Finite Group Databasebeta

← C2×#20298774 #28399902

Subgroups — C2×#20298774 #28399902

Order 14464.

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

normal subgroups only

This group has 760 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×#16813668C211yesmaximal, characteristic
2C2×#16813653C211yesmaximal, characteristic, complemented
3C2×#16813611C211yesmaximal, characteristic, complemented
4#20298774C218yesmaximal, direct product factor
5C2×#16813549C211yesmaximal, characteristic, complemented
6C2×#16813548C211yesmaximal, characteristic, complemented
7C2×#16813538C211yesmaximal, characteristic, complemented
8C2×#137×C113C211yesmaximal, characteristic, complemented
9#16813668—22no—
10#16813668—22no—
11#16813668—22no—
12#16813668C2214yes—
13C2×#14268648—21no—
14C2×#14268647—21no—
15C2×#14268647C2211yescharacteristic
16C22×D452—21no—
17C22×D452C2211yescharacteristic
18C2×D4×D113—22no—
19C2×D4×D113—21no—
20#16813653—22no—
21#16813653C2214yescomplemented
22C2×#14268637—21no—
23C2×#14268634—21no—
24C2×#14268631C2211yescharacteristic, complemented
25#16813611—22no—
26#16813611C2214yescomplemented
27C2×#14268624—22no—
28C2×C4×Dic113C2211yescharacteristic, complemented
29#16813549C2214yescomplemented
30#16813548C2214yescomplemented
31#16813538C2214yescomplemented
32C2×#41×C113C2211yescharacteristic
33C22×Q8×C113C2211yescharacteristic, complemented
34#137×C113—22no—
35#137×C113C2214yescomplemented
36C2×Q16×C113—21no—
37C2×SD16×C113—21no—
38C2×M16×C113C2211yescharacteristic, complemented
39C23×D113—22no—
40C23×D113D412yes—
41#14268648—24no—
42#14268648—24no—
43#14268647—24no—
44#14268647—21no—
45#14268647—21no—
46#14268647—24no—
47#14268647D412yes—
48#14268647C2312yes—
49C2×Dic226D411yescharacteristic
50C2×D452—22no—
51C2×D452—24no—
52C2×D452—21no—
53C2×D452—21no—
54C2×D452—21no—
55C2×D452—21no—
56C2×D452—24no—
57C2×D452D412yes—
58C2×D452D411yescharacteristic
59C2×D452C2312yes—
60C2×C22⋊D113—22no—
61C2×C22⋊D113—21no—
62D4×D113—28no—
63D4×D113—24no—
64D4×D113—28no—
65D4×D113—24no—
66C22×Dic113—21no—
67C2×C4×D113—22no—
68C2×C4×D113—21no—
69#14268637—24no—
70#14268634—24no—
71#14268631—21no—
72#14268631C2312yescomplemented
73C2×#11744751—21no—
74#14268624—28no—
75C4×Dic113—21no—
76C4×Dic113C2312yescomplemented
77#41×C113—21no—
78#41×C113—21no—
79#41×C113D412yes—
80#41×C113C2312yes—
81C2×Q8×C113—21no—
82C2×Q8×C113—21no—
83C2×Q8×C113—21no—
84C2×Q8×C113—21no—
85C2×Q8×C113—21no—
86C2×Q8×C113D411yescharacteristic
87C2×Q8×C113C2312yescomplemented
88C2×D4×C113—21no—
89C2×D4×C113D411yescharacteristic
90C22×C4×C113—21no—
91C22×C4×C113—21no—
92C22×C4×C113C2311yescharacteristic
93Q16×C113—24no—
94SD16×C113—24no—
95M16×C113—21no—
96M16×C113C2312yescomplemented
97C2×C8×C113—21no—
98C23×C113—21no—
99C23×C113C2×D411yescharacteristic
100C22×D113—22no—
101C22×D113—21no—
102C22×D113—28no—
103C22×D113—22no—
104C22×D113—44no—
105C22×D113—84no—
106C22×D113C2×D414yes—
107Q8×C113—21no—
108Q8×C113—22no—
109Q8×C113—22no—
110Q8×C113—41no—
