Finite Group Databasebeta

← C2×#20292796 #28345786

Subgroups — C2×#20292796 #28345786

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×#16812137C211yesmaximal, characteristic, complemented
3C2×#16812134C211yesmaximal, characteristic, complemented
4C2×#16812106C211yesmaximal, characteristic, complemented
5#20292796C218yesmaximal, 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#16812155C2214yescomplemented
13C2×#13133819—21no—
14C2×#13133818—21no—
15C2×#13133818C2211yescharacteristic
16C22×D428—21no—
17C22×D428—22no—
18C22×D428C2211yescharacteristic
19C22×D428C2211yescharacteristic, complemented
20C2×D4×D107—22no—
21C2×D4×D107—21no—
22#16812137—22no—
23#16812137C2214yescomplemented
24#16812134—22no—
25#16812134C2214yescomplemented
26C2×#13133807—21no—
27C2×#13133805—21no—
28C2×D856—21no—
29C2×#13133803—21no—
30C2×#13133802C2211yescharacteristic, complemented
31#16812106—22no—
32#16812106C2214yescomplemented
33#16812037C2214yescomplemented
34#16812025C2214yescomplemented
35C2×#41×C107C2211yescharacteristic
36C2×M16×C107C2211yescharacteristic, complemented
37C2×C42×C107C2211yescharacteristic, complemented
38#104×C107C2214yescomplemented
39C23×D107—22no—
40C23×D107—21no—
41C23×D107D412yes—
42#13133819—24no—
43#13133819—24no—
44#13133818—24no—
45#13133818—21no—
46#13133818—21no—
47#13133818—24no—
48#13133818D412yescomplemented
49#13133818C2312yes—
50C2×Dic214D411yescharacteristic, complemented
51C2×D428—22no—
52C2×D428—24no—
53C2×D428—21no—
54C2×D428—21no—
55C2×D428—21no—
56C2×D428—21no—
57C2×D428—24no—
58C2×D428—21no—
59C2×D428—21no—
60C2×D428—21no—
61C2×D428—21no—
62C2×D428—28no—
63C2×D428—21no—
64C2×D428—48no—
65C2×D428D412yes—
66C2×D428D411yescharacteristic, complemented
67C2×D428C2312yes—
68C2×D428C2312yescomplemented
69C2×C22⋊D107—22no—
70C2×C22⋊D107—21no—
71D4×D107—28no—
72D4×D107—24no—
73D4×D107—28no—
74D4×D107—24no—
75C2×C4×D107—22no—
76C2×C4×D107—21no—
77#13133807—24no—
78#13133805—24no—
79D856—24no—
80#13133803—24no—
81#13133802—21no—
82#13133802C2312yescomplemented
83C2×C107⋊C8—21no—
84#41×C107—21no—
85#41×C107—21no—
86#41×C107D412yescomplemented
87#41×C107C2312yes—
88C2×Q8×C107D411yescharacteristic, complemented
89C2×D4×C107—21no—
90C2×D4×C107D411yescharacteristic, complemented
91C22×C4×C107—21no—
92C22×C4×C107—21no—
93C22×C4×C107C2311yescharacteristic
94M16×C107—21no—
95M16×C107C2312yescomplemented
96C2×C8×C107—21no—
97C42×C107—21no—
98C42×C107C2312yescomplemented
99C23×C107—21no—
100C23×C107C2×D411yescharacteristic
101C22×D107—22no—
102C22×D107—21no—
103C22×D107—28no—
104C22×D107—22no—
105C22×D107—22no—
106C22×D107—44no—
107C22×D107—44no—
108C22×D107—41no—
109C22×D107—41no—
110C22×D107—84no—
111C22×D107C2×D414yes—
112Q8×C107—21no—
113Q8×C107C2×D412yescomplemented
114D4×C107—22no—
115D4×C107—22no—
116D4×C107—21no—
117D4×C107C2×D412yescomplemented
118C2×C4×C107—24no—
119C2×C4×C107—21no—
120C2×C4×C107—22no—
121C2×C4×C107—22no—
