Finite Group Databasebeta

← #28390992

Subgroups — #28390992

Order 14464.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#20299105C211yesmaximal, characteristic
2#20298582C211yesmaximal, characteristic, complemented
3#20298561C212yesmaximal, complemented
4#20298369C212yesmaximal, complemented
5#568×C113C211yesmaximal, characteristic, complemented
6C22×C4×D113C2211yescharacteristic
7C2×#14268631C2212yes—
8#16813603C412yes—
9#16813598—21no—
10#16813593—21no—
11#16813590C412yes—
12C4×C22⋊D113—22no—
13#16813581C2211yescharacteristic
14#16813541—21no—
15#16813539—21no—
16#16813534—24no—
17C2×#41×C113C2211yescharacteristic
18#131×C113—24no—
19C2×M16×C113—22no—
20C2×M16×C113C2212yes—
21C23×D113C2×C411yescharacteristic
22C2×C22⋊D113—22no—
23C22×Dic113C2×C411yescharacteristic
24C2×C4×D113—21no—
25C2×C4×D113—21no—
26C2×C4×D113—22no—
27C2×C4×D113—22no—
28#14268631—22no—
29#14268631—22no—
30C2×#11744751—22no—
31C4⋊Dic113—21no—
32#14268626—22no—
33C22⋊Dic113—22no—
34C22⋊Dic113C2×C412yes—
35#14268624—22no—
36C4×Dic113—21no—
37#41×C113—22no—
38#41×C113—22no—
39C2×Q8×C113D411yescharacteristic
40C2×D4×C113C2×C412yes—
41C2×D4×C113D411yescharacteristic
42C22×C4×C113D412yes—
43C22×C4×C113C2311yescharacteristic
44M16×C113—24no—
45M16×C113—22no—
46M16×C113—22no—
47C2×C8×C113—24no—
48C2×C8×C113D414yescomplemented
49C23×C113C22⋊C412yes—
50C23×C113C22×C411yescharacteristic
51C22×D113—21no—
52C22×D113—22no—
53C22×D113—42no—
54Q8×C113—22no—
55D4×C113—24no—
56D4×C113—22no—
57C2×C4×C113—22no—
58C2×C4×C113—22no—
59C2×C4×C113C22⋊C412yes—
60C2×C4×C113C2×D412yes—
61C2×C4×C113C2×D411yescharacteristic
62C2×C4×C113C2×D411yescharacteristic
63C2×C4×C113#4112yes—
64C22⋊D113—44no—
65C2×Dic113—22no—
66C2×Dic113—21no—
67C2×Dic113—22no—
68C4×D113—41no—
69C4×D113—42no—
70C4×D113—41no—
71C8×C113—24no—
72#11744751—42no—
73C22×C113—22no—
74C22×C113—41no—
75C22×C113—42no—
76C22×C113C2×C22⋊C411yescharacteristic
77C22×C113C4×D412yes—
78C2×D113—22no—
79C2×D113—24no—
80C2×D113—41no—
81C2×D113—42no—
82C2×D113—41no—
83C4×C113—22no—
84C4×C113#12011yescharacteristic
85C4×C113#12112yes—
86C4×C113#12311yescharacteristic
87Dic113—42no—
88Dic113—41no—
89C2×C113—21no—
90C2×C113—22no—
91C2×C113—42no—
92C2×C113#38611yescharacteristic
93D113—42no—
94#1891—1131nomaximal, Sylow
95C113#189111yescharacteristic, complemented, Sylow
96#568D11311yescharacteristic, complemented
97#525—1131no—
98#413—1131no—
99#407—1132no—
100#344—1132no—
101C2×#41C2×D11311yescharacteristic
102C23×C4—1131no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1132no—
106C2×M16C2×D11312yes—
107#123—1132no—
108C22⋊Q8—2261no—
109#121—2261no—
110#120—1132no—
111C4×D4—2262no—
112#117—1131no—
113#101—2261no—
114#100—2261no—
115C22⋊C8—2264no—
116C24—1131no—
117#41—22no—
118#41—22no—
119C2×Q8D45211yescharacteristic
120C2×D4—2262no—
121C2×D4C4×D11312yes—
122C2×D4D45211yescharacteristic
123C22×C4—1131no—
124C22×C4—2262no—
125C22×C4—2261no—
126C22×C4—2261no—
127C22×C4—2262no—
128C22×C4C22⋊D11312yes—
129C22×C4C22×D11311yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2262no—
134M16—2262no—
135C2×C8—24no—
136C2×C8—2262no—
137C2×C8C22⋊D11314yescomplemented
138C4⋊C4—2262no—
139C4⋊C4—2261no—
140C22⋊C4—1132no—
141C22⋊C4—2262no—
142C22⋊C4—2262no—
143C42—2261no—
144C23—4522no—
145C23—2261no—
146C23—2262no—
147C23#1426862412yes—
148C23C2×C4×D11311yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—4524no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—4521no—
156C2×C4—4522no—
157C2×C4—4521no—
158C2×C4—2262no—
159C2×C4—2261no—
160C2×C4—2262no—
161C2×C4#1426862412yes—
162C2×C4C2×C22⋊D11312yes—
163C2×C4C2×C22⋊D11311yescharacteristic
164C2×C4C2×D45211yescharacteristic
165C2×C4#1426864712yes—
166C8—24no—
167C8—4522no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—4521no—
172C22—4522no—
173C22—4521no—
174C22—2262no—
175C22—2264no—
176C22C2×#1426862411yescharacteristic
177C22C4×C22⋊D11312yes—
178C4—22no—
179C4—4522no—
180C4—4521no—
181C4#1681359111yescharacteristic
182C4#1681359612yes—
183C4#1681360511yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—4522no—
188C2#2029847711yescharacteristic
subgroupG/H
centerC4#16813591
commutatorC23×C113C22×C4
FrattiniC22×C4C22×D113
Fitting#568×C113C2
socleC2×C113#386
prime radical#20299105C2
cosocleC22×C4×C113C23
hypercenter#568D113
hypocenterC113#1891
LCSG ▷ C23×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20298582C2semidirect
3. #20298561C2semidirect
4. #20298369C2semidirect
5. #568×C113C2semidirect
48. C2×C8×C113D4semidirect
95. C113#1891semidirect
96. #568D113semidirect
137. C2×C8C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
#202991052
#202985822
#202985612
#202983692
#568×C1132
#1891113