Finite Group Databasebeta

← C2×#31770781 #46820213

Subgroups — C2×#31770781 #46820213

Order 34432.

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×#22575499C211yesmaximal, characteristic
2C2×#22575484C211yesmaximal, characteristic, complemented
3C2×#22575442C211yesmaximal, characteristic, complemented
4#31770781C218yesmaximal, direct product factor
5C2×#22575380C211yesmaximal, characteristic, complemented
6C2×#22575379C211yesmaximal, characteristic, complemented
7C2×#22575369C211yesmaximal, characteristic, complemented
8C2×#137×C269C211yesmaximal, characteristic, complemented
9#22575499—22no—
10#22575499—22no—
11#22575499—22no—
12#22575499C2214yes—
13C2×#16864403—21no—
14C2×#16864402—21no—
15C2×#16864402C2211yescharacteristic
16C22×D1076—21no—
17C22×D1076C2211yescharacteristic
18C2×D4×D269—22no—
19C2×D4×D269—21no—
20#22575484—22no—
21#22575484C2214yescomplemented
22C2×#16864392—21no—
23C2×#16864389—21no—
24C2×#16864386C2211yescharacteristic, complemented
25#22575442—22no—
26#22575442C2214yescomplemented
27C2×#16864379—22no—
28C2×C4×Dic269C2211yescharacteristic, complemented
29#22575380C2214yescomplemented
30#22575379C2214yescomplemented
31#22575369C2214yescomplemented
32C2×#41×C269C2211yescharacteristic
33C22×Q8×C269C2211yescharacteristic, complemented
34#137×C269—22no—
35#137×C269C2214yescomplemented
36C2×Q16×C269—21no—
37C2×SD16×C269—21no—
38C2×M16×C269C2211yescharacteristic, complemented
39C23×D269—22no—
40C23×D269D412yes—
41#16864403—24no—
42#16864403—24no—
43#16864402—24no—
44#16864402—24no—
45#16864402—21no—
46#16864402—21no—
47#16864402D412yes—
48#16864402C2312yes—
49C2×Dic538D411yescharacteristic
50C2×D1076—22no—
51C2×D1076—24no—
52C2×D1076—24no—
53C2×D1076—21no—
54C2×D1076—21no—
55C2×D1076—21no—
56C2×D1076—21no—
57C2×D1076D412yes—
58C2×D1076D411yescharacteristic
59C2×D1076C2312yes—
60C2×C22⋊D269—22no—
61C2×C22⋊D269—21no—
62D4×D269—28no—
63D4×D269—24no—
64D4×D269—28no—
65D4×D269—24no—
66C22×Dic269—21no—
67C2×C4×D269—22no—
68C2×C4×D269—21no—
69#16864392—24no—
70#16864389—24no—
71#16864386—21no—
72#16864386C2312yescomplemented
73C2×#14524130—21no—
74#16864379—28no—
75C4×Dic269—21no—
76C4×Dic269C2312yescomplemented
77#41×C269—21no—
78#41×C269—21no—
79#41×C269D412yes—
80#41×C269C2312yes—
81C2×Q8×C269—21no—
82C2×Q8×C269—21no—
83C2×Q8×C269—21no—
84C2×Q8×C269—21no—
85C2×Q8×C269—21no—
86C2×Q8×C269D411yescharacteristic
87C2×Q8×C269C2312yescomplemented
88C2×D4×C269—21no—
89C2×D4×C269D411yescharacteristic
90C22×C4×C269—21no—
91C22×C4×C269—21no—
92C22×C4×C269C2311yescharacteristic
93Q16×C269—24no—
94SD16×C269—24no—
95M16×C269—21no—
96M16×C269C2312yescomplemented
97C2×C8×C269—21no—
98C22×D269—22no—
99C22×D269—21no—
100C22×D269—28no—
101C22×D269—22no—
102C22×D269—44no—
103C22×D269—84no—
104C22×D269C2×D414yes—
105Dic538—21no—
106Dic538C2×D412yes—
107D1076—28no—
108D1076—22no—
109D1076—22no—
110D1076—21no—
111D1076—22no—
112D1076—22no—
