Finite Group Databasebeta

← #28335420

Subgroups — #28335420

Order 13696.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293253C211yesmaximal, characteristic
2#20292785C212yesmaximal, complemented
3#20292539C211yesmaximal, characteristic
4#20292365C212yesmaximal, complemented
5#323×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7C2×Dic428—22no—
8#16812088—21no—
9#16812088C2212yes—
10#16812086—22no—
11#16812086—22no—
12C2×C22⋊Dic107—21no—
13C2×C22⋊Dic107—21no—
14#16812034—22no—
15C107⋊#95—21no—
16C107⋊#95C2211yescharacteristic
17C2×C22⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19C2×Dic214—22no—
20C2×Dic214—21no—
21C2×Dic214—42no—
22C2×Dic214D412yes—
23C22×Dic107—21no—
24C22×Dic107—21no—
25C22×Dic107—21no—
26C22×Dic107—21no—
27Dic428—44no—
28C4⋊Dic107—21no—
29C4⋊Dic107D412yes—
30Dic107⋊C4—22no—
31Dic107⋊C4—22no—
32C22⋊Dic107—22no—
33C22⋊Dic107—22no—
34C22⋊Dic107—42no—
35C24×C107D411yescharacteristic
36C22×C4×C107D411yescharacteristic
37C22×C4×C107C2311yescharacteristic
38C2×C8×C107—22no—
39C22⋊C4×C107—22no—
40C23×C107—21no—
41C23×C107—21no—
42C23×C107—41no—
43C23×C107—41no—
44C23×C107Q1612yescomplemented
45C23×C107C2×D411yescharacteristic
46C2×C4×C107—21no—
47C2×C4×C107—41no—
48C2×C4×C107—41no—
49C2×C4×C107C2×D412yes—
50C2×Dic107—22no—
51C2×Dic107—22no—
52C2×Dic107—22no—
53C2×Dic107—41no—
54C2×Dic107—41no—
55C2×Dic107—41no—
56C2×Dic107—41no—
57C2×Dic107—41no—
58C2×Dic107—81no—
59C2×Dic107—81no—
60Dic214—24no—
61Dic214—42no—
62Dic214—42no—
63Dic214—81no—
64C8×C107—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—
77Dic107—42no—
78Dic107—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#543—1071no—
90#451—1072no—
91#393—1071no—
92#328—1072no—
93#323D10711yescharacteristic
94C22×Q8—1071no—
95C2×Q16—2142no—
96C22⋊Q8—1072no—
97C22⋊Q8—2142no—
98C22⋊Q8—2142no—
99C22⋊Q8—2141no—
100C2×C22⋊C4—2141no—
101C2×C22⋊C4—2141no—
102C2×C22⋊C4C2×D10711yescharacteristic
103#103—2142no—
104C22⋊C8C2×D10712yes—
105#95—1071no—
106#95—2141no—
107C24D42811yescharacteristic
108C2×Q8—1072no—
109C2×Q8—4282no—
110C2×Q8—2141no—
111C2×Q8—2142no—
112C22×C4—2141no—
113C22×C4—2141no—
114C22×C4—2141no—
115C22×C4—2141no—
116C22×C4D42811yescharacteristic
117C22×C4C22×D10711yescharacteristic
118Q16—4284no—
119C2×C8—22no—
120C4⋊C4—1072no—
121C4⋊C4—2142no—
122C4⋊C4—2141no—
123C4⋊C4—2142no—
124C22⋊C4—22no—
125C22⋊C4—4282no—
126C22⋊C4—2142no—
127C22⋊C4—2142no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic42812yescomplemented
133C23C2×D42811yescharacteristic
134Q8—8561no—
135Q8—4282no—
136Q8—4282no—
137Q8—2144no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—8561no—
142C2×C4—8561no—
143C2×C4—4281no—
144C2×C4—4281no—
145C2×C4—4281no—
146C2×C4—4281no—
147C2×C4—4281no—
148C2×C4—2142no—
149C2×C4—2142no—
150C2×C4—2142no—
151C2×C4D4×D10712yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1681207911yescharacteristic
161C22C2×Dic42811yescharacteristic
162C22#1681214211yescharacteristic
163C4—22no—
164C4—41no—
165C4—8561no—
166C4—4282no—
167C4—4282no—
168C4—4282no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2029278511yescharacteristic
173C2#2029280711yescharacteristic
174C2#2029281511yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#323×C107C2
socleC22×C107#120
prime radicalC24×C107D4
hypercenter#323D107
hypocenterC107#1602
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20292785C2semidirect
4. #20292365C2semidirect
44. C23×C107Q16semidirect
88. C107#1602semidirect
132. C23Dic428semidirect

Maximal subgroups

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

subgroupindex
#202932532
#202927852
#202925392
#202923652
#323×C1072
#1602107