Finite Group Databasebeta

← #36363946

Subgroups — #36363946

Order 22144.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#26012047C211yesmaximal, characteristic, complemented
2#124×D173C211yesmaximal, characteristic
3#26011852C212yesmaximal, complemented
4#26011849C212yesmaximal, complemented
5#26011848C211yesmaximal, characteristic, complemented
6#26011846C211yesmaximal, characteristic, complemented
7#26011845C211yesmaximal, characteristic, complemented
8#26011498C212yesmaximal, complemented
9#26011494C212yesmaximal
10C4×#14583809C211yesmaximal, characteristic, complemented
11#488×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14C2×#14583809C2212yes—
15#19149678C2212yescomplemented
16#19149671C2211yescharacteristic, complemented
17#19149668C2211yescharacteristic
18#19149667C2211yescharacteristic
19#19149666C2211yescharacteristic
20#19149665—22no—
21#19149657—22no—
22#19149644—22no—
23#19149637C2212yes—
24C22⋊C4×D173—22no—
25C42×D173C2211yescharacteristic
26#19149604C2212yescomplemented
27#19149603C2212yescomplemented
28#19149602C2212yescomplemented
29#19149601C2212yes—
30#19149600C2212yes—
31#19149599C2212yes—
32#19149589C2212yescomplemented
33C4×#13063436C2211yescharacteristic, complemented
34#125×C173C2211yescharacteristic, complemented
35#124×C173C2211yescharacteristic
36#103×C173C2212yescomplemented
37#102×C173C2212yes—
38C4×C8×C173C2211yescharacteristic, complemented
39C23×D173—21no—
40C2×Dic346—22no—
41C2×Dic346—21no—
42C2×Dic346C2311yescharacteristic
43Q8×D173—24no—
44C2×D692C2311yescharacteristic
45C2×C22⋊D173—21no—
46D4×D173—24no—
47C2×C4×D173—21no—
48C2×C4×D173D412yes—
49C2×C4×D173C2311yescharacteristic
50#14583809—24no—
51C2×#13063436C2312yes—
52C4⋊Dic173C2312yes—
53#14583805—22no—
54#14583805—22no—
55C22⋊Dic173—22no—
56#14583803—22no—
57#14583803—22no—
58#14583803—22no—
59C4×Dic173—22no—
60C4×Dic173D412yes—
61C4×Dic173C2311yescharacteristic
62C2×Q8×C173C2311yescharacteristic
63C2×D4×C173C2311yescharacteristic
64C2×C8×C173C2312yes—
65C4⋊C4×C173—22no—
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C42×C173C2311yescharacteristic
69C22×D173—21no—
70C22×D173—44no—
71C22×D173—42no—
72C22×D173C2×D411yescharacteristic
73Dic346—22no—
74Dic346—44no—
75Dic346—42no—
76D692—22no—
77C22⋊D173—42no—
78C2×Dic173—21no—
79C2×Dic173—22no—
80C2×Dic173C2×D412yes—
81C2×Dic173C2×D411yescharacteristic
82C4×D173—42no—
83C4×D173—42no—
84C4×D173#4114yes—
85#13063436—22no—
86C23×C173—21no—
87Q8×C173—22no—
88D4×C173—22no—
89C2×C4×C173—21no—
90C2×C4×C173—22no—
91C2×C4×C173C2×D412yes—
92C2×C4×C173C2411yescharacteristic
93C8×C173—22no—
94C22×C173—41no—
95C22×C173—42no—
96C22×C173C22×D411yescharacteristic
97C2×D173—22no—
98C2×D173—41no—
99C2×D173—42no—
100C2×D173—84no—
101C2×D173#12412yes—
102C4×C173—22no—
103C4×C173—41no—
104C4×C173—42no—
105C4×C173C2×#4112yes—
106Dic173—22no—
107Dic173—41no—
108Dic173—42no—
109Dic173#12412yes—
110C2×C173—81no—
111C2×C173C2×#12411yescharacteristic
112C2×C173#57512yes—
113D173—22no—
114D173—81no—
115C173#314011yescharacteristic, complemented, Sylow
116#3140—1731nomaximal, Sylow
