Finite Group Databasebeta

← C2×#20292793 #28345783

Subgroups — C2×#20292793 #28345783

Order 13696.

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

normal subgroups only

This group has 844 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, complemented
2C2×#16812135C211yesmaximal, characteristic, complemented
3C2×#16812105C211yesmaximal, characteristic, complemented
4#20292793C218yesmaximal, direct product factor
5C2×#16812036C211yesmaximal, characteristic, complemented
6C2×#16812035C211yesmaximal, characteristic, complemented
7C2×#16812025C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
9#16812155—22no—
10#16812155—22no—
11#16812155—22no—
12#16812155C2214yescomplemented
13C2×#13133819—21no—
14C2×#13133818—21no—
15C2×#13133818C2211yescharacteristic
16C22×D428—21no—
17C22×D428C2211yescharacteristic
18C22×C22⋊D107—22no—
19C2×D4×D107—22no—
20C2×D4×D107—21no—
21#16812135—22no—
22#16812135C2214yescomplemented
23C2×#13133806—21no—
24C2×#13133803—21no—
25C2×#13133802C2211yescharacteristic, complemented
26#16812105—22no—
27#16812105C2214yescomplemented
28C2×C4×Dic107C2211yescharacteristic, complemented
29#16812036C2214yescomplemented
30#16812035C2214yescomplemented
31#16812025C2214yescomplemented
32C2×#41×C107C2211yescharacteristic
33C22×D4×C107C2211yescharacteristic, complemented
34AGL(1,ℤ/8)×C107—22no—
35AGL(1,ℤ/8)×C107C2214yescomplemented
36C2×SD16×C107—21no—
37C2×D8×C107—21no—
38C2×M16×C107C2211yescharacteristic, complemented
39C23×D107—22no—
40C23×D107D412yes—
41#13133819—24no—
42#13133819—24no—
43#13133818—24no—
44#13133818—21no—
45#13133818—21no—
46#13133818—24no—
47#13133818D412yescomplemented
48#13133818C2312yes—
49C2×Dic214D411yescharacteristic, complemented
50C2×D428—22no—
51C2×D428—24no—
52C2×D428—21no—
53C2×D428—21no—
54C2×D428—21no—
55C2×D428—21no—
56C2×D428—24no—
57C2×D428D412yes—
58C2×D428D411yescharacteristic, complemented
59C2×D428C2312yes—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—21no—
62C2×C22⋊D107—28no—
63C2×C22⋊D107—48no—
64D4×D107—28no—
65D4×D107—24no—
66D4×D107—28no—
67D4×D107—24no—
68C22×Dic107—21no—
69C2×C4×D107—22no—
70C2×C4×D107—21no—
71#13133806—24no—
72#13133803—24no—
73#13133802—21no—
74#13133802C2312yescomplemented
75C2×C107⋊C8—21no—
76C4×Dic107—21no—
77C4×Dic107C2312yescomplemented
78C24×C107—21no—
79#41×C107—21no—
80#41×C107—21no—
81#41×C107D412yescomplemented
82#41×C107C2312yes—
83C2×Q8×C107D411yescharacteristic, complemented
84C2×D4×C107—21no—
85C2×D4×C107—21no—
86C2×D4×C107—21no—
87C2×D4×C107—21no—
88C2×D4×C107—21no—
89C2×D4×C107—21no—
90C2×D4×C107D411yescharacteristic, complemented
91C2×D4×C107C2312yescomplemented
92C22×C4×C107—21no—
93C22×C4×C107C2311yescharacteristic
94SD16×C107—24no—
95D8×C107—24no—
96M16×C107—21no—
97M16×C107C2312yescomplemented
98C2×C8×C107—21no—
99C23×C107—21no—
100C23×C107—22no—
101C23×C107—44no—
102C23×C107—41no—
103C23×C107—41no—
104C23×C107C2×D411yescharacteristic
105C22×D107—22no—
106C22×D107—21no—
107C22×D107—28no—
108C22×D107—22no—
109C22×D107—44no—
110C22×D107—84no—
111C22×D107C2×D414yes—
112Q8×C107—21no—
113Q8×C107C2×D412yescomplemented
114D4×C107—22no—
115D4×C107—22no—
116D4×C107—21no—
117D4×C107—22no—
118D4×C107—22no—
