Finite Group Databasebeta

← #488×D173 #36363876

Subgroups — #488×D173 #36363876

Order 22144.

412 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 125 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#125×D173C211yesmaximal, characteristic, complemented
2#124×D173C211yesmaximal, characteristic
3#26011848C211yesmaximal, characteristic, complemented
4#26011847C212yesmaximal, complemented
5#26011846C211yesmaximal, characteristic, complemented
6#26011845C211yesmaximal, characteristic, complemented
7#26011844C212yesmaximal, complemented
8#103×D173C212yesmaximal, complemented
9#102×D173C212yesmaximal
10C4×C8×D173C211yesmaximal, characteristic, complemented
11#488×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic
13C2×D4×D173C2211yescharacteristic
14C2×C8×D173C2212yes—
15#19149671C2211yescharacteristic, complemented
16#19149670C2212yescomplemented
17#19149669—22no—
18#19149668C2211yescharacteristic
19#19149667C2211yescharacteristic
20#19149666C2211yescharacteristic
21#19149665—22no—
22C4⋊C4×D173—22no—
23C4⋊C4×D173C2212yes—
24C22⋊C4×D173—22no—
25C42×D173C2211yescharacteristic
26#19149604C2212yescomplemented
27#19149603C2212yescomplemented
28#19149602C2212yescomplemented
29#19149601C2212yes—
30#19149600C2212yes—
31#19149599C2212yes—
32C4×#13063436C2211yescharacteristic, complemented
33C8×Dic173C2212yescomplemented
34#125×C173C2211yescharacteristic, complemented
35#124×C173C2211yescharacteristic
36#103×C173C2212yescomplemented
37#102×C173C2212yes—
38C4×C8×C173C2211yescharacteristic, complemented
39C23×D173—21no—
40C2×Dic346—21no—
41C2×Dic346C2311yescharacteristic
42Q8×D173—24no—
43C2×D692C2311yescharacteristic
44C2×C22⋊D173—21no—
45D4×D173—24no—
46C2×C4×D173—21no—
47C2×C4×D173—22no—
48C2×C4×D173D412yes—
49C2×C4×D173C2311yescharacteristic
50C2×#13063436C2312yes—
51C8×D173—24no—
52C4⋊Dic173—22no—
53C4⋊Dic173C2312yes—
54#14583805—22no—
55#14583805—22no—
56#14583805—22no—
57C22⋊Dic173—22no—
58#14583803—22no—
59#14583803—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—42no—
71C22×D173—44no—
72C22×D173C2×D411yescharacteristic
73Dic346—22no—
74Dic346—42no—
75D692—22no—
76C22⋊D173—42no—
77C2×Dic173—21no—
78C2×Dic173—22no—
79C2×Dic173C2×D412yes—
80C2×Dic173C2×D411yescharacteristic
81C4×D173—24no—
82C4×D173—42no—
83C4×D173—44no—
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—41no—
98C2×D173—42no—
99C2×D173—84no—
100C2×D173#12412yes—
101C2×D173#13514yes—
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×C173C2×#13512yes—
113D173—81no—
114D173#48814yesdirect product factor
115C173C2×#48811yescharacteristic, complemented, Sylow
116C2×#488—1731nomaximal, Sylow
117C2×#125—1731no—
118C2×#124—1731no—
119#488—1731no—
120#488—1731no—
121#488—1732no—
122#488—1732no—
123#488—1731no—
124#488D17311yescharacteristic, direct product factor
125C2×#103—1732no—
126C2×#102—1732no—
127C2×C4×C8—1731no—
128C22×Q8—1731no—
129C22×D4—1731no—
130C22×C8—1732no—
131#125—3462no—
132#125—1732no—
133#125—1731no—
134#125C2×D17311yescharacteristic, complemented
135#124—3462no—
136#124—1731no—
137#124—1731no—
138#124—1731no—
139#124C2×D17311yescharacteristic
140C2×C4⋊C4—3462no—
141C2×C4⋊C4—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—
152C4×C8—1732no—
153C4×C8—1731no—
154C4×C8C2×D17311yescharacteristic, complemented
155C24—3461no—
156C2×Q8—3461no—
157C2×Q8—3464no—
158C2×Q8—1731no—
159C2×Q8C22×D17311yescharacteristic
160C2×D4—3464no—
161C2×D4—3461no—
162C2×D4—1731no—
163C2×D4C22×D17311yescharacteristic
164C22×C4—3461no—
165C22×C4—3462no—
166C22×C4—1732no—
167C22×C4—1731no—
168C2×C8—3464no—
169C2×C8—1732no—
170C2×C8C22×D17312yes—
171C4⋊C4—22no—
172C4⋊C4—3462no—
173C4⋊C4—3462no—
174C4⋊C4—3462no—
175C4⋊C4—3462no—
176C4⋊C4—1732no—
177C4⋊C4C22×D17312yes—
178C22⋊C4—22no—
179C22⋊C4—3462no—
180C22⋊C4—3462no—
181C22⋊C4—3462no—
182C42—1732no—
183C42—1731no—
184C42C22×D17311yescharacteristic
185C23—21no—
186C23—3461no—
187C23—1731no—
188C23—6922no—
189C23—6924no—
190Q8—22no—
191Q8—3462no—
192Q8—6922no—
193D4—22no—
194D4—3462no—
195D4—6922no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—3464no—
199C2×C4—3461no—
200C2×C4—3462no—
201C2×C4—1732no—
202C2×C4—1731no—
203C2×C4—1734no—
204C2×C4—6924no—
205C2×C4—6922no—
206C2×C4D4×D17312yes—
207C2×C4C23×D17311yescharacteristic
208C8—22no—
209C8—3462no—
210C22—41no—
211C22—42no—
212C22—13844no—
213C22—1732no—
214C22—1734no—
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—13841no—
228C2—1734no—
229C2#124×D17311yescharacteristic
230C2#135×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#488×C173C2
socleC22×C173C22×D4
prime radicalC2×D4×D173C22
hypercenter#488D173
hypocenterC173C2×#488
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. #125×D173C2semidirect
3. #26011848C2semidirect
4. #26011847C2semidirect
5. #26011846C2semidirect
6. #26011845C2semidirect
7. #26011844C2semidirect
8. #103×D173C2semidirect
10. C4×C8×D173C2semidirect
11. #488×C173C2semidirect
15. #19149671C22semidirect
16. #19149670C22semidirect
26. #19149604C22semidirect
27. #19149603C22semidirect
28. #19149602C22semidirect
32. C4×#13063436C22semidirect
33. C8×Dic173C22semidirect
34. #125×C173C22semidirect
36. #103×C173C22semidirect
38. C4×C8×C173C22semidirect
114. D173#488direct (N × H)
115. C173C2×#488semidirect
124. #488D173direct (N × H)
134. #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
#125×D1732
#124×D1732
#260118482
#260118472
#260118462
#260118452
#260118442
#103×D1732
#102×D1732
C4×C8×D1732
#488×C1732
C2×#488173