122C2×C4×C107—22no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—28no—
130C22⋊D107—22no—
131C22⋊D107—22no—
132C2×Dic107—21no—
133D428—28no—
134D428—22no—
135D428—22no—
136D428—21no—
137D428—22no—
138D428—22no—
139D428—22no—
140D428—22no—
141D428—416no—
142D428—41no—
143D428—41no—
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—42no—
155C22×C107#12011yescharacteristic
156C22×C107#12011yescharacteristic
157C22×C107C22×D411yescharacteristic
158C2×D107—24no—
159C2×D107—22no—
160C2×D107—44no—
161C2×D107—42no—
162C2×D107—44no—
163C2×D107—44no—
164C2×D107—42no—
165C2×D107—82no—
166C2×D107—88no—
167C2×D107—82no—
168C2×D107—82no—
169C4×C107—24no—
170C4×C107—22no—
171C4×C107#12011yescharacteristic
172C4×C107#12011yescharacteristic
173C4×C107C22×D411yescharacteristic
174C4×C107C22×D411yescharacteristic
175Dic107—22no—
176C2×C107—21no—
177C2×C107—21no—
178C2×C107—42no—
179C2×C107#45312yescomplemented
180C2×C107C2×#12011yescharacteristic
181D107—44no—
182D107—42no—
183D107—82no—
184C2×#453—1071nomaximal, Sylow
185C107C2×#45311yescharacteristic, complemented, Sylow
186C2×H(5,𝔽2)—1071no—
187C2×AGL(1,ℤ/8)—1071no—
188C2×AGL(1,ℤ/8)—1071no—
189C2×#127—1071no—
190#453—1078no—
191C2×#104—1071no—
192C2×#104D10711yescharacteristic, complemented
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—1071no—
204C22×D4—2141no—
205C22×D4—2142no—
206C22×D4—2142no—
207C22×D4—2141no—
208AGL(1,ℤ/8)—1074no—
209AGL(1,ℤ/8)—1074no—
210AGL(1,ℤ/8)—2142no—
211AGL(1,ℤ/8)—2142no—
212C2×SD16—2141no—
213C2×SD16—2141no—
214C2×D8—2141no—
215C2×D8—2141no—
216C2×M16—1071no—
217C2×M16C2×D10711yescharacteristic, complemented
218#127—1074no—
219#127—2142no—
220C2×C42C2×D10711yescharacteristic, complemented
221#104—1074no—
222#104C2×D10714yescomplemented
223#100—1074no—
224C24—1072no—
225C24—2142no—
226C24—2141no—
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#41D42812yescomplemented
238#41C22×D10712yes—
239C2×Q8—1071no—
240C2×Q8D42811yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—1071no—
243C2×D4—1072no—
244C2×D4—1072no—
245C2×D4—1072no—
246C2×D4—4288no—
247C2×D4—2144no—
248C2×D4—2148no—
249C2×D4—2142no—
250C2×D4—2144no—
251C2×D4—2148no—
252C2×D4—2148no—
253C2×D4—2144no—
254C2×D4—2142no—
255C2×D4—2141no—
256C2×D4—2144no—
257C2×D4—2141no—
258C2×D4—2141no—
259C2×D4—2141no—
260C2×D4—2141no—
261C2×D4—2141no—
262C2×D4—2141no—
263C2×D4—2141no—
264C2×D4—2141no—
265C2×D4—2141no—
266C2×D4D42811yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2142no—
270C22×C4—2141no—
271C22×C4C22×D10711yescharacteristic
272SD16—2144no—
273SD16—2144no—
274D8—2144no—
275D8—2144no—
276M16—21no—
277M16—1072no—
278M16—2141no—
279M16C22×D10712yescomplemented
280C2×C8—21no—
281C2×C8—2141no—
282C42—21no—
283C42C22×D10712yescomplemented
284C23—21no—
285C23—8564no—
286C23—1074no—
287C23—4284no—
288C23—4284no—
289C23—4281no—
290C23—4281no—
291C23—2148no—
292C23—2142no—
293C23—2142no—
294C23—2141no—
295C23—2142no—
296C23D4×D10711yescharacteristic
