Finite Group Databasebeta

← #36356651

Subgroups — #36356651

Order 22144.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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×#19149677C211yesmaximal, characteristic
2#26011398C211yesmaximal, characteristic
3#26011395C212yesmaximal
4#26011388C212yesmaximal
5#26011386C212yesmaximal
6C4×#14583805C212yesmaximal
7#26011342C212yesmaximal
8#26011342C212yesmaximal
9#391×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11#19149677C418yescomplemented
12C2×C4⋊Dic173C2211yescharacteristic
13C2×#14583805C2212yes—
14C2×#14583805C2212yes—
15C2×#14583805C2212yes—
16C2×#14583805C2212yes—
17C2×#14583805C2212yes—
18C2×C4×Dic173C2212yes—
19C2×C4×Dic173C2212yes—
20C2×C4×Dic173C2212yes—
21C2×C4×Dic173C2211yescharacteristic
22C2×C4×Dic173C2211yescharacteristic
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2212yes—
26#19149586C2212yes—
27C22×Q8×C173C2211yescharacteristic
28C2×C4⋊C4×C173C2211yescharacteristic
29C2×C42×C173C2211yescharacteristic
30#95×C173C2212yes—
31#95×C173C2212yes—
32C2×Dic346—41no—
33C2×Dic346—41no—
34C2×Dic346C2×C414yes—
35C22×Dic173C2312yes—
36C22×Dic173C2312yes—
37C22×Dic173C2312yes—
38C22×Dic173C2312yes—
39C4⋊Dic173—41no—
40#14583805—24no—
41#14583805—42no—
42#14583805C2×C418yes—
43#14583805C2×C418yes—
44#14583805Q818yescomplemented
45C4×Dic173—24no—
46C4×Dic173—24no—
47C4×Dic173—42no—
48C4×Dic173—41no—
49C4×Dic173C2×C414yes—
50C2×Q8×C173—41no—
51C2×Q8×C173—41no—
52C2×Q8×C173C2×C414yes—
53C22×C4×C173C2312yes—
54C22×C4×C173C2311yescharacteristic
55C22×C4×C173C2311yescharacteristic
56C22×C4×C173C2311yescharacteristic
57C22×C4×C173C2312yes—
58C4⋊C4×C173—41no—
59C42×C173—41no—
60Dic346—44no—
61C2×Dic173—22no—
62C2×Dic173—22no—
63C2×Dic173—22no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67C2×Dic173—22no—
68C2×Dic173C22×C414yes—
69C2×Dic173C22×C414yes—
70C2×Dic173C2×Q814yes—
71C2×Dic173C2×Q814yes—
72C2×Dic173#4114yes—
73C23×C173C2411yescharacteristic
74Q8×C173—44no—
75C2×C4×C173—22no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—21no—
79C2×C4×C173—22no—
80C2×C4×C173—21no—
81C2×C4×C173—21no—
82C2×C4×C173—21no—
83C2×C4×C173—22no—
84C2×C4×C173—22no—
85C2×C4×C173—22no—
86C2×C4×C173C22×C412yes—
87C2×C4×C173C22×C412yes—
88C2×C4×C173C22×C412yes—
89C2×C4×C173C2×Q814yes—
90C22×C173C23×C411yescharacteristic
91C22×C173C22×Q812yes—
92C22×C173C2×#4112yes—
93C22×C173H(5,𝔽2)11yescharacteristic
94C22×C173H(5,𝔽2)11yescharacteristic
95C4×C173—22no—
96C4×C173—22no—
97C4×C173—22no—
98C4×C173—24no—
99C4×C173—42no—
100Dic173—24no—
101Dic173—24no—
102Dic173—42no—
103Dic173C4×Q818yes—
104C2×C173C2×C4×Q812yes—
105C2×C173#51811yescharacteristic
106C2×C173#55412yes—
107C2×C173#55511yescharacteristic
108C2×C173Q8211yescharacteristic
109C173#236911yescharacteristic, complemented, Sylow
110#2369—1731nomaximal, Sylow
111C2×C4⋊Q8—1731no—
112#391—1731no—
113#391—1732no—
114#391D17311yescharacteristic
115#389—1732no—
116#389—1732no—
117C4×C4⋊C4—1732no—
118C4×C4⋊C4—1732no—
119C4×C4⋊C4—1732no—
120C22×Q8—1731no—
121C22×Q8C2×D17311yescharacteristic
122C4⋊Q8—1738no—
123C2×C4⋊C4—1732no—
124C2×C4⋊C4—1732no—
125C2×C4⋊C4—1732no—
126C2×C4⋊C4—1732no—
127C2×C4⋊C4—1732no—
128C2×C4⋊C4—1731no—
129C2×C4⋊C4C2×D17311yescharacteristic
130C2×C42—1732no—
131C2×C42—1732no—
132C2×C42—1731no—
133C2×C42—1731no—
134C2×C42—1732no—
135C2×C42C2×D17311yescharacteristic
136#95—1732no—
137#95—1732no—
138#95—1732no—
139#95—1732no—
140#95C2×D17312yes—
141#95C2×D17312yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—1734no—
145C2×Q8—6921no—
146C2×Q8—6921no—
147C2×Q8C4×D17314yes—
148C22×C4—1732no—
149C22×C4—1732no—
150C22×C4—1732no—
151C22×C4—1732no—
152C22×C4C22×D17312yes—
153C22×C4C22×D17311yescharacteristic
154C22×C4C22×D17311yescharacteristic
155C22×C4C22×D17311yescharacteristic
156C22×C4C22×D17312yes—
157C4⋊C4—41no—
158C4⋊C4—3464no—
159C4⋊C4—1738no—
160C4⋊C4—1738no—
161C4⋊C4—1738no—
162C4⋊C4—6922no—
163C4⋊C4—6921no—
164C42—41no—
165C42—3464no—
166C42—3464no—
167C42—1734no—
168C42—6922no—
169C42—6921no—
170C23C23×D17311yescharacteristic
171Q8—44no—
172Q8—6924no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—3462no—
185C2×C4—3462no—
186C2×C4—3462no—
187C2×C4—3462no—
188C2×C4—3462no—
189C2×C4—3462no—
190C2×C4—3462no—
191C2×C4—1734no—
192C2×C4—1734no—
193C2×C4—1734no—
194C2×C4—1734no—
195C2×C4—1734no—
196C2×C4C2×C4×D17312yes—
197C2×C4C2×C4×D17312yes—
198C2×C4C2×C4×D17312yes—
199C2×C4Q8×D17314yes—
200C22C22×C4×D17311yescharacteristic
201C22C2×Q8×D17312yes—
202C22#41×D17312yes—
203C22#1914972411yescharacteristic
204C22#1914972411yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—3464no—
211C4—3464no—
212C4—1738no—
213C4—6922no—
214C2C4×Q8×D17312yes—
215C2#2601196411yescharacteristic
216C2#2601219512yes—
217C2#2601220311yescharacteristic
218C2#2601222111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#391×C173C2
socle, prime radical, cosocleC23×C173C24
hypercenter#391D173
hypocenterC173#2369
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #19149677C4semidirect
44. #14583805Q8semidirect
109. C173#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#191496772
#260113982
#260113952
#260113882
#260113862
C4×#145838052
#260113422
#391×C1732
#2369173