Finite Group Databasebeta

← #36354022

Subgroups — #36354022

Order 22144.

378 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19149638C211yesmaximal, characteristic, complemented
2C2×#19149594C212yesmaximal
3#26011354C211yesmaximal, characteristic, complemented
4#26011258C212yesmaximal, complemented
5C2×#123×C173C211yesmaximal, characteristic
6#19149638—24no—
7C2×C4⋊Dic173—21no—
8C2×C22⋊Dic173C2212yes—
9C2×C22⋊Dic173C2212yes—
10C2×C4×Dic173—21no—
11#19149594—24no—
12#19149586—22no—
13C22×D4×C173C2211yescharacteristic
14C23×C4×C173C2211yescharacteristic
15#123×C173—22no—
16#123×C173—21no—
17#123×C173—21no—
18C2×C4⋊C4×C173C412yescomplemented
19C2×C22⋊C4×C173C412yescomplemented
20C2×C22⋊C4×C173C2211yescharacteristic
21C22×Dic173—22no—
22C22×Dic173—22no—
23C4⋊Dic173—24no—
24C22⋊Dic173—22no—
25C22⋊Dic173—22no—
26C22⋊Dic173—22no—
27C22⋊Dic173D414yes—
28C4×Dic173—24no—
29C24×C173C2×C411yescharacteristic
30C24×C173C2311yescharacteristic
31C2×D4×C173—22no—
32C2×D4×C173—21no—
33C2×D4×C173—21no—
34C2×D4×C173—22no—
35C22×C4×C173—22no—
36C22×C4×C173—22no—
37C22×C4×C173C2×C411yescharacteristic
38C22×C4×C173C2×C412yes—
39C22×C4×C173C2×C412yes—
40C22×C4×C173D412yes—
41C22×C4×C173D411yescharacteristic
42C22×C4×C173D411yescharacteristic
43C4⋊C4×C173—22no—
44C4⋊C4×C173—22no—
45C22⋊C4×C173—21no—
46C22⋊C4×C173—21no—
47C22⋊C4×C173—22no—
48C22⋊C4×C173—22no—
49C2×Dic173—24no—
50C2×Dic173—44no—
51C2×Dic173—44no—
52C2×Dic173—42no—
53C23×C173—21no—
54C23×C173—22no—
55C23×C173—42no—
56C23×C173—42no—
57C23×C173C22×C411yescharacteristic
58C23×C173C2×D412yes—
59C23×C173C2×D411yescharacteristic
60C23×C173C2×D411yescharacteristic
61C23×C173#4112yes—
62D4×C173—41no—
63D4×C173—42no—
64D4×C173—41no—
65C2×C4×C173—24no—
66C2×C4×C173—21no—
67C2×C4×C173—21no—
68C2×C4×C173—24no—
69C2×C4×C173—21no—
70C2×C4×C173—21no—
71C2×C4×C173—22no—
72C2×C4×C173—21no—
73C2×C4×C173—22no—
74C2×C4×C173—42no—
75C2×C4×C173C22⋊C412yes—
76C2×C4×C173C22⋊C412yes—
77C22×C173—22no—
78C22×C173—22no—
79C22×C173—22no—
80C22×C173—42no—
81C22×C173—42no—
82C22×C173—41no—
83C22×C173—42no—
84C22×C173—82no—
85C22×C173C2×C22⋊C411yescharacteristic
86C22×C173C4×D412yes—
87C22×C173#12011yescharacteristic
88C22×C173#12112yes—
89C22×C173#12311yescharacteristic
90C4×C173—22no—
91C4×C173—22no—
92C4×C173—41no—
93C4×C173—42no—
94Dic173—44no—
95Dic173—82no—
96C2×C173—21no—
97C2×C173—22no—
98C2×C173—22no—
99C2×C173—21no—
100C2×C173—81no—
101C2×C173#38611yescharacteristic
102C2×C173#41012yes—
103C173#189911yescharacteristic, complemented, Sylow
104#1899—1731nomaximal, Sylow
105C2×#123D17311yescharacteristic
106C2×#117—1731no—
107C2×#99—1732no—
108#381—1731no—
109#342—1732no—
110C22×D4C2×D17311yescharacteristic
111C23×C4C2×D17311yescharacteristic
112#123—22no—
113#123—21no—
114#123—21no—
115#117—3464no—
116C2×C4⋊C4—3461no—
117C2×C4⋊C4Dic17312yescomplemented
118C2×C22⋊C4—1732no—
119C2×C22⋊C4—1732no—
120C2×C22⋊C4Dic17312yescomplemented
121C2×C22⋊C4C2×D17311yescharacteristic
122C2×C42—3461no—
123#99—3464no—
124#95—3462no—
125C24C2×Dic17311yescharacteristic
126C24C22×D17311yescharacteristic
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—3462no—
134C22×C4—3462no—
135C22×C4C2×Dic17311yescharacteristic
136C22×C4C2×Dic17312yes—
137C22×C4C2×Dic17312yes—
138C22×C4C22⋊D17312yes—
139C22×C4C22⋊D17311yescharacteristic
140C22×C4C22⋊D17311yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—3464no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—3462no—
149C22⋊C4—3462no—
150C22⋊C4—3462no—
151C22⋊C4—1734no—
152C42—3464no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23C22×Dic17311yescharacteristic
158C23D4×D17312yes—
159C23C2×C22⋊D17311yescharacteristic
160C23C2×C22⋊D17311yescharacteristic
161C23#1458382512yes—
162D4—41no—
163D4—42no—
164D4—41no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—3464no—
176C2×C4—6924no—
177C2×C4—6922no—
178C2×C4—6924no—
179C2×C4C22⋊Dic17312yes—
180C2×C4C22⋊Dic17312yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—41no—
188C22—82no—
189C22C2×C22⋊Dic17311yescharacteristic
190C22D4×Dic17312yes—
191C22#1914964711yescharacteristic
192C22#1914965112yes—
193C22#1914966011yescharacteristic
194C4—22no—
195C4—22no—
196C4—42no—
197C4—41no—
198C4—13842no—
199C4—6924no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2#2601137111yescharacteristic
206C2#2601146512yes—
subgroupG/H
centerC22C2×C22⋊Dic173
commutatorC23×C173C22×C4
FrattiniC24C22×D173
FittingC2×#123×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×D4×C173C22
cosocleC24×C173C23
hypercenterC2×#123D173
hypocenterC173#1899
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#19149638C2semidirect
3. #26011354C2semidirect
4. #26011258C2semidirect
18. C2×C4⋊C4×C173C4semidirect
19. C2×C22⋊C4×C173C4semidirect
103. C173#1899semidirect
117. C2×C4⋊C4Dic173semidirect
120. C2×C22⋊C4Dic173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496382
C2×#191495942
#260113542
#260112582
C2×#123×C1732
#1899173