Finite Group Databasebeta

← #36356281

Subgroups — #36356281

Order 22144.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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
2C2×Q8×Dic173C211yesmaximal, characteristic
3C2×#19149644C211yesmaximal, characteristic
4#26011397C211yesmaximal, characteristic
5#26011378C214yesmaximal
6#26011373C214yesmaximal
7#26011356C211yesmaximal, characteristic
8#26011337C211yesmaximal, characteristic
9#391×C173C211yesmaximal, characteristic
10C22×Dic346C2211yescharacteristic
11#19149677C418yescomplemented
12Q8×Dic173—24no—
13#19149644—24no—
14C2×C4⋊Dic173C2211yescharacteristic
15C2×C4⋊Dic173C2212yes—
16C2×#14583805C2212yes—
17C2×#14583805C2214yes—
18C2×C4×Dic173C2212yes—
19C2×C4×Dic173C2212yes—
20C2×C4×Dic173C2211yescharacteristic
21C2×C4×Dic173C2211yescharacteristic
22#19149586C2214yes—
23#19149586C2212yes—
24#19149586C2212yes—
25#19149586C2214yes—
26C22×Q8×C173C2211yescharacteristic
27C2×C4⋊C4×C173C2211yescharacteristic
28C2×C42×C173C2211yescharacteristic
29#95×C173C2214yes—
30C2×Dic346—22no—
31C2×Dic346—22no—
32C2×Dic346C2×C414yes—
33C22×Dic173C2312yes—
34C22×Dic173C2312yes—
35C22×Dic173C2312yes—
36C22×Dic173C2311yescharacteristic
37C22×Dic173C2311yescharacteristic
38C4⋊Dic173—24no—
39C4⋊Dic173—22no—
40#14583805—24no—
41#14583805C2×C4116yes—
42C4×Dic173—24no—
43C4×Dic173—24no—
44C4×Dic173C2×C414yes—
45C4×Dic173D414yes—
46C2×Q8×C173—22no—
47C2×Q8×C173—22no—
48C2×Q8×C173C2×C414yes—
49C22×C4×C173C2312yes—
50C22×C4×C173C2312yes—
51C22×C4×C173C2312yes—
52C22×C4×C173C2311yescharacteristic
53C4⋊C4×C173—22no—
54C42×C173—22no—
55Dic346—28no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—22no—
61C2×Dic173—22no—
62C2×Dic173—22no—
63C2×Dic173—24no—
64C2×Dic173C22×C414yes—
65C2×Dic173C22×C414yes—
66C2×Dic173C2×D412yes—
67C2×Dic173C2×D412yes—
68C23×C173C2411yescharacteristic
69Q8×C173—28no—
70C2×C4×C173—22no—
71C2×C4×C173—22no—
72C2×C4×C173—22no—
73C2×C4×C173—22no—
74C2×C4×C173—24no—
75C2×C4×C173—22no—
76C2×C4×C173—22no—
77C2×C4×C173C22×C414yes—
78C2×C4×C173C22×C412yes—
79C2×C4×C173C2×D412yes—
80C2×C4×C173#4112yes—
81C22×C173C23×C411yescharacteristic
82C22×C173C22×D411yescharacteristic
83C22×C173C2×#4111yescharacteristic
84C22×C1732−1+412yes—
85C22×C1732−1+412yes—
86C4×C173—24no—
87C4×C173—42no—
88C4×C173—42no—
89C4×C173C4×D414yes—
90Dic173—24no—
91Dic173—24no—
92Dic173—22no—
93Dic173—22no—
94Dic173—42no—
95C2×C173C2×C4×D411yescharacteristic
96C2×C173#51912yes—
97C2×C173#53611yescharacteristic
98C2×C173#53611yescharacteristic
99C2×C173#54112yes—
100C173#232711yescharacteristic, complemented, Sylow
101#2327—1731nomaximal, Sylow
102C2×C4⋊Q8—1731no—
103C2×C4×Q8—1731no—
104C2×C4×Q8—1731no—
105#391—1731no—
106#391D17311yescharacteristic
107#387—1734no—
108#387—1734no—
109#382—1731no—
110#376—1731no—
111C22×Q8—1731no—
112C22×Q8C2×D17311yescharacteristic
113C4⋊Q8—1738no—
114C4×Q8—3464no—
115C4×Q8—3464no—
116C2×C4⋊C4—1732no—
117C2×C4⋊C4—1731no—
118C2×C4⋊C4—1732no—
119C2×C4⋊C4—1734no—
120C2×C4⋊C4C2×D17311yescharacteristic
121C2×C42—1732no—
122C2×C42—1732no—
123C2×C42—1731no—
124C2×C42—1731no—
125C2×C42C2×D17311yescharacteristic
126#95—1732no—
127#95—1732no—
128#95—1734no—
129#95—1734no—
130#95C2×D17314yes—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—3462no—
134C2×Q8—3462no—
135C2×Q8—1734no—
136C2×Q8C4×D17314yes—
137C22×C4—1732no—
138C22×C4—1732no—
139C22×C4—1731no—
140C22×C4—1731no—
141C22×C4—1732no—
142C22×C4C22×D17312yes—
143C22×C4C22×D17312yes—
144C22×C4C22×D17312yes—
145C22×C4C22×D17311yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—3464no—
148C4⋊C4—3464no—
149C4⋊C4—3462no—
150C4⋊C4—17316no—
151C42—22no—
152C42—3464no—
153C42—3464no—
154C42—1734no—
155C42—1734no—
156C23C23×D17311yescharacteristic
157Q8—28no—
158Q8—3468no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—3462no—
167C2×C4—3462no—
168C2×C4—3462no—
169C2×C4—3462no—
170C2×C4—3462no—
171C2×C4—3462no—
172C2×C4—3462no—
173C2×C4—3464no—
174C2×C4—1734no—
175C2×C4—1734no—
176C2×C4—1732no—
177C2×C4—1732no—
178C2×C4C2×C4×D17314yes—
179C2×C4C2×C4×D17312yes—
180C2×C4D4×D17312yes—
181C2×C4#1458382512yes—
182C22C22×C4×D17311yescharacteristic
183C22C2×D4×D17311yescharacteristic
184C22C2×#1458382511yescharacteristic
185C22#1914972612yes—
186C22#1914972712yes—
187C4—24no—
188C4—42no—
189C4—42no—
190C4—3464no—
191C4—3464no—
192C4—3462no—
193C4—3462no—
194C4—6922no—
195C4#1914963914yes—
196C2C2×#1914963911yescharacteristic
197C2#2601196712yes—
198C2#2601206111yescharacteristic
199C2#2601206611yescharacteristic
200C2#2601210112yes—
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#391×C173C2
socle, prime radical, cosocleC23×C173C24
hypercenter#391D173
hypocenterC173#2327
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #19149677C4semidirect
100. C173#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#191496772
C2×Q8×Dic1732
C2×#191496442
#260113972
#260113782
#260113732
#260113562
#260113372
#391×C1732
#2327173