Finite Group Databasebeta

← C2×#23781844 #35223867

Subgroups — C2×#23781844 #35223867

Order 20864.

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×#19104781C211yesmaximal, characteristic
2C2×#19104768C211yesmaximal, characteristic, complemented
3C2×#19104765C211yesmaximal, characteristic, complemented
4C2×#19104723C211yesmaximal, characteristic, complemented
5#23781844C218yesmaximal, direct product factor
6C2×#19104663C211yesmaximal, characteristic, complemented
7C2×#19104651C211yesmaximal, characteristic, complemented
8C2×#104×C163C211yesmaximal, characteristic, complemented
9#19104781—22no—
10#19104781—22no—
11#19104781—22no—
12#19104781C2214yes—
13C2×#14579700—21no—
14C2×#14579699—21no—
15C2×#14579699C2211yescharacteristic
16C22×Dic326C2211yescharacteristic, complemented
17C22×D652—21no—
18C22×D652C2211yescharacteristic
19C2×D4×D163—22no—
20C2×D4×D163—21no—
21#19104768—22no—
22#19104768C2214yescomplemented
23#19104765—22no—
24#19104765C2214yescomplemented
25C2×Dic652—21no—
26C2×#14579689—21no—
27C2×#14579688—21no—
28C2×#14579687—21no—
29C2×#14579683C2211yescharacteristic, complemented
30#19104723—22no—
31#19104723C2214yescomplemented
32C2×(C2×D163)⋊C4—22no—
33#19104663C2214yescomplemented
34#19104651C2214yescomplemented
35C2×#41×C163C2211yescharacteristic
36C2×M16×C163C2211yescharacteristic, complemented
37C2×C42×C163C2211yescharacteristic, complemented
38#104×C163C2214yescomplemented
39C23×D163—22no—
40C23×D163D412yes—
41#14579700—24no—
42#14579700—24no—
43#14579699—24no—
44#14579699—24no—
45#14579699—21no—
46#14579699—21no—
47#14579699D412yes—
48#14579699C2312yes—
49C2×Dic326—21no—
50C2×Dic326—21no—
51C2×Dic326—21no—
52C2×Dic326—21no—
53C2×Dic326—21no—
54C2×Dic326D411yescharacteristic
55C2×Dic326C2312yescomplemented
56C2×D652—22no—
57C2×D652—24no—
58C2×D652—24no—
59C2×D652—21no—
60C2×D652—21no—
61C2×D652—21no—
62C2×D652—21no—
63C2×D652D412yes—
64C2×D652D411yescharacteristic
65C2×D652C2312yes—
66C2×C22⋊D163—22no—
67C2×C22⋊D163—21no—
68D4×D163—28no—
69D4×D163—24no—
70D4×D163—28no—
71D4×D163—24no—
72C22×Dic163—21no—
73C2×C4×D163—22no—
74C2×C4×D163—21no—
75Dic652—24no—
76#14579689—24no—
77#14579688—24no—
78#14579687—24no—
79#14579683—21no—
80#14579683C2312yescomplemented
81C2×C163⋊C8—21no—
82(C2×D163)⋊C4—28no—
83#41×C163—21no—
84#41×C163—21no—
85#41×C163D412yes—
86#41×C163C2312yes—
87C2×Q8×C163D411yescharacteristic
88C2×D4×C163—21no—
89C2×D4×C163D411yescharacteristic
90C22×C4×C163—21no—
91C22×C4×C163—21no—
92C22×C4×C163C2311yescharacteristic
93M16×C163—21no—
94M16×C163C2312yescomplemented
95C2×C8×C163—21no—
96C42×C163—21no—
97C42×C163C2312yescomplemented
98C22×D163—22no—
99C22×D163—21no—
100C22×D163—28no—
101C22×D163—22no—
102C22×D163—44no—
103C22×D163—84no—
104C22×D163C2×D414yes—
105Dic326—21no—
106Dic326—22no—
107Dic326—22no—
108Dic326—41no—
109Dic326—41no—
110Dic326C2×D412yes—
111D652—28no—
112D652—22no—
113D652—22no—
