Finite Group Databasebeta

← C2×#23781834 #35223861

Subgroups — C2×#23781834 #35223861

Order 20864.

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×#19104781C211yesmaximal, characteristic, complemented
2C2×#19104763C211yesmaximal, characteristic, complemented
3C2×#19104760C211yesmaximal, characteristic, complemented
4C2×#19104732C211yesmaximal, characteristic, complemented
5#23781834C218yesmaximal, 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#19104781C2214yescomplemented
13C2×#14579700—21no—
14C2×#14579699—21no—
15C2×#14579699C2211yescharacteristic
16C22×D652—21no—
17C22×D652—22no—
18C22×D652C2211yescharacteristic
19C22×D652C2211yescharacteristic, complemented
20C2×D4×D163—22no—
21C2×D4×D163—21no—
22#19104763—22no—
23#19104763C2214yescomplemented
24#19104760—22no—
25#19104760C2214yescomplemented
26C2×#14579688—21no—
27C2×#14579686—21no—
28C2×D1304—21no—
29C2×#14579684—21no—
30C2×#14579683C2211yescharacteristic, complemented
31#19104732—22no—
32#19104732C2214yescomplemented
33#19104663C2214yescomplemented
34#19104651C2214yescomplemented
35C2×#41×C163C2211yescharacteristic
36C2×M16×C163C2211yescharacteristic, complemented
37C2×C42×C163C2211yescharacteristic, complemented
38#104×C163C2214yescomplemented
39C23×D163—22no—
40C23×D163—21no—
41C23×D163D412yes—
42#14579700—24no—
43#14579700—24no—
44#14579699—24no—
45#14579699—24no—
46#14579699—21no—
47#14579699—21no—
48#14579699D412yescomplemented
49#14579699C2312yes—
50C2×Dic326D411yescharacteristic, complemented
51C2×D652—22no—
52C2×D652—24no—
53C2×D652—24no—
54C2×D652—28no—
55C2×D652—21no—
56C2×D652—21no—
57C2×D652—21no—
58C2×D652—21no—
59C2×D652—21no—
60C2×D652—21no—
61C2×D652—21no—
62C2×D652—21no—
63C2×D652—21no—
64C2×D652—48no—
65C2×D652D412yes—
66C2×D652D411yescharacteristic, complemented
67C2×D652C2312yes—
68C2×D652C2312yescomplemented
69C2×C22⋊D163—22no—
70C2×C22⋊D163—21no—
71D4×D163—28no—
72D4×D163—24no—
73D4×D163—28no—
74D4×D163—24no—
75C2×C4×D163—22no—
76C2×C4×D163—21no—
77#14579688—24no—
78#14579686—24no—
79D1304—24no—
80#14579684—24no—
81#14579683—21no—
82#14579683C2312yescomplemented
83C2×C163⋊C8—21no—
84#41×C163—21no—
85#41×C163—21no—
86#41×C163D412yescomplemented
87#41×C163C2312yes—
88C2×Q8×C163D411yescharacteristic, complemented
89C2×D4×C163—21no—
90C2×D4×C163D411yescharacteristic, complemented
91C22×C4×C163—21no—
92C22×C4×C163—21no—
93C22×C4×C163C2311yescharacteristic
94M16×C163—21no—
95M16×C163C2312yescomplemented
96C2×C8×C163—21no—
97C42×C163—21no—
98C42×C163C2312yescomplemented
99C22×D163—22no—
100C22×D163—21no—
101C22×D163—28no—
102C22×D163—22no—
103C22×D163—22no—
104C22×D163—44no—
105C22×D163—44no—
106C22×D163—41no—
107C22×D163—41no—
108C22×D163—84no—
109C22×D163C2×D414yes—
110Dic326—21no—
111Dic326C2×D412yescomplemented
112D652—28no—
113D652—22no—
114D652—22no—
115D652—21no—
116D652—22no—
117D652—22no—
118D652—22no—
119D652—22no—
120D652—416no—
121D652—41no—
122D652—41no—
123D652C2×D412yescomplemented
124C22⋊D163—28no—
125C22⋊D163—22no—
126C22⋊D163—22no—
127C2×Dic163—21no—
128C4×D163—28no—
129C4×D163—22no—
130C4×D163—22no—
131C163⋊C8—22no—
