Finite Group Databasebeta

← #28338457

Subgroups — #28338457

Order 13696.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#20292816C212yesmaximal, complemented
2#20292794C211yesmaximal, characteristic
3#20292420C211yesmaximal, characteristic, complemented
4#20292417C212yesmaximal, complemented
5#559×C107C211yesmaximal, characteristic, complemented
6#16812154C2211yescharacteristic
7#16812135—21no—
8#16812092—22no—
9#16812037—21no—
10#16812026C2211yescharacteristic
11#16812024C2212yes—
12#16812023—22no—
13#127×C107C2211yescharacteristic
14#124×C107—22no—
15#123×C107—21no—
16#123×C107C2212yes—
17C4×Q8×C107—21no—
18C4×D4×C107—22no—
19#13133818—21no—
20#13133817—41no—
21C2×C22⋊D107—22no—
22D4×D107—41no—
23#13133806—41no—
24#13133803—41no—
25#13133802—21no—
26Dic107⋊C4—42no—
27C22⋊Dic107—22no—
28(C2×D107)⋊C4—42no—
29C2×Q8×C107—21no—
30C2×D4×C107—22no—
31C2×D4×C107D411yescharacteristic
32C2×D4×C107C2311yescharacteristic
33C22×C4×C107—21no—
34C22×C4×C107D412yes—
35C4⋊C4×C107—21no—
36C4⋊C4×C107—22no—
37C22⋊C4×C107—22no—
38C22⋊C4×C107—21no—
39C22⋊C4×C107—22no—
40C22⋊C4×C107D412yes—
41C42×C107—22no—
42C42×C107D411yescharacteristic
43C23×C107—21no—
44C23×C107C2×D412yes—
45C22×D107—22no—
46Q8×C107—42no—
47D4×C107—41no—
48D4×C107—41no—
49D4×C107—44no—
50C2×C4×C107—21no—
51C2×C4×C107—22no—
52C2×C4×C107—22no—
53C2×C4×C107—42no—
54C2×C4×C107—81no—
55C2×C4×C107C2×D411yescharacteristic
56C22⋊D107—42no—
57C22⋊D107—41no—
58C2×Dic107—22no—
59C2×Dic107—42no—
60D428—21no—
61C4×D107—41no—
62Dic214—21no—
63C107⋊C8—41no—
64C22×C107—22no—
65C22×C107—42no—
66C22×C107—42no—
67C22×C107—41no—
68C22×C107#12011yescharacteristic
69C2×D107—41no—
70C2×D107—82no—
71C4×C107—22no—
72C4×C107—21no—
73C4×C107—42no—
74C4×C107—42no—
75Dic107—41no—
76Dic107—82no—
77C2×C107—21no—
78C2×C107—42no—
79C2×C107—81no—
80C2×C107B(4,𝔽2)11yescharacteristic
81D107—81no—
82#2198—1071nomaximal, Sylow
83C107#219811yescharacteristic, complemented, Sylow
84#559D10711yescharacteristic, complemented
85#458—1072no—
86#453—1071no—
87#353—1071no—
88#352—1072no—
89H(5,𝔽2)—1071no—
90AGL(1,ℤ/8)—2141no—
91#127C2×D10711yescharacteristic
92#124—22no—
93#123—21no—
94#123—2142no—
95#123C2×D10712yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—2141no—
99#100—1071no—
100#99—1072no—
101#99—2142no—
102#41—4281no—
103#41—2141no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—4281no—
107C2×D4—2142no—
108C2×D4C22⋊D10711yescharacteristic
109C2×D4C22×D10711yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D10712yes—
112SD16—4281no—
113D8—4281no—
114M16—2141no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—4282no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—4282no—
122C22⋊C4—2142no—
123C22⋊C4C22⋊D10712yes—
124C42—22no—
125C42C22⋊D10711yescharacteristic
126C23—21no—
127C23—2142no—
128C23C2×C22⋊D10712yes—
129Q8—42no—
130Q8—2141no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—4282no—
135D4—4281no—
136D4—2141no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—4281no—
143C2×C4—4282no—
144C2×C4—2142no—
145C2×C4C2×C22⋊D10711yescharacteristic
146C8—4281no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—8562no—
152C22—4281no—
153C22#1681207811yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—8562no—
159C4—4281no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—8561no—
164C2#2029281411yescharacteristic
subgroupG/H
centerC2#20292814
commutator, cosocleC2×D4×C107C23
FrattiniC2×D4C22×D107
Fitting#559×C107C2
socleC2×C107B(4,𝔽2)
prime radical#20292794C2
hypercenter#559D107
hypocenterC107#2198
LCSG ▷ C2×D4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #20292816C2semidirect
3. #20292420C2semidirect
4. #20292417C2semidirect
5. #559×C107C2semidirect
83. C107#2198semidirect
84. #559D107semidirect

Maximal subgroups

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

subgroupindex
#202928162
#202927942
#202924202
#202924172
#559×C1072
#2198107