111Q8×C113—41no—
112Q8×C113C2×D412yes—
113D4×C113—22no—
114D4×C113—22no—
115D4×C113—21no—
116D4×C113C2×D412yes—
117C2×C4×C113—21no—
118C2×C4×C113—22no—
119C2×C4×C113—22no—
120C2×C4×C113—22no—
121C2×C4×C113—41no—
122C2×C4×C113—41no—
123C2×C4×C113C2×D411yescharacteristic
124C2×C4×C113C2×D411yescharacteristic
125C2×C4×C113C2×D411yescharacteristic
126C2×C4×C113C2×D411yescharacteristic
127C2×C4×C113C2×D411yescharacteristic
128C2×C4×C113C2411yescharacteristic
129C22⋊D113—28no—
130C22⋊D113—22no—
131C22⋊D113—22no—
132C2×Dic113—21no—
133C2×Dic113—22no—
134C2×Dic113—42no—
135D452—28no—
136D452—22no—
137D452—22no—
138D452—21no—
139D452—22no—
140D452—22no—
141D452C2×D412yes—
142C4×D113—28no—
143C4×D113—22no—
144C4×D113—22no—
145Dic226—21no—
146Dic226C2×D412yes—
147C8×C113—22no—
148#11744751—22no—
149C22×C113—22no—
150C22×C113—22no—
151C22×C113—42no—
152C22×C113#12011yescharacteristic
153C22×C113#12011yescharacteristic
154C22×C113C22×D411yescharacteristic
155C2×D113—24no—
156C2×D113—22no—
157C2×D113—44no—
158C2×D113—42no—
159C2×D113—44no—
160C2×D113—82no—
161C2×D113—88no—
162C4×C113—22no—
163C4×C113—42no—
164C4×C113#12011yescharacteristic
165C4×C113#12011yescharacteristic
166C4×C113C22×D411yescharacteristic
167C4×C113C22×D411yescharacteristic
168Dic113—22no—
169Dic113—42no—
170C2×C113—21no—
171C2×C113—21no—
172C2×C113—42no—
173C2×C113#45512yescomplemented
174C2×C113C2×#12011yescharacteristic
175D113—44no—
176D113—42no—
177C2×#455—1131nomaximal, Sylow
178C113C2×#45511yescharacteristic, complemented, Sylow
179C2×H(5,𝔽2)—1131no—
180C2×#137—1131no—
181C2×#137D11311yescharacteristic, complemented
182C2×#124—1131no—
183#455—1138no—
184C2×#104—1131no—
185C2×#104—1131no—
186C2×#100—1131no—
187H(5,𝔽2)—1134no—
188H(5,𝔽2)—2262no—
189H(5,𝔽2)—2262no—
190H(5,𝔽2)—2262no—
191C2×#41—1131no—
192C2×#41—2261no—
193C2×#41—2261no—
194C2×#41C2×D11311yescharacteristic
195C22×Q8C2×D11311yescharacteristic, complemented
196C22×D4—1131no—
197C22×D4—2261no—
198C22×D4—2262no—
199C22×D4—2261no—
200#137—22no—
201#137—1134no—
202#137—2262no—
203#137C2×D11314yescomplemented
204C2×Q16—21no—
205C2×Q16—2261no—
206C2×SD16—21no—
207C2×SD16—2261no—
208C2×M16—1131no—
209C2×M16C2×D11311yescharacteristic, complemented
210#124—1134no—
211#124—2262no—
212C2×C22⋊C4—2262no—
213C2×C42—1131no—
214#104—1134no—
215#104—1134no—
216#100—1134no—
217C24—1132no—
218C24—2262no—
219#41—21no—
220#41—21no—
221#41—1132no—
222#41—1132no—
223#41—2264no—
224#41—2264no—
225#41—2264no—
226#41—2264no—
227#41—2261no—
228#41—2261no—
229#41C22⋊D11312yes—
230#41C22×D11312yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—1131no—
237C2×Q8C22⋊D11311yescharacteristic
238C2×Q8C22×D11312yescomplemented
239C2×D4—21no—
240C2×D4—1131no—
241C2×D4—1132no—
242C2×D4—1132no—
243C2×D4—2264no—
244C2×D4—2262no—
245C2×D4—2264no—
246C2×D4—2268no—
247C2×D4—2268no—
248C2×D4—2264no—
249C2×D4—2262no—
250C2×D4—2261no—
251C2×D4—2264no—
252C2×D4—2261no—
253C2×D4—2261no—
254C2×D4—2261no—
255C2×D4—2261no—