297Q8—21no—
298Q8—1072no—
299Q8—2141no—
300Q8C2×D42812yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—1072no—
305D4—42816no—
306D4—4281no—
307D4—4281no—
308D4—2148no—
309D4—2148no—
310D4—2142no—
311D4—2142no—
312D4—2141no—
313D4—2142no—
314D4—2142no—
315D4—2142no—
316D4—2142no—
317D4—2142no—
318D4—2142no—
319D4C2×D42812yescomplemented
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—2148no—
326C2×C4—2141no—
327C2×C4—2142no—
328C2×C4—2142no—
329C2×C4D4×D10711yescharacteristic
330C2×C4D4×D10711yescharacteristic
331C2×C4D4×D10711yescharacteristic
332C2×C4C2×D42811yescharacteristic
333C2×C4C2×D42811yescharacteristic
334C2×C4C23×D10711yescharacteristic
335C8—22no—
336C8—2142no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—8562no—
341C22—8562no—
342C22—8568no—
343C22—8562no—
344C22—4284no—
345C22—4284no—
346C22—4282no—
347C22—4282no—
348C22—4284no—
349C22—2142no—
350C22—2144no—
351C22#1681207911yescharacteristic
352C22#1681207911yescharacteristic
353C22C2×D4×D10711yescharacteristic
354C4—24no—
355C4—22no—
356C4—2142no—
357C4#1681207911yescharacteristic
358C4#1681207911yescharacteristic
359C4C2×D4×D10711yescharacteristic
360C4C22×D42811yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—8562no—
365C2—4282no—
366C2—4284no—
367C2#2029279612yesdirect product factor
368C2C2×#1681207911yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#104×C107C2
socleC22×C107#120
hypercenterC2×#104D107
hypocenterC107C2×#453
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
1. C2×#16812155C2semidirect
2. C2×#16812137C2semidirect
3. C2×#16812134C2semidirect
4. C2×#16812106C2semidirect
5. #20292796C2direct (N × H)
6. C2×#16812037C2semidirect
7. C2×#16812025C2semidirect
8. C2×#104×C107C2semidirect
12. #16812155C22semidirect
19. C22×D428C22semidirect
23. #16812137C22semidirect
25. #16812134C22semidirect
30. C2×#13133802C22semidirect
32. #16812106C22semidirect
33. #16812037C22semidirect
34. #16812025C22semidirect
36. C2×M16×C107C22semidirect
37. C2×C42×C107C22semidirect
38. #104×C107C22semidirect
48. #13133818D4semidirect
50. C2×Dic214D4semidirect
66. C2×D428D4semidirect
68. C2×D428C23semidirect
82. #13133802C23semidirect
86. #41×C107D4semidirect
88. C2×Q8×C107D4semidirect
90. C2×D4×C107D4semidirect
95. M16×C107C23semidirect
98. C42×C107C23semidirect
113. Q8×C107C2×D4semidirect
117. D4×C107C2×D4semidirect
144. D428C2×D4semidirect
149. Dic214C2×D4semidirect
179. C2×C107#453semidirect
185. C107C2×#453semidirect
192. C2×#104D107semidirect
217. C2×M16C2×D107semidirect
220. C2×C42C2×D107semidirect
222. #104C2×D107semidirect
237. #41D428semidirect
240. C2×Q8D428semidirect
266. C2×D4D428semidirect
279. M16C22×D107semidirect
283. C42C22×D107semidirect
300. Q8C2×D428semidirect
319. D4C2×D428semidirect
367. C2#20292796direct (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×#168121372
C2×#168121342
C2×#168121062
#202927962
C2×#168120372
C2×#168120252
C2×#104×C1072
C2×#453107