Finite Group Databasebeta

← #28336890

Subgroups — #28336890

Order 13696.

432 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C23×C107⋊C8C211yesmaximal, characteristic, complemented
2C2×#16812030C212yesmaximal, complemented
3#20292487C211yesmaximal, characteristic, complemented
4#20292389C212yesmaximal, complemented
5C2×#121×C107C211yesmaximal, characteristic
6C22×C107⋊C8—24no—
7C22×C107⋊C8—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—
21C2×C107⋊C8—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—42no—
35C2×D4×C107—42no—
36C2×D4×C107C2×C412yes—
37C22×C4×C107—22no—
38C22×C4×C107—22no—
39C22×C4×C107—21no—
40C22×C4×C107C2×C412yes—
41C22×C4×C107D411yescharacteristic
42C22×C4×C107D412yes—
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—21no—
45C4⋊C4×C107C2×C412yes—
46C22⋊C4×C107—24no—
47C23×C107—22no—
48C23×C107—22no—
49C23×C107—22no—
50C23×C107—42no—
51C23×C107—41no—
52C23×C107—41no—
53C23×C107—42no—
54C23×C107C22⋊C412yes—
55C23×C107D812yes—
56C23×C107SD1612yes—
57C23×C107C2×D411yescharacteristic
58D4×C107—24no—
59D4×C107—48no—
60D4×C107—41no—
61D4×C107—41no—
62C2×C4×C107—22no—
63C2×C4×C107—24no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—22no—
67C2×C4×C107—41no—
68C2×C4×C107—41no—
69C2×C4×C107C22⋊C412yes—
70C2×C4×C107C22×C411yescharacteristic
71C2×C4×C107C2×D412yes—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107#4112yes—
74C2×Dic107—42no—
75C2×Dic107—42no—
76C2×Dic107—42no—
77C107⋊C8—24no—
78C22×C107—22no—
79C22×C107—22no—
80C22×C107—24no—
81C22×C107—24no—
82C22×C107—42no—
83C22×C107—42no—
84C22×C107—42no—
85C22×C107—82no—
86C22×C107—82no—
87C22×C107#10214yescomplemented
88C22×C107C2×C22⋊C411yescharacteristic
89C22×C107#12112yes—
90C22×C107C2×D811yescharacteristic
91C22×C107C2×SD1611yescharacteristic
92C22×C107#13512yes—
93C4×C107—22no—
94C4×C107—42no—
95C4×C107C4×D412yes—
96C4×C107#12011yescharacteristic
97C4×C107#12311yescharacteristic
98Dic107—82no—
99C2×C107—24no—
100C2×C107—82no—
101C2×C107#38611yescharacteristic
102C2×C107C2×#10211yescharacteristic
103C2×C107#41611yescharacteristic
104C2×C107#46512yes—
105C2×C107#46612yes—
106#1893—1071nomaximal, Sylow
107C107#189311yescharacteristic, complemented, Sylow
108C23×C8—1071no—
109C2×#121D10711yescharacteristic
110C2×#102—1072no—
111#380—1071no—
112#340—1072no—
113C22×D4—21no—
114C22×D4C2×D10711yescharacteristic
115C23×C4C2×D10711yescharacteristic
116C22×C8—1072no—
117C22×C8—2144no—
118C22×C8—2142no—
119#121—22no—
120#121Dic10714yescomplemented
121C2×C4⋊C4—1072no—
122C2×C4⋊C4C2×D10711yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—2141no—
125#102—2144no—
126#95—2141no—
127C24—21no—
128C24C22⋊D10711yescharacteristic
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×Dic10712yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—2142no—
140C22×C4C2×Dic10712yes—
141C22×C4C22⋊D10711yescharacteristic
142C22×C4C22⋊D10712yes—
143C22×C4C22×D10711yescharacteristic
144C2×C8—1074no—
145C2×C8—2148no—
146C2×C8—2142no—
147C2×C8—2142no—
148C4⋊C4—21no—
149C4⋊C4—2142no—
150C4⋊C4—2142no—
151C4⋊C4C2×Dic10712yes—
152C22⋊C4—24no—
153C22⋊C4—4282no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—42no—
158C23—41no—
159C23—41no—
160C23—42no—
161C23C22⋊Dic10712yes—
162C23#1313380312yes—
163C23#1313380612yes—
164C23C2×C22⋊D10711yescharacteristic
165D4—24no—
166D4—48no—
167D4—41no—
168D4—41no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—4282no—
177C2×C4—4282no—
178C2×C4—4282no—
179C2×C4C22⋊Dic10712yes—
180C2×C4C22×Dic10711yescharacteristic
181C2×C4D4×D10712yes—
182C2×C4C2×C22⋊D10711yescharacteristic
183C2×C4#1313381712yes—
184C8—2144no—
185C22—22no—
186C22—22no—
187C22—24no—
188C22—24no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—82no—
193C22—82no—
194C22#1681203014yescomplemented
195C22C2×C22⋊Dic10711yescharacteristic
196C22#1681208112yes—
197C22C2×#1313380311yescharacteristic
198C22C2×#1313380611yescharacteristic
199C22#1681212912yes—
200C4—22no—
201C4—42no—
202C4—8562no—
203C4D4×Dic10712yes—
204C4#1681207711yescharacteristic
205C4#1681209011yescharacteristic
206C2—24no—
207C2—82no—
208C2#2029250711yescharacteristic
209C2C2×#1681203011yescharacteristic
210C2#2029262311yescharacteristic
211C2#2029285412yes—
212C2#2029286112yes—
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#1893
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. C23×C107⋊C8C2semidirect
2. C2×#16812030C2semidirect
3. #20292487C2semidirect
4. #20292389C2semidirect
17. #121×C107C4semidirect
24. C2×C107⋊C8D4semidirect
87. C22×C107#102semidirect
107. C107#1893semidirect
120. #121Dic107semidirect
194. C22#16812030semidirect

Maximal subgroups

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

subgroupindex
C23×C107⋊C82
C2×#168120302
#202924872
#202923892
C2×#121×C1072
#1893107