Finite Group Databasebeta

← C2×#26011637 #36362765

Subgroups — C2×#26011637 #36362765

Order 22144.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#14583825C211yesmaximal, characteristic, complemented
2C22×#14583815C211yesmaximal, characteristic, complemented
3C22×#14583811C211yesmaximal, characteristic, complemented
4C2×#19149654C211yesmaximal, characteristic, complemented
5#26011637C218yesmaximal, direct product factor
6C2×#19149602C211yesmaximal, characteristic, complemented
7C2×#19149591C211yesmaximal, characteristic, complemented
8C2×#102×C173C211yesmaximal, characteristic, complemented
9C2×#14583825—24no—
10C2×#14583825—24no—
11C2×#14583825—22no—
12C2×#14583825C2214yescomplemented
13C22×Dic346C2211yescharacteristic
14C22×D692—21no—
15C22×D692C2211yescharacteristic
16C22×C22⋊D173—21no—
17C23×Dic173—21no—
18C22×C4×D173C2211yescharacteristic
19C2×#14583815—24no—
20C2×#14583815C2214yescomplemented
21C2×#14583811—24no—
22C2×#14583811C2214yescomplemented
23C22×#13063436C2211yescharacteristic
24#19149654—22no—
25#19149654C2214yescomplemented
26C2×#14583803—21no—
27#19149602C2214yescomplemented
28#19149591C2214yescomplemented
29C22×D4×C173C2211yescharacteristic
30C22×C8×C173C2211yescharacteristic
31C2×C4⋊C4×C173C2211yescharacteristic
32#102×C173C2214yescomplemented
33C23×D173—21no—
34C23×D173D411yescharacteristic
35#14583825—216no—
36#14583825—28no—
37#14583825—28no—
38C2×Dic346—21no—
39C2×Dic346—22no—
40C2×Dic346D414yescomplemented
41C2×Dic346C2312yes—
42C2×D692—24no—
43C2×D692—22no—
44C2×D692—22no—
45C2×D692—21no—
46C2×D692—44no—
47C2×D692C2312yes—
48C2×C22⋊D173—28no—
49C2×C22⋊D173—24no—
50C22×Dic173—28no—
51C22×Dic173—24no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—22no—
55C2×C4×D173—22no—
56C2×C4×D173D412yes—
57C2×C4×D173C2312yes—
58#14583815—28no—
59#14583811—28no—
60C2×#13063436—22no—
61C2×#13063436C2312yes—
62#14583803—24no—
63C24×C173—21no—
64C2×D4×C173—21no—
65C2×D4×C173—22no—
66C2×D4×C173D414yescomplemented
67C2×D4×C173C2312yes—
68C22×C4×C173—21no—
69C22×C4×C173C2311yescharacteristic
70C2×C8×C173—22no—
71C2×C8×C173C2312yes—
72C4⋊C4×C173—21no—
73C4⋊C4×C173C2312yes—
74C22×D173—22no—
75C22×D173—22no—
76C22×D173—22no—
77C22×D173—22no—
78C22×D173—42no—
79C22×D173—42no—
80C22×D173—42no—
81C22×D173C2×D412yes—
82Dic346—24no—
83Dic346—22no—
84Dic346C2×D414yescomplemented
85D692—22no—
86D692—22no—
87D692—48no—
88D692—42no—
89C22⋊D173—216no—
90C2×Dic173—216no—
91C2×Dic173—22no—
92C2×Dic173—24no—
93C2×Dic173—22no—
94C2×Dic173C2×D412yes—
95C4×D173—24no—
96C4×D173—22no—
97C4×D173—22no—
98#13063436—22no—
99C23×C173—24no—
100C23×C173—22no—
101C23×C173—42no—
102C23×C173—42no—
103C23×C173C2×D411yescharacteristic
104D4×C173—24no—
105D4×C173—22no—
106D4×C173C2×D414yescomplemented
107C2×C4×C173—22no—
108C2×C4×C173—42no—
109C2×C4×C173C2×D411yescharacteristic
110C2×C4×C173C2×D412yes—
111C2×C4×C173C2×D412yes—
112C2×C4×C173C2411yescharacteristic
113C8×C173—22no—
114C22×C173—24no—
115C22×C173—42no—
116C22×C173—42no—
117C22×C173—48no—
118C22×C173#12012yes—
119C22×C173#13512yes—
120C22×C173AGL(1,ℤ/8)12yes—
121C22×C173C22×D411yescharacteristic
122C2×D173—22no—
123C2×D173—22no—
124C2×D173—22no—
125C2×D173—44no—
126C2×D173—42no—
127C2×D173—42no—
128C2×D173—42no—
129C2×D173—84no—
130C4×C173—42no—
131C4×C173#12012yes—
132C4×C173C22×D411yescharacteristic
133C4×C173C22×D411yescharacteristic
134Dic173—22no—
135Dic173—24no—
136C2×C173—44no—
137C2×C173#44914yescomplemented
138C2×C173C2×#12011yescharacteristic
139C2×C173C2×#13511yescharacteristic
140C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
141D173—42no—
142D173—82no—
143C173C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1731nomaximal, Sylow
145C22×#41—1731no—
146C22×SD16—1731no—