117#533—1731no—
118C2×#124—1731no—
119#489—1732no—
120#489—1732no—
121#488—1731no—
122#488—1731no—
123#488—1731no—
124#488D17311yescharacteristic, complemented
125#419—1732no—
126#418—1732no—
127C4×M16—1731no—
128C22×Q8—1731no—
129C22×D4—1731no—
130C2×M16—1732no—
131C4⋊Q8—1732no—
132#125—1731no—
133#125C2×D17311yescharacteristic, complemented
134#124—3462no—
135#124—1731no—
136#124—1731no—
137#124—1731no—
138#124C2×D17311yescharacteristic
139C22⋊Q8—3462no—
140C4×Q8—3462no—
141#117—1732no—
142C2×C22⋊C4—3462no—
143C2×C42—1731no—
144#103—1732no—
145#103—1732no—
146#103—1732no—
147#103C2×D17312yescomplemented
148#102—1732no—
149#102—1732no—
150#102—1732no—
151#102C2×D17312yes—
152#97—1732no—
153C4×C8—1731no—
154C4×C8C2×D17311yescharacteristic, complemented
155C24—3461no—
156C2×Q8—3462no—
157C2×Q8—3461no—
158C2×Q8—3464no—
159C2×Q8—1731no—
160C2×Q8C22×D17311yescharacteristic
161C2×D4—3464no—
162C2×D4—3461no—
163C2×D4—1731no—
164C2×D4C22×D17311yescharacteristic
165C22×C4—3461no—
166C22×C4—1731no—
167C22×C4—1732no—
168M16—3464no—
169C2×C8—1732no—
170C2×C8C22×D17312yes—
171C4⋊C4—22no—
172C4⋊C4—3462no—
173C4⋊C4—3462no—
174C4⋊C4—1732no—
175C4⋊C4C22×D17312yes—
176C22⋊C4—22no—
177C22⋊C4—3462no—
178C22⋊C4—3462no—
179C22⋊C4—3462no—
180C22⋊C4—3462no—
181C42—3462no—
182C42—1732no—
183C42—1731no—
184C42C22×D17311yescharacteristic
185C23—21no—
186C23—3461no—
187C23—1731no—
188C23—6924no—
189C23—6922no—
190Q8—22no—
191Q8—3462no—
192Q8—6924no—
193Q8—6922no—
194D4—22no—
195D4—3462no—
196D4—6922no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3462no—
200C2×C4—3461no—
201C2×C4—1732no—
202C2×C4—1731no—
203C2×C4—1734no—
204C2×C4—6922no—
205C2×C4—6922no—
206C2×C4D4×D17312yes—
207C2×C4C23×D17311yescharacteristic
208C8—22no—
209C8—3462no—
210C22—41no—
211C22—42no—
212C22—3462no—
213C22—13844no—
214C22—1732no—
215C22—6921no—
216C22—6922no—
217C22C2×D4×D17311yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—3462no—
222C4—1732no—
223C4—6921no—
224C4—6922no—
225C4#41×D17312yes—
226C2—81no—
227C2—3462no—
228C2—13841no—
229C2#124×D17311yescharacteristic
230C2#2601228812yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#488×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#488D173
hypocenterC173#3140
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #26012047C2semidirect
3. #26011852C2semidirect
4. #26011849C2semidirect
5. #26011848C2semidirect
6. #26011846C2semidirect
7. #26011845C2semidirect
8. #26011498C2semidirect
10. C4×#14583809C2semidirect
11. #488×C173C2semidirect
15. #19149678C22semidirect
16. #19149671C22semidirect
26. #19149604C22semidirect
27. #19149603C22semidirect
28. #19149602C22semidirect
32. #19149589C22semidirect
33. C4×#13063436C22semidirect
34. #125×C173C22semidirect
36. #103×C173C22semidirect
38. C4×C8×C173C22semidirect
115. C173#3140semidirect
124. #488D173semidirect
133. #125C2×D173semidirect
147. #103C2×D173semidirect
154. C4×C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120472
#124×D1732
#260118522
#260118492
#260118482
#260118462
#260118452
#260114982
#260114942
C4×#145838092
#488×C1732
#3140173