Finite Group Databasebeta

← #36363806

Subgroups — #36363806

Order 22144.

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

normal subgroups only

This group has 418 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#26012096C211yesmaximal, characteristic, complemented
2#26011815C212yesmaximal, complemented
3#26011791C212yesmaximal, complemented
4#26011686C212yesmaximal, complemented
5#26011625C212yesmaximal, complemented
6#26011601C212yesmaximal, complemented
7C4×#14583811C212yesmaximal, complemented
8#26011552C211yesmaximal, characteristic, complemented
9#518×C173C211yesmaximal, characteristic, complemented
10C2×#14583811C2212yes—
11C2×#14583811C2212yes—
12#19149676C2211yescharacteristic, complemented
13#19149671C2211yescharacteristic, complemented
14#19149664—22no—
15#19149653C2212yes—
16#19149652—22no—
17C4×D692C2212yescomplemented
18#19149612C2212yescomplemented
19#19149610C2212yescomplemented
20C4⋊#13063436C2212yescomplemented
21#19149600C2212yes—
22#19149600C2212yes—
23#19149599C2212yescomplemented
24#19149599C2212yescomplemented
25C22⋊#13063436C2212yes—
26#19149590C2211yescharacteristic, complemented
27C4×#13063436C2211yescharacteristic, complemented
28C22×D4×C173C2211yescharacteristic
29C4×D4×C173—22no—
30C4×D4×C173C2212yes—
31C4×D4×C173C2212yes—
32#117×C173C2211yescharacteristic
33#117×C173C2211yescharacteristic
34C2×C22⋊C4×C173—22no—
35C2×D692—22no—
36C2×D692C2312yes—
37C2×C22⋊D173—22no—
38C2×C4×D173—22no—
39#14583811—24no—
40#14583811—24no—
41C2×#13063436C2312yes—
42C2×#13063436C2312yes—
43C4⋊Dic173C2312yes—
44#14583805—22no—
45#14583803—22no—
46#14583803—22no—
47C24×C173—21no—
48C2×D4×C173—24no—
49C2×D4×C173—21no—
50C2×D4×C173C2312yes—
51C22×C4×C173—22no—
52C22×C4×C173—22no—
53C22×C4×C173C2311yescharacteristic
54C4⋊C4×C173D412yes—
55C4⋊C4×C173C2312yes—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—22no—
58C22⋊C4×C173—22no—
59C22⋊C4×C173D412yes—
60C42×C173—21no—
61C42×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63C22×D173—22no—
64D692—22no—
65D692—22no—
66D692—82no—
67C22⋊D173—82no—
68C2×Dic173—22no—
69C4×D173—82no—
70#13063436—22no—
71#13063436—22no—
72C23×C173—24no—
73C23×C173—22no—
74C23×C173—82no—
75C23×C173C2×D411yescharacteristic
76D4×C173—24no—
77D4×C173—22no—
78D4×C173#4114yes—
79C2×C4×C173—22no—
80C2×C4×C173—22no—
81C2×C4×C173—82no—
82C2×C4×C173—82no—
83C2×C4×C173C2×D412yes—
84C2×C4×C173C2×D412yes—
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2411yescharacteristic
87C22×C173—21no—
88C22×C173—24no—
89C22×C173—22no—
90C22×C173—42no—
91C22×C173—42no—
92C22×C173—84no—
93C22×C173C22×D411yescharacteristic
94C2×D173—42no—
95C2×D173—42no—
96C2×D173—42no—
97C4×C173—21no—
98C4×C173—22no—
99C4×C173—22no—
100C4×C173—42no—
101C4×C173C2×#4112yes—
102Dic173—42no—
103C2×C173—41no—
104C2×C173—44no—
105C2×C173#52511yescharacteristic
106C2×C173#57612yes—
107D173—82no—
108C173#312211yescharacteristic, complemented, Sylow
109#3122—1731nomaximal, Sylow
110#540—1731no—
111#518D17311yescharacteristic, complemented
112#482—1732no—
113#478—1732no—
114#459—1732no—
115#447—1732no—
116#442—1732no—
117C4×D8—1732no—
118#433—1731no—
119C22×D4C2×D17311yescharacteristic
120C2×D8—1732no—
121C2×D8—1732no—
122#127—1731no—
123#125—1731no—
124#123—3462no—
125#121—3462no—
126#121—1732no—
127C4×D4—22no—
128C4×D4—1732no—
129C4×D4C2×D17312yes—
130C4×D4C2×D17312yes—
131#117C2×D17311yescharacteristic
132#117C2×D17311yescharacteristic
133C2×C22⋊C4—22no—
134#107—1732no—
135#106—1732no—
136C4⋊C8—1732no—
137#102—1732no—
138#102—1732no—
139#102—1732no—
140#102—1732no—
141C22⋊C8—1732no—
142#97—1731no—
143C4×C8—1731no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3462no—
148C2×D4—3462no—
149C2×D4—1732no—
150C2×D4C22×D17312yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—3462no—
154C22×C4C22×D17311yescharacteristic
155D8—3464no—
156D8—3464no—
157C2×C8—1732no—
158C2×C8—1732no—
159C4⋊C4—3462no—
160C4⋊C4—1732no—
161C4⋊C4C22⋊D17312yes—
162C4⋊C4C22×D17312yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—3462no—
167C22⋊C4—3462no—
168C22⋊C4C22⋊D17312yes—
169C42—21no—
170C42C22×D17311yescharacteristic
171C42C22×D17311yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—3462no—
176C23C2×C22⋊D17311yescharacteristic
177D4—24no—
178D4—22no—
179D4—3462no—
180D4—3462no—
181D4—13842no—
182D4—13842no—
183D4#1458382614yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—3462no—
189C2×C4—13842no—
190C2×C4C2×C22⋊D17311yescharacteristic
191C2×C4C2×C22⋊D17312yes—
192C2×C4C2×C22⋊D17312yes—
193C2×C4C23×D17311yescharacteristic
194C8—3462no—
195C8—3462no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—6922no—
203C22—6922no—
204C22—6922no—
205C22C22×C22⋊D17311yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—6922no—
211C4C2×#1458382612yes—
212C2—41no—
213C2—44no—
214C2—13842no—
215C2#2601200211yescharacteristic
216C2#2601229412yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#518×C173C2
socleC22×C173C22×D4
hypercenter#518D173
hypocenterC173#3122
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #26012096C2semidirect
2. #26011815C2semidirect
3. #26011791C2semidirect
4. #26011686C2semidirect
5. #26011625C2semidirect
6. #26011601C2semidirect
7. C4×#14583811C2semidirect
8. #26011552C2semidirect
9. #518×C173C2semidirect
12. #19149676C22semidirect
13. #19149671C22semidirect
17. C4×D692C22semidirect
18. #19149612C22semidirect
19. #19149610C22semidirect
20. C4⋊#13063436C22semidirect
23. #19149599C22semidirect
24. #19149599C22semidirect
26. #19149590C22semidirect
27. C4×#13063436C22semidirect
108. C173#3122semidirect
111. #518D173semidirect

Maximal subgroups

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

subgroupindex
#260120962
#260118152
#260117912
#260116862
#260116252
#260116012
C4×#145838112
#260115522
#518×C1732
#3122173