Finite Group Databasebeta

← C2×#22581132 #31815735

Subgroups — C2×#22581132 #31815735

Order 17792.

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×#17984888C211yesmaximal, characteristic
2C2×#17984875C211yesmaximal, characteristic, complemented
3C2×#17984872C211yesmaximal, characteristic, complemented
4C2×#17984830C211yesmaximal, characteristic, complemented
5#22581132C218yesmaximal, 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#17984888C2214yes—
13C2×#14555786—21no—
14C2×#14555785—21no—
15C2×#14555785C2211yescharacteristic
16C22×Dic278C2211yescharacteristic, complemented
17C22×D556—21no—
18C22×D556C2211yescharacteristic
19C2×D4×D139—22no—
20C2×D4×D139—21no—
21#17984875—22no—
22#17984875C2214yescomplemented
23#17984872—22no—
24#17984872C2214yescomplemented
25C2×Dic556—21no—
26C2×#14555775—21no—
27C2×#14555774—21no—
28C2×#14555773—21no—
29C2×#14555769C2211yescharacteristic, complemented
30#17984830—22no—
31#17984830C2214yescomplemented
32C2×(C2×D139)⋊C4—22no—
33#17984770C2214yescomplemented
34#17984758C2214yescomplemented
35C2×#41×C139C2211yescharacteristic
36C2×M16×C139C2211yescharacteristic, complemented
37C2×C42×C139C2211yescharacteristic, complemented
38#104×C139C2214yescomplemented
39C23×D139—22no—
40C23×D139D412yes—
41#14555786—24no—
42#14555786—24no—
43#14555785—24no—
44#14555785—24no—
45#14555785—21no—
46#14555785—21no—
47#14555785D412yes—
48#14555785C2312yes—
49C2×Dic278—21no—
50C2×Dic278—21no—
51C2×Dic278—21no—
52C2×Dic278—21no—
53C2×Dic278—21no—
54C2×Dic278D411yescharacteristic
55C2×Dic278C2312yescomplemented
56C2×D556—22no—
57C2×D556—24no—
58C2×D556—24no—
59C2×D556—21no—
60C2×D556—21no—
61C2×D556—21no—
62C2×D556—21no—
63C2×D556D412yes—
64C2×D556D411yescharacteristic
65C2×D556C2312yes—
66C2×C22⋊D139—22no—
67C2×C22⋊D139—21no—
68D4×D139—28no—
69D4×D139—24no—
70D4×D139—28no—
71D4×D139—24no—
72C22×Dic139—21no—
73C2×C4×D139—22no—
74C2×C4×D139—21no—
75Dic556—24no—
76#14555775—24no—
77#14555774—24no—
78#14555773—24no—
79#14555769—21no—
80#14555769C2312yescomplemented
81C2×C139⋊C8—21no—
82(C2×D139)⋊C4—28no—
83#41×C139—21no—
84#41×C139—21no—
85#41×C139D412yes—
86#41×C139C2312yes—
87C2×Q8×C139D411yescharacteristic
88C2×D4×C139—21no—
89C2×D4×C139D411yescharacteristic
90C22×C4×C139—21no—
91C22×C4×C139—21no—
92C22×C4×C139C2311yescharacteristic
93M16×C139—21no—
94M16×C139C2312yescomplemented
95C2×C8×C139—21no—
96C42×C139—21no—
97C42×C139C2312yescomplemented
98C22×D139—22no—
99C22×D139—21no—
100C22×D139—28no—
101C22×D139—22no—
102C22×D139—44no—
103C22×D139—84no—
104C22×D139C2×D414yes—
105Dic278—21no—
106Dic278—22no—
107Dic278—22no—
108Dic278—41no—
109Dic278—41no—
110Dic278C2×D412yes—
111D556—28no—
112D556—22no—
113D556—22no—
114D556—21no—
115D556—22no—
116D556—22no—
117D556C2×D412yes—
118C22⋊D139—28no—
119C22⋊D139—22no—
120C22⋊D139—22no—
121C2×Dic139—21no—
122C2×Dic139—22no—
123C2×Dic139—41no—
124C2×Dic139—41no—
125C4×D139—28no—
