Finite Group Databasebeta

← #28346066

Subgroups — #28346066

Order 13696.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133801C211yesmaximal, characteristic, complemented
2#20293134C212yesmaximal, complemented
3#20292882C212yesmaximal, complemented
4#20292877C212yesmaximal, complemented
5#20292870C214yesmaximal, complemented
6#20292665C211yesmaximal, characteristic, complemented
7#20292623C212yesmaximal, complemented
8C2×#135×C107C211yesmaximal, characteristic, complemented
9C22×C4×D107C2211yescharacteristic
10C2×#13133801—24no—
11C2×#13133801C2212yes—
12C2×#13133801C2212yes—
13C22×C107⋊C8C2211yescharacteristic
14#16812090—22no—
15#16812085C2214yes—
16#16812080C2214yes—
17#16812077—22no—
18C4×C22⋊D107—24no—
19#16812068C2212yes—
20#16812043C2212yescomplemented
21#16812041C2212yescomplemented
22#16812033C2214yescomplemented
23#16812030C2214yescomplemented
24C2×#41×C107C2212yes—
25#135×C107—24no—
26C2×Q16×C107C2211yescharacteristic, complemented
27C2×SD16×C107C2212yescomplemented
28C2×D8×C107C2211yescharacteristic, complemented
29C22×C8×C107C2211yescharacteristic
30C23×D107D411yescharacteristic
31C2×C22⋊D107—24no—
32C22×Dic107D411yescharacteristic
33C2×C4×D107—22no—
34C2×C4×D107—22no—
35C2×C4×D107D412yes—
36C2×C4×D107C2312yes—
37#13133801—24no—
38#13133801—24no—
39C2×C107⋊C8—22no—
40C2×C107⋊C8C2312yes—
41C4⋊Dic107C2314yes—
42Dic107⋊C4—24no—
43C22⋊Dic107—22no—
44C22⋊Dic107—24no—
45(C2×D107)⋊C4—24no—
46C4×Dic107—22no—
47#41×C107—24no—
48#41×C107—24no—
49C2×Q8×C107C2312yes—
50C2×D4×C107—22no—
51C2×D4×C107C2312yes—
52C22×C4×C107—22no—
53C22×C4×C107C2311yescharacteristic
54Q16×C107—22no—
55SD16×C107—24no—
56D8×C107—22no—
57C2×C8×C107D414yescomplemented
58C2×C8×C107C2311yescharacteristic
59C2×C8×C107C2311yescharacteristic
60C23×C107—22no—
61C23×C107C2×D411yescharacteristic
62C22×D107—22no—
63C22×D107—42no—
64C22×D107C2×D412yes—
65Q8×C107—22no—
66Q8×C107—22no—
67D4×C107—22no—
68D4×C107—22no—
69D4×C107—44no—
70C2×C4×C107—22no—
71C2×C4×C107—44no—
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2×D412yes—
74C2×C4×C107#4112yes—
75C2×C4×C107C2411yescharacteristic
76C22⋊D107—48no—
77C2×Dic107—22no—
78C2×Dic107—24no—
79C2×Dic107C2×D412yes—
80C4×D107—24no—
81C4×D107—22no—
82C4×D107—22no—
83C8×C107C2×D412yes—
84C8×C107C2×D412yes—
85C107⋊C8—22no—
86C22×C107—22no—
87C22×C107—42no—
88C22×C107—44no—
89C22×C107#12112yes—
90C22×C107C22×D411yescharacteristic
91C2×D107—22no—
92C2×D107—24no—
93C2×D107—44no—
94C4×C107—42no—
95C4×C107#12112yes—
96C4×C107C22×D411yescharacteristic
97C4×C107C2×#4111yescharacteristic
98Dic107—22no—
99Dic107—44no—
100C2×C107—22no—
101C2×C107—82no—
102C2×C107C2×#12111yescharacteristic
103C2×C107#57512yes—
104D107—42no—
105#3054—1071nomaximal, Sylow
106C107#305411yescharacteristic, complemented, Sylow
107C2×#135D10711yescharacteristic, complemented
108C22×M16—1071no—
109#525—1072no—
110#470—1072no—
