Finite Group Databasebeta

← #28335084

Subgroups — #28335084

Order 13696.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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×#16812097C211yesmaximal, characteristic
2#20292684C211yesmaximal, characteristic, complemented
3#20292366C214yesmaximal, complemented
4#432×C107C211yesmaximal, characteristic, complemented
5C22×Dic214C411yescharacteristic
6C22×D428C411yescharacteristic
7#16812097—21no—
8#16812097—21no—
9#16812097C412yes—
10#16812097C2212yes—
11C2×(C2×D107)⋊C4—21no—
12C2×(C2×D107)⋊C4—21no—
13C4⋊(C107⋊C8)—21no—
14C22⋊(C107⋊C8)—21no—
15#16812020C2212yescomplemented
16C2×C42×C107C2211yescharacteristic
17C4⋊C8×C107—21no—
18C22⋊C8×C107—21no—
19#97×C107C2212yescomplemented
20C23×D107—21no—
21C2×Dic214—21no—
22C2×Dic214—42no—
23C2×Dic214C2×C412yes—
24C2×D428—21no—
25C2×D428—42no—
26C2×D428C2×C412yes—
27C22×Dic107—21no—
28C2×C107⋊C8—22no—
29(C2×D107)⋊C4—22no—
30(C2×D107)⋊C4—22no—
31(C2×D107)⋊C4—22no—
32(C2×D107)⋊C4—22no—
33C22×C4×C107C2×C411yescharacteristic
34C22×C4×C107D411yescharacteristic
35C22×C4×C107D411yescharacteristic
36C2×C8×C107—22no—
37C42×C107C2×C411yescharacteristic
38C42×C107D411yescharacteristic
39C42×C107D411yescharacteristic
40C42×C107C2311yescharacteristic
41C23×C107C22⋊C411yescharacteristic
42C22×D107—22no—
43C22×D107—42no—
44C22×D107—42no—
45C22×D107—42no—
46C2×C4×C107—22no—
47C2×C4×C107—41no—
48C2×C4×C107—41no—
49C2×C4×C107C22⋊C411yescharacteristic
50C2×C4×C107C22⋊C411yescharacteristic
51C2×C4×C107C22⋊C411yescharacteristic
52C2×C4×C107C22×C411yescharacteristic
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107C2×D411yescharacteristic
55C2×Dic107—22no—
56C2×Dic107—42no—
57D428—42no—
58D428—42no—
59Dic214—42no—
60Dic214—42no—
61C8×C107—42no—
62C107⋊C8—42no—
63C22×C107—21no—
64C22×C107—21no—
65C22×C107#10012yes—
66C22×C107C2×C22⋊C411yescharacteristic
67C2×D107—42no—
68C2×D107—42no—
69C2×D107—42no—
70C2×D107—84no—
71C4×C107—21no—
72C4×C107—21no—
73C4×C107—21no—
74C4×C107—21no—
75C4×C107—21no—
76C4×C107—21no—
77Dic107—42no—
78C2×C107—22no—
79C2×C107C2×#10011yescharacteristic
80C2×C107#42111yescharacteristic
81C2×C107#42111yescharacteristic
82D107—82no—
83#1533—1071nomaximal, Sylow
84C107#153311yescharacteristic, complemented, Sylow
85C2×#124—1071no—
86#432—1071no—
87#432D10711yescharacteristic, complemented
88#329—1074no—
89C22×Q8—1071no—
90C22×D4—1071no—
91#124—1072no—
92#124—1072no—
93#124—2141no—
94#124—2141no—
95C2×C22⋊C4—2141no—
96C2×C22⋊C4—2141no—
97C2×C42C2×D10711yescharacteristic
98C4⋊C8—21no—
99C4⋊C8—2141no—
100C22⋊C8—21no—
101C22⋊C8—2141no—
102#97—1072no—
103#97C2×D10712yescomplemented
104C24—2141no—
105C2×Q8—1072no—
106C2×Q8—4282no—
107C2×Q8—2141no—
108C2×D4—1072no—
109C2×D4—4282no—
110C2×D4—2141no—
111C22×C4—2141no—
112C22×C4C4×D10711yescharacteristic
113C22×C4D42811yescharacteristic
114C22×C4C22⋊D10711yescharacteristic
115C2×C8—22no—
116C2×C8—2142no—
117C22⋊C4—2142no—
118C22⋊C4—2142no—
119C22⋊C4—2142no—
120C22⋊C4—2142no—
121C42C4×D10711yescharacteristic
122C42D42811yescharacteristic
123C42C22⋊D10711yescharacteristic
124C42C22×D10711yescharacteristic
125C23—4282no—
126C23—4282no—
127C23—4282no—
128C23—2142no—
129C23(C2×D107)⋊C411yescharacteristic
130Q8—4282no—
131Q8—4282no—
132D4—4282no—
133D4—4282no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—4282no—
138C2×C4—2142no—
139C2×C4(C2×D107)⋊C411yescharacteristic
140C2×C4(C2×D107)⋊C411yescharacteristic
141C2×C4(C2×D107)⋊C411yescharacteristic
142C2×C4C2×C4×D10711yescharacteristic
143C2×C4C2×C22⋊D10711yescharacteristic
144C2×C4C2×D42811yescharacteristic
145C8—42no—
146C8—4282no—
147C22—21no—
148C22—21no—
149C22—8564no—
150C22—4282no—
151C22—4282no—
152C22—4282no—
153C22#1681202512yes—
154C22C2×(C2×D107)⋊C411yescharacteristic
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—4282no—
162C2—22no—
163C2—8562no—
164C2C2×#1681202511yescharacteristic
165C2#2029264311yescharacteristic
166C2#2029264511yescharacteristic
subgroupG/H
centerC22C2×(C2×D107)⋊C4
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
Fitting#432×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC22×D428C4
cosocleC42×C107C23
hypercenter#432D107
hypocenterC107#1533
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #20292684C2semidirect
3. #20292366C2semidirect
4. #432×C107C2semidirect
15. #16812020C22semidirect
19. #97×C107C22semidirect
84. C107#1533semidirect
87. #432D107semidirect
103. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120972
#202926842
#202923662
#432×C1072
#1533107