Finite Group Databasebeta

← #28346912

Subgroups — #28346912

Order 13696.

656 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#16812133C214yesmaximal, complemented
2#20293175C211yesmaximal, characteristic, complemented
3#127×D107C211yesmaximal, characteristic, complemented
4#20293015C214yesmaximal, complemented
5#20293003C211yesmaximal, characteristic, complemented
6#20293000C211yesmaximal, characteristic, complemented
7#20292999C211yesmaximal, characteristic, complemented
8C4×#13133801C211yesmaximal, characteristic, complemented
9#493×C107C211yesmaximal, characteristic, complemented
10C2×#13133817C2212yes—
11C2×D4×D107—22no—
12C2×D4×D107C2212yes—
13#16812133—216no—
14C2×#13133807C2214yescomplemented
15C2×#13133806C2214yescomplemented
16C2×#13133803C2214yescomplemented
17C2×#13133801C2212yes—
18#16812109C2211yescharacteristic, complemented
19#16812106C2211yescharacteristic
20#16812105—22no—
21#16812105C2212yes—
22#16812096C2212yescomplemented
23#16812080—22no—
24D4×Dic107—22no—
25C42×D107C2211yescharacteristic
26#16812019C2212yescomplemented
27C4×C107⋊C8C2211yescharacteristic, complemented
28C2×D8×C107C2214yescomplemented
29#127×C107C2211yescharacteristic
30#127×C107C2211yescharacteristic, complemented
31C4×C8×C107C2211yescharacteristic, complemented
32C23×D107—22no—
33#13133817—28no—
34C2×Dic214C2312yes—
35C2×D428—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C22⋊D107—22no—
39C2×C22⋊D107—24no—
40D4×D107—28no—
41D4×D107—48no—
42C22×Dic107—22no—
43C2×C4×D107D412yes—
44C2×C4×D107C2311yescharacteristic
45#13133807—28no—
46#13133806—28no—
47#13133803—28no—
48#13133801D418yescomplemented
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—22no—
51C22⋊Dic107—24no—
52C4×Dic107D412yes—
53C4×Dic107C2311yescharacteristic
54C2×D4×C107—22no—
55C2×D4×C107—22no—
56C2×D4×C107C2312yes—
57C2×D4×C107C2312yes—
58D8×C107—28no—
59C2×C8×C107C2312yes—
60C42×C107C2311yescharacteristic
61C23×C107—22no—
62C23×C107—22no—
63C22×D107—22no—
64C22×D107—48no—
65C22×D107—44no—
66C22×D107C2×D411yescharacteristic
67D4×C107—24no—
68D4×C107—24no—
69D4×C107—44no—
70D4×C107—44no—
71C2×C4×C107C2×D412yes—
72C2×C4×C107C2411yescharacteristic
73C22⋊D107—48no—
74C22⋊D107—44no—
75C22⋊D107—44no—
76C2×Dic107—22no—
77C2×Dic107—44no—
78C2×Dic107C2×D412yes—
79C2×Dic107C2×D411yescharacteristic
80D428—24no—
81D428—44no—
82C4×D107—24no—
83C4×D107C2×D414yes—
84Dic214—24no—
85C8×C107C2×D414yescomplemented
86C107⋊C8C2×D414yescomplemented
87C22×C107—42no—
88C22×C107—42no—
89C22×C107—44no—
90C22×C107—44no—
91C22×C107C22×D411yescharacteristic
92C2×D107—22no—
93C2×D107—42no—
94C2×D107—44no—
95C2×D107—88no—
96C2×D107#12712yes—
97C4×C107AGL(1,ℤ/8)14yescomplemented
98C4×C107C22×D412yes—
99Dic107—22no—
100Dic107—42no—
101Dic107#12712yes—
102C2×C107—82no—
103C2×C107—82no—
104C2×C107C2×#12711yescharacteristic
105C2×C107C2×AGL(1,ℤ/8)12yes—
106D107—22no—
107D107—82no—
108#3151—1071nomaximal, Sylow
109C107#315111yescharacteristic, complemented, Sylow
110C2×AGL(1,ℤ/8)—1074no—
111#532—1071no—
112C2×#127—1071no—
