Finite Group Databasebeta

← #36362923

Subgroups — #36362923

Order 22144.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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×#19149724C211yesmaximal, characteristic, complemented
2#26012291C212yesmaximal, complemented
3#26012058C211yesmaximal, characteristic, complemented
4#26011671C214yesmaximal, complemented
5#26011661C214yesmaximal, complemented
6#26011507C212yesmaximal, complemented
7C2×#100×C173C211yesmaximal, characteristic, complemented
8#19149724—22no—
9#19149724—24no—
10#19149724C2214yes—
11C2×#14583826C2212yes—
12C2×#14583826C2211yescharacteristic, complemented
13C2×#14583825—22no—
14C22×C22⋊D173—22no—
15C2×D4×D173—22no—
16#19149712—22no—
17#19149712C2214yescomplemented
18#19149707—22no—
19#19149707C2214yescomplemented
20#19149701—24no—
21#19149675C2212yescomplemented
22#19149666C2212yescomplemented
23#19149650—22no—
24#19149648—22no—
25#19149638C2211yescharacteristic, complemented
26#19149605C2218yescomplemented
27C22×D4×C173C2211yescharacteristic
28C2×M16×C173C2212yescomplemented
29#100×C173C2214yescomplemented
30C23×D173—22no—
31#14583826—22no—
32#14583826—22no—
33#14583826—22no—
34#14583826—22no—
35#14583826D414yescomplemented
36#14583826C2314yes—
37#14583825—28no—
38#14583825—28no—
39C2×Dic346D412yescomplemented
40C2×Dic346C2311yescharacteristic, complemented
41C2×D692D412yescomplemented
42C2×D692C2311yescharacteristic, complemented
43C2×C22⋊D173—24no—
44C2×C22⋊D173—28no—
45C2×C22⋊D173—22no—
46C2×C22⋊D173—28no—
47C2×C22⋊D173—21no—
48C2×C22⋊D173—24no—
49D4×D173—28no—
50D4×D173—28no—
51C22×Dic173—22no—
52C2×C4×D173—22no—
53C2×C4×D173—21no—
54Dic692—24no—
55#14583815—24no—
56#14583815—24no—
57D1384—24no—
58C4⋊Dic173—21no—
59C22⋊Dic173—21no—
60C22⋊Dic173—24no—
61C4×Dic173C2312yescomplemented
62C24×C173D412yes—
63C2×D4×C173—22no—
64C2×D4×C173—22no—
65C2×D4×C173D412yes—
66C2×D4×C173C2312yes—
67C22×C4×C173C2311yescharacteristic
68M16×C173—22no—
69M16×C173C2314yescomplemented
70C2×C8×C173—22no—
71C22×D173—22no—
72C22×D173—28no—
73C22×D173—21no—
74C22×D173—84no—
75Dic346—22no—
76Dic346—22no—
77Dic346C2×D414yes—
78D692—22no—
79D692—22no—
80D692C2×D414yes—
81C22⋊D173—216no—
82C22⋊D173—24no—
83C22⋊D173—24no—
84C22⋊D173—41no—
85C22⋊D173—41no—
86C22⋊D173—84no—
87C2×Dic173—28no—
88C2×Dic173—22no—
89C2×Dic173—21no—
90C2×Dic173—22no—
91C4×D173—24no—
92C4×D173—24no—
93C4×D173—42no—
94C23×C173—22no—
95C23×C173—22no—
96C23×C173—82no—
97C23×C173C2×D414yes—
98C23×C173C2×D411yescharacteristic
99D4×C173—24no—
100D4×C173—24no—
101C2×C4×C173C2×D412yes—
102C2×C4×C173C2×D412yes—
103C2×C4×C173C2×D411yescharacteristic
104C2×C4×C173C2411yescharacteristic
105C8×C173—24no—
106C22×C173—24no—
107C22×C173—41no—
108C22×C173—44no—
109C22×C173—44no—
110C22×C173—82no—
111C22×C173#12011yescharacteristic
112C22×C173#12011yescharacteristic
113C22×C173C22×D411yescharacteristic
114C2×D173—24no—
115C2×D173—44no—
116C2×D173—42no—
117C2×D173—41no—
118C2×D173—88no—
119C4×C173#12012yes—
120C4×C173C22×D412yes—
121Dic173—24no—
122Dic173—42no—
123Dic173—41no—
124C2×C173—21no—
125C2×C173—21no—
126C2×C173—21no—
127C2×C173—44no—
128C2×C173C2×#12011yescharacteristic
129D173—44no—
130D173—81no—
131C173#301911yescharacteristic, complemented, Sylow
132#3019—1731nomaximal, Sylow
133C2×H(5,𝔽2)—1731no—
134#575—1732no—
135#535—1731no—
136#455—1734no—
