Finite Group Databasebeta

← #36356748

Subgroups — #36356748

Order 22144.

408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#19149671C211yesmaximal, characteristic
2#26011388C212yesmaximal
3#26011386C212yesmaximal
4#26011378C212yesmaximal
5#26011376C212yesmaximal
6#26011374C212yesmaximal
7#26011372C212yesmaximal
8#26011357C211yesmaximal, characteristic
9#382×C173C211yesmaximal, characteristic
10#19149671C418yescomplemented
11C2×C4⋊Dic173C2211yescharacteristic
12C2×C4⋊Dic173C2211yescharacteristic
13C2×C4⋊Dic173C2211yescharacteristic
14C2×C4⋊Dic173C2211yescharacteristic
15C2×#14583805C2212yes—
16C2×#14583805C2212yes—
17C2×#14583805C2212yes—
18C2×#14583805C2212yes—
19C2×C4×Dic173C2212yes—
20C2×C4×Dic173C2212yes—
21#19149586C2212yes—
22#19149586C2212yes—
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2212yes—
26#19149586C2212yes—
27C2×C4⋊C4×C173C2211yescharacteristic
28C2×C4⋊C4×C173C2211yescharacteristic
29C2×C42×C173C2211yescharacteristic
30#95×C173C2212yes—
31#95×C173C2212yes—
32C22×Dic173C2312yes—
33C22×Dic173C2312yes—
34C22×Dic173C2312yes—
35C22×Dic173C2312yes—
36C4⋊Dic173—41no—
37C4⋊Dic173—41no—
38C4⋊Dic173C2×C414yes—
39C4⋊Dic173C2×C414yes—
40#14583805—42no—
41#14583805—42no—
42#14583805C2×C418yes—
43#14583805C2×C418yes—
44C4×Dic173—42no—
45C4×Dic173—42no—
46C22×C4×C173C2312yes—
47C22×C4×C173C2312yes—
48C22×C4×C173C2311yescharacteristic
49C22×C4×C173C2311yescharacteristic
50C22×C4×C173C2311yescharacteristic
51C4⋊C4×C173—41no—
52C4⋊C4×C173—41no—
53C42×C173C2×C414yes—
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×Dic173C22×C414yes—
65C2×Dic173C22×C414yes—
66C23×C173C2411yescharacteristic
67C2×C4×C173—21no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—21no—
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×C173—22no—
79C2×C4×C173C22×C412yes—
80C2×C4×C173C22×C412yes—
81C2×C4×C173C22×C412yes—
82C22×C173C23×C411yescharacteristic
83C22×C173H(5,𝔽2)11yescharacteristic
84C22×C173H(5,𝔽2)11yescharacteristic
85C22×C1732−1+411yescharacteristic
86C22×C1732−1+411yescharacteristic
87C22×C1732−1+411yescharacteristic
88C22×C1732−1+411yescharacteristic
89C4×C173—22no—
90C4×C173—22no—
91C4×C173—22no—
92C4×C173—42no—
93C4×C173—42no—
94Dic173—24no—
95Dic173—24no—
96Dic173—42no—
97Dic173—42no—
98C2×C173#51911yescharacteristic
99C2×C173#52011yescharacteristic
100C2×C173#52011yescharacteristic
101C2×C173#56211yescharacteristic
102C2×C173#56311yescharacteristic
103C2×C173#56311yescharacteristic
104C2×C173#56411yescharacteristic
105C173#238211yescharacteristic, complemented, Sylow
106#2382—1731nomaximal, Sylow
107C2×#125—1731no—
108#389—1732no—
109#389—1732no—
110#387—1732no—
111#387—1732no—
112#387—1732no—
113#387—1732no—
114#382—1731no—
115#382D17311yescharacteristic
116#125—1738no—
117C2×C4⋊C4—1732no—
118C2×C4⋊C4—1732no—
119C2×C4⋊C4—1731no—
120C2×C4⋊C4—1731no—
121C2×C4⋊C4—1732no—
122C2×C4⋊C4—1732no—
123C2×C4⋊C4—1731no—
124C2×C4⋊C4—1731no—
125C2×C4⋊C4C2×D17311yescharacteristic
126C2×C4⋊C4C2×D17311yescharacteristic
127C2×C42—1732no—
128C2×C42—1732no—
129C2×C42C2×D17311yescharacteristic
130#95—1732no—
131#95—1732no—
132#95—1732no—
133#95—1732no—
134#95—1732no—
135#95—1732no—
136#95C2×D17312yes—
137#95C2×D17312yes—
138C22×C4—1732no—
139C22×C4—1732no—
140C22×C4—1732no—
141C22×C4—1732no—
142C22×C4C22×D17312yes—
143C22×C4C22×D17312yes—
144C22×C4C22×D17311yescharacteristic
145C22×C4C22×D17311yescharacteristic
146C22×C4C22×D17311yescharacteristic
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—1738no—
150C4⋊C4—1738no—
151C4⋊C4—1734no—
152C4⋊C4—1734no—
153C4⋊C4—6922no—
154C4⋊C4—6922no—
155C4⋊C4—6921no—
156C4⋊C4—6921no—
157C42—6922no—
158C42—6922no—
159C42C4×D17314yes—
160C23C23×D17311yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—3462no—
174C2×C4—3462no—
175C2×C4—3462no—
176C2×C4—3462no—
177C2×C4—3462no—
178C2×C4—3462no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4—3462no—
182C2×C4—3462no—
183C2×C4—1734no—
184C2×C4—1734no—
185C2×C4C2×C4×D17312yes—
186C2×C4C2×C4×D17312yes—
187C2×C4C2×C4×D17312yes—
188C22C22×C4×D17311yescharacteristic
189C22#1914972411yescharacteristic
190C22#1914972511yescharacteristic
191C22#1914972611yescharacteristic
192C22#1914972611yescharacteristic
193C22#1914972611yescharacteristic
194C22#1914972711yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—3464no—
201C4—3464no—
202C4—6922no—
203C4—6922no—
204C2#2601196811yescharacteristic
205C2#2601197111yescharacteristic
206C2#2601197211yescharacteristic
207C2#2601223511yescharacteristic
208C2#2601224111yescharacteristic
209C2#2601224511yescharacteristic
210C2C173⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#382×C173C2
socle, prime radical, cosocleC23×C173C24
hypercenter#382D173
hypocenterC173#2382
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
10. #19149671C4semidirect
105. C173#2382semidirect

Maximal subgroups

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

subgroupindex
C2×#191496712
#260113882
#260113862
#260113782
#260113762
#260113742
#260113722
#260113572
#382×C1732
#2382173