Finite Group Databasebeta

← #28399979

Subgroups — #28399979

Order 14464.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#20299434C211yesmaximal, characteristic
2#20299172C212yesmaximal, complemented
3#20298787C218yesmaximal, complemented
4#20298572C211yesmaximal, characteristic, complemented
5#20298571C212yesmaximal, complemented
6#536×C113C211yesmaximal, characteristic, complemented
7#16813670—21no—
8#16813669—22no—
9#16813668—22no—
10#16813667—21no—
11#16813667C2214yes—
12C2×#14268647—22no—
13C2×#14268647C2212yes—
14#41×D113—24no—
15#16813620—22no—
16#16813611—22no—
17#16813605C2218yescomplemented
18#16813598—24no—
19#16813581C2212yescomplemented
20#16813537C2214yescomplemented
21#16813536C2218yescomplemented
22C2×#41×C113C2211yescharacteristic
23C22×Q8×C113C2211yescharacteristic, complemented
24C4⋊Q8×C113—21no—
25#124×C113—21no—
26#123×C113C2214yescomplemented
27C22⋊Q8×C113—22no—
28#117×C113C2211yescharacteristic, complemented
29#14268648—24no—
30#14268647—24no—
31#14268647—24no—
32#14268647—22no—
33#14268647—22no—
34#14268647—24no—
35#14268647—22no—
36#14268647—22no—
37#14268646—24no—
38#14268646—28no—
39C2×Dic226—22no—
40C2×Dic226D412yes—
41Q8×D113—24no—
42C2×D452—22no—
43C2×D452D412yes—
44C2×C22⋊D113—24no—
45C2×C22⋊D113C2314yes—
46D4×D113—24no—
47D4×D113—28no—
48C2×C4×D113—24no—
49C2×C4×D113D414yes—
50C4⋊Dic113—22no—
51#14268626—28no—
52C22⋊Dic113C2314yes—
53#14268624—28no—
54C4×Dic113—22no—
55#41×C113—22no—
56#41×C113—22no—
57C2×Q8×C113—22no—
58C2×Q8×C113—22no—
59C2×Q8×C113—21no—
60C2×Q8×C113—21no—
61C2×Q8×C113D411yescharacteristic
62C2×D4×C113D411yescharacteristic
63C2×D4×C113C2312yes—
64C22×C4×C113D412yes—
65C22×C4×C113C2312yes—
66C22×C4×C113C2311yescharacteristic
67C4⋊C4×C113—21no—
68C4⋊C4×C113—24no—
69C22⋊C4×C113—24no—
70C22⋊C4×C113C2312yes—
71C42×C113—21no—
72C23×C113C2×D412yes—
73C23×C113C2411yescharacteristic
74C22×D113—24no—
75C22×D113C2×D414yes—
76Q8×C113—22no—
77Q8×C113—41no—
78Q8×C113—42no—
79Q8×C113—41no—
80D4×C113—24no—
81D4×C113—22no—
82C2×C4×C113—22no—
83C2×C4×C113—22no—
84C2×C4×C113—22no—
85C2×C4×C113—22no—
86C2×C4×C113—24no—
87C2×C4×C113C2×D412yes—
88C2×C4×C113C2×D412yes—
89C2×C4×C113C2×D412yes—
90C2×C4×C113C2×D411yescharacteristic
91C2×C4×C113C2×D411yescharacteristic
92C22⋊D113—28no—
93C22⋊D113—24no—
94C22⋊D113—24no—
95C2×Dic113—24no—
96C2×Dic113—24no—
97C2×Dic113C2×D414yes—
98D452—24no—
99D452—22no—
100D452—22no—
101C4×D113—28no—
102C4×D113—24no—
103C4×D113—24no—
104Dic226—24no—
105Dic226—22no—
106Dic226—22no—
107C22×C113—22no—
108C22×C113—41no—
109C22×C113—42no—
110C22×C113C22×D412yes—
111C22×C113C22×D411yescharacteristic
112C2×D113—24no—
113C2×D113—44no—
114C2×D113—84no—
115C4×C113—22no—
116C4×C113—42no—
117C4×C113—42no—
118C4×C113#12011yescharacteristic
119C4×C113#12012yes—
120C4×C113#12011yescharacteristic
121Dic113—24no—
122Dic113—44no—
123C2×C113—22no—
124C2×C113—21no—
125C2×C113—42no—
126C2×C113C2×#12011yescharacteristic
127D113—44no—
128#3028—1131nomaximal, Sylow
129C113#302811yescharacteristic, complemented, Sylow
130#585—1131no—
131#536—1132no—
132#536D11311yescharacteristic, complemented
