Finite Group Databasebeta

← #28338461

Subgroups — #28338461

Order 13696.

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

normal subgroups only

This group has 270 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#20293369C211yesmaximal, characteristic, complemented
2#20292814C212yesmaximal, complemented
3#20292422C211yesmaximal, characteristic, complemented
4#20292415C212yesmaximal, complemented
5#455×C107C211yesmaximal, characteristic, complemented
6#16812098—21no—
7#16812097—21no—
8#16812095—22no—
9#16812079—22no—
10#16812077—22no—
11#16812077C2212yes—
12#16812024—22no—
13#16812024C2212yes—
14H(5,𝔽2)×C107C2211yescharacteristic
15#137×C107—21no—
16#124×C107C2211yescharacteristic
17#104×C107—21no—
18#100×C107C2211yescharacteristic
19C23×D107D412yescomplemented
20C2×Dic214—21no—
21C2×D428—21no—
22C2×D428—21no—
23C2×C22⋊D107—22no—
24C2×C22⋊D107—42no—
25C22⋊Dic107—22no—
26C22⋊Dic107D412yescomplemented
27(C2×D107)⋊C4—22no—
28(C2×D107)⋊C4—22no—
29(C2×D107)⋊C4—22no—
30(C2×D107)⋊C4—42no—
31C4×Dic107—21no—
32#41×C107—21no—
33#41×C107—41no—
34C2×Q8×C107D411yescharacteristic, complemented
35C2×D4×C107—22no—
36C2×D4×C107—41no—
37C2×D4×C107C2311yescharacteristic
38Q16×C107—41no—
39SD16×C107—41no—
40M16×C107—21no—
41C22⋊C4×C107—22no—
42C42×C107D411yescharacteristic, complemented
43C23×C107—22no—
44C23×C107C2×D412yes—
45C22×D107—22no—
46C22×D107—42no—
47C22×D107—82no—
48Q8×C107—21no—
49Q8×C107—41no—
50D4×C107—21no—
51D4×C107—41no—
52D4×C107—42no—
53D4×C107—41no—
54C2×C4×C107—22no—
55C2×C4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107—41no—
58C2×C4×C107C2×D411yescharacteristic
59C22⋊D107—82no—
60C22⋊D107—84no—
61C2×Dic107—22no—
62C2×Dic107—42no—
63D428—81no—
64D428—81no—
65Dic214—81no—
66C8×C107—41no—
67C22×C107—22no—
68C22×C107—42no—
69C22×C107—41no—
70C22×C107—82no—
71C22×C107#12011yescharacteristic
72C2×D107—42no—
73C2×D107—42no—
74C2×D107—44no—
75C2×D107—42no—
76C2×D107—82no—
77C4×C107—21no—
78C4×C107—41no—
79C4×C107—41no—
80C4×C107—41no—
81Dic107—42no—
82Dic107—82no—
83C2×C107—21no—
84C2×C107—42no—
85C2×C107—81no—
86C2×C107B(4,𝔽2)11yescharacteristic
87D107—82no—
88#2199—1071nomaximal, Sylow
89C107#219911yescharacteristic, complemented, Sylow
90H(3,𝔽4)—1071no—
91B(4,𝔽2)—1072no—
92#455D10711yescharacteristic, complemented
93#354—1071no—
94AΣL(1,𝔽16)—1072no—
95H(5,𝔽2)C2×D10711yescharacteristic
96#137—21no—
97#124—2142no—
98#124—2141no—
99#124—2141no—
100#124C2×D10711yescharacteristic
101#120—1072no—
102#120—2142no—
103#120—2142no—
104#104—21no—
105#100C2×D10711yescharacteristic
106#99—1072no—
107#99—2142no—
108C24—1072no—
109#41—21no—
110#41—41no—
111C2×Q8—2141no—
112C2×Q8C22⋊D10711yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—4282no—
116C2×D4—2141no—
117C2×D4—2141no—
118C2×D4—2142no—
119C2×D4C22×D10711yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—1072no—
125C22⋊C4—4282no—
126C22⋊C4—2142no—
127C22⋊C4—2142no—
128C22⋊C4—2142no—
129C22⋊C4—2142no—
130C42—2141no—
131C42C22⋊D10711yescharacteristic, complemented
132C23—22no—
133C23—8562no—
134C23—4282no—
135C23—2142no—
136C23D4×D10712yes—
137Q8—21no—
138Q8—41no—
139Q8—8561no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—8561no—
145D4—8561no—
146D4—8564no—
147D4—8562no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—4282no—
153C2×C4—2142no—
154C2×C4C2×C22⋊D10711yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—8562no—
161C22—4282no—
162C22—4282no—
163C22—4284no—
164C22—4282no—
165C22#1681207711yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—8562no—
171C4—4282no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—8562no—
176C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×D4×C107C23
FrattiniC2×D4C22×D107
Fitting#455×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#455D107
hypocenterC107#2199
LCSG ▷ C2×D4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #20293369C2semidirect
2. #20292814C2semidirect
3. #20292422C2semidirect
4. #20292415C2semidirect
5. #455×C107C2semidirect
19. C23×D107D4semidirect
26. C22⋊Dic107D4semidirect
34. C2×Q8×C107D4semidirect
42. C42×C107D4semidirect
89. C107#2199semidirect
92. #455D107semidirect
112. C2×Q8C22⋊D107semidirect
131. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202933692
#202928142
#202924222
#202924152
#455×C1072
#2199107