Finite Group Databasebeta

← C2×#20292806 #28345792

Subgroups — C2×#20292806 #28345792

Order 13696.

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×#16812155C211yesmaximal, characteristic
2C2×#16812142C211yesmaximal, characteristic, complemented
3C2×#16812139C211yesmaximal, characteristic, complemented
4C2×#16812097C211yesmaximal, characteristic, complemented
5#20292806C218yesmaximal, direct product factor
6C2×#16812037C211yesmaximal, characteristic, complemented
7C2×#16812025C211yesmaximal, characteristic, complemented
8C2×#104×C107C211yesmaximal, characteristic, complemented
9#16812155—22no—
10#16812155—22no—
11#16812155—22no—
12#16812155C2214yes—
13C2×#13133819—21no—
14C2×#13133818—21no—
15C2×#13133818C2211yescharacteristic
16C22×Dic214C2211yescharacteristic, complemented
17C22×D428—21no—
18C22×D428C2211yescharacteristic
19C2×D4×D107—22no—
20C2×D4×D107—21no—
21#16812142—22no—
22#16812142C2214yescomplemented
23#16812139—22no—
24#16812139C2214yescomplemented
25C2×Dic428—21no—
26C2×#13133808—21no—
27C2×#13133807—21no—
28C2×#13133806—21no—
29C2×#13133802C2211yescharacteristic, complemented
30#16812097—22no—
31#16812097C2214yescomplemented
32C2×(C2×D107)⋊C4—22no—
33#16812037C2214yescomplemented
34#16812025C2214yescomplemented
35C2×#41×C107C2211yescharacteristic
36C2×M16×C107C2211yescharacteristic, complemented
37C2×C42×C107C2211yescharacteristic, complemented
38#104×C107C2214yescomplemented
39C23×D107—22no—
40C23×D107D412yes—
41#13133819—24no—
42#13133819—24no—
43#13133818—24no—
44#13133818—21no—
45#13133818—21no—
46#13133818—24no—
47#13133818D412yes—
48#13133818C2312yes—
49C2×Dic214—21no—
50C2×Dic214—21no—
51C2×Dic214—21no—
52C2×Dic214—21no—
53C2×Dic214—21no—
54C2×Dic214D411yescharacteristic
55C2×Dic214C2312yescomplemented
56C2×D428—22no—
57C2×D428—24no—
58C2×D428—21no—
59C2×D428—21no—
60C2×D428—21no—
61C2×D428—21no—
62C2×D428—24no—
63C2×D428D412yes—
64C2×D428D411yescharacteristic
65C2×D428C2312yes—
66C2×C22⋊D107—22no—
67C2×C22⋊D107—21no—
68D4×D107—28no—
69D4×D107—24no—
70D4×D107—28no—
71D4×D107—24no—
72C22×Dic107—21no—
73C2×C4×D107—22no—
74C2×C4×D107—21no—
75Dic428—24no—
76#13133808—24no—
77#13133807—24no—
78#13133806—24no—
79#13133802—21no—
80#13133802C2312yescomplemented
81C2×C107⋊C8—21no—
82(C2×D107)⋊C4—28no—
83#41×C107—21no—
84#41×C107—21no—
85#41×C107D412yes—
86#41×C107C2312yes—
87C2×Q8×C107D411yescharacteristic
88C2×D4×C107—21no—
89C2×D4×C107D411yescharacteristic
90C22×C4×C107—21no—
91C22×C4×C107—21no—
92C22×C4×C107C2311yescharacteristic
93M16×C107—21no—
94M16×C107C2312yescomplemented
95C2×C8×C107—21no—
96C42×C107—21no—
97C42×C107C2312yescomplemented
98C23×C107—21no—
99C23×C107C2×D411yescharacteristic
100C22×D107—22no—
101C22×D107—21no—
102C22×D107—28no—
103C22×D107—22no—
104C22×D107—44no—
105C22×D107—84no—
106C22×D107C2×D414yes—
107Q8×C107—21no—
108Q8×C107C2×D412yes—
109D4×C107—22no—
110D4×C107—22no—
111D4×C107—21no—