114D652—21no—
115D652—22no—
116D652—22no—
117D652C2×D412yes—
118C22⋊D163—28no—
119C22⋊D163—22no—
120C22⋊D163—22no—
121C2×Dic163—21no—
122C2×Dic163—22no—
123C2×Dic163—41no—
124C2×Dic163—41no—
125C4×D163—28no—
126C4×D163—22no—
127C4×D163—22no—
128C163⋊C8—22no—
129C23×C163—21no—
130C23×C163C2×D411yescharacteristic
131Q8×C163—21no—
132Q8×C163C2×D412yes—
133D4×C163—22no—
134D4×C163—22no—
135D4×C163—21no—
136D4×C163C2×D412yes—
137C2×C4×C163—21no—
138C2×C4×C163—22no—
139C2×C4×C163—22no—
140C2×C4×C163—22no—
141C2×C4×C163—42no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C8×C163—22no—
149C22×C163—22no—
150C22×C163—22no—
151C22×C163—42no—
152C22×C163#12011yescharacteristic
153C22×C163#12011yescharacteristic
154C22×C163C22×D411yescharacteristic
155C2×D163—24no—
156C2×D163—22no—
157C2×D163—44no—
158C2×D163—42no—
159C2×D163—44no—
160C2×D163—82no—
161C2×D163—88no—
162C4×C163—22no—
163C4×C163—42no—
164C4×C163#12011yescharacteristic
165C4×C163#12011yescharacteristic
166C4×C163C22×D411yescharacteristic
167C4×C163C22×D411yescharacteristic
168Dic163—22no—
169Dic163—42no—
170C2×C163—21no—
171C2×C163—21no—
172C2×C163—42no—
173C2×C163#45512yescomplemented
174C2×C163C2×#12011yescharacteristic
175D163—44no—
176D163—42no—
177C163C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1631nomaximal, Sylow
179C2×H(5,𝔽2)—1631no—
180C2×#137—1631no—
181C2×#137—1631no—
182C2×#124—1631no—
183#455—1638no—
184C2×#104—1631no—
185C2×#104D16311yescharacteristic, complemented
186C2×#100—1631no—
187H(5,𝔽2)—3262no—
188H(5,𝔽2)—3262no—
189H(5,𝔽2)—3262no—
190H(5,𝔽2)—1634no—
191C2×#41—3261no—
192C2×#41—3261no—
193C2×#41—1631no—
194C2×#41C2×D16311yescharacteristic
195C22×Q8—1631no—
196C22×D4—3261no—
197C22×D4—3262no—
198C22×D4—3261no—
199C22×D4—1631no—
200#137—3262no—
201#137—3262no—
202#137—1634no—
203#137—1634no—
204C2×Q16—3261no—
205C2×Q16—3261no—
206C2×SD16—3261no—
207C2×SD16—3261no—
208C2×M16—1631no—
209C2×M16C2×D16311yescharacteristic, complemented
210#124—3262no—
211#124—1634no—
212C2×C22⋊C4—3262no—
213C2×C42C2×D16311yescharacteristic, complemented
214#104—1634no—
215#104C2×D16314yescomplemented
216#100—1634no—
217C24—3262no—
218C24—1632no—
219#41—21no—
220#41—21no—
221#41—3264no—
222#41—3264no—
223#41—3264no—
224#41—3264no—
225#41—3261no—
226#41—3261no—
227#41—1632no—
228#41—1632no—
229#41D65212yes—
230#41C22×D16312yes—
231C2×Q8—3261no—
232C2×Q8—3261no—
233C2×Q8—3261no—
234C2×Q8—3261no—
235C2×Q8—3261no—
236C2×Q8—1631no—
237C2×Q8—1632no—
238C2×Q8D65211yescharacteristic
239C2×D4—21no—
240C2×D4—3262no—
241C2×D4—3264no—
242C2×D4—3264no—
243C2×D4—3268no—
244C2×D4—3264no—
245C2×D4—3268no—
246C2×D4—3262no—
247C2×D4—3261no—
248C2×D4—3264no—
249C2×D4—3261no—
250C2×D4—3261no—
251C2×D4—3261no—
252C2×D4—3261no—
253C2×D4—1632no—
254C2×D4—1631no—
255C2×D4—1632no—
256C2×D4D65211yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3262no—
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4C22×D16311yescharacteristic
