Finite Group Databasebeta

← C2×#20298787 #28399961

Subgroups — C2×#20298787 #28399961

Order 14464.

794 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16813667C211yesmaximal, characteristic
2C2×#16813605C212yesmaximal, complemented
3#20298787C218yesmaximal, direct product factor
4C2×#16813537C211yesmaximal, characteristic, complemented
5C2×#16813536C212yesmaximal, complemented
6C2×#123×C113C211yesmaximal, characteristic, complemented
7#16813667—24no—
8#16813667—22no—
9#16813667C2214yes—
10C2×#14268647—21no—
11C2×#14268646—22no—
12C22×C22⋊D113—21no—
13C22×C22⋊D113C2212yes—
14C2×D4×D113—22no—
15#16813605—24no—
16#16813605C2218yescomplemented
17C2×#14268626—22no—
18C2×C22⋊Dic113C2212yescomplemented
19C2×#14268624—22no—
20#16813537C2214yescomplemented
21#16813536C2218yescomplemented
22C22×D4×C113C2211yescharacteristic
23C23×C4×C113C2211yescharacteristic, complemented
24#123×C113—22no—
25#123×C113C2214yescomplemented
26C2×C4⋊C4×C113—21no—
27C2×C22⋊C4×C113—21no—
28C2×C22⋊C4×C113C2211yescharacteristic, complemented
29C23×D113—21no—
30C23×D113D412yes—
31#14268647—24no—
32#14268647—24no—
33#14268646—28no—
34#14268646—28no—
35C2×Dic226—21no—
36C2×D452—21no—
37C2×C22⋊D113—22no—
38C2×C22⋊D113—24no—
39C2×C22⋊D113—22no—
40C2×C22⋊D113—22no—
41C2×C22⋊D113—24no—
42C2×C22⋊D113—22no—
43C2×C22⋊D113—22no—
44C2×C22⋊D113D414yes—
45C2×C22⋊D113C2314yes—
46D4×D113—28no—
47D4×D113—28no—
48C22×Dic113—21no—
49C22×Dic113—22no—
50C22×Dic113D412yes—
51C2×C4×D113—22no—
52#14268626—28no—
53C22⋊Dic113—22no—
54C22⋊Dic113C2314yescomplemented
55#14268624—28no—
56C24×C113D411yescharacteristic
57C24×C113C2311yescharacteristic
58C2×D4×C113—22no—
59C2×D4×C113—22no—
60C2×D4×C113D412yes—
61C2×D4×C113C2312yes—
62C22×C4×C113—22no—
63C22×C4×C113—22no—
64C22×C4×C113—21no—
65C22×C4×C113—21no—
66C22×C4×C113—21no—
67C22×C4×C113D411yescharacteristic
68C22×C4×C113C2312yescomplemented
69C4⋊C4×C113—24no—
70C22⋊C4×C113—24no—
71C22⋊C4×C113—21no—
72C22⋊C4×C113C2312yescomplemented
73C23×C113—21no—
74C23×C113—21no—
75C23×C113—42no—
76C23×C113—42no—
77C23×C113C2×D412yes—
78C23×C113C2×D412yes—
79C23×C113C2×D411yescharacteristic
80C23×C113C2×D412yes—
81C23×C113C2×D411yescharacteristic
82C23×C113C2411yescharacteristic
83C22×D113—22no—
84C22×D113—24no—
85C22×D113—22no—
86C22×D113—44no—
87C22×D113—82no—
88C22×D113C2×D414yes—
89D4×C113—24no—
90D4×C113—24no—
91C2×C4×C113—21no—
92C2×C4×C113—21no—
93C2×C4×C113—22no—
94C2×C4×C113—22no—
95C2×C4×C113—24no—
96C2×C4×C113—42no—
97C2×C4×C113—42no—
98C2×C4×C113—41no—
99C2×C4×C113—41no—
100C2×C4×C113C2×D412yes—
101C22⋊D113—28no—
102C22⋊D113—24no—
103C22⋊D113—24no—
104C22⋊D113—24no—
105C22⋊D113—24no—
106C2×Dic113—24no—
107C2×Dic113—22no—
108C2×Dic113—22no—
109C2×Dic113—24no—
110C2×Dic113—44no—
111C2×Dic113C2×D414yes—
112D452—24no—
113C4×D113—28no—
114Dic226—24no—
115C22×C113—24no—
116C22×C113—22no—
117C22×C113—22no—
118C22×C113—42no—
119C22×C113—41no—
120C22×C113—41no—
121C22×C113—42no—
122C22×C113—42no—
123C22×C113—81no—
124C22×C113#12012yes—
125C22×C113#12011yescharacteristic
126C22×C113#12011yescharacteristic
127C22×C113C22×D412yes—
128C22×C113C22×D411yescharacteristic
129C2×D113—24no—
130C2×D113—44no—
131C2×D113—44no—
132C2×D113—82no—
133C2×D113—84no—
134C4×C113—22no—
135C4×C113—42no—
136C4×C113—42no—
137Dic113—24no—
138Dic113—44no—
139C2×C113—22no—
140C2×C113—21no—
141C2×C113—22no—
142C2×C113—21no—
143C2×C113—42no—
144C2×C113#45812yescomplemented
145C2×C113C2×#12011yescharacteristic
146D113—44no—
147C2×#458—1131nomaximal, Sylow
148C113C2×#45811yescharacteristic, complemented, Sylow
149C2×H(5,𝔽2)—1131no—
150C2×#123—1132no—
