Finite Group Databasebeta

← #28344533

Subgroups — #28344533

Order 13696.

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

normal subgroups only

This group has 456 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×#16812103C211yesmaximal, characteristic, complemented
2#20292560C213yesmaximal, complemented
3#20292550C213yesmaximal, complemented
4#20292549C213yesmaximal, complemented
5#20292543C213yesmaximal
6#20292497C211yesmaximal, characteristic, complemented
7C2×#126×C107C211yesmaximal, characteristic, complemented
8C22×C22⋊D107C2211yescharacteristic
9#16812103—24no—
10C2×C4⋊Dic107C2213yes—
11C2×C4⋊Dic107C2213yes—
12C2×Dic107⋊C4C2213yes—
13C2×Dic107⋊C4C2213yes—
14C2×C22⋊Dic107C2213yes—
15C2×(C2×D107)⋊C4C2213yes—
16C2×(C2×D107)⋊C4C2213yes—
17C107⋊#95C2213yes—
18C107⋊#95C2213yes—
19#126×C107—24no—
20C2×C4⋊C4×C107C2213yes—
21C2×C22⋊C4×C107C2213yes—
22C2×C42×C107C2211yescharacteristic
23C23×D107C2311yescharacteristic
24C2×C22⋊D107—43no—
25C22×Dic107C2313yes—
26C22×Dic107C2313yes—
27C22×Dic107C2311yescharacteristic
28C4⋊Dic107—43no—
29C4⋊Dic107—43no—
30Dic107⋊C4—26no—
31Dic107⋊C4—43no—
32C22⋊Dic107—43no—
33(C2×D107)⋊C4—26no—
34(C2×D107)⋊C4—43no—
35C24×C107C2311yescharacteristic
36C22×C4×C107C2313yes—
37C22×C4×C107C2313yes—
38C4⋊C4×C107—26no—
39C22⋊C4×C107—26no—
40C42×C107D414yescomplemented
41C23×C107—21no—
42C23×C107—23no—
43C23×C107—23no—
44C23×C107C2411yescharacteristic
45C22×D107—21no—
46C22×D107—23no—
47C22×D107—23no—
48C2×C4×C107—23no—
49C2×C4×C107—23no—
50C2×C4×C107—23no—
51C2×C4×C107—23no—
52C2×C4×C107—23no—
53C2×C4×C107C2×D416yes—
54C22⋊D107—44no—
55C2×Dic107—23no—
56C2×Dic107—23no—
57C2×Dic107—23no—
58C2×Dic107—23no—
59C2×Dic107—23no—
60C2×Dic107—23no—
61C2×Dic107—23no—
62C22×C107—43no—
63C22×C107—41no—
64C22×C107—43no—
65C22×C107C22×D411yescharacteristic
66C22×C107H(5,𝔽2)13yes—
67C22×C1072−1+413yes—
68C2×D107—43no—
69C2×D107—41no—
70C2×D107—43no—
71C4×C107—26no—
72C4×C107—43no—
73Dic107—43no—
74Dic107—43no—
75Dic107—41no—
76C2×C107—81no—
77C2×C107#53713yes—
78C2×C107#56011yescharacteristic
79C2×C107#56313yes—
80D107—81no—
81#2823—1071nomaximal, Sylow
82C107#282311yescharacteristic, complemented, Sylow
83C2×#126—1071no—
84C2×#126D10711yescharacteristic, complemented
85#398—1073no—
86#396—1073no—
87#396—1073no—
88#394—1073no—
89#384—1071no—
90C22×D4—1071no—
91#126—24no—
92#126—2144no—
93C2×C4⋊C4—1073no—
94C2×C4⋊C4—1073no—
95C2×C4⋊C4—1073no—
96C2×C4⋊C4—1073no—
97C2×C4⋊C4C2×D10713yes—
98C2×C22⋊C4—1073no—
99C2×C22⋊C4—1073no—
100C2×C22⋊C4—1073no—
101C2×C22⋊C4C2×D10713yes—
102C2×C42C2×D10711yescharacteristic
103#95—1073no—
104#95—1073no—
105C24—1071no—
106C24C22×D10711yescharacteristic
107C2×D4—4283no—
108C22×C4—1071no—
109C22×C4—1073no—
110C22×C4—1073no—
111C22×C4C22×D10713yes—
112C22×C4C22×D10713yes—
113C4⋊C4—26no—
114C4⋊C4—4283no—
115C4⋊C4—4283no—
116C4⋊C4—4283no—
117C4⋊C4—2146no—
118C22⋊C4—26no—
119C22⋊C4—4283no—
120C22⋊C4—4283no—
121C22⋊C4—2146no—
122C42C22⋊D10714yescomplemented
123C23—21no—
124C23—23no—
125C23—23no—
126C23—2141no—
127C23—2143no—
128C23—2143no—
129C23C23×D10711yescharacteristic
130D4—4284no—
131C2×C4—23no—
132C2×C4—23no—
133C2×C4—23no—
134C2×C4—23no—
135C2×C4—23no—
136C2×C4—2143no—
137C2×C4—2143no—
138C2×C4—2143no—
139C2×C4—2143no—
140C2×C4—2143no—
141C2×C4—2143no—
142C2×C4—2143no—
143C2×C4C2×C22⋊D10716yes—
144C22—43no—
145C22—43no—
146C22—41no—
147C22—4283no—
148C22—4283no—
149C22—4281no—
150C22C22×C22⋊D10711yescharacteristic
151C22#1681215513yes—
152C22#1681215613yes—
153C4—26no—
154C4—43no—
155C4—4283no—
156C4—4281no—
157C4—4283no—
158C2—81no—
159C2—8561no—
160C2#2029320513yes—
161C2#2029336811yescharacteristic
162C2#2029337913yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#126×C107C2
prime radicalC22×C22⋊D107C22
hypercenterC2×#126D107
hypocenterC107#2823
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#16812103C2semidirect
2. #20292560C2semidirect
3. #20292550C2semidirect
4. #20292549C2semidirect
6. #20292497C2semidirect
7. C2×#126×C107C2semidirect
40. C42×C107D4semidirect
82. C107#2823semidirect
84. C2×#126D107semidirect
122. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121032
#202925602
#202925502
#202925492
#202925432
#202924972
C2×#126×C1072
#2823107