133#458—1138no—
134#410—1131no—
135#410—1132no—
1362−1+4—2262no—
1372−1+4—2261no—
138H(5,𝔽2)—1134no—
139H(5,𝔽2)—2262no—
140H(5,𝔽2)—2261no—
141C2×#41—1132no—
142C2×#41—2264no—
143C2×#41—2262no—
144C2×#41C2×D11311yescharacteristic
145C22×Q8C2×D11311yescharacteristic, complemented
146C4⋊Q8—21no—
147C4⋊Q8—2262no—
148#124—21no—
149#124—2262no—
150#123—1138no—
151#123C2×D11314yescomplemented
152C22⋊Q8—22no—
153C22⋊Q8—2264no—
154#117—1132no—
155#117C2×D11311yescharacteristic, complemented
156#99—1134no—
157#99—1138no—
158#41—22no—
159#41—22no—
160#41—2264no—
161#41—2264no—
162#41—2268no—
163#41—2264no—
164#41—2264no—
165#41—2262no—
166#41—2262no—
167#41—2262no—
168#41—2262no—
169#41—2264no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—1132no—
175C2×Q8—2262no—
176C2×Q8—2264no—
177C2×Q8C22⋊D11311yescharacteristic
178C2×D4—1134no—
179C2×D4—1132no—
180C2×D4—2262no—
181C2×D4—2268no—
182C2×D4—2264no—
183C2×D4—2264no—
184C2×D4C22⋊D11311yescharacteristic
185C2×D4C22×D11312yes—
186C22×C4—1134no—
187C22×C4—2264no—
188C22×C4C22⋊D11312yes—
189C22×C4C22×D11312yes—
190C22×C4C22×D11311yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—24no—
193C4⋊C4—2268no—
194C4⋊C4—2262no—
195C22⋊C4—24no—
196C22⋊C4—1134no—
197C22⋊C4—2268no—
198C22⋊C4C22×D11312yes—
199C42—21no—
200C42—2262no—
201C23—1134no—
202C23—2264no—
203C23C2×C22⋊D11312yes—
204C23C23×D11311yescharacteristic
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—42no—
209Q8—2264no—
210Q8—2262no—
211Q8—2262no—
212D4—24no—
213D4—22no—
214D4—2264no—
215D4—2268no—
216D4—2264no—
217D4—2264no—
218D4—2262no—
219D4—2262no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—1134no—
226C2×C4—2268no—
227C2×C4—2264no—
228C2×C4—2264no—
229C2×C4—2264no—
230C2×C4—2264no—
231C2×C4D4×D11312yes—
232C2×C4D4×D11312yes—
233C2×C4C2×C22⋊D11312yes—
234C2×C4C2×C22⋊D11311yescharacteristic
235C2×C4C2×C22⋊D11311yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—9044no—
240C22—4524no—
241C22—2264no—
242C22C2×D4×D11312yes—
243C22C22×C22⋊D11311yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—4524no—
248C4—2264no—
249C4#1681359011yescharacteristic
250C4#1681359011yescharacteristic
251C4#1681359012yes—
252C2—21no—
253C2—22no—
254C2—42no—
255C2—4524no—
256C2C2×#1681359011yescharacteristic
subgroupG/H
centerC2C2×#16813590
commutator, cosocleC23×C113C24
FrattiniC23C23×D113
Fitting#536×C113C2
socleC2×C113C2×#120
prime radical#20299434C2
hypercenter#536D113
hypocenterC113#3028
LCSG ▷ C23×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #20299172C2semidirect
3. #20298787C2semidirect
4. #20298572C2semidirect
5. #20298571C2semidirect
6. #536×C113C2semidirect
17. #16813605C22semidirect
19. #16813581C22semidirect
20. #16813537C22semidirect
21. #16813536C22semidirect
23. C22×Q8×C113C22semidirect
26. #123×C113C22semidirect
28. #117×C113C22semidirect
129. C113#3028semidirect
132. #536D113semidirect
145. C22×Q8C2×D113semidirect
151. #123C2×D113semidirect
155. #117C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202994342
#202991722
#202987872
#202985722
#202985712
#536×C1132
#3028113