Finite Group Databasebeta

← #28338477

Subgroups — #28338477

Order 13696.

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

normal subgroups only

This group has 248 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#20292800C212yesmaximal, complemented
3#20292424C211yesmaximal, characteristic, complemented
4#20292423C212yesmaximal, complemented
5#454×C107C211yesmaximal, characteristic, complemented
6#16812134—22no—
7#16812098C2212yes—
8#16812097—21no—
9#16812095—22no—
10#16812079—22no—
11#16812077—21no—
12#16812036—22no—
13#16812028C2212yescomplemented
142−1+4×C107C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C107—21no—
16#124×C107C2211yescharacteristic
17#104×C107—21no—
18#101×C107C2211yescharacteristic, complemented
19C23×D107—21no—
20C2×Dic214—21no—
21C2×D428—21no—
22C2×D428D412yes—
23C2×C22⋊D107—22no—
24#13133805—42no—
25#13133803—42no—
26#13133802—22no—
27C22⋊Dic107—21no—
28(C2×D107)⋊C4—22no—
29(C2×D107)⋊C4—22no—
30(C2×D107)⋊C4—22no—
31C4×Dic107D412yes—
32#41×C107—22no—
33#41×C107—21no—
34#41×C107—41no—
35C2×Q8×C107—41no—
36C2×Q8×C107C2311yescharacteristic
37C2×D4×C107D411yescharacteristic
38SD16×C107—41no—
39D8×C107—41no—
40M16×C107—21no—
41C22⋊C4×C107—22no—
42C42×C107D411yescharacteristic
43C23×C107—21no—
44C22×D107—22no—
45C22×D107—82no—
46C22×D107—81no—
47Q8×C107—21no—
48Q8×C107—22no—
49Q8×C107—41no—
50D4×C107—21no—
51D4×C107—22no—
52D4×C107—41no—
53D4×C107—41no—
54C2×C4×C107—21no—
55C2×C4×C107—42no—
56C2×C4×C107—41no—
57C2×C4×C107C2×D412yes—
58C2×C4×C107C2×D411yescharacteristic
59C22⋊D107—82no—
60C2×Dic107—22no—
61D428—42no—
62D428—81no—
63Dic214—81no—
64C8×C107—41no—
65C107⋊C8—42no—
66C22×C107—42no—
67C22×C107—41no—
68C22×C107—41no—
69C22×C107#12011yescharacteristic
70C2×D107—42no—
71C2×D107—42no—
72C2×D107—42no—
73C2×D107—84no—
74C4×C107—22no—
75C4×C107—21no—
76C4×C107—41no—
77C4×C107—41no—
78Dic107—42no—
79C2×C107—21no—
80C2×C107—81no—
81C2×C107—81no—
82C2×C107B(4,𝔽2)11yescharacteristic
83D107—82no—
84#2202—1071nomaximal, Sylow
85C107#220211yescharacteristic, complemented, Sylow
86H(3,𝔽4)—1071no—
87#454—1072no—
88#454D10711yescharacteristic, complemented
89#355—1071no—
90#355—1072no—
912−1+4C2×D10711yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—2142no—
94#124—1072no—
95#124—2142no—
96#124—2141no—
97#124C2×D10711yescharacteristic
98#120—2142no—
99#120—2141no—
100#104—21no—
101#104—2142no—
102#101—1072no—
103#101C2×D10711yescharacteristic, complemented
104C24—2141no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—2141no—
110C2×Q8C22×D10711yescharacteristic
111C2×D4—1072no—
112C2×D4—2141no—
113C2×D4—2142no—
114C2×D4C22⋊D10711yescharacteristic
115SD16—41no—
116SD16—4282no—
117D8—41no—
118D8—4282no—
119M16—21no—
120M16—2142no—
121C22⋊C4—22no—
122C22⋊C4—2142no—
123C22⋊C4—2141no—
124C22⋊C4—2142no—
125C22⋊C4—2142no—
126C42—1072no—
127C42C22⋊D10711yescharacteristic
128C23—21no—
129C23—8562no—
130C23—8561no—
131C23—2142no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—8561no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—8561no—
141D4—8562no—
142D4—4282no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—2142no—
147C2×C4D4×D10712yes—
148C2×C4C2×C22⋊D10711yescharacteristic
149C8—41no—
150C8—4282no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—8564no—
155C22—4282no—
156C22—4282no—
157C22—4282no—
158C22#1681207711yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—4282no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—8562no—
168C2#2029281311yescharacteristic
subgroupG/H
centerC2#20292813
commutator, cosocleC2×Q8×C107C23
FrattiniC2×Q8C22×D107
Fitting#454×C107C2
socleC2×C107B(4,𝔽2)
hypercenter#454D107
hypocenterC107#2202
LCSG ▷ C2×Q8×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #20293369C2semidirect
2. #20292800C2semidirect
3. #20292424C2semidirect
4. #20292423C2semidirect
5. #454×C107C2semidirect
13. #16812028C22semidirect
14. 2−1+4×C107C22semidirect
18. #101×C107C22semidirect
85. C107#2202semidirect
88. #454D107semidirect
91. 2−1+4C2×D107semidirect
103. #101C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933692
#202928002
#202924242
#202924232
#454×C1072
#2202107