113D1076C2×D412yes—
114C22⋊D269—28no—
115C22⋊D269—22no—
116C22⋊D269—22no—
117C2×Dic269—21no—
118C2×Dic269—22no—
119C2×Dic269—42no—
120C4×D269—28no—
121C4×D269—22no—
122C4×D269—22no—
123#14524130—22no—
124C23×C269—21no—
125C23×C269C2×D411yescharacteristic
126Q8×C269—21no—
127Q8×C269—22no—
128Q8×C269—22no—
129Q8×C269—41no—
130Q8×C269—41no—
131Q8×C269C2×D412yes—
132D4×C269—22no—
133D4×C269—22no—
134D4×C269—21no—
135D4×C269C2×D412yes—
136C2×C4×C269—21no—
137C2×C4×C269—22no—
138C2×C4×C269—22no—
139C2×C4×C269—22no—
140C2×C4×C269—41no—
141C2×C4×C269—41no—
142C2×C4×C269C2×D411yescharacteristic
143C2×C4×C269C2×D411yescharacteristic
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C8×C269—22no—
149C22×C269—22no—
150C22×C269—22no—
151C22×C269—42no—
152C22×C269#12011yescharacteristic
153C22×C269#12011yescharacteristic
154C22×C269C22×D411yescharacteristic
155C2×D269—24no—
156C2×D269—22no—
157C2×D269—44no—
158C2×D269—42no—
159C2×D269—44no—
160C2×D269—82no—
161C2×D269—88no—
162C4×C269—22no—
163C4×C269—42no—
164C4×C269#12011yescharacteristic
165C4×C269#12011yescharacteristic
166C4×C269C22×D411yescharacteristic
167C4×C269C22×D411yescharacteristic
168Dic269—22no—
169Dic269—42no—
170C2×C269—21no—
171C2×C269—21no—
172C2×C269—42no—
173C2×C269#45512yescomplemented
174C2×C269C2×#12011yescharacteristic
175D269—44no—
176D269—42no—
177C269C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—2691nomaximal, Sylow
179C2×H(5,𝔽2)—2691no—
180C2×#137—2691no—
181C2×#137D26911yescharacteristic, complemented
182C2×#124—2691no—
183#455—2698no—
184C2×#104—2691no—
185C2×#104—2691no—
186C2×#100—2691no—
187H(5,𝔽2)—2694no—
188H(5,𝔽2)—5382no—
189H(5,𝔽2)—5382no—
190H(5,𝔽2)—5382no—
191C2×#41—2691no—
192C2×#41—5381no—
193C2×#41—5381no—
194C2×#41C2×D26911yescharacteristic
195C22×Q8C2×D26911yescharacteristic, complemented
196C22×D4—2691no—
197C22×D4—5381no—
198C22×D4—5382no—
199C22×D4—5381no—
200#137—22no—
201#137—2694no—
202#137—5382no—
203#137C2×D26914yescomplemented
204C2×Q16—21no—
205C2×Q16—5381no—
206C2×SD16—21no—
207C2×SD16—5381no—
208C2×M16—2691no—
209C2×M16C2×D26911yescharacteristic, complemented
210#124—2694no—
211#124—5382no—
212C2×C22⋊C4—5382no—
213C2×C42—2691no—
214#104—2694no—
215#104—2694no—
216#100—2694no—
217C24—2692no—
218C24—5382no—
219#41—21no—
220#41—21no—
221#41—2692no—
222#41—2692no—
223#41—5384no—
224#41—5384no—
225#41—5384no—
226#41—5384no—
227#41—5381no—
228#41—5381no—
229#41C22⋊D26912yes—
230#41C22×D26912yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—2691no—
237C2×Q8C22⋊D26911yescharacteristic
238C2×Q8C22×D26912yescomplemented
239C2×D4—21no—
240C2×D4—2691no—
241C2×D4—2692no—
242C2×D4—2692no—
243C2×D4—5384no—
244C2×D4—5382no—
245C2×D4—5384no—
246C2×D4—5388no—
247C2×D4—5388no—
248C2×D4—5384no—
249C2×D4—5382no—
250C2×D4—5381no—
251C2×D4—5384no—
252C2×D4—5381no—
