Finite Group Databasebeta

← #28406014

Subgroups — #28406014

Order 14464.

288 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 70 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#20299566C211yesmaximal, characteristic
2#20299540C211yesmaximal, characteristic
3#20299452C214yesmaximal, complemented
4C2×#16813598C211yesmaximal, characteristic
5C2×#14268655—21no—
6C2×C4⋊#90673—21no—
7C2×C22⋊#90673—21no—
8#16813679C2212yes—
9C22⋊#11744753—21no—
10(C2×#90673)⋊C4C2212yes—
11C22×C4×D113C2211yescharacteristic
12#16813598—21no—
13#16813598—21no—
14#16813598C412yescomplemented
15#16813598C2212yes—
16C2×C4⋊Dic113C411yescharacteristic, complemented
17C2×#14268626—21no—
18C2×#14268624—21no—
19C22×Q8×C113C411yescharacteristic, complemented
20C22×#90673—22no—
21#14268655—42no—
22C2×#11744753—22no—
23C4⋊#90673—42no—
24C22⋊#90673—42no—
25C23×D113D411yescharacteristic
26C22×Dic113—21no—
27C22×Dic113D411yescharacteristic
28C2×C4×D113—42no—
29C2×C4×D113C2×C411yescharacteristic
30C2×C4×D113D411yescharacteristic
31C2×C4×D113D411yescharacteristic
32C2×C4×D113C2311yescharacteristic
33C4⋊Dic113—21no—
34C4⋊Dic113C2×C412yes—
35#14268626—22no—
36#14268626—22no—
37#14268624—22no—
38#14268624—22no—
39C2×Q8×C113—21no—
40C2×Q8×C113—42no—
41C2×Q8×C113C2×C412yes—
42C22×C4×C113—21no—
43C22×C4×C113C2×C411yescharacteristic
44C23×C113C22⋊C411yescharacteristic
45C22×D113—41no—
46C22×D113—41no—
47C22×D113C22⋊C411yescharacteristic
48C22×D113SD1612yes—
49C22×D113Q1612yes—
50C22×D113C2×D411yescharacteristic
51C2×#90673—42no—
52C2×#90673—42no—
53C2×#90673—42no—
54Q8×C113—42no—
55Q8×C113—42no—
56C2×C4×C113—22no—
57C2×C4×C113—22no—
58C2×C4×C113—42no—
59C2×C4×C113C22⋊C411yescharacteristic
60C2×C4×C113C22×C411yescharacteristic
61C2×Dic113—22no—
62C2×Dic113—41no—
63C2×Dic113—42no—
64C2×Dic113C22⋊C411yescharacteristic
65C2×Dic113C2×D411yescharacteristic
66C4×D113—22no—
67C4×D113—22no—
68C4×D113—41no—
69C4×D113—41no—
70#11744753—42no—
71C22×C113—21no—
72C22×C113—21no—
73C22×C113#10312yes—
74C22×C113C2×C22⋊C411yescharacteristic
75C2×D113—21no—
76C2×D113—21no—
77C2×D113—21no—
78C2×D113—21no—
79C2×D113—42no—
80C2×D113—42no—
81C2×D113#10312yes—
82C2×D113C2×SD1611yescharacteristic
83C2×D113C2×Q1611yescharacteristic
84#90673—82no—
85C4×C113—21no—
86C4×C113—42no—
87C4×C113#9912yes—
88Dic113—21no—
89Dic113—21no—
90Dic113—42no—
91C2×C113—22no—
92C2×C113C2×#9911yescharacteristic
93C2×C113C2×#10311yescharacteristic
94C2×C113#42111yescharacteristic
95D113—21no—
96D113—22no—
97D113—21no—
98#1520—1131nomaximal, Sylow
99C113#152011yescharacteristic, complemented, Sylow
100C2×C22⋊Q8—1131no—
101C2×C22⋊C8—1131no—
102#380—1131no—
103#328—1134no—
104C22×Q8#9067311yescharacteristic, complemented
105C23×C4—1131no—
106C22×C8—2261no—
107C22⋊Q8—1132no—
108C22⋊Q8—1132no—
109C22⋊Q8—2261no—
110C22⋊Q8—2261no—
111C2×C4⋊C4—1131no—
112C2×C4⋊C4—2261no—
113C2×C4⋊C4—2261no—
114C2×C22⋊C4—2261no—
115C2×C22⋊C4—2261no—
116C22⋊C8—1132no—
117C22⋊C8—2261no—
118#95—1132no—
119C24—1131no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×#9067312yes—
123C22×C4—21no—
124C22×C4—1131no—
125C22×C4—1131no—
126C22×C4—1131no—
127C22×C4—1131no—
128C22×C4—1131no—
129C22×C4—4522no—
130C22×C4—2262no—
131C22×C4—2261no—
132C22×C4C2×#9067311yescharacteristic
133C2×C8—4522no—
134C2×C8—2262no—
135C4⋊C4—1132no—
136C4⋊C4—4522no—
137C4⋊C4—2262no—
138C4⋊C4—2262no—
139C4⋊C4—2261no—
140C22⋊C4—4522no—
141C22⋊C4—2262no—
142C22⋊C4—2262no—
143C23—1132no—
144C23—1132no—
145C23—1131no—
146C23—1131no—
147C23—4521no—
148C23—4521no—
149C23C22⋊#9067311yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—1131no—
156C2×C4—1131no—
157C2×C4—4521no—
158C2×C4—4521no—
159C2×C4—4522no—
160C2×C4—4522no—
161C2×C4—4522no—
162C2×C4—4522no—
163C2×C4—4521no—
164C2×C4—2262no—
165C2×C4—2262no—
166C2×C4—2262no—
167C2×C4C22⋊#9067311yescharacteristic
168C2×C4C22×#9067311yescharacteristic
169C8—4522no—
170C22—21no—
171C22—21no—
172C22—1131no—
173C22—1132no—
174C22—1131no—
175C22—4522no—
176C22—4522no—
177C22—2261no—
178C22—2261no—
179C22—2261no—
180C22—2261no—
181C22#1681368512yes—
182C22C2×C22⋊#9067311yescharacteristic
183C4—21no—
184C4—42no—
185C4—9042no—
186C4—4522no—
187C4—2261no—
188C4—2261no—
189C4#1681368012yes—
190C2—22no—
191C2—2261no—
192C2—2262no—
193C2—2261no—
194C2C2×#1681368011yescharacteristic
195C2C2×#1681368511yescharacteristic
196C2#2029959711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#90673
commutatorC2×C4×C113C22×C4
FrattiniC2×C4C22×#90673
FittingC22×Q8×C113C4
socleC22×C113C2×C22⋊C4
prime radicalC23×D113D4
cosocleC2×C4×D113C23
hypercenterC22×Q8#90673
hypocenterC113#1520
LCSG ▷ C2×C4×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20299452C2semidirect
14. #16813598C4semidirect
16. C2×C4⋊Dic113C4semidirect
19. C22×Q8×C113C4semidirect
99. C113#1520semidirect
104. C22×Q8#90673semidirect

Maximal subgroups

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

subgroupindex
#202995662
#202995402
#202994522
C2×#168135982
#1520113