132C23×C163—21no—
133C23×C163C2×D411yescharacteristic
134Q8×C163—21no—
135Q8×C163C2×D412yescomplemented
136D4×C163—22no—
137D4×C163—22no—
138D4×C163—21no—
139D4×C163C2×D412yescomplemented
140C2×C4×C163—24no—
141C2×C4×C163—21no—
142C2×C4×C163—22no—
143C2×C4×C163—22no—
144C2×C4×C163—22no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2411yescharacteristic
151C8×C163—22no—
152C22×C163—22no—
153C22×C163—22no—
154C22×C163—42no—
155C22×C163#12011yescharacteristic
156C22×C163#12011yescharacteristic
157C22×C163C22×D411yescharacteristic
158C2×D163—24no—
159C2×D163—22no—
160C2×D163—44no—
161C2×D163—42no—
162C2×D163—44no—
163C2×D163—44no—
164C2×D163—42no—
165C2×D163—82no—
166C2×D163—88no—
167C2×D163—82no—
168C2×D163—82no—
169C4×C163—24no—
170C4×C163—22no—
171C4×C163#12011yescharacteristic
172C4×C163#12011yescharacteristic
173C4×C163C22×D411yescharacteristic
174C4×C163C22×D411yescharacteristic
175Dic163—22no—
176C2×C163—21no—
177C2×C163—21no—
178C2×C163—42no—
179C2×C163#45312yescomplemented
180C2×C163C2×#12011yescharacteristic
181D163—44no—
182D163—42no—
183D163—82no—
184C163C2×#45311yescharacteristic, complemented, Sylow
185C2×#453—1631nomaximal, Sylow
186C2×H(5,𝔽2)—1631no—
187C2×AGL(1,ℤ/8)—1631no—
188C2×AGL(1,ℤ/8)—1631no—
189C2×#127—1631no—
190#453—1638no—
191C2×#104—1631no—
192C2×#104D16311yescharacteristic, complemented
193C2×#100—1631no—
194H(5,𝔽2)—3262no—
195H(5,𝔽2)—3262no—
196H(5,𝔽2)—3262no—
197H(5,𝔽2)—1634no—
198C2×#41—3261no—
199C2×#41—3261no—
200C2×#41—1631no—
201C2×#41C2×D16311yescharacteristic
202C22×D4—3261no—
203C22×D4—3262no—
204C22×D4—3262no—
205C22×D4—3261no—
206C22×D4—1631no—
207C22×D4—1631no—
208AGL(1,ℤ/8)—3262no—
209AGL(1,ℤ/8)—3262no—
210AGL(1,ℤ/8)—1634no—
211AGL(1,ℤ/8)—1634no—
212C2×SD16—3261no—
213C2×SD16—3261no—
214C2×D8—3261no—
215C2×D8—3261no—
216C2×M16—1631no—
217C2×M16C2×D16311yescharacteristic, complemented
218#127—3262no—
219#127—1634no—
220C2×C42C2×D16311yescharacteristic, complemented
221#104—1634no—
222#104C2×D16314yescomplemented
223#100—1634no—
224C24—3262no—
225C24—3261no—
226C24—1632no—
227#41—21no—
228#41—21no—
229#41—3264no—
230#41—3264no—
231#41—3264no—
232#41—3264no—
233#41—3261no—
234#41—3261no—
235#41—1632no—
236#41—1632no—
237#41D65212yescomplemented
238#41C22×D16312yes—
239C2×Q8—1631no—
240C2×Q8D65211yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—3262no—
243C2×D4—3264no—
244C2×D4—3264no—
245C2×D4—3268no—
246C2×D4—3268no—
247C2×D4—3264no—
248C2×D4—3268no—
249C2×D4—3262no—
250C2×D4—3261no—
251C2×D4—3264no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—3261no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—6528no—
262C2×D4—1632no—
263C2×D4—1631no—
264C2×D4—1632no—
265C2×D4—1632no—
266C2×D4D65211yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—3262no—
270C22×C4—3261no—
271C22×C4C22×D16311yescharacteristic
272SD16—3264no—
273SD16—3264no—
274D8—3264no—
275D8—3264no—
276M16—21no—
277M16—3261no—
278M16—1632no—
279M16C22×D16312yescomplemented
280C2×C8—21no—
281C2×C8—3261no—
282C42—21no—
283C42C22×D16312yescomplemented
284C23—21no—
285C23—13044no—
286C23—3268no—
287C23—3262no—