263Q16—3264no—
264Q16—3264no—
265SD16—3264no—
266SD16—3264no—
267M16—21no—
268M16—3261no—
269M16—1632no—
270M16C22×D16312yescomplemented
271C2×C8—21no—
272C2×C8—3261no—
273C22⋊C4—3268no—
274C42—21no—
275C42C22×D16312yescomplemented
276C23—21no—
277C23—13044no—
278C23—3268no—
279C23—3262no—
280C23—3262no—
281C23—3261no—
282C23—6524no—
283C23—1634no—
284C23D4×D16311yescharacteristic
285Q8—21no—
286Q8—3261no—
287Q8—3262no—
288Q8—3262no—
289Q8—6521no—
290Q8—6521no—
291Q8—1632no—
292Q8C2×D65212yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3268no—
297D4—3268no—
298D4—3262no—
299D4—3262no—
300D4—3262no—
301D4—3262no—
302D4—3262no—
303D4—3261no—
304D4—3262no—
305D4—1632no—
306D4C2×D65212yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—3268no—
313C2×C4—3261no—
314C2×C4—3262no—
315C2×C4—3262no—
316C2×C4—3262no—
317C2×C4—6521no—
318C2×C4—6521no—
319C2×C4D4×D16311yescharacteristic
320C2×C4D4×D16311yescharacteristic
321C2×C4D4×D16311yescharacteristic
322C2×C4C2×D65211yescharacteristic
323C2×C4C2×D65211yescharacteristic
324C2×C4C23×D16311yescharacteristic
325C8—22no—
326C8—3262no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—13042no—
331C22—13048no—
332C22—3264no—
333C22—3262no—
334C22—6524no—
335C22—6522no—
336C22—6524no—
337C22#1910470511yescharacteristic
338C22#1910470511yescharacteristic
339C22C2×D4×D16311yescharacteristic
340C4—22no—
341C4—42no—
342C4—3262no—
343C4—6522no—
344C4#1910470511yescharacteristic
345C4#1910470511yescharacteristic
346C4C2×D4×D16311yescharacteristic
347C4C22×D65211yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—6522no—
352C2—6524no—
353C2#2378184412yesdirect product factor
354C2C2×#1910470511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#104×C163C2
socleC22×C163#120
prime radicalC2×#19104781C2
hypercenterC2×#104D163
hypocenterC163C2×#455
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#19104768C2semidirect
3. C2×#19104765C2semidirect
4. C2×#19104723C2semidirect
5. #23781844C2direct (N × H)
6. C2×#19104663C2semidirect
7. C2×#19104651C2semidirect
8. C2×#104×C163C2semidirect
16. C22×Dic326C22semidirect
22. #19104768C22semidirect
24. #19104765C22semidirect
29. C2×#14579683C22semidirect
31. #19104723C22semidirect
33. #19104663C22semidirect
34. #19104651C22semidirect
36. C2×M16×C163C22semidirect
37. C2×C42×C163C22semidirect
38. #104×C163C22semidirect
55. C2×Dic326C23semidirect
80. #14579683C23semidirect
94. M16×C163C23semidirect
97. C42×C163C23semidirect
173. C2×C163#455semidirect
177. C163C2×#455semidirect
185. C2×#104D163semidirect
209. C2×M16C2×D163semidirect
213. C2×C42C2×D163semidirect
215. #104C2×D163semidirect
270. M16C22×D163semidirect
275. C42C22×D163semidirect
353. C2#23781844direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191047812
C2×#191047682
C2×#191047652
C2×#191047232
#237818442
C2×#191046632
C2×#191046512
C2×#104×C1632
C2×#455163