113#496—1074no—
114#493—1071no—
115#493D10711yescharacteristic, complemented
116#492—1071no—
117#492—1071no—
118C4×M16—1071no—
119C2×#41—1072no—
120C22×D4—1072no—
121C22×D4—2142no—
122AGL(1,ℤ/8)—21416no—
123C2×SD16—1074no—
124C2×SD16—1074no—
125C2×D8—1074no—
126C2×D8C2×D10714yescomplemented
127C2×M16—1072no—
128C4⋊Q8—1071no—
129#127—1072no—
130#127—1071no—
131#127—2142no—
132#127C2×D10711yescharacteristic
133#127C2×D10711yescharacteristic, complemented
134#124—1072no—
135#121—2142no—
136C4×D4—2142no—
137C2×C42—1071no—
138#97—1072no—
139C4×C8—1071no—
140C4×C8C2×D10711yescharacteristic, complemented
141C24—2142no—
142#41—2148no—
143C2×Q8—1072no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—1072no—
147C2×D4—4288no—
148C2×D4—2148no—
149C2×D4—2144no—
150C2×D4—2142no—
151C2×D4—2142no—
152C2×D4—2142no—
153C2×D4C22×D10712yes—
154C2×D4C22×D10712yes—
155C22×C4—1071no—
156C22×C4—1072no—
157C22×C4—2142no—
158SD16—2148no—
159SD16—2148no—
160D8—28no—
161D8—2148no—
162M16—1078no—
163C2×C8—1072no—
164C2×C8C22×D10712yes—
165C4⋊C4—2142no—
166C22⋊C4—2144no—
167C42—1072no—
168C42—1071no—
169C42C22×D10711yescharacteristic
170C23—22no—
171C23—22no—
172C23—1071no—
173C23—4284no—
174C23—4288no—
175C23—2142no—
176Q8—2144no—
177D4—24no—
178D4—24no—
179D4—44no—
180D4—44no—
181D4—4284no—
182D4—4284no—
183D4—4288no—
184D4—4284no—
185D4—2144no—
186C2×C4—1071no—
187C2×C4—1074no—
188C2×C4—1072no—
189C2×C4—4284no—
190C2×C4—2142no—
191C2×C4—2144no—
192C2×C4D4×D10712yes—
193C2×C4C23×D10711yescharacteristic
194C8—1074no—
195C8D4×D10714yescomplemented
196C22—44no—
197C22—44no—
198C22—42no—
199C22—42no—
200C22—8568no—
201C22—1072no—
202C22—4284no—
203C22—4282no—
204C22—2142no—
205C22C2×D4×D10711yescharacteristic
206C4—1072no—
207C4—4282no—
208C4—2142no—
209C4#1681213314yescomplemented
210C4C2×D4×D10712yes—
211C2—82no—
212C2—82no—
213C2—8562no—
214C2—2142no—
215C2#127×D10711yescharacteristic
216C2C2×#1681213312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#493×C107C2
socleC22×C107C22×D4
hypercenter#493D107
hypocenterC107#3151
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293175C2semidirect
3. #127×D107C2semidirect
4. #20293015C2semidirect
5. #20293003C2semidirect
6. #20293000C2semidirect
7. #20292999C2semidirect
8. C4×#13133801C2semidirect
9. #493×C107C2semidirect
14. C2×#13133807C22semidirect
15. C2×#13133806C22semidirect
16. C2×#13133803C22semidirect
18. #16812109C22semidirect
22. #16812096C22semidirect
26. #16812019C22semidirect
27. C4×C107⋊C8C22semidirect
28. C2×D8×C107C22semidirect
30. #127×C107C22semidirect
31. C4×C8×C107C22semidirect
48. #13133801D4semidirect
85. C8×C107C2×D4semidirect
86. C107⋊C8C2×D4semidirect
97. C4×C107AGL(1,ℤ/8)semidirect
109. C107#3151semidirect
115. #493D107semidirect
126. C2×D8C2×D107semidirect
133. #127C2×D107semidirect
140. C4×C8C2×D107semidirect
195. C8D4×D107semidirect
209. C4#16812133semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202931752
#127×D1072
#202930152
#202930032
#202930002
#202929992
C4×#131338012
#493×C1072
#3151107