288C23—3262no—
289C23—3261no—
290C23—3262no—
291C23—6524no—
292C23—6524no—
293C23—6521no—
294C23—6521no—
295C23—1634no—
296C23D4×D16311yescharacteristic
297Q8—21no—
298Q8—3261no—
299Q8—1632no—
300Q8C2×D65212yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—3268no—
305D4—3268no—
306D4—3262no—
307D4—3262no—
308D4—3262no—
309D4—3262no—
310D4—3262no—
311D4—3262no—
312D4—3262no—
313D4—3261no—
314D4—3262no—
315D4—65216no—
316D4—6521no—
317D4—6521no—
318D4—1632no—
319D4C2×D65212yescomplemented
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—3268no—
326C2×C4—3261no—
327C2×C4—3262no—
328C2×C4—3262no—
329C2×C4D4×D16311yescharacteristic
330C2×C4D4×D16311yescharacteristic
331C2×C4D4×D16311yescharacteristic
332C2×C4C2×D65211yescharacteristic
333C2×C4C2×D65211yescharacteristic
334C2×C4C23×D16311yescharacteristic
335C8—22no—
336C8—3262no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—13042no—
341C22—13042no—
342C22—13048no—
343C22—13042no—
344C22—3264no—
345C22—3262no—
346C22—6524no—
347C22—6524no—
348C22—6522no—
349C22—6522no—
350C22—6524no—
351C22#1910470511yescharacteristic
352C22#1910470511yescharacteristic
353C22C2×D4×D16311yescharacteristic
354C4—24no—
355C4—22no—
356C4—3262no—
357C4#1910470511yescharacteristic
358C4#1910470511yescharacteristic
359C4C2×D4×D16311yescharacteristic
360C4C22×D65211yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—13042no—
365C2—6522no—
366C2—6524no—
367C2#2378183412yesdirect product factor
368C2C2×#1910470511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#104×C163C2
socleC22×C163#120
hypercenterC2×#104D163
hypocenterC163C2×#453
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
1. C2×#19104781C2semidirect
2. C2×#19104763C2semidirect
3. C2×#19104760C2semidirect
4. C2×#19104732C2semidirect
5. #23781834C2direct (N × H)
6. C2×#19104663C2semidirect
7. C2×#19104651C2semidirect
8. C2×#104×C163C2semidirect
12. #19104781C22semidirect
19. C22×D652C22semidirect
23. #19104763C22semidirect
25. #19104760C22semidirect
30. C2×#14579683C22semidirect
32. #19104732C22semidirect
33. #19104663C22semidirect
34. #19104651C22semidirect
36. C2×M16×C163C22semidirect
37. C2×C42×C163C22semidirect
38. #104×C163C22semidirect
48. #14579699D4semidirect
50. C2×Dic326D4semidirect
66. C2×D652D4semidirect
68. C2×D652C23semidirect
82. #14579683C23semidirect
86. #41×C163D4semidirect
88. C2×Q8×C163D4semidirect
90. C2×D4×C163D4semidirect
95. M16×C163C23semidirect
98. C42×C163C23semidirect
111. Dic326C2×D4semidirect
123. D652C2×D4semidirect
135. Q8×C163C2×D4semidirect
139. D4×C163C2×D4semidirect
179. C2×C163#453semidirect
184. C163C2×#453semidirect
192. C2×#104D163semidirect
217. C2×M16C2×D163semidirect
220. C2×C42C2×D163semidirect
222. #104C2×D163semidirect
237. #41D652semidirect
240. C2×Q8D652semidirect
266. C2×D4D652semidirect
279. M16C22×D163semidirect
283. C42C22×D163semidirect
300. Q8C2×D652semidirect
319. D4C2×D652semidirect
367. C2#23781834direct (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×#191047632
C2×#191047602
C2×#191047322
#237818342
C2×#191046632
C2×#191046512
C2×#104×C1632
C2×#453163