119D4×C107—41no—
120D4×C107—41no—
121D4×C107C2×D412yescomplemented
122C2×C4×C107—21no—
123C2×C4×C107—22no—
124C2×C4×C107—22no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—28no—
132C22⋊D107—22no—
133C22⋊D107—22no—
134C22⋊D107—416no—
135C2×Dic107—24no—
136C2×Dic107—21no—
137C2×Dic107—22no—
138D428—28no—
139D428—22no—
140D428—22no—
141D428—21no—
142D428—22no—
143D428—22no—
144D428C2×D412yescomplemented
145C4×D107—28no—
146C4×D107—22no—
147C4×D107—22no—
148Dic214—21no—
149Dic214C2×D412yescomplemented
150C8×C107—22no—
151C107⋊C8—22no—
152C22×C107—22no—
153C22×C107—22no—
154C22×C107—44no—
155C22×C107—42no—
156C22×C107—42no—
157C22×C107—82no—
158C22×C107—82no—
159C22×C107#12011yescharacteristic
160C22×C107#12011yescharacteristic
161C22×C107C22×D411yescharacteristic
162C2×D107—24no—
163C2×D107—22no—
164C2×D107—44no—
165C2×D107—42no—
166C2×D107—44no—
167C2×D107—82no—
168C2×D107—88no—
169C4×C107—22no—
170C4×C107#12011yescharacteristic
171C4×C107#12011yescharacteristic
172C4×C107C22×D411yescharacteristic
173C4×C107C22×D411yescharacteristic
174Dic107—24no—
175Dic107—22no—
176C2×C107—21no—
177C2×C107—21no—
178C2×C107—42no—
179C2×C107—82no—
180C2×C107#45312yescomplemented
181C2×C107C2×#12011yescharacteristic
182D107—44no—
183D107—42no—
184C2×#453—1071nomaximal, Sylow
185C107C2×#45311yescharacteristic, complemented, Sylow
186C2×H(5,𝔽2)—1071no—
187C2×AGL(1,ℤ/8)—1071no—
188C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
189C2×#127—1071no—
190#453—1078no—
191C2×#104—1071no—
192C2×#104—1071no—
193C2×#100—1071no—
194H(5,𝔽2)—1074no—
195H(5,𝔽2)—2142no—
196H(5,𝔽2)—2142no—
197H(5,𝔽2)—2142no—
198C2×#41—1071no—
199C2×#41—2141no—
200C2×#41—2141no—
201C2×#41C2×D10711yescharacteristic
202C22×D4—1071no—
203C22×D4—2141no—
204C22×D4—2142no—
205C22×D4—2142no—
206C22×D4—2141no—
207C22×D4C2×D10711yescharacteristic, complemented
208AGL(1,ℤ/8)—22no—
209AGL(1,ℤ/8)—1074no—
210AGL(1,ℤ/8)—2142no—
211AGL(1,ℤ/8)C2×D10714yescomplemented
212C2×SD16—21no—
213C2×SD16—2141no—
214C2×D8—21no—
215C2×D8—2141no—
216C2×M16—1071no—
217C2×M16C2×D10711yescharacteristic, complemented
218#127—1074no—
219#127—2142no—
220C2×C42—1071no—
221#104—1074no—
222#104—1074no—
223#100—1074no—
224C24—21no—
225C24—1072no—
226C24—2142no—
227#41—21no—
228#41—21no—
229#41—1072no—
230#41—1072no—
231#41—2144no—
232#41—2144no—
233#41—2144no—
234#41—2144no—
235#41—2141no—
236#41—2141no—
237#41C22⋊D10712yescomplemented
238#41C22×D10712yes—
239C2×Q8—1071no—
240C2×Q8C22⋊D10711yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—1071no—
248C2×D4—1072no—
249C2×D4—1072no—
250C2×D4—4288no—
251C2×D4—2144no—
252C2×D4—2142no—
253C2×D4—2144no—
254C2×D4—2148no—
255C2×D4—2148no—
256C2×D4—2144no—
257C2×D4—2148no—
258C2×D4—2142no—
259C2×D4—2141no—
260C2×D4—2144no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2141no—
264C2×D4—2141no—
265C2×D4C22⋊D10711yescharacteristic, complemented
266C2×D4C22×D10712yescomplemented
267C22×C4—21no—
268C22×C4—2142no—
269C22×C4—2141no—
270C22×C4—2141no—
271C22×C4C22×D10711yescharacteristic
272SD16—24no—
273SD16—2144no—
274D8—24no—
275D8—2144no—
276M16—21no—
277M16—1072no—
278M16—2141no—
279M16C22×D10712yescomplemented
280C2×C8—21no—
281C2×C8—2141no—
