Finite Group Databasebeta

← #28337623

Subgroups — #28337623

Order 13696.

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#20293258C211yesmaximal, characteristic
2#20292637C212yesmaximal
3#20292532C211yesmaximal, characteristic
4#20292388C212yesmaximal
5#343×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7#16812108—22no—
8#16812100—22no—
9#16812088—22no—
10#16812068C2212yes—
11C2×C4×Dic107C2211yescharacteristic
12#16812034—22no—
13#16812034—22no—
14C107⋊#95—22no—
15C2×M16×C107C2212yes—
16C2×C4⋊C4×C107C2211yescharacteristic
17C2×Dic214—22no—
18C2×Dic214—22no—
19C2×Dic214—21no—
20C2×Dic214—22no—
21C2×Dic214Q812yescomplemented
22C22×Dic107—21no—
23C22×Dic107—21no—
24C4⋊Dic107—22no—
25C4⋊Dic107D412yes—
26C4⋊Dic107D412yes—
27Dic107⋊C4—24no—
28C22⋊Dic107—22no—
29C4×Dic107—22no—
30C4×Dic107—22no—
31C22×C4×C107D412yes—
32C22×C4×C107C2311yescharacteristic
33M16×C107—22no—
34M16×C107—22no—
35C2×C8×C107—22no—
36C4⋊C4×C107—22no—
37C23×C107C2×D411yescharacteristic
38C2×C4×C107—22no—
39C2×C4×C107—42no—
40C2×C4×C107C2×D412yes—
41C2×C4×C107C2×Q811yescharacteristic
42C2×C4×C107#4111yescharacteristic
43C2×C4×C107#4112yes—
44C2×Dic107—22no—
45C2×Dic107—22no—
46C2×Dic107—22no—
47C2×Dic107—22no—
48C2×Dic107—41no—
49C2×Dic107—41no—
50C2×Dic107—41no—
51C2×Dic107—41no—
52Dic214—24no—
53Dic214—42no—
54Dic214—82no—
55C8×C107—42no—
56C22×C107—21no—
57C22×C107—21no—
58C22×C107#12011yescharacteristic
59C22×C107C22⋊Q811yescharacteristic
60C22×C107#12411yescharacteristic
61C4×C107—42no—
62C4×C107C22⋊Q812yes—
63C4×C107#12412yes—
64Dic107—42no—
65Dic107—42no—
66Dic107—42no—
67Dic107—42no—
68C2×C107—21no—
69C2×C107—21no—
70C2×C107#39311yescharacteristic
71C2×C107#45511yescharacteristic
72C2×C107#45611yescharacteristic
73#2028—1071nomaximal, Sylow
74C107#202811yescharacteristic, complemented, Sylow
75#544—1071no—
76#419—1072no—
77#391—1071no—
78#343D10711yescharacteristic
79#339—1072no—
80C22×Q8—1071no—
81C2×M16C2×D10712yes—
82C4⋊Q8—2142no—
83#125—2142no—
84C22⋊Q8—2142no—
85#117—1072no—
86C2×C4⋊C4C2×D10711yescharacteristic
87C2×C42—1071no—
88#103—2142no—
89#103—2142no—
90#95—2142no—
91C2×Q8—1072no—
92C2×Q8—2142no—
93C2×Q8—2142no—
94C2×Q8—2142no—
95C2×Q8—2141no—
96C22×C4—2141no—
97C22×C4—2141no—
98C22×C4D42812yes—
99C22×C4C22×D10711yescharacteristic
100M16—22no—
101M16—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—1072no—
105C4⋊C4—1072no—
106C4⋊C4—2142no—
107C4⋊C4—2144no—
108C22⋊C4—2142no—
109C42—2142no—
110C42—2142no—
111C23C2×D42811yescharacteristic
112Q8—8562no—
113Q8—4282no—
114Q8—2144no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—4281no—
118C2×C4—4281no—
119C2×C4—4281no—
120C2×C4—4281no—
121C2×C4—2142no—
122C2×C4—2142no—
123C2×C4—2142no—
124C2×C4—2142no—
125C2×C4D4×D10712yes—
126C2×C4Q8×D10711yescharacteristic
127C2×C4#1313381711yescharacteristic
128C2×C4#1313381812yes—
129C8—42no—
130C22—21no—
131C22—21no—
132C22#1681207911yescharacteristic
133C22#1681208911yescharacteristic
134C22#1681209711yescharacteristic
135C4—42no—
136C4—4282no—
137C4—4282no—
138C4—4282no—
139C4—4282no—
140C4#1681208712yes—
141C4#1681209512yes—
142C2—21no—
143C2—21no—
144C2#2029254011yescharacteristic
145C2#2029280711yescharacteristic
146C2#2029281211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#343×C107C2
socleC22×C107#120
prime radicalC23×C107C2×D4
hypercenter#343D107
hypocenterC107#2028
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
21. C2×Dic214Q8semidirect
74. C107#2028semidirect

Maximal subgroups

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

subgroupindex
#202932582
#202926372
#202925322
#202923882
#343×C1072
#2028107