Finite Group Databasebeta

← C2×#22581122 #31815729

Subgroups — C2×#22581122 #31815729

Order 17792.

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×#17984888C211yesmaximal, characteristic, complemented
2C2×#17984870C211yesmaximal, characteristic, complemented
3C2×#17984867C211yesmaximal, characteristic, complemented
4C2×#17984839C211yesmaximal, characteristic, complemented
5#22581122C218yesmaximal, direct product factor
6C2×#17984770C211yesmaximal, characteristic, complemented
7C2×#17984758C211yesmaximal, characteristic, complemented
8C2×#104×C139C211yesmaximal, characteristic, complemented
9#17984888—22no—
10#17984888—22no—
11#17984888—22no—
12#17984888C2214yescomplemented
13C2×#14555786—21no—
14C2×#14555785—21no—
15C2×#14555785C2211yescharacteristic
16C22×D556—21no—
17C22×D556—22no—
18C22×D556C2211yescharacteristic
19C22×D556C2211yescharacteristic, complemented
20C2×D4×D139—22no—
21C2×D4×D139—21no—
22#17984870—22no—
23#17984870C2214yescomplemented
24#17984867—22no—
25#17984867C2214yescomplemented
26C2×#14555774—21no—
27C2×#14555772—21no—
28C2×D1112—21no—
29C2×#14555770—21no—
30C2×#14555769C2211yescharacteristic, complemented
31#17984839—22no—
32#17984839C2214yescomplemented
33#17984770C2214yescomplemented
34#17984758C2214yescomplemented
35C2×#41×C139C2211yescharacteristic
36C2×M16×C139C2211yescharacteristic, complemented
37C2×C42×C139C2211yescharacteristic, complemented
38#104×C139C2214yescomplemented
39C23×D139—22no—
40C23×D139—21no—
41C23×D139D412yes—
42#14555786—24no—
43#14555786—24no—
44#14555785—24no—
45#14555785—24no—
46#14555785—21no—
47#14555785—21no—
48#14555785D412yescomplemented
49#14555785C2312yes—
50C2×Dic278D411yescharacteristic, complemented
51C2×D556—22no—
52C2×D556—24no—
53C2×D556—24no—
54C2×D556—28no—
55C2×D556—21no—
56C2×D556—21no—
57C2×D556—21no—
58C2×D556—21no—
59C2×D556—21no—
60C2×D556—21no—
61C2×D556—21no—
62C2×D556—21no—
63C2×D556—21no—
64C2×D556—48no—
65C2×D556D412yes—
66C2×D556D411yescharacteristic, complemented
67C2×D556C2312yes—
68C2×D556C2312yescomplemented
69C2×C22⋊D139—22no—
70C2×C22⋊D139—21no—
71D4×D139—28no—
72D4×D139—24no—
73D4×D139—28no—
74D4×D139—24no—
75C2×C4×D139—22no—
76C2×C4×D139—21no—
77#14555774—24no—
78#14555772—24no—
79D1112—24no—
80#14555770—24no—
81#14555769—21no—
82#14555769C2312yescomplemented
83C2×C139⋊C8—21no—
84#41×C139—21no—
85#41×C139—21no—
86#41×C139D412yescomplemented
87#41×C139C2312yes—
88C2×Q8×C139D411yescharacteristic, complemented
89C2×D4×C139—21no—
90C2×D4×C139D411yescharacteristic, complemented
91C22×C4×C139—21no—
92C22×C4×C139—21no—
93C22×C4×C139C2311yescharacteristic
94M16×C139—21no—
95M16×C139C2312yescomplemented
96C2×C8×C139—21no—
97C42×C139—21no—
98C42×C139C2312yescomplemented
99C22×D139—22no—
100C22×D139—21no—
101C22×D139—28no—
102C22×D139—22no—
103C22×D139—22no—
104C22×D139—44no—
105C22×D139—44no—
106C22×D139—41no—
107C22×D139—41no—
108C22×D139—84no—
109C22×D139C2×D414yes—
110Dic278—21no—
111Dic278C2×D412yescomplemented
112D556—28no—
113D556—22no—
114D556—22no—
115D556—21no—
116D556—22no—
117D556—22no—
118D556—22no—
119D556—22no—
120D556—416no—
121D556—41no—
122D556—41no—
123D556C2×D412yescomplemented
124C22⋊D139—28no—
125C22⋊D139—22no—
126C22⋊D139—22no—
127C2×Dic139—21no—
128C4×D139—28no—
