Finite Group Databasebeta

← #28344501

Subgroups — #28344501

Order 13696.

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

normal subgroups only

This group has 630 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×#16812106C211yesmaximal, characteristic, complemented
2#20292544C216yesmaximal, complemented
3#20292536C216yesmaximal, complemented
4#20292495C211yesmaximal, characteristic, complemented
5C2×#126×C107C211yesmaximal, characteristic, complemented
6C22×D428C2216yes—
7C22×C22⋊D107C2214yes—
8#16812106—24no—
9C2×(C2×D107)⋊C4C22112yes—
10C107⋊#95C2216yes—
11#126×C107—24no—
12C2×C4⋊C4×C107C2213yes—
13C2×C22⋊C4×C107C2213yes—
14C2×C42×C107C2211yescharacteristic
15C23×D107C2314yes—
16C2×D428—212no—
17C2×D428—412no—
18C2×C22⋊D107—412no—
19C22×Dic107C2314yes—
20(C2×D107)⋊C4—412no—
21C24×C107C2311yescharacteristic
22C22×C4×C107C2313yes—
23C22×C4×C107C2313yes—
24C4⋊C4×C107—26no—
25C22⋊C4×C107—26no—
26C42×C107D414yescomplemented
27C23×C107—21no—
28C23×C107—26no—
29C23×C107C2411yescharacteristic
30C22×D107—24no—
31C22×D107—212no—
32C22×D107—212no—
33C2×C4×C107—26no—
34C2×C4×C107—23no—
35C2×C4×C107—26no—
36C2×C4×C107C2×D416yes—
37C22⋊D107—416no—
38C2×Dic107—212no—
39D428—424no—
40C22×C107—43no—
41C22×C107—44no—
42C22×C107C22×D411yescharacteristic
43C22×C107H(5,𝔽2)16yes—
44C2×D107—412no—
45C2×D107—44no—
46C2×D107—412no—
47C4×C107—26no—
48C4×C107—43no—
49Dic107—44no—
50C2×C107—81no—
51C2×C107#53513yes—
52C2×C107#56014yes—
53D107—84no—
54#2818—1071nomaximal, Sylow
55C107#281811yescharacteristic, complemented, Sylow
56C2×#127—1071no—
57C2×#126D10711yescharacteristic, complemented
58#394—1076no—
59#392—1076no—
60#383—1071no—
61C22×D4—1074no—
62C22×D4—1076no—
63#127—2144no—
64#126—24no—
65C2×C4⋊C4C2×D10713yes—
66C2×C22⋊C4—10712no—
67C2×C22⋊C4C2×D10713yes—
68C2×C42C2×D10711yescharacteristic
69#95—1076no—
70C24—1074no—
71C24C22×D10711yescharacteristic
72C2×D4—42812no—
73C2×D4—42812no—
74C2×D4—21412no—
75C22×C4—1074no—
76C22×C4C22×D10713yes—
77C22×C4C22×D10713yes—
78C4⋊C4—26no—
79C22⋊C4—26no—
80C22⋊C4—42812no—
81C42C22⋊D10714yescomplemented
82C23—21no—
83C23—26no—
84C23—2144no—
85C23—21412no—
86C23—21412no—
87C23C23×D10711yescharacteristic
88D4—42824no—
89D4—42816no—
90C2×C4—26no—
91C2×C4—23no—
92C2×C4—26no—
93C2×C4—21412no—
94C2×C4C2×C22⋊D10716yes—
95C22—43no—
96C22—44no—
97C22—42812no—
98C22—4284no—
99C22—42812no—
100C22C22×C22⋊D10711yescharacteristic
101C22#1681215516yes—
102C4—26no—
103C4—43no—
104C4—4284no—
105C2—81no—
106C2—8564no—
107C2#2029319113yes—
108C2#2029336814yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#126×C107C2
hypercenterC2×#126D107
hypocenterC107#2818
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16812106C2semidirect
2. #20292544C2semidirect
3. #20292536C2semidirect
4. #20292495C2semidirect
5. C2×#126×C107C2semidirect
26. C42×C107D4semidirect
55. C107#2818semidirect
57. C2×#126D107semidirect
81. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121062
#202925442
#202925362
#202924952
C2×#126×C1072
#2818107