Finite Group Databasebeta

← #36363849

Subgroups — #36363849

Order 22144.

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

normal subgroups only

This group has 382 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#26011968C211yesmaximal, characteristic, complemented
2#26011812C212yesmaximal, complemented
3#26011795C212yesmaximal, complemented
4#26011688C212yesmaximal, complemented
5#26011629C212yesmaximal, complemented
6#26011591C212yesmaximal, complemented
7C4×#14583814C212yesmaximal, complemented
8#26011552C211yesmaximal, characteristic, complemented
9#540×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11C2×#14583814C2212yes—
12C2×#14583814C2212yes—
13C4×Dic346C2212yes—
14C4×Dic346C2212yes—
15#19149644—22no—
16#19149638C2211yescharacteristic
17#19149635—22no—
18#19149612C2212yes—
19#19149610C2212yes—
20C4⋊#13063436C2212yes—
21#19149602C2212yes—
22#19149602C2212yes—
23#19149600C2212yes—
24#19149600C2212yes—
25C22⋊#13063436C2212yes—
26#19149590C2211yescharacteristic
27C4×#13063436C2211yescharacteristic
28#127×C173C2211yescharacteristic
29#125×C173C2211yescharacteristic
30#123×C173—22no—
31#121×C173—22no—
32#121×C173C2212yes—
33C4×D4×C173C2212yes—
34#117×C173C2211yescharacteristic
35C2×Dic346—24no—
36C2×Dic346—21no—
37C2×Dic346C2312yes—
38C22×Dic173—21no—
39#14583814—24no—
40#14583814—24no—
41C2×#13063436C2312yes—
42C2×#13063436C2312yes—
43C4⋊Dic173C2312yes—
44#14583805—22no—
45#14583805—22no—
46C22⋊Dic173—21no—
47C4×Dic173—22no—
48C4×Dic173—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
63Dic346—24no—
64Dic346—22no—
65Dic346#4114yes—
66C2×Dic173—24no—
67C2×Dic173—22no—
68C2×Dic173—22no—
69#13063436—22no—
70#13063436—22no—
71C23×C173—22no—
72C23×C173C2×D411yescharacteristic
73D4×C173—22no—
74D4×C173—22no—
75D4×C173—82no—
76D4×C173—82no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—82no—
80C2×C4×C173C2×D411yescharacteristic
81C2×C4×C173C2×D412yes—
82C2×C4×C173C2×D412yes—
83C2×C4×C173C2411yescharacteristic
84C22×C173—21no—
85C22×C173—22no—
86C22×C173—42no—
87C22×C173—42no—
88C22×C173—42no—
89C22×C173C22×D411yescharacteristic
90C4×C173—21no—
91C4×C173—22no—
92C4×C173—22no—
93C4×C173—42no—
94C4×C173C2×#4112yes—
95Dic173—24no—
96Dic173—42no—
97C2×C173—41no—
98C2×C173—82no—
99C2×C173#52511yescharacteristic
100C2×C173#57712yes—
101C173#312611yescharacteristic, complemented, Sylow
102#3126—1731nomaximal, Sylow
103#540D17311yescharacteristic
104#519—1731no—
105#482—1732no—
106#479—1732no—
107#460—1732no—
108#448—1732no—
109#440—1732no—
110C4×SD16—1732no—
111#433—1731no—
112C22×Q8—1731no—
113C2×SD16—1732no—
114C2×SD16—1732no—
115#127C2×D17311yescharacteristic
116#125C2×D17311yescharacteristic
117#123—22no—
118#121—22no—
119#121C2×D17312yes—
120C4×Q8—3462no—
121C4×Q8—1732no—
122C4×Q8—1732no—
123C4×D4C2×D17312yes—
124#117—3462no—
125#117—1731no—
126#117C2×D17311yescharacteristic
127#107—1732no—
128#106—1732no—
129C4⋊C8—1732no—
130#103—1732no—
131#103—1732no—
132#102—1732no—
133#102—1732no—
134C22⋊C8—1732no—
135#97—1731no—
136C4×C8—1731no—
137C2×Q8—3464no—
138C2×Q8—3461no—
139C2×Q8—1732no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22×D17312yes—
143C22×C4—22no—
144C22×C4—3461no—
145C22×C4C22×D17311yescharacteristic
146SD16—3464no—
147SD16—3464no—
148C2×C8—1732no—
149C2×C8—1732no—
150C4⋊C4—22no—
151C4⋊C4—3462no—
152C4⋊C4—3462no—
153C4⋊C4—1732no—
154C4⋊C4C22⋊D17312yes—
155C4⋊C4C22×D17312yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—3461no—
159C22⋊C4C22⋊D17312yes—
160C42—3462no—
161C42—3462no—
162C42—3461no—
163C42C22×D17311yescharacteristic
164C42C22×D17311yescharacteristic
165C23—22no—
166C23C2×C22⋊D17311yescharacteristic
167Q8—3464no—
168Q8—3462no—
169Q8—1734no—
170D4—22no—
171D4—22no—
172D4—82no—
173D4—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—82no—
177C2×C4—3464no—
178C2×C4—3462no—
179C2×C4—3462no—
180C2×C4C2×C22⋊D17312yes—
181C2×C4C2×C22⋊D17312yes—
182C2×C4C2×C22⋊D17311yescharacteristic
183C2×C4C23×D17311yescharacteristic
184C8—3462no—
185C8—3462no—
186C22—21no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C22×C22⋊D17311yescharacteristic
192C4—21no—
193C4—22no—
194C4—22no—
195C4—42no—
196C4—3464no—
197C4—6922no—
198C4#41×D17312yes—
199C2—41no—
200C2—82no—
201C2#2601199811yescharacteristic
202C2#2601230212yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#540×C173C2
socleC22×C173C22×D4
hypercenter#540D173
hypocenterC173#3126
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. #26011968C2semidirect
2. #26011812C2semidirect
3. #26011795C2semidirect
4. #26011688C2semidirect
5. #26011629C2semidirect
6. #26011591C2semidirect
7. C4×#14583814C2semidirect
8. #26011552C2semidirect
101. C173#3126semidirect

Maximal subgroups

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

subgroupindex
#260119682
#260118122
#260117952
#260116882
#260116292
#260115912
C4×#145838142
#260115522
#540×C1732
#3126173