282C42—1072no—
283C42—2141no—
284C23—21no—
285C23—22no—
286C23—44no—
287C23—41no—
288C23—41no—
289C23—8564no—
290C23—1074no—
291C23—4284no—
292C23—2148no—
293C23—2142no—
294C23—2141no—
295C23—2142no—
296C23D4×D10711yescharacteristic
297Q8—21no—
298Q8—1072no—
299Q8—2141no—
300Q8C2×C22⋊D10712yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—1072no—
309D4—42816no—
310D4—2148no—
311D4—2148no—
312D4—2142no—
313D4—2142no—
314D4—2141no—
315D4—2142no—
316D4—2142no—
317D4—2142no—
318D4—2142no—
319D4C2×C22⋊D10712yescomplemented
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—2148no—
324C2×C4—2144no—
325C2×C4—2141no—
326C2×C4—2142no—
327C2×C4—2142no—
328C2×C4—2142no—
329C2×C4D4×D10711yescharacteristic
330C2×C4D4×D10711yescharacteristic
331C2×C4D4×D10711yescharacteristic
332C2×C4C2×C22⋊D10711yescharacteristic
333C2×C4C2×C22⋊D10711yescharacteristic
334C2×C4C23×D10711yescharacteristic
335C8—22no—
336C8—2142no—
337C22—22no—
338C22—22no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22—82no—
344C22—8562no—
345C22—8568no—
346C22—4284no—
347C22—4282no—
348C22—4284no—
349C22—2142no—
350C22—2144no—
351C22#1681207711yescharacteristic
352C22#1681207711yescharacteristic
353C22C2×D4×D10711yescharacteristic
354C4—22no—
355C4—2144no—
356C4—2142no—
357C4#1681207711yescharacteristic
358C4#1681207711yescharacteristic
359C4C2×D4×D10711yescharacteristic
360C4C22×C22⋊D10711yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—82no—
365C2—4282no—
366C2—4284no—
367C2#2029279312yesdirect product factor
368C2C2×#1681207711yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107#120
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107C2×#453
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812155C2semidirect
2. C2×#16812135C2semidirect
3. C2×#16812105C2semidirect
4. #20292793C2direct (N × H)
5. C2×#16812036C2semidirect
6. C2×#16812035C2semidirect
7. C2×#16812025C2semidirect
8. C2×AGL(1,ℤ/8)×C107C2semidirect
12. #16812155C22semidirect
22. #16812135C22semidirect
25. C2×#13133802C22semidirect
27. #16812105C22semidirect
28. C2×C4×Dic107C22semidirect
29. #16812036C22semidirect
30. #16812035C22semidirect
31. #16812025C22semidirect
33. C22×D4×C107C22semidirect
35. AGL(1,ℤ/8)×C107C22semidirect
38. C2×M16×C107C22semidirect
47. #13133818D4semidirect
49. C2×Dic214D4semidirect
58. C2×D428D4semidirect
74. #13133802C23semidirect
77. C4×Dic107C23semidirect
81. #41×C107D4semidirect
83. C2×Q8×C107D4semidirect
90. C2×D4×C107D4semidirect
91. C2×D4×C107C23semidirect
97. M16×C107C23semidirect
113. Q8×C107C2×D4semidirect
121. D4×C107C2×D4semidirect
144. D428C2×D4semidirect
149. Dic214C2×D4semidirect
180. C2×C107#453semidirect
185. C107C2×#453semidirect
188. C2×AGL(1,ℤ/8)D107semidirect
207. C22×D4C2×D107semidirect
211. AGL(1,ℤ/8)C2×D107semidirect
217. C2×M16C2×D107semidirect
237. #41C22⋊D107semidirect
240. C2×Q8C22⋊D107semidirect
265. C2×D4C22⋊D107semidirect
266. C2×D4C22×D107semidirect
279. M16C22×D107semidirect
300. Q8C2×C22⋊D107semidirect
319. D4C2×C22⋊D107semidirect
367. C2#20292793direct (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×#168121352
C2×#168121052
#202927932
C2×#168120362
C2×#168120352
C2×#168120252
C2×AGL(1,ℤ/8)×C1072
C2×#453107