Finite Group Databasebeta

← #28335435

Subgroups — #28335435

Order 13696.

238 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 57 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#20292782C212yesmaximal
2#20292547C211yesmaximal, characteristic
3#20292364C212yesmaximal
4#20292353C211yesmaximal, characteristic
5#544×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7C2×#13133808—22no—
8C2×Dic107⋊C4—21no—
9#16812032—22no—
10C22⋊(C107⋊C8)C2212yes—
11C107⋊#95—21no—
12C107⋊#95C2211yescharacteristic
13C4⋊Q8×C107—22no—
14#125×C107—22no—
15C22⋊Q8×C107—21no—
16C22⋊Q8×C107C2212yes—
17#117×C107—21no—
18C2×C4⋊C4×C107C2211yescharacteristic
19C2×Dic214—22no—
20C2×Dic214—42no—
21C22×Dic107—21no—
22C22×Dic107—21no—
23#13133808—44no—
24C2×C107⋊C8—22no—
25Dic107⋊C4—42no—
26C2×Q8×C107—21no—
27C2×Q8×C107D412yes—
28C22×C4×C107D411yescharacteristic
29C22×C4×C107D411yescharacteristic
30C22×C4×C107C2311yescharacteristic
31C4⋊C4×C107—21no—
32C4⋊C4×C107—21no—
33C4⋊C4×C107—21no—
34C4⋊C4×C107—22no—
35C4⋊C4×C107—22no—
36C4⋊C4×C107—22no—
37C4⋊C4×C107D412yes—
38C22⋊C4×C107—22no—
39C42×C107—22no—
40C23×C107C2×D411yescharacteristic
41Q8×C107—42no—
42Q8×C107—81no—
43C2×C4×C107—21no—
44C2×C4×C107—21no—
45C2×C4×C107—22no—
46C2×C4×C107—22no—
47C2×C4×C107—21no—
48C2×C4×C107—41no—
49C2×C4×C107—41no—
50C2×C4×C107Q1612yescomplemented
51C2×C4×C107C2×D412yes—
52C2×Dic107—22no—
53C2×Dic107—22no—
54C2×Dic107—41no—
55C2×Dic107—41no—
56C2×Dic107—41no—
57C2×Dic107—41no—
58Dic214—24no—
59Dic214—42no—
60C107⋊C8—42no—
61C22×C107—21no—
62C22×C107—21no—
63C22×C107#12011yescharacteristic
64C22×C107C2×Q1611yescharacteristic
65C22×C107#13711yescharacteristic
66C4×C107—22no—
67C4×C107—22no—
68C4×C107—42no—
69C4×C107—41no—
70C4×C107—42no—
71Dic107—42no—
72Dic107—42no—
73C2×C107—21no—
74C2×C107—21no—
75C2×C107#45111yescharacteristic
76C2×C107#45611yescharacteristic
77C2×C107B(4,𝔽2)11yescharacteristic
78#1605—1071nomaximal, Sylow
79C107#160511yescharacteristic, complemented, Sylow
80#544D10711yescharacteristic
81#451—1072no—
82#395—1071no—
83#328—1072no—
84#324—1071no—
85C22×Q8—1071no—
86C2×Q16—2142no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C2×D10712yes—
91#117—21no—
92C2×C4⋊C4—2141no—
93C2×C4⋊C4C2×D10711yescharacteristic
94#103—2142no—
95C22⋊C8—1072no—
96#95—1071no—
97#95—2141no—
98C2×Q8—21no—
99C2×Q8—4282no—
100C2×Q8—2142no—
101C2×Q8C22⋊D10712yes—
102C22×C4—2141no—
103C22×C4—2141no—
104C22×C4C22⋊D10711yescharacteristic
105C22×C4C22⋊D10711yescharacteristic
106C22×C4C22×D10711yescharacteristic
107Q16—4284no—
108C2×C8—2142no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—4282no—
116C4⋊C4C22⋊D10712yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×C22⋊D10711yescharacteristic
120Q8—42no—
121Q8—81no—
122Q8—4282no—
123Q8—2144no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—4281no—
132C2×C4—4281no—
133C2×C4—4281no—
134C2×C4—4281no—
135C2×C4—2142no—
136C2×C4—2142no—
137C2×C4#1313380812yescomplemented
138C2×C4C2×C22⋊D10712yes—
139C8—4282no—
140C22—21no—
141C22—21no—
142C22#1681207811yescharacteristic
143C22C2×#1313380811yescharacteristic
144C22#1681214011yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—4282no—
151C4—4282no—
152C2—21no—
153C2—21no—
154C2#2029278311yescharacteristic
155C2#2029281011yescharacteristic
156C2#2029281411yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#544×C107C2
socleC22×C107#120
prime radicalC23×C107C2×D4
hypercenter#544D107
hypocenterC107#1605
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
50. C2×C4×C107Q16semidirect
79. C107#1605semidirect
137. C2×C4#13133808semidirect

Maximal subgroups

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

subgroupindex
#202927822
#202925472
#202923642
#202923532
#544×C1072
#1605107