Finite Group Databasebeta

← C2×#26011631 #36362752

Subgroups — C2×#26011631 #36362752

Order 22144.

792 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 173 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×Q8×D173C211yesmaximal, characteristic, complemented
2C22×#14583815C211yesmaximal, characteristic, complemented
3C22×#14583813C211yesmaximal, characteristic, complemented
4C2×#19149654C211yesmaximal, characteristic
5#26011631C218yesmaximal, direct product factor
6C2×#19149602C211yesmaximal, characteristic, complemented
7C2×#19149591C211yesmaximal, characteristic, complemented
8C2×#103×C173C211yesmaximal, characteristic, complemented
9C22×Dic346—21no—
10C22×Dic346C2211yescharacteristic
11C2×Q8×D173—24no—
12C2×Q8×D173—24no—
13C2×Q8×D173—22no—
14C2×Q8×D173C2214yescomplemented
15C22×D692—21no—
16C22×D692C2211yescharacteristic
17C22×C4×D173—21no—
18C22×C4×D173C2211yescharacteristic
19C2×#14583815—24no—
20C2×#14583815C2214yescomplemented
21C2×#14583813—24no—
22C2×#14583813C2214yescomplemented
23C22×#13063436C2211yescharacteristic
24#19149654—22no—
25#19149654C2214yes—
26C2×#14583803—21no—
27#19149602C2214yescomplemented
28#19149591C2214yescomplemented
29C22×Q8×C173C2211yescharacteristic
30C22×C8×C173C2211yescharacteristic
31C2×C4⋊C4×C173C2211yescharacteristic
32#103×C173C2214yescomplemented
33C23×D173—21no—
34C23×D173D411yescharacteristic
35C2×Dic346—28no—
36C2×Dic346—24no—
37C2×Dic346—21no—
38C2×Dic346—22no—
39C2×Dic346D414yescomplemented
40C2×Dic346C2312yes—
41Q8×D173—216no—
42Q8×D173—28no—
43Q8×D173—28no—
44C2×D692—24no—
45C2×D692—22no—
46C2×D692—22no—
47C2×D692—21no—
48C2×D692—44no—
49C2×D692C2312yes—
50C22×Dic173—21no—
51C22×Dic173D411yescharacteristic
52C2×C4×D173—28no—
53C2×C4×D173—24no—
54C2×C4×D173—22no—
55C2×C4×D173—22no—
56C2×C4×D173D412yes—
57C2×C4×D173C2312yes—
58#14583815—28no—
59#14583813—28no—
60C2×#13063436—22no—
61C2×#13063436C2312yes—
62#14583803—24no—
63C2×Q8×C173—22no—
64C2×Q8×C173—21no—
65C2×Q8×C173D414yescomplemented
66C2×Q8×C173C2312yes—
67C22×C4×C173—21no—
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—42no—
78C22×D173—42no—
79C22×D173—42no—
80C22×D173SD1614yescomplemented
81C22×D173C2×D412yes—
82Dic346—216no—
83Dic346—24no—
84Dic346—22no—
85Dic346C2×D414yescomplemented
86D692—22no—
87D692—22no—
88D692—48no—
89D692—42no—
90C2×Dic173—24no—
91C2×Dic173—22no—
92C2×Dic173—22no—
93C2×Dic173C2×D412yes—
94C4×D173—216no—
95C4×D173—24no—
96C4×D173—22no—
97C4×D173—22no—
98#13063436—22no—
99C23×C173C2×D411yescharacteristic
100Q8×C173—24no—
101Q8×C173—22no—
102Q8×C173C2×D414yescomplemented
103C2×C4×C173—24no—
104C2×C4×C173—22no—
105C2×C4×C173—22no—
106C2×C4×C173—42no—
107C2×C4×C173C2×D411yescharacteristic
108C2×C4×C173C2×D412yes—
109C2×C4×C173C2×D412yes—
110C2×C4×C173C2411yescharacteristic
111C8×C173—22no—
112C22×C173#12012yes—
113C22×C173C2×SD1612yes—
114C22×C173#13712yes—
115C22×C173C22×D411yescharacteristic
116C2×D173—22no—
117C2×D173—22no—
118C2×D173—28no—
119C2×D173—42no—
120C2×D173—42no—
121C2×D173—42no—
122C2×D173—84no—
123C2×D173C2×SD1614yescomplemented
124C4×C173—24no—
125C4×C173—42no—
126C4×C173#12012yes—
127C4×C173C22×D411yescharacteristic
128C4×C173C22×D411yescharacteristic
129Dic173—24no—
130Dic173—22no—
131C2×C173#44814yescomplemented
132C2×C173C2×#12011yescharacteristic
133C2×C173C22×SD1611yescharacteristic
134C2×C173C2×#13711yescharacteristic
135D173—24no—
136D173—82no—
137C173C2×#44811yescharacteristic, complemented, Sylow
138C2×#448—1731nomaximal, Sylow
139C23×Q8—1731no—
140C22×SD16—1731no—
141C22×SD16—1731no—
142C2×#121—1731no—
143#448—1738no—
144C2×#103—1731no—