129C4×D139—22no—
130C4×D139—22no—
131C139⋊C8—22no—
132C23×C139—21no—
133C23×C139C2×D411yescharacteristic
134Q8×C139—21no—
135Q8×C139C2×D412yescomplemented
136D4×C139—22no—
137D4×C139—22no—
138D4×C139—21no—
139D4×C139C2×D412yescomplemented
140C2×C4×C139—24no—
141C2×C4×C139—21no—
142C2×C4×C139—22no—
143C2×C4×C139—22no—
144C2×C4×C139—22no—
145C2×C4×C139C2×D411yescharacteristic
146C2×C4×C139C2×D411yescharacteristic
147C2×C4×C139C2×D411yescharacteristic
148C2×C4×C139C2×D411yescharacteristic
149C2×C4×C139C2×D411yescharacteristic
150C2×C4×C139C2411yescharacteristic
151C8×C139—22no—
152C22×C139—22no—
153C22×C139—22no—
154C22×C139—42no—
155C22×C139#12011yescharacteristic
156C22×C139#12011yescharacteristic
157C22×C139C22×D411yescharacteristic
158C2×D139—24no—
159C2×D139—22no—
160C2×D139—44no—
161C2×D139—42no—
162C2×D139—44no—
163C2×D139—44no—
164C2×D139—42no—
165C2×D139—82no—
166C2×D139—88no—
167C2×D139—82no—
168C2×D139—82no—
169C4×C139—24no—
170C4×C139—22no—
171C4×C139#12011yescharacteristic
172C4×C139#12011yescharacteristic
173C4×C139C22×D411yescharacteristic
174C4×C139C22×D411yescharacteristic
175Dic139—22no—
176C2×C139—21no—
177C2×C139—21no—
178C2×C139—42no—
179C2×C139#45312yescomplemented
180C2×C139C2×#12011yescharacteristic
181D139—44no—
182D139—42no—
183D139—82no—
184C139C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—1391nomaximal, Sylow
186C2×H(5,𝔽2)—1391no—
187C2×AGL(1,ℤ/8)—1391no—
188C2×AGL(1,ℤ/8)—1391no—
189C2×#127—1391no—
190#453—1398no—
191C2×#104—1391no—
192C2×#104D13911yescharacteristic, complemented
193C2×#100—1391no—
194H(5,𝔽2)—2782no—
195H(5,𝔽2)—2782no—
196H(5,𝔽2)—2782no—
197H(5,𝔽2)—1394no—
198C2×#41—2781no—
199C2×#41—2781no—
200C2×#41—1391no—
201C2×#41C2×D13911yescharacteristic
202C22×D4—2781no—
203C22×D4—2782no—
204C22×D4—2782no—
205C22×D4—2781no—
206C22×D4—1391no—
207C22×D4—1391no—
208AGL(1,ℤ/8)—2782no—
209AGL(1,ℤ/8)—2782no—
210AGL(1,ℤ/8)—1394no—
211AGL(1,ℤ/8)—1394no—
212C2×SD16—2781no—
213C2×SD16—2781no—
214C2×D8—2781no—
215C2×D8—2781no—
216C2×M16—1391no—
217C2×M16C2×D13911yescharacteristic, complemented
218#127—2782no—
219#127—1394no—
220C2×C42C2×D13911yescharacteristic, complemented
221#104—1394no—
222#104C2×D13914yescomplemented
223#100—1394no—
224C24—2782no—
225C24—2781no—
226C24—1392no—
227#41—21no—
228#41—21no—
229#41—2784no—
230#41—2784no—
231#41—2784no—
232#41—2784no—
233#41—2781no—
234#41—2781no—
235#41—1392no—
236#41—1392no—
237#41D55612yescomplemented
238#41C22×D13912yes—
239C2×Q8—1391no—
240C2×Q8D55611yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—2782no—
243C2×D4—2784no—
244C2×D4—2784no—
245C2×D4—2788no—
246C2×D4—2788no—
247C2×D4—2784no—
248C2×D4—2788no—
249C2×D4—2782no—
250C2×D4—2781no—
251C2×D4—2784no—
252C2×D4—2781no—
253C2×D4—2781no—
254C2×D4—2781no—
255C2×D4—2781no—
256C2×D4—2781no—
257C2×D4—2781no—
258C2×D4—2781no—
259C2×D4—2781no—
260C2×D4—2781no—
261C2×D4—5568no—
262C2×D4—1392no—
263C2×D4—1391no—
264C2×D4—1392no—
265C2×D4—1392no—
266C2×D4D55611yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2782no—
270C22×C4—2781no—
271C22×C4C22×D13911yescharacteristic
272SD16—2784no—
273SD16—2784no—
274D8—2784no—
275D8—2784no—
276M16—21no—
277M16—2781no—
278M16—1392no—
279M16C22×D13912yescomplemented
280C2×C8—21no—
281C2×C8—2781no—
