Finite Group Databasebeta

← #36361836

Subgroups — #36361836

Order 22144.

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

normal subgroups only

This group has 356 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#26011432C212yesmaximal
2#26011431C212yesmaximal
3#26011430C211yesmaximal, characteristic
4#26011412C212yesmaximal
5#26011412C212yesmaximal
6#26011411C212yesmaximal
7#26011356C211yesmaximal, characteristic
8#26011355C212yesmaximal
9#400×C173C211yesmaximal, characteristic
10C22×Dic346C2212yes—
11C2×C4⋊Dic173C2211yescharacteristic
12C2×C4⋊Dic173C2212yes—
13C2×C4⋊Dic173C2211yescharacteristic
14C2×#14583805C2212yes—
15C2×#14583805C2212yes—
16C2×#14583805C2212yes—
17C2×C4×Dic173C2211yescharacteristic
18#19149586C2212yes—
19#19149586C2212yes—
20#19149586C2212yes—
21#19149586C2212yes—
22#19149586C2212yes—
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2211yescharacteristic
26C2×C4⋊C4×C173C2211yescharacteristic
27C2×C4⋊C4×C173C2211yescharacteristic
28#95×C173C2212yes—
29#95×C173C2212yes—
30#95×C173C2211yescharacteristic
31C2×Dic346—42no—
32C2×Dic346—42no—
33C2×Dic346—42no—
34C22×Dic173C2312yes—
35C22×Dic173C2312yes—
36C22×Dic173C2312yes—
37C22×Dic173C2312yes—
38C4⋊Dic173—41no—
39C4⋊Dic173—41no—
40C4⋊Dic173—42no—
41#14583805—42no—
42#14583805—42no—
43#14583805—42no—
44C4×Dic173Q814yescomplemented
45C22×C4×C173C2311yescharacteristic
46C22×C4×C173C2311yescharacteristic
47C22×C4×C173C2312yes—
48C22×C4×C173C2312yes—
49C22×C4×C173C2311yescharacteristic
50C4⋊C4×C173—41no—
51C4⋊C4×C173—41no—
52Dic346—48no—
53C2×Dic173—22no—
54C2×Dic173—22no—
55C2×Dic173—22no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—22no—
61C2×Dic173—22no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173C2×Q814yes—
65C23×C173C2411yescharacteristic
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—22no—
69C2×C4×C173—22no—
70C2×C4×C173—22no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—22no—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173C2×Q812yes—
79C22×C173C22×Q811yescharacteristic
80C22×C173H(5,𝔽2)11yescharacteristic
81C22×C173H(5,𝔽2)12yes—
82C22×C1732−1+411yescharacteristic
83C22×C1732−1+412yes—
84C4×C173—22no—
85C4×C173—42no—
86C4×C173—42no—
87C4×C173—41no—
88C4×C173—41no—
89Dic173—24no—
90Dic173—42no—
91Dic173—42no—
92Dic173—42no—
93C2×C173#54411yescharacteristic
94C2×C173#54412yes—
95C2×C173H(3,𝔽4)11yescharacteristic
96C2×C173#56312yes—
97C2×C173#56311yescharacteristic
98C173#284411yescharacteristic, complemented, Sylow
99#2844—1731nomaximal, Sylow
100#400—1731no—
101#400—1732no—
102#400—1732no—
103#400D17311yescharacteristic
104#395—1732no—
105#395—1732no—
106#395—1732no—
107#382—1731no—
108#382—1732no—
109C22×Q8—1732no—
110C2×C4⋊C4—1732no—
111C2×C4⋊C4—1732no—
112C2×C4⋊C4—1732no—
113C2×C4⋊C4—1731no—
114C2×C4⋊C4—1731no—
115C2×C4⋊C4—1732no—
116C2×C4⋊C4C2×D17311yescharacteristic
117C2×C4⋊C4C2×D17311yescharacteristic
118C2×C42—1731no—
119#95—1732no—
120#95—1732no—
121#95—1732no—
122#95—1732no—
123#95—1732no—
124#95—1732no—
125#95—1732no—
126#95—1731no—
127#95C2×D17312yes—
128#95C2×D17312yes—
129#95C2×D17311yescharacteristic
130C2×Q8—6922no—
131C2×Q8—6922no—
132C2×Q8—6922no—
133C22×C4—1732no—
134C22×C4—1732no—
135C22×C4—1732no—
136C22×C4—1732no—
137C22×C4C22×D17311yescharacteristic
138C22×C4C22×D17311yescharacteristic
139C22×C4C22×D17312yes—
140C22×C4C22×D17312yes—
141C22×C4C22×D17311yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—6922no—
145C4⋊C4—6922no—
146C4⋊C4—6922no—
147C4⋊C4—6921no—
148C4⋊C4—6921no—
149C4⋊C4—6922no—
150C42—1734no—
151C23C23×D17311yescharacteristic
152Q8—6928no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—3462no—
166C2×C4—3462no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3462no—
171C2×C4—3462no—
172C2×C4—3462no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—1734no—
177C2×C4Q8×D17312yes—
178C22C2×Q8×D17311yescharacteristic
179C22#1914972411yescharacteristic
180C22#1914972512yes—
181C22#1914972611yescharacteristic
182C22#1914972612yes—
183C4—22no—
184C4—42no—
185C4—42no—
186C4—41no—
187C4—41no—
188C4—3464no—
189C4—6922no—
190C4—6922no—
191C4—6922no—
192C2#2601212011yescharacteristic
193C2#2601212112yes—
194C2#2601223311yescharacteristic
195C2#2601224312yes—
196C2#2601224411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, prime radical, cosocleC23×C173C24
Fitting#400×C173C2
hypercenter#400D173
hypocenterC173#2844
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
44. C4×Dic173Q8semidirect
98. C173#2844semidirect

Maximal subgroups

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

subgroupindex
#260114322
#260114312
#260114302
#260114122
#260114112
#260113562
#260113552
#400×C1732
#2844173