126C4×D139—22no—
127C4×D139—22no—
128C139⋊C8—22no—
129C23×C139—21no—
130C23×C139C2×D411yescharacteristic
131Q8×C139—21no—
132Q8×C139C2×D412yes—
133D4×C139—22no—
134D4×C139—22no—
135D4×C139—21no—
136D4×C139C2×D412yes—
137C2×C4×C139—21no—
138C2×C4×C139—22no—
139C2×C4×C139—22no—
140C2×C4×C139—22no—
141C2×C4×C139—42no—
142C2×C4×C139C2×D411yescharacteristic
143C2×C4×C139C2×D411yescharacteristic
144C2×C4×C139C2×D411yescharacteristic
145C2×C4×C139C2×D411yescharacteristic
146C2×C4×C139C2×D411yescharacteristic
147C2×C4×C139C2411yescharacteristic
148C8×C139—22no—
149C22×C139—22no—
150C22×C139—22no—
151C22×C139—42no—
152C22×C139#12011yescharacteristic
153C22×C139#12011yescharacteristic
154C22×C139C22×D411yescharacteristic
155C2×D139—24no—
156C2×D139—22no—
157C2×D139—44no—
158C2×D139—42no—
159C2×D139—44no—
160C2×D139—82no—
161C2×D139—88no—
162C4×C139—22no—
163C4×C139—42no—
164C4×C139#12011yescharacteristic
165C4×C139#12011yescharacteristic
166C4×C139C22×D411yescharacteristic
167C4×C139C22×D411yescharacteristic
168Dic139—22no—
169Dic139—42no—
170C2×C139—21no—
171C2×C139—21no—
172C2×C139—42no—
173C2×C139#45512yescomplemented
174C2×C139C2×#12011yescharacteristic
175D139—44no—
176D139—42no—
177C139C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1391nomaximal, Sylow
179C2×H(5,𝔽2)—1391no—
180C2×#137—1391no—
181C2×#137—1391no—
182C2×#124—1391no—
183#455—1398no—
184C2×#104—1391no—
185C2×#104D13911yescharacteristic, complemented
186C2×#100—1391no—
187H(5,𝔽2)—2782no—
188H(5,𝔽2)—2782no—
189H(5,𝔽2)—2782no—
190H(5,𝔽2)—1394no—
191C2×#41—2781no—
192C2×#41—2781no—
193C2×#41—1391no—
194C2×#41C2×D13911yescharacteristic
195C22×Q8—1391no—
196C22×D4—2781no—
197C22×D4—2782no—
198C22×D4—2781no—
199C22×D4—1391no—
200#137—2782no—
201#137—2782no—
202#137—1394no—
203#137—1394no—
204C2×Q16—2781no—
205C2×Q16—2781no—
206C2×SD16—2781no—
207C2×SD16—2781no—
208C2×M16—1391no—
209C2×M16C2×D13911yescharacteristic, complemented
210#124—2782no—
211#124—1394no—
212C2×C22⋊C4—2782no—
213C2×C42C2×D13911yescharacteristic, complemented
214#104—1394no—
215#104C2×D13914yescomplemented
216#100—1394no—
217C24—2782no—
218C24—1392no—
219#41—21no—
220#41—21no—
221#41—2784no—
222#41—2784no—
223#41—2784no—
224#41—2784no—
225#41—2781no—
226#41—2781no—
227#41—1392no—
228#41—1392no—
229#41D55612yes—
230#41C22×D13912yes—
231C2×Q8—2781no—
232C2×Q8—2781no—
233C2×Q8—2781no—
234C2×Q8—2781no—
235C2×Q8—2781no—
236C2×Q8—1391no—
237C2×Q8—1392no—
238C2×Q8D55611yescharacteristic
239C2×D4—21no—
240C2×D4—2782no—
241C2×D4—2784no—
242C2×D4—2784no—
243C2×D4—2788no—
244C2×D4—2784no—
245C2×D4—2788no—
246C2×D4—2782no—
247C2×D4—2781no—
248C2×D4—2784no—
249C2×D4—2781no—
250C2×D4—2781no—
251C2×D4—2781no—
252C2×D4—2781no—
253C2×D4—1392no—
254C2×D4—1391no—
255C2×D4—1392no—
256C2×D4D55611yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2782no—
260C22×C4—2781no—
261C22×C4—2781no—
262C22×C4C22×D13911yescharacteristic
263Q16—2784no—