256C2×D4C22⋊D11311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2262no—
260C22×C4—2261no—
261C22×C4—2261no—
262C22×C4C22×D11311yescharacteristic
263Q16—24no—
264Q16—2264no—
265SD16—24no—
266SD16—2264no—
267M16—21no—
268M16—1132no—
269M16—2261no—
270M16C22×D11312yescomplemented
271C2×C8—21no—
272C2×C8—2261no—
273C22⋊C4—2268no—
274C42—1132no—
275C42—2261no—
276C23—21no—
277C23—1134no—
278C23—9044no—
279C23—4524no—
280C23—2268no—
281C23—2262no—
282C23—2261no—
283C23—2262no—
284C23D4×D11311yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—1132no—
291Q8—2261no—
292Q8C2×C22⋊D11312yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—1132no—
297D4—2268no—
298D4—2268no—
299D4—2262no—
300D4—2262no—
301D4—2261no—
302D4—2262no—
303D4—2262no—
304D4—2262no—
305D4—2262no—
306D4C2×C22⋊D11312yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—4522no—
314C2×C4—2268no—
315C2×C4—2262no—
316C2×C4—2261no—
317C2×C4—2262no—
318C2×C4—2262no—
319C2×C4D4×D11311yescharacteristic
320C2×C4D4×D11311yescharacteristic
321C2×C4D4×D11311yescharacteristic
322C2×C4C2×C22⋊D11311yescharacteristic
323C2×C4C2×C22⋊D11311yescharacteristic
324C2×C4C23×D11311yescharacteristic
325C8—22no—
326C8—2262no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—9042no—
331C22—9048no—
332C22—4524no—
333C22—4522no—
334C22—4524no—
335C22—2262no—
336C22—2264no—
337C22#1681359011yescharacteristic
338C22#1681359011yescharacteristic
339C22C2×D4×D11311yescharacteristic
340C4—22no—
341C4—42no—
342C4—4522no—
343C4—2262no—
344C4#1681359011yescharacteristic
345C4#1681359011yescharacteristic
346C4C2×D4×D11311yescharacteristic
347C4C22×C22⋊D11311yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—4522no—
352C2—4524no—
353C2#2029877412yesdirect product factor
354C2C2×#1681359011yescharacteristic
subgroupG/H
centerC22#16813590
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×#137×C113C2
socleC22×C113#120
prime radicalC2×#16813668C2
hypercenterC2×#137D113
hypocenterC113C2×#455
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16813653C2semidirect
3. C2×#16813611C2semidirect
4. #20298774C2direct (N × H)
5. C2×#16813549C2semidirect
6. C2×#16813548C2semidirect
7. C2×#16813538C2semidirect
8. C2×#137×C113C2semidirect
21. #16813653C22semidirect
24. C2×#14268631C22semidirect
26. #16813611C22semidirect
28. C2×C4×Dic113C22semidirect
29. #16813549C22semidirect
30. #16813548C22semidirect
31. #16813538C22semidirect
33. C22×Q8×C113C22semidirect
35. #137×C113C22semidirect
38. C2×M16×C113C22semidirect
72. #14268631C23semidirect
76. C4×Dic113C23semidirect
87. C2×Q8×C113C23semidirect
96. M16×C113C23semidirect
173. C2×C113#455semidirect
178. C113C2×#455semidirect
181. C2×#137D113semidirect
195. C22×Q8C2×D113semidirect
203. #137C2×D113semidirect
209. C2×M16C2×D113semidirect
238. C2×Q8C22×D113semidirect
270. M16C22×D113semidirect
353. C2#20298774direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136682
C2×#168136532
C2×#168136112
#202987742
C2×#168135492
C2×#168135482
C2×#168135382
C2×#137×C1132
C2×#455113