112D4×C107C2×D412yes—
113C2×C4×C107—21no—
114C2×C4×C107—22no—
115C2×C4×C107—22no—
116C2×C4×C107—22no—
117C2×C4×C107—42no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C22⋊D107—28no—
125C22⋊D107—22no—
126C22⋊D107—22no—
127C2×Dic107—21no—
128C2×Dic107—22no—
129C2×Dic107—41no—
130C2×Dic107—41no—
131D428—28no—
132D428—22no—
133D428—22no—
134D428—21no—
135D428—22no—
136D428—22no—
137D428C2×D412yes—
138C4×D107—28no—
139C4×D107—22no—
140C4×D107—22no—
141Dic214—21no—
142Dic214—22no—
143Dic214—22no—
144Dic214—41no—
145Dic214—41no—
146Dic214C2×D412yes—
147C8×C107—22no—
148C107⋊C8—22no—
149C22×C107—22no—
150C22×C107—22no—
151C22×C107—42no—
152C22×C107#12011yescharacteristic
153C22×C107#12011yescharacteristic
154C22×C107C22×D411yescharacteristic
155C2×D107—24no—
156C2×D107—22no—
157C2×D107—44no—
158C2×D107—42no—
159C2×D107—44no—
160C2×D107—82no—
161C2×D107—88no—
162C4×C107—22no—
163C4×C107—42no—
164C4×C107#12011yescharacteristic
165C4×C107#12011yescharacteristic
166C4×C107C22×D411yescharacteristic
167C4×C107C22×D411yescharacteristic
168Dic107—22no—
169Dic107—42no—
170C2×C107—21no—
171C2×C107—21no—
172C2×C107—42no—
173C2×C107#45512yescomplemented
174C2×C107C2×#12011yescharacteristic
175D107—44no—
176D107—42no—
177C2×#455—1071nomaximal, Sylow
178C107C2×#45511yescharacteristic, complemented, Sylow
179C2×H(5,𝔽2)—1071no—
180C2×#137—1071no—
181C2×#137—1071no—
182C2×#124—1071no—
183#455—1078no—
184C2×#104—1071no—
185C2×#104D10711yescharacteristic, complemented
186C2×#100—1071no—
187H(5,𝔽2)—1074no—
188H(5,𝔽2)—2142no—
189H(5,𝔽2)—2142no—
190H(5,𝔽2)—2142no—
191C2×#41—1071no—
192C2×#41—2141no—
193C2×#41—2141no—
194C2×#41C2×D10711yescharacteristic
195C22×Q8—1071no—
196C22×D4—1071no—
197C22×D4—2141no—
198C22×D4—2142no—
199C22×D4—2141no—
200#137—1074no—
201#137—1074no—
202#137—2142no—
203#137—2142no—
204C2×Q16—2141no—
205C2×Q16—2141no—
206C2×SD16—2141no—
207C2×SD16—2141no—
208C2×M16—1071no—
209C2×M16C2×D10711yescharacteristic, complemented
210#124—1074no—
211#124—2142no—
212C2×C22⋊C4—2142no—
213C2×C42C2×D10711yescharacteristic, complemented
214#104—1074no—
215#104C2×D10714yescomplemented
216#100—1074no—
217C24—1072no—
218C24—2142no—
219#41—21no—
220#41—21no—
221#41—1072no—
222#41—1072no—
223#41—2144no—
224#41—2144no—
225#41—2144no—
226#41—2144no—
227#41—2141no—
228#41—2141no—
229#41D42812yes—
230#41C22×D10712yes—
231C2×Q8—1071no—
232C2×Q8—1072no—
233C2×Q8—2141no—
234C2×Q8—2141no—
235C2×Q8—2141no—
236C2×Q8—2141no—
237C2×Q8—2141no—
238C2×Q8D42811yescharacteristic
239C2×D4—21no—
240C2×D4—1071no—
241C2×D4—1072no—
242C2×D4—1072no—
243C2×D4—2144no—
244C2×D4—2142no—
245C2×D4—2144no—
246C2×D4—2148no—
247C2×D4—2148no—
248C2×D4—2144no—
249C2×D4—2142no—
250C2×D4—2141no—
251C2×D4—2144no—
252C2×D4—2141no—
253C2×D4—2141no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4D42811yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2142no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