151C2×#123D11311yescharacteristic, complemented
152#458—1138no—
153C2×#99—1131no—
154C2×#99—1132no—
155H(5,𝔽2)—1134no—
156H(5,𝔽2)—2264no—
157H(5,𝔽2)—2262no—
158C2×#41—2262no—
159C2×#41—2261no—
160C22×D4—1132no—
161C22×D4—2262no—
162C22×D4—2261no—
163C22×D4C2×D11311yescharacteristic
164C23×C4C2×D11311yescharacteristic, complemented
165#123—22no—
166#123—1138no—
167#123—2264no—
168#123C2×D11314yescomplemented
169C2×C4⋊C4—21no—
170C2×C4⋊C4—2262no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—1132no—
173C2×C22⋊C4—2262no—
174C2×C22⋊C4C2×D11311yescharacteristic, complemented
175#99—1134no—
176#99—1138no—
177C24—1132no—
178C24—2261no—
179C24C22⋊D11311yescharacteristic
180C24C22×D11311yescharacteristic
181#41—2264no—
182#41—2268no—
183#41—2268no—
184#41—2264no—
185C2×Q8—2261no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—1134no—
189C2×D4—1134no—
190C2×D4—2261no—
191C2×D4—2268no—
192C2×D4—2268no—
193C2×D4—2262no—
194C2×D4—2264no—
195C2×D4—2262no—
196C2×D4—2262no—
197C2×D4—2262no—
198C2×D4—2262no—
199C2×D4—2264no—
200C2×D4C22⋊D11312yes—
201C2×D4C22×D11312yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—1132no—
208C22×C4—2262no—
209C22×C4—2262no—
210C22×C4—2261no—
211C22×C4C22⋊D11311yescharacteristic
212C22×C4C22×D11312yescomplemented
213C4⋊C4—24no—
214C4⋊C4—2268no—
215C22⋊C4—24no—
216C22⋊C4—21no—
217C22⋊C4—1134no—
218C22⋊C4—2268no—
219C22⋊C4—2262no—
220C22⋊C4C22×D11312yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—1134no—
226C23—9042no—
227C23—4524no—
228C23—2264no—
229C23—2262no—
230C23—2262no—
231C23D4×D11312yes—
232C23D4×D11312yes—
233C23C2×C22⋊D11311yescharacteristic
234C23C2×C22⋊D11312yes—
235C23C2×C22⋊D11311yescharacteristic
236C23C23×D11311yescharacteristic
237Q8—2264no—
238D4—24no—
239D4—24no—
240D4—2264no—
241D4—2268no—
242D4—2264no—
243D4—2264no—
244D4—2264no—
245D4—2264no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—1134no—
256C2×C4—4524no—
257C2×C4—2268no—
258C2×C4—2264no—
259C2×C4—2264no—
260C2×C4—2262no—
261C2×C4—2262no—
262C2×C4C2×C22⋊D11312yes—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—9042no—
273C22—9044no—
274C22—4524no—
275C22—4524no—
276C22—2264no—
277C22#1681359011yescharacteristic
278C22#1681359012yes—
279C22#1681359011yescharacteristic
280C22C2×D4×D11312yes—
281C22C22×C22⋊D11311yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—4524no—
286C4—2264no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—4524no—
293C2#2029878712yesdirect product factor
294C2C2×#1681359011yescharacteristic
subgroupG/H
centerC22#16813590
commutator, cosocleC23×C113C24
FrattiniC23C23×D113
FittingC2×#123×C113C2
socleC22×C113#120
prime radicalC2×#16813667C2
hypercenterC2×#123D113
hypocenterC113C2×#458
LCSG ▷ C23×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16813605C2semidirect
3. #20298787C2direct (N × H)
4. C2×#16813537C2semidirect
5. C2×#16813536C2semidirect
6. C2×#123×C113C2semidirect
16. #16813605C22semidirect
18. C2×C22⋊Dic113C22semidirect
20. #16813537C22semidirect
21. #16813536C22semidirect
23. C23×C4×C113C22semidirect
25. #123×C113C22semidirect
28. C2×C22⋊C4×C113C22semidirect
54. C22⋊Dic113C23semidirect
68. C22×C4×C113C23semidirect
72. C22⋊C4×C113C23semidirect
144. C2×C113#458semidirect
148. C113C2×#458semidirect
151. C2×#123D113semidirect
164. C23×C4C2×D113semidirect
168. #123C2×D113semidirect
174. C2×C22⋊C4C2×D113semidirect
212. C22×C4C22×D113semidirect
220. C22⋊C4C22×D113semidirect
293. C2#20298787direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136672
C2×#168136052
#202987872
C2×#168135372
C2×#168135362
C2×#123×C1132
C2×#458113