Finite Group Databasebeta

← #36363801

Subgroups — #36363801

Order 22144.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#26011965C211yesmaximal, characteristic, complemented
2#26011812C212yesmaximal, complemented
3#26011788C212yesmaximal, complemented
4#26011683C212yesmaximal, complemented
5#26011624C212yesmaximal, complemented
6#26011601C212yesmaximal, complemented
7C4×#14583811C212yesmaximal, complemented
8#26011552C211yesmaximal, characteristic, complemented
9#540×C173C211yesmaximal, characteristic, complemented
10C22×D692C2211yescharacteristic
11C2×#14583811C2212yes—
12C2×#14583811C2212yes—
13C4×D692C2212yes—
14C4×D692C2212yes—
15#19149641—22no—
16#19149638C2211yescharacteristic
17C22⋊C4×D173—22no—
18#19149612C2212yescomplemented
19#19149610C2212yescomplemented
20C4⋊#13063436C2212yescomplemented
21#19149600C2212yescomplemented
22#19149600C2212yescomplemented
23#19149599C2212yes—
24#19149599C2212yes—
25C22⋊#13063436C2212yes—
26#19149590C2211yescharacteristic, complemented
27C4×#13063436C2211yescharacteristic, complemented
28#127×C173C2211yescharacteristic, complemented
29#125×C173C2211yescharacteristic, complemented
30#123×C173—22no—
31#121×C173—22no—
32#121×C173C2212yes—
33C4×D4×C173C2212yescomplemented
34#117×C173C2211yescharacteristic
35C23×D173—21no—
36C2×D692—24no—
37C2×D692—21no—
38C2×D692C2312yes—
39C2×C4×D173—22no—
40C2×C4×D173—22no—
41#14583811—24no—
42#14583811—24no—
43C2×#13063436C2312yes—
44C2×#13063436C2312yes—
45C4⋊Dic173C2312yes—
46C22⋊Dic173—21no—
47#14583803—22no—
48#14583803—22no—
49C4×Dic173—21no—
50C2×D4×C173—22no—
51C2×D4×C173—22no—
52C2×D4×C173C2312yes—
53C22×C4×C173—22no—
54C22×C4×C173C2311yescharacteristic
55C4⋊C4×C173—22no—
56C4⋊C4×C173D412yes—
57C4⋊C4×C173C2312yes—
58C22⋊C4×C173—22no—
59C22⋊C4×C173—22no—
60C22⋊C4×C173D412yes—
61C42×C173C2311yescharacteristic
62C42×C173C2311yescharacteristic
63C22×D173—22no—
64C22×D173—24no—
65C22×D173—82no—
66D692—24no—
67D692—22no—
68D692#4114yes—
69C2×Dic173—22no—
70C4×D173—82no—
71C4×D173—82no—
72#13063436—22no—
73#13063436—22no—
74C23×C173—22no—
75C23×C173C2×D411yescharacteristic
76D4×C173—22no—
77D4×C173—22no—
78D4×C173—82no—
79D4×C173—82no—
80C2×C4×C173—22no—
81C2×C4×C173—22no—
82C2×C4×C173—82no—
83C2×C4×C173C2×D412yes—
84C2×C4×C173C2×D412yes—
85C2×C4×C173C2×D411yescharacteristic
86C2×C4×C173C2411yescharacteristic
87C22×C173—21no—
88C22×C173—22no—
89C22×C173—42no—
90C22×C173—42no—
91C22×C173—42no—
92C22×C173C22×D411yescharacteristic
93C2×D173—24no—
94C2×D173—42no—
95C2×D173—42no—
96C2×D173—84no—
97C4×C173—21no—
98C4×C173—22no—
99C4×C173—22no—
100C4×C173—42no—
101C4×C173C2×#4112yes—
102Dic173—42no—
103C2×C173—41no—
104C2×C173—82no—
105C2×C173#52511yescharacteristic
106C2×C173#57612yes—
107D173—44no—
108C173#312211yescharacteristic, complemented, Sylow
109#3122—1731nomaximal, Sylow
110#540D17311yescharacteristic, complemented
111#518—1731no—
112#482—1732no—
113#478—1732no—
114#459—1732no—
115#447—1732no—
116#442—1732no—
117C4×D8—1732no—
118#433—1731no—
119C22×D4—1731no—
120C2×D8—1732no—
121C2×D8—1732no—
122#127C2×D17311yescharacteristic, complemented
123#125C2×D17311yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D17312yes—
127C4×D4—3462no—
128C4×D4—1732no—
129C4×D4—1732no—
130C4×D4C2×D17312yescomplemented
131#117—1731no—
132#117C2×D17311yescharacteristic
133C2×C22⋊C4—3462no—
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—3461no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—3464no—
148C2×D4—3461no—
149C2×D4—1732no—
150C2×D4C22×D17312yes—
151C22×C4—22no—
152C22×C4—3462no—
153C22×C4—3462no—
154C22×C4C22×D17311yescharacteristic
155D8—3464no—
156D8—3464no—
157C2×C8—1732no—
158C2×C8—1732no—
159C4⋊C4—22no—
160C4⋊C4—1732no—
161C4⋊C4C22⋊D17312yes—
162C4⋊C4C22×D17312yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—3461no—
166C22⋊C4—3462no—
167C22⋊C4—3462no—
168C22⋊C4C22⋊D17312yes—
169C42—3461no—
170C42C22×D17311yescharacteristic
171C42C22×D17311yescharacteristic
172C23—22no—
173C23—3464no—
174C23—3462no—
175C23—13842no—
176C23C2×C22⋊D17311yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—3464no—
182D4—3462no—
183D4—1734no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—3462no—
188C2×C4—13842no—
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—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—3464no—
202C22—13844no—
203C22—6922no—
204C22—6922no—
205C22C22×C22⋊D17311yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—6922no—
211C4#41×D17312yes—
212C2—41no—
213C2—82no—
214C2—6924no—
215C2#2601199811yescharacteristic
216C2#2601229412yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#540×C173C2
socleC22×C173C22×D4
hypercenter#540D173
hypocenterC173#3122
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #26011965C2semidirect
2. #26011812C2semidirect
3. #26011788C2semidirect
4. #26011683C2semidirect
5. #26011624C2semidirect
6. #26011601C2semidirect
7. C4×#14583811C2semidirect
8. #26011552C2semidirect
9. #540×C173C2semidirect
18. #19149612C22semidirect
19. #19149610C22semidirect
20. C4⋊#13063436C22semidirect
21. #19149600C22semidirect
22. #19149600C22semidirect
26. #19149590C22semidirect
27. C4×#13063436C22semidirect
28. #127×C173C22semidirect
29. #125×C173C22semidirect
33. C4×D4×C173C22semidirect
108. C173#3122semidirect
110. #540D173semidirect
122. #127C2×D173semidirect
123. #125C2×D173semidirect
130. C4×D4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260119652
#260118122
#260117882
#260116832
#260116242
#260116012
C4×#145838112
#260115522
#540×C1732
#3122173