Finite Group Databasebeta

← #28336900

Subgroups — #28336900

Order 13696.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×#13133802C211yesmaximal, characteristic, complemented
2C2×#16812030C212yesmaximal, complemented
3#20292487C211yesmaximal, characteristic, complemented
4#20292389C212yesmaximal, complemented
5C2×#121×C107C211yesmaximal, characteristic
6C2×#13133802—24no—
7C2×#13133802—22no—
8C22×C107⋊C8C2212yes—
9C2×C4⋊Dic107C2212yes—
10C2×C22⋊Dic107—21no—
11#16812030—24no—
12C107⋊#95—21no—
13C22×D4×C107—21no—
14C22×D4×C107C2211yescharacteristic
15C23×C4×C107C2211yescharacteristic
16#121×C107—22no—
17#121×C107C414yescomplemented
18C2×C4⋊C4×C107C2211yescharacteristic
19C2×C22⋊C4×C107—21no—
20C22×Dic107—22no—
21#13133802—28no—
22C2×C107⋊C8—22no—
23C2×C107⋊C8—22no—
24C2×C107⋊C8D414yescomplemented
25C4⋊Dic107—22no—
26C4⋊Dic107—22no—
27C22⋊Dic107—42no—
28C24×C107—21no—
29C24×C107D411yescharacteristic
30C2×D4×C107—24no—
31C2×D4×C107—22no—
32C2×D4×C107—21no—
33C2×D4×C107—22no—
34C2×D4×C107—44no—
35C2×D4×C107C2×C412yes—
36C22×C4×C107—22no—
37C22×C4×C107—22no—
38C22×C4×C107—21no—
39C22×C4×C107C2×C412yes—
40C22×C4×C107D411yescharacteristic
41C22×C4×C107D412yes—
42C22×C4×C107C2311yescharacteristic
43C4⋊C4×C107—21no—
44C4⋊C4×C107C2×C412yes—
45C22⋊C4×C107—24no—
46C23×C107—22no—
47C23×C107—22no—
48C23×C107—21no—
49C23×C107—21no—
50C23×C107—22no—
51C23×C107—42no—
52C23×C107—42no—
53C23×C107—41no—
54C23×C107—41no—
55C23×C107C22⋊C412yes—
56C23×C107C2×D411yescharacteristic
57D4×C107—24no—
58D4×C107—48no—
59D4×C107—41no—
60D4×C107—41no—
61C2×C4×C107—22no—
62C2×C4×C107—24no—
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—42no—
67C2×C4×C107C22⋊C412yes—
68C2×C4×C107C22×C411yescharacteristic
69C2×C4×C107C2×D412yes—
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107#4112yes—
72C2×Dic107—42no—
73C2×Dic107—42no—
74C2×Dic107—42no—
75C107⋊C8—24no—
76C22×C107—22no—
77C22×C107—22no—
78C22×C107—22no—
79C22×C107—42no—
80C22×C107—42no—
81C22×C107—42no—
82C22×C107—42no—
83C22×C107—42no—
84C22×C107—82no—
85C22×C107—82no—
86C22×C107C2×C22⋊C411yescharacteristic
87C22×C107#12112yes—
88C22×C107AGL(1,ℤ/8)11yescharacteristic
89C22×C107AGL(1,ℤ/8)12yes—
90C22×C107#13711yescharacteristic
91C4×C107—22no—
92C4×C107—42no—
93C4×C107C4×D412yes—
94C4×C107#12011yescharacteristic
95C4×C107#12311yescharacteristic
96Dic107—82no—
97C2×C107—42no—
98C2×C107—82no—
99C2×C107#38611yescharacteristic
100C2×C107#41711yescharacteristic
101C2×C107#41811yescharacteristic
102C2×C107#46812yes—
103C2×C107#46912yes—
104#1895—1071nomaximal, Sylow
105C107#189511yescharacteristic, complemented, Sylow
106C22×M16—1071no—
107C2×#121D10711yescharacteristic
108C2×#102—1072no—
109#380—1071no—
110#340—1072no—
111C22×D4—21no—
112C22×D4C2×D10711yescharacteristic
113C23×C4C2×D10711yescharacteristic
114C2×M16—2144no—
115C2×M16—2142no—
116C22×C8—1072no—
117#121—22no—
118#121Dic10714yescomplemented
119C2×C4⋊C4—1072no—
120C2×C4⋊C4C2×D10711yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—2141no—
123#102—2144no—
124#95—2141no—
125C24—21no—
126C24C22⋊D10711yescharacteristic
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4C2×Dic10712yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—2142no—
137C22×C4C2×Dic10712yes—
138C22×C4C22⋊D10711yescharacteristic
139C22×C4C22⋊D10712yes—
140C22×C4C22×D10711yescharacteristic
141M16—2148no—
142C2×C8—1074no—
143C2×C8—2142no—
144C2×C8—2142no—
145C4⋊C4—21no—
146C4⋊C4—2142no—
147C4⋊C4—2142no—
148C4⋊C4C2×Dic10712yes—
149C22⋊C4—24no—
150C22⋊C4—4282no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—42no—
157C23—41no—
158C23—42no—
159C23—41no—
160C23C22⋊Dic10712yes—
161C23C2×C22⋊D10711yescharacteristic
162D4—24no—
163D4—48no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—4282no—
173C2×C4—4282no—
174C2×C4—4282no—
175C2×C4C22⋊Dic10712yes—
176C2×C4C22×Dic10711yescharacteristic
177C2×C4D4×D10712yes—
178C2×C4C2×C22⋊D10711yescharacteristic
179C2×C4#1313381712yes—
180C8—2144no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22C2×C22⋊Dic10711yescharacteristic
192C22#1681208112yes—
193C22#1681213411yescharacteristic
194C22#1681213512yes—
195C22#1681213911yescharacteristic
196C4—22no—
197C4—42no—
198C4—8562no—
199C4D4×Dic10712yes—
200C4#1681207711yescharacteristic
201C4#1681209011yescharacteristic
202C2—42no—
203C2—82no—
204C2#2029250711yescharacteristic
205C2#2029262811yescharacteristic
206C2#2029263211yescharacteristic
207C2#2029287112yes—
208C2#2029287812yes—
subgroupG/H
centerC23C2×C22⋊D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting, prime radicalC2×#121×C107C2
socleC23×C107C2×D4
cosocleC22×C4×C107C23
hypercenterC2×#121D107
hypocenterC107#1895
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13133802C2semidirect
2. C2×#16812030C2semidirect
3. #20292487C2semidirect
4. #20292389C2semidirect
17. #121×C107C4semidirect
24. C2×C107⋊C8D4semidirect
105. C107#1895semidirect
118. #121Dic107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338022
C2×#168120302
#202924872
#202923892
C2×#121×C1072
#1895107