Finite Group Databasebeta

← #36363936

Subgroups — #36363936

Order 22144.

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

normal subgroups only

This group has 484 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#26012047C211yesmaximal, characteristic, complemented
2#127×D173C211yesmaximal, characteristic
3#26011857C211yesmaximal, characteristic, complemented
4#26011856C211yesmaximal, characteristic, complemented
5#26011855C211yesmaximal, characteristic, complemented
6#26011836C214yesmaximal, complemented
7#26011494C214yesmaximal, complemented
8#26011443C211yesmaximal, characteristic, complemented
9#490×C173C211yesmaximal, characteristic, complemented
10C2×D4×D173—22no—
11C2×D4×D173C2212yes—
12C2×#14583809C2212yes—
13#19149678C2212yescomplemented
14#19149676C2211yescharacteristic
15#19149675—22no—
16#19149675C2212yes—
17#19149671C2211yescharacteristic, complemented
18#19149657—22no—
19#19149644—22no—
20#19149637C2212yes—
21C42×D173C2211yescharacteristic
22#19149601C2214yescomplemented
23#19149600C2214yescomplemented
24#19149599C2214yescomplemented
25#19149590C2211yescharacteristic, complemented
26C8×Dic173C2212yescomplemented
27#127×C173C2211yescharacteristic
28#125×C173C2211yescharacteristic, complemented
29#102×C173C2214yescomplemented
30#97×C173C2211yescharacteristic, complemented
31C23×D173—22no—
32C2×Dic346—22no—
33C2×D692—22no—
34C2×D692C2312yes—
35C2×C22⋊D173—22no—
36C2×C22⋊D173—24no—
37D4×D173—28no—
38D4×D173—48no—
39C2×C4×D173D412yes—
40C2×C4×D173C2311yescharacteristic
41#14583809—24no—
42C2×#13063436C2312yes—
43C4⋊Dic173C2312yes—
44#14583805—22no—
45#14583805—22no—
46#14583803—22no—
47C4×Dic173—22no—
48C4×Dic173D412yes—
49C4×Dic173C2311yescharacteristic
50C2×D4×C173—22no—
51C2×D4×C173C2312yes—
52C2×C8×C173C2312yes—
53C4⋊C4×C173—22no—
54C4⋊C4×C173C2312yes—
55C42×C173C2311yescharacteristic
56C22×D173—22no—
57C22×D173—48no—
58C22×D173—44no—
59C22×D173C2×D411yescharacteristic
60Dic346—44no—
61D692—24no—
62D692—44no—
63C22⋊D173—44no—
64C22⋊D173—48no—
65C2×Dic173—22no—
66C2×Dic173C2×D411yescharacteristic
67C2×Dic173C2×D412yes—
68C4×D173—24no—
69C4×D173#4114yes—
70#13063436—22no—
71C23×C173—22no—
72D4×C173—24no—
73D4×C173—44no—
74C2×C4×C173—22no—
75C2×C4×C173C2×D412yes—
76C2×C4×C173C2411yescharacteristic
77C8×C173—22no—
78C22×C173—42no—
79C22×C173—44no—
80C22×C173C22×D411yescharacteristic
81C2×D173—22no—
82C2×D173—42no—
83C2×D173—44no—
84C2×D173—88no—
85C2×D173#12412yes—
86C4×C173—22no—
87C4×C173—42no—
88C4×C173C2×#4112yes—
89Dic173—42no—
90Dic173#12412yes—
91Dic173AGL(1,ℤ/8)14yes—
92C2×C173—82no—
93C2×C173C2×#12411yescharacteristic
94C2×C173C2×AGL(1,ℤ/8)12yes—
95D173—22no—
96D173—82no—
97C173#313811yescharacteristic, complemented, Sylow
98#3138—1731nomaximal, Sylow
99#533—1731no—
100C2×#127—1731no—
101#490—1731no—
102#490—1731no—
103#490—1731no—
104#490D17311yescharacteristic, complemented
105#486—1734no—
106#418—1734no—
107C4×M16—1731no—
108C22×D4—3462no—
109C22×D4—1732no—
110C2×M16—1732no—
111C4⋊Q8—1732no—
112#127—3462no—
113#127—1732no—
114#127—1731no—
115#127C2×D17311yescharacteristic
116#125—1731no—
117#125C2×D17311yescharacteristic, complemented
118C22⋊Q8—3462no—
119C4×Q8—3462no—
120#117—1732no—
121C2×C42—1731no—
122#102—1734no—
123#102—1734no—
124#102—1734no—
125#102C2×D17314yescomplemented
126#97—1731no—
127#97C2×D17311yescharacteristic, complemented
128C4×C8—1732no—
129C24—3462no—
130C2×Q8—3462no—
131C2×D4—22no—
132C2×D4—3468no—
133C2×D4—3462no—
134C2×D4—3464no—
135C2×D4—3462no—
136C2×D4—1732no—
137C2×D4—6928no—
138C2×D4C22×D17312yes—
139C22×C4—1732no—
140C22×C4—1731no—
141M16—3464no—
142C2×C8—1732no—
143C2×C8C22×D17312yes—
144C4⋊C4—22no—
145C4⋊C4—3462no—
146C4⋊C4—3462no—
147C4⋊C4—1732no—
148C4⋊C4C22×D17312yes—
149C22⋊C4—3462no—
150C42—3462no—
151C42—1732no—
152C42—1731no—
153C42C22×D17311yescharacteristic
154C23—22no—
155C23—3462no—
156C23—1731no—
157C23—6928no—
158C23—6924no—
159Q8—6924no—
160D4—24no—
161D4—44no—
162D4—3464no—
163D4—6924no—
164D4—6924no—
165D4—6928no—
166C2×C4—22no—
167C2×C4—3464no—
168C2×C4—3462no—
169C2×C4—1731no—
170C2×C4—1732no—
171C2×C4—1734no—
172C2×C4D4×D17312yes—
173C2×C4C23×D17311yescharacteristic
174C8—22no—
175C8—3462no—
176C22—42no—
177C22—44no—
178C22—3462no—
179C22—13848no—
180C22—1732no—
181C22—6922no—
182C22—6924no—
183C22C2×D4×D17311yescharacteristic
184C4—22no—
185C4—42no—
186C4—1732no—
187C4—1734no—
188C4—6922no—
189C4#41×D17312yes—
190C2—82no—
191C2—3462no—
192C2—13842no—
193C2#124×D17311yescharacteristic
194C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#490×C173C2
socleC22×C173C22×D4
prime radical#127×D173C2
hypercenter#490D173
hypocenterC173#3138
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #26012047C2semidirect
3. #26011857C2semidirect
4. #26011856C2semidirect
5. #26011855C2semidirect
6. #26011836C2semidirect
7. #26011494C2semidirect
8. #26011443C2semidirect
9. #490×C173C2semidirect
13. #19149678C22semidirect
17. #19149671C22semidirect
22. #19149601C22semidirect
23. #19149600C22semidirect
24. #19149599C22semidirect
25. #19149590C22semidirect
26. C8×Dic173C22semidirect
28. #125×C173C22semidirect
29. #102×C173C22semidirect
30. #97×C173C22semidirect
97. C173#3138semidirect
104. #490D173semidirect
117. #125C2×D173semidirect
125. #102C2×D173semidirect
127. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120472
#127×D1732
#260118572
#260118562
#260118552
#260118362
#260114942
#260114432
#490×C1732
#3138173