Finite Group Databasebeta

← #28337629

Subgroups — #28337629

Order 13696.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#20293253C211yesmaximal, characteristic
2#20292488C211yesmaximal, characteristic
3#20292488C212yesmaximal
4#20292393C212yesmaximal
5#342×C107C211yesmaximal, characteristic
6C23×Dic107C2211yescharacteristic
7#16812088—22no—
8#16812086—24no—
9C2×C22⋊Dic107C2211yescharacteristic
10C2×C22⋊Dic107C2212yes—
11C107⋊#95—24no—
12C107⋊#95—22no—
13C2×C22⋊C4×C107C2212yes—
14C2×C22⋊C4×C107C2211yescharacteristic
15C2×Dic214—22no—
16C22×Dic107—22no—
17C22×Dic107—21no—
18C22×Dic107—21no—
19C22×Dic107—22no—
20C22×Dic107—21no—
21C22×Dic107D414yes—
22C22×Dic107Q812yescomplemented
23C4⋊Dic107—22no—
24Dic107⋊C4—24no—
25C22⋊Dic107—22no—
26C22⋊Dic107—22no—
27C22⋊Dic107—22no—
28C24×C107C2311yescharacteristic
29C22×C4×C107—22no—
30C22×C4×C107D412yes—
31C22⋊C4×C107—22no—
32C22⋊C4×C107—22no—
33C22⋊C4×C107—22no—
34C23×C107—42no—
35C23×C107C2×D412yes—
36C23×C107C2×D411yescharacteristic
37C23×C107C2×Q811yescharacteristic
38C23×C107#4111yescharacteristic
39C23×C107#4112yes—
40C2×C4×C107—22no—
41C2×C4×C107—42no—
42C2×C4×C107—44no—
43C2×C4×C107—42no—
44C2×Dic107—24no—
45C2×Dic107—24no—
46C2×Dic107—22no—
47C2×Dic107—41no—
48C2×Dic107—42no—
49C2×Dic107—41no—
50C2×Dic107—44no—
51C2×Dic107—42no—
52C2×Dic107—41no—
53Dic214—82no—
54C22×C107—24no—
55C22×C107—22no—
56C22×C107—41no—
57C22×C107—41no—
58C22×C107—42no—
59C22×C107#12011yescharacteristic
60C22×C107C22⋊Q812yes—
61C22×C107C22⋊Q811yescharacteristic
62C22×C107#12412yes—
63C22×C107#12411yescharacteristic
64C4×C107—42no—
65C4×C107—82no—
66Dic107—42no—
67Dic107—44no—
68Dic107—81no—
69C2×C107—21no—
70C2×C107—22no—
71C2×C107—22no—
72C2×C107—21no—
73C2×C107#39311yescharacteristic
74C2×C107#45812yes—
75#2030—1071nomaximal, Sylow
76C107#203011yescharacteristic, complemented, Sylow
77#543—1071no—
78#381—1072no—
79#381—1071no—
80#342—1072no—
81#342D10711yescharacteristic
82C23×C4—1071no—
83C22⋊Q8—2144no—
84C22⋊Q8—2142no—
85C2×C22⋊C4—1072no—
86C2×C22⋊C4—1071no—
87C2×C22⋊C4C2×D10712yes—
88C2×C22⋊C4C2×D10711yescharacteristic
89#95—2144no—
90#95—2142no—
91C24C22×D10711yescharacteristic
92C2×Q8—2142no—
93C22×C4—22no—
94C22×C4—1074no—
95C22×C4—1072no—
96C22×C4—2141no—
97C22×C4—2142no—
98C22×C4—2141no—
99C22×C4—2141no—
100C22×C4—2142no—
101C22×C4D42812yes—
102C4⋊C4—2144no—
103C4⋊C4—2142no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—2142no—
108C22⋊C4—2142no—
109C22⋊C4—2142no—
110C23—42no—
111C23D4×D10712yes—
112C23C2×D42811yescharacteristic
113C23Q8×D10711yescharacteristic
114C23#1313381711yescharacteristic
115C23#1313381812yes—
116Q8—8562no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—42no—
121C2×C4—4282no—
122C2×C4—4281no—
123C2×C4—4281no—
124C2×C4—4282no—
125C2×C4—4281no—
126C2×C4—4284no—
127C2×C4—2144no—
128C2×C4—2144no—
129C2×C4—2142no—
130C22—24no—
131C22—22no—
132C22—41no—
133C22—41no—
134C22—42no—
135C22#1681207911yescharacteristic
136C22#1681208712yes—
137C22#1681208911yescharacteristic
138C22#1681209512yes—
139C22#1681209711yescharacteristic
140C4—42no—
141C4—82no—
142C4—8561no—
143C4—4282no—
144C4—4284no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#2029254011yescharacteristic
150C2#2029281812yes—
subgroupG/H
centerC22#16812079
commutator, prime radical, cosocleC24×C107C23
FrattiniC24C22×D107
Fitting#342×C107C2
socleC22×C107#120
hypercenter#342D107
hypocenterC107#2030
LCSG ▷ C24×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
22. C22×Dic107Q8semidirect
76. C107#2030semidirect

Maximal subgroups

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

subgroupindex
#202932532
#202924882
#202923932
#342×C1072
#2030107