Finite Group Databasebeta

← #36356364

Subgroups — #36356364

Order 22144.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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#26011378C212yesmaximal
4#26011376C212yesmaximal
5#26011374C212yesmaximal
6#26011373C212yesmaximal
7C4×C4⋊Dic173C211yesmaximal, characteristic
8#26011342C212yesmaximal
9#383×C173C211yesmaximal, characteristic
10#19149671C418yescomplemented
11C2×C4⋊Dic173C2212yes—
12C2×C4⋊Dic173C2211yescharacteristic
13C2×C4⋊Dic173C2211yescharacteristic
14C2×#14583805C2212yes—
15C2×#14583805C2212yes—
16C2×#14583805C2212yes—
17C2×#14583805C2212yes—
18C2×C4×Dic173C2212yes—
19C2×C4×Dic173C2211yescharacteristic
20C2×C4×Dic173C2212yes—
21C2×C4×Dic173C2211yescharacteristic
22#19149586C2212yes—
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2212yes—
26#19149586C2212yes—
27C2×C42×C173C2211yescharacteristic
28#95×C173C2212yes—
29#95×C173C2212yes—
30#95×C173C2211yescharacteristic
31#95×C173C2211yescharacteristic
32C22×Dic173C2312yes—
33C22×Dic173C2312yes—
34C22×Dic173C2312yes—
35C22×Dic173C2312yes—
36C4⋊Dic173—24no—
37C4⋊Dic173C2×C414yes—
38C4⋊Dic173C2×C414yes—
39#14583805—24no—
40#14583805—42no—
41#14583805C2×C418yes—
42#14583805C2×C418yes—
43C4×Dic173—24no—
44C4×Dic173—41no—
45C4×Dic173—41no—
46C4×Dic173—42no—
47C22×C4×C173C2312yes—
48C22×C4×C173C2312yes—
49C22×C4×C173C2311yescharacteristic
50C22×C4×C173C2311yescharacteristic
51C22×C4×C173C2311yescharacteristic
52C42×C173C2×C414yes—
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×Dic173C22×C414yes—
63C2×Dic173C22×C414yes—
64C2×Dic173#4114yes—
65C23×C173C2411yescharacteristic
66C2×C4×C173—22no—
67C2×C4×C173—21no—
68C2×C4×C173—22no—
69C2×C4×C173—21no—
70C2×C4×C173—22no—
71C2×C4×C173—22no—
72C2×C4×C173—22no—
73C2×C4×C173—22no—
74C2×C4×C173—22no—
75C2×C4×C173—22no—
76C2×C4×C173C22×C412yes—
77C2×C4×C173C22×C412yes—
78C2×C4×C173C22×C412yes—
79C22×C173C23×C411yescharacteristic
80C22×C173C2×#4112yes—
81C22×C173H(5,𝔽2)11yescharacteristic
82C22×C1732−1+412yes—
83C22×C1732−1+411yescharacteristic
84C4×C173—22no—
85C4×C173—22no—
86C4×C173—22no—
87C4×C173—42no—
88C4×C173—42no—
89Dic173—24no—
90Dic173—24no—
91Dic173—24no—
92Dic173—42no—
93C2×C173C4×#4111yescharacteristic
94C2×C173#51911yescharacteristic
95C2×C173#52011yescharacteristic
96C2×C173#53912yes—
97C2×C173#54112yes—
98C173#233611yescharacteristic, complemented, Sylow
99#2336—1731nomaximal, Sylow
100C2×#125—1731no—
101#389—1732no—
102#387—1732no—
103#387—1732no—
104#387—1732no—
105#387—1732no—
106#383D17311yescharacteristic
107C4×C4⋊C4—1731no—
108C4×C4⋊C4—1732no—
109#125—1738no—
110C2×C4⋊C4—1732no—
111C2×C4⋊C4—1732no—
112C2×C4⋊C4—1732no—
113C2×C4⋊C4—1732no—
114C2×C4⋊C4—1731no—
115C2×C4⋊C4—1731no—
116C2×C4⋊C4—1732no—
117C2×C42—1732no—
118C2×C42—1731no—
119C2×C42—1731no—
120C2×C42—1732no—
121C2×C42C2×D17311yescharacteristic
122#95—1732no—
123#95—1732no—
124#95—1732no—
125#95—1732no—
126#95—1732no—
127#95C2×D17312yes—
128#95C2×D17311yescharacteristic
129#95C2×D17312yes—
130#95C2×D17311yescharacteristic
131C22×C4—1732no—
132C22×C4—1732no—
133C22×C4—1732no—
134C22×C4—1732no—
135C22×C4C22×D17312yes—
136C22×C4C22×D17312yes—
137C22×C4C22×D17311yescharacteristic
138C22×C4C22×D17311yescharacteristic
139C22×C4C22×D17311yescharacteristic
140C4⋊C4—3464no—
141C4⋊C4—3464no—
142C4⋊C4—1738no—
143C4⋊C4—1738no—
144C4⋊C4—1734no—
145C4⋊C4—1734no—
146C4⋊C4—6922no—
147C42—3464no—
148C42—6921no—
149C42—6921no—
150C42—6922no—
151C42C4×D17314yes—
152C23C23×D17311yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—3462no—
164C2×C4—3462no—
165C2×C4—3462no—
166C2×C4—3462no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3462no—
171C2×C4—3462no—
172C2×C4—1734no—
173C2×C4—1734no—
174C2×C4—1734no—
175C2×C4C2×C4×D17312yes—
176C2×C4C2×C4×D17312yes—
177C2×C4C2×C4×D17312yes—
178C22C22×C4×D17311yescharacteristic
179C22#41×D17312yes—
180C22#1914972511yescharacteristic
181C22#1914972612yes—
182C22#1914972611yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—3464no—
189C4—3464no—
190C4—3464no—
191C4—6922no—
192C2#2601195911yescharacteristic
193C2#2601196811yescharacteristic
194C2#2601197111yescharacteristic
195C2#2601208212yes—
196C2#2601209812yes—
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#383×C173C2
socle, prime radical, cosocleC23×C173C24
hypercenter#383D173
hypocenterC173#2336
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
10. #19149671C4semidirect
98. C173#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#191496712
#260113882
#260113782
#260113762
#260113742
#260113732
C4×C4⋊Dic1732
#260113422
#383×C1732
#2336173