Finite Group Databasebeta

← #28346394

Subgroups — #28346394

Order 13696.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#20293138C211yesmaximal, characteristic
2C2×#16812068C211yesmaximal, characteristic, complemented
3#20292969C214yesmaximal, complemented
4#20292952C214yesmaximal, complemented
5#20292669C212yesmaximal, complemented
6#20292629C212yesmaximal, complemented
7#407×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812092—21no—
10#16812088C2212yes—
11#16812083C2212yes—
12#16812078—21no—
13C4×C22⋊D107—22no—
14#16812068—22no—
15#16812068C2211yescharacteristic
16#16812068C2214yes—
17C2×C4⋊Dic107C2212yes—
18C2×C22⋊Dic107—21no—
19C2×C4×Dic107—21no—
20#16812043C2214yescomplemented
21#16812041C2214yescomplemented
22#16812034C2214yescomplemented
23#16812031C2214yescomplemented
24C23×C4×C107C2211yescharacteristic
25C2×M16×C107C2212yes—
26C22⋊C8×C107C2214yes—
27#13133818—24no—
28C2×Dic214C2311yescharacteristic
29C2×D428C2311yescharacteristic
30C2×C22⋊D107—22no—
31C2×C22⋊D107—21no—
32C22×Dic107—22no—
33C2×C4×D107—21no—
34C4⋊Dic107—22no—
35C4⋊Dic107C2314yes—
36C4⋊Dic107C2312yes—
37Dic107⋊C4—22no—
38C22⋊Dic107—24no—
39C22⋊Dic107—21no—
40C22⋊Dic107—22no—
41(C2×D107)⋊C4—22no—
42C4×Dic107—24no—
43C4×Dic107—21no—
44C24×C107D411yescharacteristic
45C22×C4×C107—24no—
46C22×C4×C107D411yescharacteristic
47C22×C4×C107D412yes—
48C22×C4×C107C2312yes—
49C22×C4×C107C2311yescharacteristic
50M16×C107—24no—
51C2×C8×C107C2314yes—
52C23×C107—22no—
53C23×C107—42no—
54C23×C107C2×D412yes—
55C23×C107C2×D411yescharacteristic
56C22×D107—21no—
57C2×C4×C107—22no—
58C2×C4×C107—24no—
59C2×C4×C107—24no—
60C2×C4×C107C2×D412yes—
61C2×C4×C107C2×D411yescharacteristic
62C2×C4×C107#4114yes—
63C2×C4×C107C2411yescharacteristic
64C22⋊D107—44no—
65C22⋊D107—42no—
66C2×Dic107—24no—
67C2×Dic107—22no—
68C2×Dic107—21no—
69C2×Dic107—44no—
70D428—22no—
71C4×D107—42no—
72Dic214—22no—
73C8×C107—24no—
74C22×C107—22no—
75C22×C107—22no—
76C22×C107—22no—
77C22×C107—22no—
78C22×C107—44no—
79C22×C107#12312yes—
80C22×C107C22×D411yescharacteristic
81C2×D107—41no—
82C2×D107—42no—
83C4×C107—22no—
84C4×C107#12312yes—
85C4×C107C2×#4112yes—
86Dic107—41no—
87Dic107—44no—
88Dic107—42no—
89C2×C107—22no—
90C2×C107—42no—
91C2×C107C2×#12311yescharacteristic
92C2×C107#57512yes—
93D107—81no—
94#3093—1071nomaximal, Sylow
95C107#309311yescharacteristic, complemented, Sylow
96#525—1071no—
97C2×#117—1071no—
98#485—1074no—
99#482—1074no—
100#428—1072no—
101#417—1072no—
102#407D10711yescharacteristic, complemented
103C2×#41—1071no—
104C23×C4C2×D10711yescharacteristic
105C2×M16C2×D10712yes—
106#123—2141no—
107C22⋊Q8—1072no—
108#121—1072no—
109#120—2141no—
110C4×D4—2142no—
111#117—1074no—
112#117—1071no—
113#117—2142no—
114C2×C4⋊C4—1072no—
115C2×C22⋊C4—2141no—
116C2×C42—2141no—
117#107—1074no—
118#106—1074no—
119#103—1074no—
120#102—1074no—
121C22⋊C8C2×D10714yes—
122C24D42811yescharacteristic
123#41—2144no—
124C2×Q8—1071no—
125C2×D4—1071no—
126C2×D4—2142no—
127C2×D4—2141no—
128C22×C4—24no—
129C22×C4—2142no—
130C22×C4—2141no—
131C22×C4D42811yescharacteristic
132C22×C4D42812yes—
133C22×C4C22×D10712yes—
134C22×C4C22×D10711yescharacteristic
135M16—24no—
136C2×C8C22×D10714yes—
137C4⋊C4—1074no—
138C4⋊C4—1072no—
139C4⋊C4—2142no—
140C4⋊C4—2142no—
141C22⋊C4—2144no—
142C22⋊C4—2141no—
143C22⋊C4—2142no—
144C22⋊C4—2142no—
145C42—2144no—
146C42—2141no—
147C23—22no—
148C23—42no—
149C23—2141no—
150C23C2×D42812yes—
151C23C2×D42811yescharacteristic
152Q8—2142no—
153D4—4284no—
154D4—4282no—
155D4—2142no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—4284no—
160C2×C4—4282no—
161C2×C4—2144no—
162C2×C4—2142no—
163C2×C4—2141no—
164C2×C4C2×D42812yes—
165C2×C4C2×D42811yescharacteristic
166C2×C4#1313381714yes—
167C2×C4C23×D10711yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—4281no—
175C22—4282no—
176C22#1681209312yes—
177C22C22×D42811yescharacteristic
178C4—22no—
179C4—4281no—
180C4—4284no—
181C4—4282no—
182C4#1681209312yes—
183C4C2×#1313381712yes—
184C2—22no—
185C2—42no—
186C2—8561no—
187C2C2×#1681209311yescharacteristic
188C2#2029342712yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenter#407D107
hypocenterC107#3093
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
2. C2×#16812068C2semidirect
3. #20292969C2semidirect
4. #20292952C2semidirect
5. #20292669C2semidirect
6. #20292629C2semidirect
7. #407×C107C2semidirect
20. #16812043C22semidirect
21. #16812041C22semidirect
22. #16812034C22semidirect
23. #16812031C22semidirect
95. C107#3093semidirect
102. #407D107semidirect

Maximal subgroups

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

subgroupindex
#202931382
C2×#168120682
#202929692
#202929522
#202926692
#202926292
#407×C1072
#3093107