Finite Group Databasebeta

← #27196306

Subgroups — #27196306

Order 13184.

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

normal subgroups only

This group has 246 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#20268422C211yesmaximal, characteristic
2#20267801C212yesmaximal
3#20267696C211yesmaximal, characteristic
4#20267552C212yesmaximal
5#343×C103C211yesmaximal, characteristic
6C22×Dic206C2211yescharacteristic
7#15693020—22no—
8#15693012—22no—
9#15693000—22no—
10#15692980C2212yes—
11C2×C4×Dic103C2211yescharacteristic
12#15692946—22no—
13#15692946—22no—
14C103⋊#95—22no—
15C2×M16×C103C2212yes—
16C2×C4⋊C4×C103C2211yescharacteristic
17C2×Dic206—22no—
18C2×Dic206—22no—
19C2×Dic206—21no—
20C2×Dic206—22no—
21C2×Dic206Q812yescomplemented
22C22×Dic103—21no—
23C22×Dic103—21no—
24C4⋊Dic103—22no—
25C4⋊Dic103D412yes—
26C4⋊Dic103D412yes—
27Dic103⋊C4—24no—
28C22⋊Dic103—22no—
29C4×Dic103—22no—
30C4×Dic103—22no—
31C22×C4×C103D412yes—
32C22×C4×C103C2311yescharacteristic
33M16×C103—22no—
34M16×C103—22no—
35C2×C8×C103—22no—
36C4⋊C4×C103—22no—
37C23×C103C2×D411yescharacteristic
38C2×C4×C103—22no—
39C2×C4×C103—42no—
40C2×C4×C103C2×D412yes—
41C2×C4×C103C2×Q811yescharacteristic
42C2×C4×C103#4111yescharacteristic
43C2×C4×C103#4112yes—
44C2×Dic103—22no—
45C2×Dic103—22no—
46C2×Dic103—22no—
47C2×Dic103—22no—
48C2×Dic103—41no—
49C2×Dic103—41no—
50C2×Dic103—41no—
51C2×Dic103—41no—
52Dic206—24no—
53Dic206—42no—
54Dic206—82no—
55C8×C103—42no—
56C22×C103—21no—
57C22×C103—21no—
58C22×C103#12011yescharacteristic
59C22×C103C22⋊Q811yescharacteristic
60C22×C103#12411yescharacteristic
61C4×C103—42no—
62C4×C103C22⋊Q812yes—
63C4×C103#12412yes—
64Dic103—42no—
65Dic103—42no—
66Dic103—42no—
67Dic103—42no—
68C2×C103—21no—
69C2×C103—21no—
70C2×C103#39311yescharacteristic
71C2×C103#45511yescharacteristic
72C2×C103#45611yescharacteristic
73#2028—1031nomaximal, Sylow
74C103#202811yescharacteristic, complemented, Sylow
75#544—1031no—
76#419—1032no—
77#391—1031no—
78#343D10311yescharacteristic
79#339—1032no—
80C22×Q8—1031no—
81C2×M16C2×D10312yes—
82C4⋊Q8—2062no—
83#125—2062no—
84C22⋊Q8—2062no—
85#117—1032no—
86C2×C4⋊C4C2×D10311yescharacteristic
87C2×C42—1031no—
88#103—2062no—
89#103—2062no—
90#95—2062no—
91C2×Q8—1032no—
92C2×Q8—2062no—
93C2×Q8—2062no—
94C2×Q8—2062no—
95C2×Q8—2061no—
96C22×C4—2061no—
97C22×C4—2061no—
98C22×C4D41212yes—
99C22×C4C22×D10311yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—1032no—
105C4⋊C4—1032no—
106C4⋊C4—2064no—
107C4⋊C4—2062no—
108C22⋊C4—2062no—
109C42—2062no—
110C42—2062no—
111C23C2×D41211yescharacteristic
112Q8—8242no—
113Q8—4122no—
114Q8—2064no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—4121no—
118C2×C4—4121no—
119C2×C4—4121no—
120C2×C4—4121no—
121C2×C4—2062no—
122C2×C4—2062no—
123C2×C4—2062no—
124C2×C4—2062no—
125C2×C4D4×D10312yes—
126C2×C4Q8×D10311yescharacteristic
127C2×C4#1311010411yescharacteristic
128C2×C4#1311010512yes—
129C8—42no—
130C22—21no—
131C22—21no—
132C22#1569299111yescharacteristic
133C22#1569300111yescharacteristic
134C22#1569300911yescharacteristic
135C4—42no—
136C4—4122no—
137C4—4122no—
138C4—4122no—
139C4—4122no—
140C4#1569299912yes—
141C4#1569300712yes—
142C2—21no—
143C2—21no—
144C2#2026770411yescharacteristic
145C2#2026797111yescharacteristic
146C2#2026797611yescharacteristic
subgroupG/H
centerC22#15692991
commutator, cosocleC22×C4×C103C23
FrattiniC22×C4C22×D103
Fitting#343×C103C2
socleC22×C103#120
prime radicalC23×C103C2×D4
hypercenter#343D103
hypocenterC103#2028
LCSG ▷ C22×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
21. C2×Dic206Q8semidirect
74. C103#2028semidirect

Maximal subgroups

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

subgroupindex
#202684222
#202678012
#202676962
#202675522
#343×C1032
#2028103