Finite Group Databasebeta

← #28406871

Subgroups — #28406871

Order 14464.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#16813683C212yesmaximal, complemented
2#20299509C212yesmaximal, complemented
3#20299507C212yesmaximal, complemented
4#20299174C211yesmaximal, characteristic
5C2×#14268656—22no—
6#16813683—22no—
7#16813683C2214yes—
8C2×#14268647C411yescharacteristic
9C22×Dic226C2211yescharacteristic
10#16813622C411yescharacteristic
11#16813622C2211yescharacteristic
12#16813610C411yescharacteristic
13#16813610C2211yescharacteristic
14#16813606—22no—
15#16813602—22no—
16#117×C113C411yescharacteristic
17#14268656—24no—
18#14268656—42no—
19C2×#11744753—42no—
20#14268647—42no—
21C2×Dic226—42no—
22C2×Dic226C2×C411yescharacteristic
23C2×Dic226C2×C411yescharacteristic
24C2×Dic226D412yes—
25C2×Dic226C2311yescharacteristic
26C2×D452C2×C411yescharacteristic
27C2×C22⋊D113—21no—
28C22×Dic113D412yes—
29C2×C4×D113—21no—
30C4⋊Dic113—22no—
31C4⋊Dic113—22no—
32#14268624—22no—
33#14268624—22no—
34C22×C4×C113C2×C411yescharacteristic
35C4⋊C4×C113—21no—
36C22⋊C4×C113—21no—
37C42×C113C2×C411yescharacteristic
38C42×C113C2×C411yescharacteristic
39C23×C113C22⋊C411yescharacteristic
40C22×D113—21no—
41C2×C4×C113—22no—
42C2×C4×C113—21no—
43C2×C4×C113—21no—
44C2×C4×C113C22⋊C411yescharacteristic
45C2×C4×C113C22×C411yescharacteristic
46C22⋊D113—42no—
47C2×Dic113—22no—
48C2×Dic113—22no—
49C2×Dic113—21no—
50C2×Dic113C22⋊C412yes—
51C2×Dic113C2×D412yes—
52D452—41no—
53C4×D113—42no—
54Dic226—41no—
55Dic226—42no—
56Dic226—41no—
57Dic226—41no—
58#11744753—44no—
59C22×C113—41no—
60C22×C113#9912yes—
61C22×C113C2×C22⋊C411yescharacteristic
62C2×D113—42no—
63C2×D113—41no—
64C4×C113—21no—
65C4×C113—21no—
66C4×C113—41no—
67C4×C113—41no—
68Dic113—22no—
69Dic113—22no—
70Dic113—41no—
71C2×C113—21no—
72C2×C113—22no—
73C2×C113C2×#9911yescharacteristic
74D113—81no—
75#2133—1131nomaximal, Sylow
76C113#213311yescharacteristic, complemented, Sylow
77#536—1131no—
78C2×#101—1132no—
79#356—1132no—
80#355—1132no—
81C2×#41—1131no—
82C22×Q8—1131no—
83C2×M16—2262no—
84C4⋊Q8—1131no—
85C4⋊Q8—1131no—
86#124—1131no—
87#124—1131no—
88#123—2262no—
89C22⋊Q8—2262no—
90#117#9067311yescharacteristic
91#101—1134no—
92#101—2262no—
93#41—4522no—
94C2×Q8—1132no—
95C2×Q8—1131no—
96C2×Q8—1131no—
97C2×Q8—1131no—
98C2×Q8—4522no—
99C2×D4—1131no—
100C2×D4—2261no—
101C22×C4—1132no—
102C22×C4—2261no—
103C22×C4C2×#9067311yescharacteristic
104M16—4522no—
105M16—2264no—
106C2×C8—4522no—
107C4⋊C4—21no—
108C4⋊C4—2262no—
109C4⋊C4—2262no—
110C22⋊C4—21no—
111C22⋊C4—2262no—
112C22⋊C4—2262no—
113C42C2×#9067311yescharacteristic
114C42C2×#9067311yescharacteristic
115C23—2261no—
116C23C22⋊#9067311yescharacteristic
117Q8—4521no—
118Q8—4522no—
119Q8—4521no—
120Q8—4521no—
121D4—4521no—
122D4—4522no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—1132no—
127C2×C4—1132no—
128C2×C4—4522no—
129C2×C4—2261no—
130C2×C4—2262no—
131C2×C4—2262no—
132C2×C4C22⋊#9067311yescharacteristic
133C2×C4C22×#9067311yescharacteristic
134C8—4524no—
135C22—41no—
136C22—4522no—
137C22—4521no—
138C22#1681368112yes—
139C22C2×C22⋊#9067311yescharacteristic
140C4—21no—
141C4—21no—
142C4—41no—
143C4—41no—
144C4—4521no—
145C4—2262no—
146C4—2262no—
147C2—21no—
148C2—22no—
149C2—9041no—
150C2C2×#1681368111yescharacteristic
subgroupG/H
centerC2C2×#16813681
commutatorC2×C4×C113C22×C4
FrattiniC2×C4C22×#90673
Fitting#117×C113C4
socleC2×C113C2×#99
prime radicalC2×#14268647C4
cosocleC2×Dic226C23
hypercenter#117#90673
hypocenterC113#2133
LCSG ▷ C2×C4×C113 ▷ C22×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#16813683C2semidirect
2. #20299509C2semidirect
3. #20299507C2semidirect
76. C113#2133semidirect

Maximal subgroups

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

subgroupindex
C2×#168136832
#202995092
#202995072
#202991742
#2133113