145C2×#103D17311yescharacteristic, complemented
146C2×C22⋊C8—1731no—
147C22×Q8—3461no—
148C22×Q8—3464no—
149C22×Q8—3464no—
150C22×Q8—3462no—
151C22×Q8—1734no—
152C22×Q8—1731no—
153C22×Q8C2×D17311yescharacteristic
154C22×D4—3461no—
155C22×D4—1731no—
156C23×C4—3461no—
157C23×C4—1731no—
158C2×SD16—3464no—
159C2×SD16—3464no—
160C2×SD16—1734no—
161C2×SD16—1734no—
162C22×C8—1731no—
163C22×C8C2×D17311yescharacteristic
164#121—3462no—
165#121—1734no—
166C2×C4⋊C4C2×D17311yescharacteristic
167C2×C22⋊C4—3461no—
168#103—1734no—
169#103C2×D17314yescomplemented
170C22⋊C8—1734no—
171C24—3461no—
172C24—1731no—
173C2×Q8—22no—
174C2×Q8—21no—
175C2×Q8—3468no—
176C2×Q8—3464no—
177C2×Q8—34616no—
178C2×Q8—3468no—
179C2×Q8—3468no—
180C2×Q8—3461no—
181C2×Q8—3462no—
182C2×Q8—1734no—
183C2×Q8—1732no—
184C2×Q8D69214yescomplemented
185C2×Q8C22×D17312yes—
186C2×D4—3464no—
187C2×D4—3462no—
188C2×D4—3461no—
189C2×D4—3462no—
190C2×D4—1732no—
191C2×D4—6924no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—3468no—
195C22×C4—3464no—
196C22×C4—3461no—
197C22×C4—3462no—
198C22×C4—3462no—
199C22×C4—1731no—
200C22×C4—1732no—
201C22×C4—1732no—
202C22×C4C22×D17311yescharacteristic
203SD16—3468no—
204SD16—3468no—
205C2×C8—22no—
206C2×C8—3462no—
207C2×C8—1732no—
208C2×C8C22×D17312yes—
209C4⋊C4—21no—
210C4⋊C4C22×D17312yes—
211C22⋊C4—3464no—
212C23—3462no—
213C23—3462no—
214C23—3462no—
215C23—1734no—
216C23—1732no—
217C23—6922no—
218C23—6922no—
219C23—6922no—
220C23D4×D17311yescharacteristic
221Q8—24no—
222Q8—22no—
223Q8—34616no—
224Q8—3464no—
225Q8—3462no—
226Q8—1734no—
227Q8C2×D69214yescomplemented
228D4—3462no—
229D4—3462no—
230D4—6928no—
231D4—6922no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—34616no—
237C2×C4—3464no—
238C2×C4—3462no—
239C2×C4—3462no—
240C2×C4—3464no—
241C2×C4—3462no—
242C2×C4—3462no—
243C2×C4—1732no—
244C2×C4D4×D17311yescharacteristic
245C2×C4D4×D17312yes—
246C2×C4C2×D69212yes—
247C2×C4C23×D17311yescharacteristic
248C8—22no—
249C8—3462no—
250C22—3462no—
251C22—3462no—
252C22—3468no—
253C22—13844no—
254C22—1734no—
255C22—6922no—
256C22—6922no—
257C22—6922no—
258C22#1914964912yes—
259C22SD16×D17312yes—
260C22#1914971112yes—
261C22C2×D4×D17311yescharacteristic
262C4—24no—
263C4—42no—
264C4—3464no—
265C4—3462no—
266C4#1914964912yes—
267C4C2×D4×D17311yescharacteristic
268C4C22×D69211yescharacteristic
269C2—3464no—
270C2—13842no—
271C2#2601163114yesdirect product factor
272C2C2×#1914964911yescharacteristic
273C2C2×SD16×D17311yescharacteristic
274C2C2×#1914971111yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#103×C173C2
socleC23×C173C2×D4
prime radicalC2×#19149654C2
hypercenterC2×#103D173
hypocenterC173C2×#448
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D173C2semidirect
2. C22×#14583815C2semidirect
3. C22×#14583813C2semidirect
5. #26011631C2direct (N × H)
6. C2×#19149602C2semidirect
7. C2×#19149591C2semidirect
8. C2×#103×C173C2semidirect
14. C2×Q8×D173C22semidirect
20. C2×#14583815C22semidirect
22. C2×#14583813C22semidirect
27. #19149602C22semidirect
28. #19149591C22semidirect
32. #103×C173C22semidirect
39. C2×Dic346D4semidirect
65. C2×Q8×C173D4semidirect
80. C22×D173SD16semidirect
85. Dic346C2×D4semidirect
102. Q8×C173C2×D4semidirect
123. C2×D173C2×SD16semidirect
131. C2×C173#448semidirect
137. C173C2×#448semidirect
145. C2×#103D173semidirect
169. #103C2×D173semidirect
184. C2×Q8D692semidirect
227. Q8C2×D692semidirect
271. C2#26011631direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1732
C22×#145838152
C22×#145838132
C2×#191496542
#260116312
C2×#191496022
C2×#191495912
C2×#103×C1732
C2×#448173