Finite Group Databasebeta

← #28335418

Subgroups — #28335418

Order 13696.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#20292782C212yesmaximal, complemented
2#20292539C211yesmaximal, characteristic
3#20292364C212yesmaximal, complemented
4#20292351C211yesmaximal, characteristic
5#543×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7C2×#13133808—22no—
8C2×C22⋊Dic107—21no—
9#16812032—22no—
10C22⋊(C107⋊C8)C2212yes—
11C107⋊#95—21no—
12C107⋊#95C2211yescharacteristic
13C22⋊Q8×C107—22no—
14C22⋊Q8×C107—21no—
15C22⋊Q8×C107—22no—
16C22⋊Q8×C107C2212yes—
17C2×C22⋊C4×C107—21no—
18C2×C22⋊C4×C107C2211yescharacteristic
19C2×Dic214—22no—
20C2×Dic214—42no—
21C22×Dic107—21no—
22C22×Dic107—21no—
23#13133808—44no—
24C2×C107⋊C8—22no—
25C22⋊Dic107—42no—
26C24×C107D411yescharacteristic
27C2×Q8×C107—21no—
28C2×Q8×C107D412yes—
29C22×C4×C107—21no—
30C22×C4×C107—21no—
31C22×C4×C107D411yescharacteristic
32C22×C4×C107C2311yescharacteristic
33C4⋊C4×C107—21no—
34C4⋊C4×C107—22no—
35C4⋊C4×C107—22no—
36C4⋊C4×C107D412yes—
37C22⋊C4×C107—22no—
38C22⋊C4×C107—22no—
39C22⋊C4×C107—22no—
40C23×C107—21no—
41C23×C107—21no—
42C23×C107—41no—
43C23×C107—41no—
44C23×C107Q1612yescomplemented
45C23×C107C2×D411yescharacteristic
46Q8×C107—42no—
47Q8×C107—81no—
48C2×C4×C107—22no—
49C2×C4×C107—21no—
50C2×C4×C107—22no—
51C2×C4×C107—41no—
52C2×C4×C107—41no—
53C2×C4×C107—81no—
54C2×C4×C107—81no—
55C2×C4×C107C2×D412yes—
56C2×Dic107—22no—
57C2×Dic107—41no—
58C2×Dic107—41no—
59C2×Dic107—41no—
60C2×Dic107—41no—
61C2×Dic107—41no—
62Dic214—24no—
63Dic214—42no—
64C107⋊C8—42no—
65C22×C107—21no—
66C22×C107—21no—
67C22×C107—22no—
68C22×C107—42no—
69C22×C107—41no—
70C22×C107—41no—
71C22×C107—81no—
72C22×C107#12011yescharacteristic
73C22×C107C2×Q1611yescharacteristic
74C22×C107#13711yescharacteristic
75C4×C107—22no—
76C4×C107—41no—
77C4×C107—42no—
78C4×C107—42no—
79Dic107—42no—
80Dic107—81no—
81C2×C107—21no—
82C2×C107—21no—
83C2×C107—42no—
84C2×C107#45111yescharacteristic
85C2×C107#45511yescharacteristic
86C2×C107B(4,𝔽2)11yescharacteristic
87#1602—1071nomaximal, Sylow
88C107#160211yescharacteristic, complemented, Sylow
89#543D10711yescharacteristic
90#451—1072no—
91#393—1071no—
92#328—1072no—
93#323—1071no—
94C22×Q8—1071no—
95C2×Q16—2142no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D10712yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—2141no—
102C2×C22⋊C4C2×D10711yescharacteristic
103#103—2142no—
104C22⋊C8—1072no—
105#95—1071no—
106#95—2141no—
107C24C22⋊D10711yescharacteristic
108C2×Q8—21no—
109C2×Q8—4282no—
110C2×Q8—2142no—
111C2×Q8C22⋊D10712yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2141no—
115C22×C4—2141no—
116C22×C4C22⋊D10711yescharacteristic
117C22×C4C22×D10711yescharacteristic
118Q16—4284no—
119C2×C8—2142no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D10712yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—4282no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1313380812yescomplemented
133C23C2×C22⋊D10711yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—4282no—
137Q8—2144no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—4281no—
146C2×C4—4281no—
147C2×C4—4281no—
148C2×C4—4281no—
149C2×C4—4281no—
150C2×C4—2142no—
151C2×C4C2×C22⋊D10712yes—
152C8—4282no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1681207811yescharacteristic
161C22C2×#1313380811yescharacteristic
162C22#1681214011yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—8561no—
168C4—4282no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2029278311yescharacteristic
173C2#2029280511yescharacteristic
174C2#2029281411yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#543×C107C2
socleC22×C107#120
prime radicalC24×C107D4
hypercenter#543D107
hypocenterC107#1602
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20292782C2semidirect
3. #20292364C2semidirect
44. C23×C107Q16semidirect
88. C107#1602semidirect
132. C23#13133808semidirect

Maximal subgroups

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

subgroupindex
#202927822
#202925392
#202923642
#202923512
#543×C1072
#1602107