263Q16—2144no—
264Q16—2144no—
265SD16—2144no—
266SD16—2144no—
267M16—21no—
268M16—1072no—
269M16—2141no—
270M16C22×D10712yescomplemented
271C2×C8—21no—
272C2×C8—2141no—
273C22⋊C4—2148no—
274C42—21no—
275C42C22×D10712yescomplemented
276C23—21no—
277C23—8564no—
278C23—1074no—
279C23—4284no—
280C23—2148no—
281C23—2142no—
282C23—2141no—
283C23—2142no—
284C23D4×D10711yescharacteristic
285Q8—21no—
286Q8—1072no—
287Q8—4281no—
288Q8—4281no—
289Q8—2142no—
290Q8—2142no—
291Q8—2141no—
292Q8C2×D42812yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—1072no—
297D4—2148no—
298D4—2148no—
299D4—2142no—
300D4—2142no—
301D4—2141no—
302D4—2142no—
303D4—2142no—
304D4—2142no—
305D4—2142no—
306D4C2×D42812yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—4281no—
313C2×C4—4281no—
314C2×C4—2148no—
315C2×C4—2142no—
316C2×C4—2141no—
317C2×C4—2142no—
318C2×C4—2142no—
319C2×C4D4×D10711yescharacteristic
320C2×C4D4×D10711yescharacteristic
321C2×C4D4×D10711yescharacteristic
322C2×C4C2×D42811yescharacteristic
323C2×C4C2×D42811yescharacteristic
324C2×C4C23×D10711yescharacteristic
325C8—22no—
326C8—2142no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—8562no—
331C22—8568no—
332C22—4284no—
333C22—4282no—
334C22—4284no—
335C22—2142no—
336C22—2144no—
337C22#1681207911yescharacteristic
338C22#1681207911yescharacteristic
339C22C2×D4×D10711yescharacteristic
340C4—22no—
341C4—42no—
342C4—4282no—
343C4—2142no—
344C4#1681207911yescharacteristic
345C4#1681207911yescharacteristic
346C4C2×D4×D10711yescharacteristic
347C4C22×D42811yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—4282no—
352C2—4284no—
353C2#2029280612yesdirect product factor
354C2C2×#1681207911yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#104×C107C2
socleC22×C107#120
prime radicalC2×#16812155C2
hypercenterC2×#104D107
hypocenterC107C2×#455
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16812142C2semidirect
3. C2×#16812139C2semidirect
4. C2×#16812097C2semidirect
5. #20292806C2direct (N × H)
6. C2×#16812037C2semidirect
7. C2×#16812025C2semidirect
8. C2×#104×C107C2semidirect
16. C22×Dic214C22semidirect
22. #16812142C22semidirect
24. #16812139C22semidirect
29. C2×#13133802C22semidirect
31. #16812097C22semidirect
33. #16812037C22semidirect
34. #16812025C22semidirect
36. C2×M16×C107C22semidirect
37. C2×C42×C107C22semidirect
38. #104×C107C22semidirect
55. C2×Dic214C23semidirect
80. #13133802C23semidirect
94. M16×C107C23semidirect
97. C42×C107C23semidirect
173. C2×C107#455semidirect
178. C107C2×#455semidirect
185. C2×#104D107semidirect
209. C2×M16C2×D107semidirect
213. C2×C42C2×D107semidirect
215. #104C2×D107semidirect
270. M16C22×D107semidirect
275. C42C22×D107semidirect
353. C2#20292806direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168121552
C2×#168121422
C2×#168121392
C2×#168120972
#202928062
C2×#168120372
C2×#168120252
C2×#104×C1072
C2×#455107