Finite Group Databasebeta

← #28337619

Subgroups — #28337619

Order 13696.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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#20292635C212yesmaximal
2#20292532C211yesmaximal, characteristic
3#20292396C211yesmaximal, characteristic
4#20292386C212yesmaximal
5#544×C107C211yesmaximal, characteristic
6C22×Dic214C2211yescharacteristic
7C2×#13133802C2212yes—
8C2×C4×Dic107C2211yescharacteristic
9#16812032—22no—
10#16812032—22no—
11C107⋊#95—22no—
12C4⋊Q8×C107—22no—
13#125×C107—22no—
14C22⋊Q8×C107—22no—
15#117×C107C2212yes—
16C2×C4⋊C4×C107C2211yescharacteristic
17C2×Dic214—22no—
18C2×Dic214—22no—
19C2×Dic214—21no—
20C2×Dic214Q812yescomplemented
21C22×Dic107—21no—
22C22×Dic107—21no—
23#13133802—22no—
24#13133802—22no—
25C2×C107⋊C8—22no—
26C4×Dic107—22no—
27C2×Q8×C107—22no—
28C22×C4×C107D412yes—
29C22×C4×C107C2311yescharacteristic
30C4⋊C4×C107—22no—
31C4⋊C4×C107—24no—
32C4⋊C4×C107—22no—
33C4⋊C4×C107D412yes—
34C4⋊C4×C107D412yes—
35C22⋊C4×C107—22no—
36C42×C107—22no—
37C23×C107C2×D411yescharacteristic
38Q8×C107—82no—
39C2×C4×C107—22no—
40C2×C4×C107—22no—
41C2×C4×C107—22no—
42C2×C4×C107—42no—
43C2×C4×C107C2×D412yes—
44C2×C4×C107C2×Q811yescharacteristic
45C2×C4×C107#4112yes—
46C2×C4×C107#4111yescharacteristic
47C2×Dic107—22no—
48C2×Dic107—22no—
49C2×Dic107—41no—
50C2×Dic107—41no—
51C2×Dic107—41no—
52C2×Dic107—41no—
53Dic214—24no—
54Dic214—42no—
55C107⋊C8—42no—
56C22×C107—21no—
57C22×C107—21no—
58C22×C107#12011yescharacteristic
59C22×C107C22⋊Q811yescharacteristic
60C22×C107#12411yescharacteristic
61C4×C107—42no—
62C4×C107—42no—
63C4×C107—42no—
64C4×C107C22⋊Q812yes—
65C4×C107#12412yes—
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#544D10711yescharacteristic
76#419—1072no—
77#391—1071no—
78#343—1071no—
79#339—1072no—
80C22×Q8—1071no—
81C2×M16—1072no—
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—22no—
85#117C2×D10712yes—
86C2×C4⋊C4C2×D10711yescharacteristic
87C2×C42—1071no—
88#103—2142no—
89#103—2142no—
90#95—2142no—
91C2×Q8—22no—
92C2×Q8—1072no—
93C2×Q8—2142no—
94C2×Q8—2142no—
95C2×Q8—2141no—
96C22×C4—2141no—
97C22×C4—2141no—
98C22×C4C22⋊D10712yes—
99C22×C4C22×D10711yescharacteristic
100M16—2142no—
101M16—2142no—
102C2×C8—2142no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4C22⋊D10712yes—
107C4⋊C4C22⋊D10712yes—
108C22⋊C4—22no—
109C42—22no—
110C42—2142no—
111C23C2×C22⋊D10711yescharacteristic
112Q8—82no—
113Q8—4282no—
114Q8—2144no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—4281no—
120C2×C4—4281no—
121C2×C4—4281no—
122C2×C4—4281no—
123C2×C4—2142no—
124C2×C4—2142no—
125C2×C4C2×C22⋊D10712yes—
126C2×C4Q8×D10711yescharacteristic
127C2×C4#1313381912yes—
128C2×C4#1313381911yescharacteristic
129C8—4282no—
130C22—21no—
131C22—21no—
132C22#1681207811yescharacteristic
133C22#1681208511yescharacteristic
134C22#1681209811yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—4282no—
139C4—4282no—
140C4#1681208512yes—
141C4#1681209812yes—
142C2—21no—
143C2—21no—
144C2#2029253811yescharacteristic
145C2#2029280511yescharacteristic
146C2#2029281011yescharacteristic
subgroupG/H
centerC22#16812078
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#544×C107C2
socleC22×C107#120
prime radicalC23×C107C2×D4
hypercenter#544D107
hypocenterC107#2028
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. 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
#202926352
#202925322
#202923962
#202923862
#544×C1072
#2028107