Finite Group Databasebeta

← #28390542

Subgroups — #28390542

Order 14464.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16813668C211yesmaximal, characteristic
2#20298616C212yesmaximal, complemented
3#20298602C212yesmaximal, complemented
4#20298582C211yesmaximal, characteristic, complemented
5#410×C113C211yesmaximal, characteristic, complemented
6#16813668—24no—
7#16813668—24no—
8C2×#14268648—21no—
9C2×#14268647C412yescomplemented
10C2×#14268647C2211yescharacteristic
11C22×D452C412yescomplemented
12C22×D452C2211yescharacteristic
13C2×D4×D113—21no—
14C2×D4×D113—22no—
15C2×#14268631C2212yescomplemented
16#16813550—24no—
17#16813542—24no—
18#16813541—21no—
19#16813539—21no—
20C2×#41×C113C2211yescharacteristic
21#117×C113C2212yescomplemented
22#99×C113—22no—
23C23×D113—22no—
24C23×D113—21no—
25#14268648—28no—
26#14268647—22no—
27#14268647—24no—
28#14268647—24no—
29#14268647D414yes—
30C2×Dic226C2×C411yescharacteristic
31C2×D452—22no—
32C2×D452—21no—
33C2×D452—22no—
34C2×D452—24no—
35C2×D452—24no—
36C2×D452C2×C411yescharacteristic
37C2×D452C2×C412yes—
38C2×D452D414yes—
39C2×C22⋊D113—22no—
40C2×C22⋊D113—21no—
41D4×D113—216no—
42D4×D113—28no—
43C2×C4×D113—22no—
44C2×C4×D113—21no—
45#14268631—22no—
46#14268631—22no—
47C2×#11744751—22no—
48#41×C113—22no—
49#41×C113—22no—
50C2×Q8×C113D411yescharacteristic
51C2×D4×C113D411yescharacteristic
52C2×D4×C113D412yes—
53C22×C4×C113C2×C412yes—
54C22×C4×C113C2311yescharacteristic
55C4⋊C4×C113—22no—
56C22⋊C4×C113—22no—
57C42×C113—22no—
58C23×C113C22⋊C412yes—
59C23×C113C2×D411yescharacteristic
60C22×D113—22no—
61C22×D113—21no—
62C22×D113—28no—
63C22×D113—24no—
64C22×D113—82no—
65C22×D113—84no—
66Q8×C113—22no—
67D4×C113—24no—
68D4×C113—22no—
69C2×C4×C113—22no—
70C2×C4×C113—22no—
71C2×C4×C113—42no—
72C2×C4×C113C22⋊C412yes—
73C2×C4×C113C22×C411yescharacteristic
74C2×C4×C113C2×D412yes—
75C2×C4×C113C2×D412yes—
76C2×C4×C113C2×D411yescharacteristic
77C22⋊D113—28no—
78C22⋊D113—24no—
79C2×Dic113—21no—
80D452—28no—
81D452—21no—
82D452—22no—
83D452—24no—
84D452C22⋊C412yes—
85D452C22⋊C414yes—
86C4×D113—28no—
87C4×D113—24no—
88Dic226—21no—
89Dic226C22⋊C412yes—
90#11744751—42no—
91C22×C113—22no—
92C22×C113—41no—
93C22×C113—42no—
94C22×C113C2×C22⋊C411yescharacteristic
95C22×C113#12012yes—
96C2×D113—24no—
97C2×D113—22no—
98C2×D113—44no—
99C2×D113—42no—
100C2×D113—84no—
101C2×D113—88no—
102C4×C113—22no—
103C4×C113—82no—
104C4×C113C2×C22⋊C411yescharacteristic
105C4×C113C2×C22⋊C411yescharacteristic
106C4×C113#12012yes—
107Dic113—22no—
108C2×C113—22no—
109C2×C113—21no—
110C2×C113—42no—
111C2×C113#37911yescharacteristic
112D113—44no—
113D113—42no—
114#1795—1131nomaximal, Sylow
115C113#179511yescharacteristic, complemented, Sylow