111#469—1072no—
112#468—1074no—
113#427—1071no—
114#416—1072no—
115C2×#41C2×D10712yes—
116C23×C4—1071no—
117#135—24no—
118C2×Q16C2×D10711yescharacteristic, complemented
119C2×SD16C2×D10712yescomplemented
120C2×D8C2×D10711yescharacteristic, complemented
121C2×M16—1072no—
122C2×M16—1072no—
123C2×M16—2144no—
124C22×C8—1071no—
125C22×C8C2×D10711yescharacteristic
126#123—2142no—
127C22⋊Q8—1074no—
128#121—1074no—
129#120—2142no—
130C4×D4—2144no—
131#117—1072no—
132#107—1072no—
133#106—1072no—
134#103—1074no—
135#102—1074no—
136C24—1071no—
137#41—24no—
138#41—24no—
139C2×Q8C22×D10712yes—
140C2×D4—22no—
141C2×D4—2144no—
142C2×D4C22×D10712yes—
143C22×C4—22no—
144C22×C4—1071no—
145C22×C4—1072no—
146C22×C4—1072no—
147C22×C4—2142no—
148C22×C4—2142no—
149C22×C4C22×D10711yescharacteristic
150Q16—22no—
151SD16—24no—
152D8—22no—
153M16—2144no—
154M16—2144no—
155C2×C8—1072no—
156C2×C8—2142no—
157C2×C8C22⋊D10714yescomplemented
158C2×C8C22×D10711yescharacteristic
159C2×C8C22×D10711yescharacteristic
160C4⋊C4—1074no—
161C4⋊C4—2144no—
162C22⋊C4—2144no—
163C22⋊C4—2144no—
164C22⋊C4—2142no—
165C42—2142no—
166C23—22no—
167C23—1072no—
168C23—4282no—
169C23—2142no—
170C23D4×D10711yescharacteristic
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—4288no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—1072no—
180C2×C4—2144no—
181C2×C4—2142no—
182C2×C4—2144no—
183C2×C4—2142no—
184C2×C4—2142no—
185C2×C4D4×D10711yescharacteristic
186C2×C4C2×C22⋊D10712yes—
187C2×C4#1313381712yes—
188C2×C4C23×D10711yescharacteristic
189C8—2142no—
190C8C2×C22⋊D10712yes—
191C8C2×C22⋊D10712yes—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—4284no—
196C22—2144no—
197C22—2142no—
198C22#1681208012yes—
199C22C2×D4×D10711yescharacteristic
200C4—42no—
201C4—4284no—
202C4—2142no—
203C4#1681208012yes—
204C4C22×C22⋊D10711yescharacteristic
205C4C2×#1313381711yescharacteristic
206C2—22no—
207C2—82no—
208C2—4282no—
209C2C2×#1681208011yescharacteristic
210C2#2029342412yes—
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#135×C107C2
socleC22×C107C22×D4
hypercenterC2×#135D107
hypocenterC107#3054
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×#13133801C2semidirect
2. #20293134C2semidirect
3. #20292882C2semidirect
4. #20292877C2semidirect
5. #20292870C2semidirect
6. #20292665C2semidirect
7. #20292623C2semidirect
8. C2×#135×C107C2semidirect
20. #16812043C22semidirect
21. #16812041C22semidirect
22. #16812033C22semidirect
23. #16812030C22semidirect
26. C2×Q16×C107C22semidirect
27. C2×SD16×C107C22semidirect
28. C2×D8×C107C22semidirect
57. C2×C8×C107D4semidirect
106. C107#3054semidirect
107. C2×#135D107semidirect
118. C2×Q16C2×D107semidirect
119. C2×SD16C2×D107semidirect
120. C2×D8C2×D107semidirect
157. C2×C8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338012
#202931342
#202928822
#202928772
#202928702
#202926652
#202926232
C2×#135×C1072
#3054107