282C42—21no—
283C42C22×D13912yescomplemented
284C23—21no—
285C23—2788no—
286C23—2782no—
287C23—2781no—
288C23—2782no—
289C23—2782no—
290C23—5564no—
291C23—5564no—
292C23—5561no—
293C23—5561no—
294C23—11124no—
295C23—1394no—
296C23D4×D13911yescharacteristic
297Q8—21no—
298Q8—2781no—
299Q8—1392no—
300Q8C2×D55612yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—2788no—
305D4—2788no—
306D4—2782no—
307D4—2782no—
308D4—2782no—
309D4—2781no—
310D4—2782no—
311D4—2782no—
312D4—2782no—
313D4—2782no—
314D4—2782no—
315D4—55616no—
316D4—5561no—
317D4—5561no—
318D4—1392no—
319D4C2×D55612yescomplemented
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—2788no—
326C2×C4—2781no—
327C2×C4—2782no—
328C2×C4—2782no—
329C2×C4D4×D13911yescharacteristic
330C2×C4D4×D13911yescharacteristic
331C2×C4D4×D13911yescharacteristic
332C2×C4C2×D55611yescharacteristic
333C2×C4C2×D55611yescharacteristic
334C2×C4C23×D13911yescharacteristic
335C8—22no—
336C8—2782no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—2784no—
341C22—2782no—
342C22—5564no—
343C22—5564no—
344C22—5564no—
345C22—5562no—
346C22—5562no—
347C22—11122no—
348C22—11122no—
349C22—11128no—
350C22—11122no—
351C22#1798481211yescharacteristic
352C22#1798481211yescharacteristic
353C22C2×D4×D13911yescharacteristic
354C4—24no—
355C4—22no—
356C4—2782no—
357C4#1798481211yescharacteristic
358C4#1798481211yescharacteristic
359C4C2×D4×D13911yescharacteristic
360C4C22×D55611yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—5564no—
365C2—5562no—
366C2—11122no—
367C2#2258112212yesdirect product factor
368C2C2×#1798481211yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×#104×C139C2
socleC22×C139#120
hypercenterC2×#104D139
hypocenterC139C2×#453
LCSG ▷ C2×C4×C139 ▷ C2×C139 ▷ C139
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#17984888C2semidirect
2. C2×#17984870C2semidirect
3. C2×#17984867C2semidirect
4. C2×#17984839C2semidirect
5. #22581122C2direct (N × H)
6. C2×#17984770C2semidirect
7. C2×#17984758C2semidirect
8. C2×#104×C139C2semidirect
12. #17984888C22semidirect
19. C22×D556C22semidirect
23. #17984870C22semidirect
25. #17984867C22semidirect
30. C2×#14555769C22semidirect
32. #17984839C22semidirect
33. #17984770C22semidirect
34. #17984758C22semidirect
36. C2×M16×C139C22semidirect
37. C2×C42×C139C22semidirect
38. #104×C139C22semidirect
48. #14555785D4semidirect
50. C2×Dic278D4semidirect
66. C2×D556D4semidirect
68. C2×D556C23semidirect
82. #14555769C23semidirect
86. #41×C139D4semidirect
88. C2×Q8×C139D4semidirect
90. C2×D4×C139D4semidirect
95. M16×C139C23semidirect
98. C42×C139C23semidirect
111. Dic278C2×D4semidirect
123. D556C2×D4semidirect
135. Q8×C139C2×D4semidirect
139. D4×C139C2×D4semidirect
179. C2×C139#453semidirect
184. C139C2×#453semidirect
192. C2×#104D139semidirect
217. C2×M16C2×D139semidirect
220. C2×C42C2×D139semidirect
222. #104C2×D139semidirect
237. #41D556semidirect
240. C2×Q8D556semidirect
266. C2×D4D556semidirect
279. M16C22×D139semidirect
283. C42C22×D139semidirect
300. Q8C2×D556semidirect
319. D4C2×D556semidirect
367. C2#22581122direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179848882
C2×#179848702
C2×#179848672
C2×#179848392
#225811222
C2×#179847702
C2×#179847582
C2×#104×C1392
C2×#453139