253C2×D4—5381no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4C22⋊D26911yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—5382no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4C22×D26911yescharacteristic
263Q16—24no—
264Q16—5384no—
265SD16—24no—
266SD16—5384no—
267M16—21no—
268M16—2692no—
269M16—5381no—
270M16C22×D26912yescomplemented
271C2×C8—21no—
272C2×C8—5381no—
273C22⋊C4—5388no—
274C42—2692no—
275C42—5381no—
276C23—21no—
277C23—2694no—
278C23—5388no—
279C23—5381no—
280C23—5382no—
281C23—5382no—
282C23—10764no—
283C23—21524no—
284C23D4×D26911yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—2692no—
291Q8—5381no—
292Q8C2×C22⋊D26912yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—2692no—
297D4—5388no—
298D4—5388no—
299D4—5382no—
300D4—5382no—
301D4—5381no—
302D4—5382no—
303D4—5382no—
304D4—5382no—
305D4—5382no—
306D4C2×C22⋊D26912yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—5388no—
314C2×C4—5381no—
315C2×C4—5382no—
316C2×C4—5382no—
317C2×C4—5382no—
318C2×C4—10762no—
319C2×C4D4×D26911yescharacteristic
320C2×C4D4×D26911yescharacteristic
321C2×C4D4×D26911yescharacteristic
322C2×C4C2×C22⋊D26911yescharacteristic
323C2×C4C2×C22⋊D26911yescharacteristic
324C2×C4C23×D26911yescharacteristic
325C8—22no—
326C8—5382no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—5382no—
331C22—5384no—
332C22—10764no—
333C22—10764no—
334C22—10762no—
335C22—21522no—
336C22—21528no—
337C22#2257542111yescharacteristic
338C22#2257542111yescharacteristic
339C22C2×D4×D26911yescharacteristic
340C4—22no—
341C4—42no—
342C4—5382no—
343C4—10762no—
344C4#2257542111yescharacteristic
345C4#2257542111yescharacteristic
346C4C2×D4×D26911yescharacteristic
347C4C22×C22⋊D26911yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—10764no—
352C2—10762no—
353C2#3177078112yesdirect product factor
354C2C2×#2257542111yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269#120
prime radicalC2×#22575499C2
hypercenterC2×#137D269
hypocenterC269C2×#455
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#22575484C2semidirect
3. C2×#22575442C2semidirect
4. #31770781C2direct (N × H)
5. C2×#22575380C2semidirect
6. C2×#22575379C2semidirect
7. C2×#22575369C2semidirect
8. C2×#137×C269C2semidirect
21. #22575484C22semidirect
24. C2×#16864386C22semidirect
26. #22575442C22semidirect
28. C2×C4×Dic269C22semidirect
29. #22575380C22semidirect
30. #22575379C22semidirect
31. #22575369C22semidirect
33. C22×Q8×C269C22semidirect
35. #137×C269C22semidirect
38. C2×M16×C269C22semidirect
72. #16864386C23semidirect
76. C4×Dic269C23semidirect
87. C2×Q8×C269C23semidirect
96. M16×C269C23semidirect
173. C2×C269#455semidirect
177. C269C2×#455semidirect
181. C2×#137D269semidirect
195. C22×Q8C2×D269semidirect
203. #137C2×D269semidirect
209. C2×M16C2×D269semidirect
238. C2×Q8C22×D269semidirect
270. M16C22×D269semidirect
353. C2#31770781direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754842
C2×#225754422
#317707812
C2×#225753802
C2×#225753792
C2×#225753692
C2×#137×C2692
C2×#455269