264Q16—2784no—
265SD16—2784no—
266SD16—2784no—
267M16—21no—
268M16—2781no—
269M16—1392no—
270M16C22×D13912yescomplemented
271C2×C8—21no—
272C2×C8—2781no—
273C22⋊C4—2788no—
274C42—21no—
275C42C22×D13912yescomplemented
276C23—21no—
277C23—2788no—
278C23—2782no—
279C23—2781no—
280C23—2782no—
281C23—5564no—
282C23—11124no—
283C23—1394no—
284C23D4×D13911yescharacteristic
285Q8—21no—
286Q8—2781no—
287Q8—2782no—
288Q8—2782no—
289Q8—5561no—
290Q8—5561no—
291Q8—1392no—
292Q8C2×D55612yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—2788no—
297D4—2788no—
298D4—2782no—
299D4—2782no—
300D4—2782no—
301D4—2781no—
302D4—2782no—
303D4—2782no—
304D4—2782no—
305D4—1392no—
306D4C2×D55612yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—2788no—
313C2×C4—2781no—
314C2×C4—2782no—
315C2×C4—2782no—
316C2×C4—2782no—
317C2×C4—5561no—
318C2×C4—5561no—
319C2×C4D4×D13911yescharacteristic
320C2×C4D4×D13911yescharacteristic
321C2×C4D4×D13911yescharacteristic
322C2×C4C2×D55611yescharacteristic
323C2×C4C2×D55611yescharacteristic
324C2×C4C23×D13911yescharacteristic
325C8—22no—
326C8—2782no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—2784no—
331C22—2782no—
332C22—5564no—
333C22—5564no—
334C22—5562no—
335C22—11122no—
336C22—11128no—
337C22#1798481211yescharacteristic
338C22#1798481211yescharacteristic
339C22C2×D4×D13911yescharacteristic
340C4—22no—
341C4—42no—
342C4—2782no—
343C4—5562no—
344C4#1798481211yescharacteristic
345C4#1798481211yescharacteristic
346C4C2×D4×D13911yescharacteristic
347C4C22×D55611yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—5564no—
352C2—5562no—
353C2#2258113212yesdirect product factor
354C2C2×#1798481211yescharacteristic
subgroupG/H
centerC22#17984812
commutator, cosocleC2×C4×C139C24
FrattiniC2×C4C23×D139
FittingC2×#104×C139C2
socleC22×C139#120
prime radicalC2×#17984888C2
hypercenterC2×#104D139
hypocenterC139C2×#455
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
2. C2×#17984875C2semidirect
3. C2×#17984872C2semidirect
4. C2×#17984830C2semidirect
5. #22581132C2direct (N × H)
6. C2×#17984770C2semidirect
7. C2×#17984758C2semidirect
8. C2×#104×C139C2semidirect
16. C22×Dic278C22semidirect
22. #17984875C22semidirect
24. #17984872C22semidirect
29. C2×#14555769C22semidirect
31. #17984830C22semidirect
33. #17984770C22semidirect
34. #17984758C22semidirect
36. C2×M16×C139C22semidirect
37. C2×C42×C139C22semidirect
38. #104×C139C22semidirect
55. C2×Dic278C23semidirect
80. #14555769C23semidirect
94. M16×C139C23semidirect
97. C42×C139C23semidirect
173. C2×C139#455semidirect
177. C139C2×#455semidirect
185. C2×#104D139semidirect
209. C2×M16C2×D139semidirect
213. C2×C42C2×D139semidirect
215. #104C2×D139semidirect
270. M16C22×D139semidirect
275. C42C22×D139semidirect
353. C2#22581132direct (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×#179848752
C2×#179848722
C2×#179848302
#225811322
C2×#179847702
C2×#179847582
C2×#104×C1392
C2×#455139