Finite Group Databasebeta

← #28336860

Subgroups — #28336860

Order 13696.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#20293102C211yesmaximal, characteristic, complemented
2#20292632C212yesmaximal, complemented
3C2×#16812026C211yesmaximal, characteristic, complemented
4#20292387C212yesmaximal, complemented
5C2×#127×C107C211yesmaximal, characteristic
6C2×#13133802C2212yes—
7D4×Dic107—24no—
8#16812068C2212yes—
9C2×C22⋊Dic107—22no—
10#16812030—24no—
11#16812026—22no—
12C22×D4×C107—22no—
13C22×D4×C107C2211yescharacteristic
14C22×D4×C107C2211yescharacteristic
15#127×C107—22no—
16#127×C107C414yescomplemented
17C2×C42×C107C2211yescharacteristic
18C22×Dic107—24no—
19#13133802—22no—
20#13133802—22no—
21C2×C107⋊C8—22no—
22C4⋊Dic107D414yescomplemented
23C22⋊Dic107—24no—
24C22⋊Dic107—22no—
25C4×Dic107—22no—
26C24×C107—21no—
27C24×C107D412yes—
28C2×D4×C107—28no—
29C2×D4×C107—22no—
30C2×D4×C107—22no—
31C2×D4×C107—21no—
32C2×D4×C107—22no—
33C2×D4×C107—22no—
34C2×D4×C107—48no—
35C2×D4×C107C2×C412yes—
36C2×D4×C107C2×C412yescomplemented
37C2×D4×C107D412yes—
38C22×C4×C107—21no—
39C22×C4×C107C2311yescharacteristic
40C42×C107—21no—
41C42×C107C2×C412yescomplemented
42C23×C107—22no—
43C23×C107—22no—
44C23×C107—22no—
45C23×C107—44no—
46C23×C107—44no—
47C23×C107—41no—
48C23×C107—41no—
49C23×C107C22⋊C414yes—
50C23×C107C2×D411yescharacteristic
51D4×C107—24no—
52D4×C107—24no—
53D4×C107—24no—
54D4×C107—416no—
55D4×C107—42no—
56C2×C4×C107—24no—
57C2×C4×C107—22no—
58C2×C4×C107C22×C411yescharacteristic
59C2×C4×C107C2×D412yes—
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107#4112yes—
62C2×Dic107—24no—
63C2×Dic107—84no—
64C107⋊C8—42no—
65C22×C107—22no—
66C22×C107—24no—
67C22×C107—44no—
68C22×C107—44no—
69C22×C107—44no—
70C22×C107—42no—
71C22×C107—82no—
72C22×C107—82no—
73C22×C107—82no—
74C22×C107C2×C22⋊C411yescharacteristic
75C22×C107#12011yescharacteristic
76C22×C107#12311yescharacteristic
77C4×C107—24no—
78C4×C107C4×D412yes—
79C4×C107#12112yes—
80Dic107—44no—
81C2×C107—21no—
82C2×C107—21no—
83C2×C107—44no—
84C2×C107—82no—
85C2×C107#38611yescharacteristic
86C2×C107#45312yes—
87#1887—1071nomaximal, Sylow
88C107#188711yescharacteristic, complemented, Sylow
89C2×#127D10711yescharacteristic
90#518—1071no—
91#418—1072no—
92C2×#100—1071no—
93#339—1072no—
94C22×D4—22no—
95C22×D4C2×D10711yescharacteristic
96C22×D4C2×D10711yescharacteristic
97C2×M16—1072no—
98#127—22no—
99#127Dic10714yescomplemented
100C4×D4—2144no—
101#117—1072no—
102C2×C22⋊C4—2142no—
103C2×C42C2×D10711yescharacteristic
104#102—2144no—
105#100—2142no—
106C24—21no—
107C24C22⋊D10712yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic10712yes—
116C2×D4C2×Dic10712yescomplemented
117C2×D4C22⋊D10712yes—
118C22×C4—21no—
119C22×C4—2144no—
120C22×C4C22×D10711yescharacteristic
121M16—2142no—
122M16—2142no—
123C2×C8—2142no—
124C4⋊C4—1074no—
125C22⋊C4—2144no—
126C22⋊C4—2142no—
127C42—21no—
128C42—2142no—
129C42C2×Dic10712yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic10714yes—
138C23C2×C22⋊D10711yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—8564no—
147C2×C4—2144no—
148C2×C4C22×Dic10711yescharacteristic
149C2×C4D4×D10712yes—
150C2×C4C2×C22⋊D10711yescharacteristic
151C2×C4#1313381712yes—
152C8—4282no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic10711yescharacteristic
163C22#1681207711yescharacteristic
164C22#1681209011yescharacteristic
165C4—24no—
166C4—4284no—
167C4D4×Dic10712yes—
168C4#1681208112yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#2029250711yescharacteristic
174C2#2029279412yes—
subgroupG/H
centerC22#16812077
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting, prime radicalC2×#127×C107C2
socleC22×C107#120
cosocleC22×C4×C107C23
hypercenterC2×#127D107
hypocenterC107#1887
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20293102C2semidirect
2. #20292632C2semidirect
3. C2×#16812026C2semidirect
4. #20292387C2semidirect
16. #127×C107C4semidirect
22. C4⋊Dic107D4semidirect
36. C2×D4×C107C2×C4semidirect
41. C42×C107C2×C4semidirect
88. C107#1887semidirect
99. #127Dic107semidirect
116. C2×D4C2×Dic107semidirect
129. C42C2×Dic107semidirect

Maximal subgroups

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

subgroupindex
#202931022
#202926322
C2×#168120262
#202923872
C2×#127×C1072
#1887107