137#453—1734no—
138#421—1732no—
139C2×#100D17311yescharacteristic, complemented
140H(5,𝔽2)—3464no—
141H(5,𝔽2)—3462no—
142H(5,𝔽2)—1734no—
143C2×#41—3462no—
144C2×#41—1731no—
145C2×#41—1732no—
146C22×D4—3462no—
147C22×D4—3462no—
148C22×D4C2×D17311yescharacteristic
149#137—3462no—
150#137—1734no—
151AGL(1,ℤ/8)—3462no—
152AGL(1,ℤ/8)—1734no—
153#135—3464no—
154C2×M16C2×D17312yescomplemented
155#127—1732no—
156#124—1732no—
157#121—3462no—
158#120—3462no—
159#117—1731no—
160#104—1738no—
161#100C2×D17314yescomplemented
162C24—3462no—
163C24D69212yes—
164#41—3468no—
165#41—3468no—
166#41—3462no—
167#41—3462no—
168#41—3462no—
169#41—3462no—
170#41—1734no—
171#41—1734no—
172C2×Q8—1732no—
173C2×Q8—1731no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—3468no—
177C2×D4—3468no—
178C2×D4—3464no—
179C2×D4—3468no—
180C2×D4—3462no—
181C2×D4—3468no—
182C2×D4—3464no—
183C2×D4—3461no—
184C2×D4—1732no—
185C2×D4—1731no—
186C2×D4D69212yes—
187C2×D4C22×D17312yes—
188C22×C4—3462no—
189C22×C4—3462no—
190C22×C4—3461no—
191C22×C4C22×D17311yescharacteristic
192Q16—3464no—
193SD16—3464no—
194SD16—3464no—
195D8—3464no—
196M16—22no—
197M16C22×D17314yescomplemented
198C2×C8—22no—
199C4⋊C4—3461no—
200C22⋊C4—3461no—
201C22⋊C4—3464no—
202C42—1732no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—3468no—
207C23—3462no—
208C23—3461no—
209C23—13844no—
210C23C2×D69214yes—
211C23C2×D69211yescharacteristic
212Q8—3462no—
213Q8—3462no—
214Q8—1734no—
215D4—24no—
216D4—24no—
217D4—34616no—
218D4—3464no—
219D4—3464no—
220D4—3462no—
221D4—3462no—
222D4—13844no—
223D4—1734no—
224D4—6921no—
225D4—6921no—
226C2×C4—3468no—
227C2×C4—3462no—
228C2×C4—3464no—
229C2×C4—3464no—
230C2×C4—3462no—
231C2×C4—3461no—
232C2×C4—6922no—
233C2×C4D4×D17312yes—
234C2×C4D4×D17312yes—
235C2×C4C2×D69211yescharacteristic
236C2×C4C23×D17311yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—3464no—
244C22—13848no—
245C22—6922no—
246C22—6924no—
247C22—6921no—
248C22#1914964911yescharacteristic
249C22#1914964911yescharacteristic
250C22C22×D69211yescharacteristic
251C4—3464no—
252C4—6922no—
253C4—6921no—
254C4#1914964912yes—
255C4C2×D4×D17312yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—13841no—
261C2—6924no—
262C2C2×#1914964911yescharacteristic
subgroupG/H
centerC2C2×#19149649
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#100×C173C2
socleC2×C173C2×#120
hypercenterC2×#100D173
hypocenterC173#3019
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#19149724C2semidirect
2. #26012291C2semidirect
3. #26012058C2semidirect
4. #26011671C2semidirect
5. #26011661C2semidirect
6. #26011507C2semidirect
7. C2×#100×C173C2semidirect
12. C2×#14583826C22semidirect
17. #19149712C22semidirect
19. #19149707C22semidirect
21. #19149675C22semidirect
22. #19149666C22semidirect
25. #19149638C22semidirect
26. #19149605C22semidirect
28. C2×M16×C173C22semidirect
29. #100×C173C22semidirect
35. #14583826D4semidirect
39. C2×Dic346D4semidirect
40. C2×Dic346C23semidirect
41. C2×D692D4semidirect
42. C2×D692C23semidirect
61. C4×Dic173C23semidirect
69. M16×C173C23semidirect
131. C173#3019semidirect
139. C2×#100D173semidirect
154. C2×M16C2×D173semidirect
161. #100C2×D173semidirect
197. M16C22×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497242
#260122912
#260120582
#260116712
#260116612
#260115072
C2×#100×C1732
#3019173