—
177C22—4282no—
178C22—2144no—
179C22#1681207711yescharacteristic
180C4—22no—
181C4—21no—
182C4—41no—
183C4—8562no—
184C4—4282no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—81no—
189C2—4284no—
190C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×D4×C107C23
FrattiniC2×D4C22×D107
Fitting#453×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#453D107
hypocenterC107#2196
LCSG ▷ C2×D4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20293262C2semidirect
2. #20292814C2semidirect
3. #20292420C2semidirect
4. #20292415C2semidirect
5. #453×C107C2semidirect
13. #16812024C22semidirect
14. H(5,𝔽2)×C107C22semidirect
18. #100×C107C22semidirect
20. C23×D107D4semidirect
30. C22⋊Dic107D4semidirect
38. C2×D4×C107D4semidirect
43. C42×C107D4semidirect
96. C107#2196semidirect
99. #453D107semidirect
102. H(5,𝔽2)C2×D107semidirect
112. #100C2×D107semidirect
127. C2×D4C22⋊D107semidirect
139. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202932622
#202928142
#202924202
#202924152
#453×C1072
#2196107