116C2×H(5,𝔽2)—1131no—
117#421—1132no—
118#418—1132no—
119#413—1131no—
120#410D11311yescharacteristic, complemented
121H(5,𝔽2)—2264no—
122H(5,𝔽2)—2264no—
123C2×#41—1132no—
124C2×#41—1131no—
125C2×#41—2261no—
126C2×#41C2×D11311yescharacteristic
127C22×D4—1132no—
128C22×D4—1131no—
129C22×D4—2261no—
130C22×D4—2262no—
131C2×M16—1132no—
132#117C2×D11312yescomplemented
133#104—2264no—
134#102—2264no—
135#101—2261no—
136#100—2261no—
137#99—22no—
138C24—2262no—
139C24—2261no—
140#41—22no—
141#41—22no—
142#41—1134no—
143#41—2264no—
144#41—2264no—
145#41—2268no—
146#41—2262no—
147C2×Q8—1131no—
148C2×Q8D45211yescharacteristic
149C2×D4—1132no—
150C2×D4—1131no—
151C2×D4—1134no—
152C2×D4—2264no—
153C2×D4—2264no—
154C2×D4—22616no—
155C2×D4—2268no—
156C2×D4—2262no—
157C2×D4—2261no—
158C2×D4—2262no—
159C2×D4—2262no—
160C2×D4—2261no—
161C2×D4D45211yescharacteristic
162C2×D4C22⋊D11312yes—
163C22×C4—2262no—
164C22×C4—2261no—
165C22×C4C4×D11312yes—
166C22×C4C22×D11311yescharacteristic
167M16—2262no—
168M16—2262no—
169C2×C8—2262no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—9042no—
174C23—9044no—
175C23—2268no—
176C23—2264no—
177C23—2262no—
178C23—2261no—
179C23#1426862412yes—
180C23C2×C22⋊D11311yescharacteristic
181Q8—22no—
182Q8—1132no—
183Q8—2261no—
184D4—24no—
185D4—22no—
186D4—1134no—
187D4—1132no—
188D4—2268no—
189D4—2268no—
190D4—2264no—
191D4—2261no—
192D4—2262no—
193D4—2264no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—2268no—
198C2×C4—2261no—
199C2×C4—2264no—
200C2×C4#1426862412yes—
201C2×C4C2×C4×D11311yescharacteristic
202C2×C4D4×D11312yes—
203C2×C4D4×D11312yes—
204C2×C4C2×D45211yescharacteristic
205C8—4522no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—9044no—
210C22—9048no—
211C22—4524no—
212C22—4522no—
213C22—2264no—
214C22—2262no—
215C22C2×#1426862411yescharacteristic
216C22#1681359012yes—
217C4—22no—
218C4—82no—
219C4—2262no—
220C4C22⋊C4×D11311yescharacteristic
221C4C22⋊C4×D11311yescharacteristic
222C4#1681359212yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—4524no—
227C2—4522no—
228C2#2029845311yescharacteristic
subgroupG/H
centerC2#20298453
commutatorC2×C4×C113C22×C4
FrattiniC22×C4C22×D113
Fitting#410×C113C2
socleC2×C113#379
prime radicalC2×#16813668C2
cosocleC22×C4×C113C23
hypercenter#410D113
hypocenterC113#1795
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20298616C2semidirect
3. #20298602C2semidirect
4. #20298582C2semidirect
5. #410×C113C2semidirect
9. C2×#14268647C4semidirect
11. C22×D452C4semidirect
15. C2×#14268631C22semidirect
21. #117×C113C22semidirect
115. C113#1795semidirect
120. #410D113semidirect
132. #117C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136682
#202986162
#202986022
#202985822
#410×C1132
#1795113