Finite Group Databasebeta

← #28345698

Subgroups — #28345698

Order 13696.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 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×Dic214C211yesmaximal, characteristic, complemented
2C2×#16812142C212yesmaximal, complemented
3#20293138C211yesmaximal, characteristic
4#20292785C214yesmaximal, complemented
5#20292768C214yesmaximal, complemented
6#20292637C212yesmaximal, complemented
7#407×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×Dic214—28no—
10C22×Dic214—22no—
11C22×Dic214—21no—
12C22×Dic214C2214yes—
13C22×Dic214C2212yes—
14C23×Dic107—21no—
15#16812142—28no—
16C2×Dic428C2214yescomplemented
17C2×#13133807C2214yescomplemented
18#16812092—21no—
19#16812088C2212yes—
20#16812083C2212yes—
21#16812078—21no—
22C4×C22⋊D107—22no—
23#16812068C2211yescharacteristic
24#16812034C2218yescomplemented
25C23×C4×C107C2211yescharacteristic
26C2×M16×C107C2212yes—
27C22⋊C8×C107C2214yes—
28#13133818—24no—
29C2×Dic214—28no—
30C2×Dic214—216no—
31C2×Dic214—24no—
32C2×Dic214—24no—
33C2×Dic214—22no—
34C2×Dic214—216no—
35C2×Dic214D418yescomplemented
36C2×Dic214C2311yescharacteristic
37C2×Dic214C2314yes—
38C2×D428C2311yescharacteristic
39C2×C22⋊D107—22no—
40C2×C22⋊D107—21no—
41C22×Dic107—28no—
42C22×Dic107—24no—
43C22×Dic107—22no—
44C2×C4×D107—21no—
45Dic428—28no—
46#13133807—28no—
47C4⋊Dic107C2312yes—
48Dic107⋊C4—22no—
49C22⋊Dic107—21no—
50C22⋊Dic107—22no—
51(C2×D107)⋊C4—22no—
52C4×Dic107—21no—
53C24×C107D411yescharacteristic
54C22×C4×C107—24no—
55C22×C4×C107D412yes—
56C22×C4×C107D411yescharacteristic
57C22×C4×C107C2312yes—
58C22×C4×C107C2311yescharacteristic
59M16×C107—24no—
60C2×C8×C107C2314yes—
61C23×C107—22no—
62C23×C107—24no—
63C23×C107C2×D411yescharacteristic
64C23×C107C2×D412yes—
65C22×D107—21no—
66C2×C4×C107—22no—
67C2×C4×C107—24no—
68C2×C4×C107—24no—
69C2×C4×C107C2×D414yes—
70C2×C4×C107C2×D412yes—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2411yescharacteristic
73C22⋊D107—42no—
74C22⋊D107—44no—
75C2×Dic107—216no—
76C2×Dic107—28no—
77C2×Dic107—24no—
78C2×Dic107—22no—
79C2×Dic107—21no—
80D428—22no—
81C4×D107—42no—
82Dic214—216no—
83Dic214—28no—
84Dic214—24no—
85Dic214—22no—
86Dic214C2×D418yes—
87C8×C107—24no—
88C22×C107—28no—
89C22×C107—22no—
90C22×C107—22no—
91C22×C107—22no—
92C22×C107#12012yes—
93C22×C107#13714yescomplemented
94C22×C107C22×D411yescharacteristic
95C2×D107—41no—
96C2×D107—42no—
97C4×C107—22no—
98C4×C107#12012yes—
99C4×C107C22×D412yes—
100Dic107—28no—
101Dic107—41no—
102Dic107—42no—
103C2×C107—22no—
104C2×C107—24no—
105C2×C107C2×#12011yescharacteristic
106C2×C107C2×#13712yes—
107D107—81no—
108#3004—1071nomaximal, Sylow
109C107#300411yescharacteristic, complemented, Sylow
110C23×Q8—1071no—
111C2×#137—1072no—
112#525—1071no—
113#451—1074no—
114#448—1074no—
115#419—1072no—
116#407D10711yescharacteristic, complemented
117C2×#41—1071no—
118C22×Q8—1072no—
119C22×Q8—1074no—
120C22×Q8—2148no—
121C22×Q8—2142no—
122C22×Q8—2141no—
123C23×C4—2141no—
124C23×C4C2×D10711yescharacteristic
125#137—2148no—
126C2×Q16—1074no—
127C2×SD16—1074no—
128C2×M16C2×D10712yes—
129#123—2141no—
130C22⋊Q8—1072no—
131#121—1072no—
132#120—2141no—
133C4×D4—2142no—
134#117—1071no—
135#103—1078no—
136C22⋊C8C2×D10714yes—
137C24D42811yescharacteristic
138#41—2144no—
139C2×Q8—1078no—
140C2×Q8—1074no—
141C2×Q8—1071no—
142C2×Q8—21416no—
143C2×Q8—2148no—
144C2×Q8—21416no—
145C2×Q8—2144no—
146C2×Q8—2144no—
147C2×Q8—2142no—
148C2×D4—1071no—
149C2×D4—2142no—
150C2×D4—2141no—
151C22×C4—24no—
152C22×C4—2141no—
153C22×C4—2148no—
154C22×C4—2144no—
155C22×C4—2142no—
156C22×C4D42812yes—
157C22×C4D42811yescharacteristic
158C22×C4C22×D10712yes—
159C22×C4C22×D10711yescharacteristic
160Q16—2148no—
161SD16—2148no—
162M16—24no—
163C2×C8C22×D10714yes—
164C4⋊C4—1072no—
165C4⋊C4—2142no—
166C22⋊C4—2141no—
167C22⋊C4—2142no—
168C22⋊C4—2142no—
169C42—2141no—
170C23—22no—
171C23—24no—
172C23—2141no—
173C23C2×D42811yescharacteristic
174C23C2×D42812yes—
175Q8—1078no—
176Q8—21416no—
177Q8—2142no—
178Q8—2148no—
179Q8—2144no—
180D4—4282no—
181D4—4284no—
182D4—2142no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—4282no—
187C2×C4—2142no—
188C2×C4—2141no—
189C2×C4—21416no—
190C2×C4—2148no—
191C2×C4—2144no—
192C2×C4D4×D10714yes—
193C2×C4C2×D42812yes—
194C2×C4C2×D42811yescharacteristic
195C2×C4C23×D10711yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—4282no—
202C22—4281no—
203C22#1681207912yes—
204C22#1681214214yescomplemented
205C22C22×D42811yescharacteristic
206C4—22no—
207C4—4282no—
208C4—4281no—
209C4—2148no—
210C4#1681207912yes—
211C4C2×D4×D10712yes—
212C2—22no—
213C2—24no—
214C2—8561no—
215C2C2×#1681207911yescharacteristic
216C2C2×#1681214212yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenter#407D107
hypocenterC107#3004
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×Dic214C2semidirect
2. C2×#16812142C2semidirect
4. #20292785C2semidirect
5. #20292768C2semidirect
6. #20292637C2semidirect
7. #407×C107C2semidirect
16. C2×Dic428C22semidirect
17. C2×#13133807C22semidirect
24. #16812034C22semidirect
35. C2×Dic214D4semidirect
93. C22×C107#137semidirect
109. C107#3004semidirect
116. #407D107semidirect
204. C22#16812142semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2142
C2×#168121422
#202931382
#202927852
#202927682
#202926372
#407×C1072
#3004107