147C22×D8—1731no—
148C2×#121—1731no—
149#449—1738no—
150C2×#103—1731no—
151C2×#102D17311yescharacteristic, complemented
152C2×C22⋊C8—1731no—
153C2×#41—3464no—
154C2×#41—3464no—
155C2×#41—3462no—
156C2×#41—1734no—
157C22×Q8—1731no—
158C22×D4—3461no—
159C22×D4—3461no—
160C22×D4—1731no—
161C22×D4C2×D17311yescharacteristic
162C23×C4—3461no—
163C23×C4—1731no—
164C2×SD16—3464no—
165C2×SD16—1734no—
166C2×D8—3464no—
167C2×D8—1734no—
168C22×C8—1731no—
169C22×C8C2×D17311yescharacteristic
170#121—3462no—
171#121—1734no—
172C2×C4⋊C4C2×D17311yescharacteristic
173C2×C22⋊C4—3461no—
174#103—1734no—
175#102C2×D17314yescomplemented
176C22⋊C8—1734no—
177C24—21no—
178C24—3461no—
179C24—1731no—
180#41—34616no—
181#41—3468no—
182#41—3468no—
183C2×Q8—3461no—
184C2×Q8—3462no—
185C2×Q8—1734no—
186C2×Q8—1732no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—3464no—
190C2×D4—3468no—
191C2×D4—3464no—
192C2×D4—3462no—
193C2×D4—3461no—
194C2×D4—3462no—
195C2×D4—1732no—
196C2×D4—6924no—
197C2×D4D69214yescomplemented
198C2×D4C22×D17312yes—
199C22×C4—21no—
200C22×C4—3468no—
201C22×C4—3461no—
202C22×C4—3464no—
203C22×C4—3462no—
204C22×C4—3462no—
205C22×C4—1731no—
206C22×C4—1732no—
207C22×C4—1732no—
208C22×C4C22×D17311yescharacteristic
209SD16—3468no—
210D8—3468no—
211C2×C8—22no—
212C2×C8—3462no—
213C2×C8—1732no—
214C2×C8C22×D17312yes—
215C4⋊C4—21no—
216C4⋊C4C22×D17312yes—
217C22⋊C4—3464no—
218C23—24no—
219C23—22no—
220C23—42no—
221C23—42no—
222C23—3462no—
223C23—3462no—
224C23—3462no—
225C23—3462no—
226C23—1732no—
227C23—6922no—
228C23—6922no—
229C23—6922no—
230C23D4×D17311yescharacteristic
231Q8—3464no—
232Q8—3462no—
233Q8—1734no—
234D4—24no—
235D4—22no—
236D4—34616no—
237D4—3462no—
238D4—3462no—
239D4—6928no—
240D4—6922no—
241D4C2×D69214yescomplemented
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—34616no—
245C2×C4—3462no—
246C2×C4—3464no—
247C2×C4—3462no—
248C2×C4—3464no—
249C2×C4—3462no—
250C2×C4—3462no—
251C2×C4—1732no—
252C2×C4D4×D17311yescharacteristic
253C2×C4D4×D17312yes—
254C2×C4C2×D69212yes—
255C2×C4C23×D17311yescharacteristic
256C8—22no—
257C8—3462no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—48no—
262C22—3462no—
263C22—3462no—
264C22—3462no—
265C22—13844no—
266C22—6922no—
267C22—6922no—
268C22—6922no—
269C22—6924no—
270C22#1914964912yes—
271C22#1914970012yes—
272C22#1914970312yes—
273C22C2×D4×D17311yescharacteristic
274C4—42no—
275C4—3462no—
276C4—3464no—
277C4#1914964912yes—
278C4C2×D4×D17311yescharacteristic
279C4C22×D69211yescharacteristic
280C2—44no—
281C2—13842no—
282C2—6922no—
283C2#2601163714yesdirect product factor
284C2C2×#1914964911yescharacteristic
285C2C2×#1914970011yescharacteristic
286C2C2×#1914970311yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#102×C173C2
socleC23×C173C2×D4
hypercenterC2×#102D173
hypocenterC173C2×#449
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14583825C2semidirect
2. C22×#14583815C2semidirect
3. C22×#14583811C2semidirect
4. C2×#19149654C2semidirect
5. #26011637C2direct (N × H)
6. C2×#19149602C2semidirect
7. C2×#19149591C2semidirect
8. C2×#102×C173C2semidirect
12. C2×#14583825C22semidirect
20. C2×#14583815C22semidirect
22. C2×#14583811C22semidirect
25. #19149654C22semidirect
27. #19149602C22semidirect
28. #19149591C22semidirect
32. #102×C173C22semidirect
40. C2×Dic346D4semidirect
66. C2×D4×C173D4semidirect
84. Dic346C2×D4semidirect
106. D4×C173C2×D4semidirect
137. C2×C173#449semidirect
143. C173C2×#449semidirect
151. C2×#102D173semidirect
175. #102C2×D173semidirect
197. C2×D4D692semidirect
241. D4C2×D692semidirect
283. C2#26011637direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838252
C22×#145838152
C22×#145838112
C2×#191496542
#260116372
C2×#191496022
C2×#191495912
C2×#102×C1732
C2×#449173