Finite Group Databasebeta

← #36354021

Subgroups — #36354021

Order 22144.

378 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 93 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×#19149662C211yesmaximal, characteristic
2C2×#19149593C212yesmaximal
3#26011354C211yesmaximal, characteristic, complemented
4#26011257C212yesmaximal, complemented
5C2×#117×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173C2211yescharacteristic
7#19149662—21no—
8#19149662—21no—
9#19149662—22no—
10C2×#14583805C412yescomplemented
11C2×C22⋊Dic173C2212yes—
12C2×C22⋊Dic173C2211yescharacteristic
13C2×#14583803C412yescomplemented
14#19149593—24no—
15#19149586—22no—
16C23×C4×C173C2211yescharacteristic
17#117×C173—24no—
18C2×C4⋊C4×C173—21no—
19C2×C22⋊C4×C173C2212yes—
20C2×C42×C173—21no—
21C23×D173C2×C411yescharacteristic
22C2×C22⋊D173—22no—
23C2×C22⋊D173—21no—
24C2×C22⋊D173—21no—
25C2×C22⋊D173—22no—
26C22×Dic173—22no—
27C22×Dic173C2×C411yescharacteristic
28C22×Dic173C2×C412yes—
29#14583805—22no—
30#14583805—22no—
31C22⋊Dic173—21no—
32C22⋊Dic173—21no—
33C22⋊Dic173—22no—
34C22⋊Dic173—22no—
35#14583803—22no—
36#14583803—22no—
37C24×C173C2311yescharacteristic
38C22×C4×C173—22no—
39C22×C4×C173—22no—
40C22×C4×C173—22no—
41C22×C4×C173C2×C412yes—
42C22×C4×C173D411yescharacteristic
43C22×C4×C173D411yescharacteristic
44C22×C4×C173D412yes—
45C4⋊C4×C173—24no—
46C22⋊C4×C173—22no—
47C22⋊C4×C173D414yes—
48C42×C173—24no—
49C22×D173—21no—
50C22×D173—22no—
51C22×D173—42no—
52C22⋊D173—41no—
53C22⋊D173—42no—
54C22⋊D173—41no—
55C2×Dic173—21no—
56C2×Dic173—22no—
57C2×Dic173—22no—
58C2×Dic173—42no—
59C2×Dic173—44no—
60C2×Dic173—42no—
61C23×C173—42no—
62C23×C173C22×C411yescharacteristic
63C23×C173C2×D411yescharacteristic
64C23×C173C2×D412yes—
65C23×C173C2×D411yescharacteristic
66C23×C173#4112yes—
67C2×C4×C173—24no—
68C2×C4×C173—21no—
69C2×C4×C173—21no—
70C2×C4×C173—24no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173—24no—
74C2×C4×C173—44no—
75C2×C4×C173C22⋊C412yes—
76C2×C4×C173C22⋊C412yes—
77C22×C173—22no—
78C22×C173—22no—
79C22×C173—22no—
80C22×C173—42no—
81C22×C173—42no—
82C22×C173C2×C22⋊C411yescharacteristic
83C22×C173C4×D412yes—
84C22×C173#12011yescharacteristic
85C22×C173#12112yes—
86C22×C173#12311yescharacteristic
87C2×D173—41no—
88C2×D173—42no—
89C2×D173—82no—
90C4×C173—22no—
91C4×C173—22no—
92C4×C173—44no—
93Dic173—41no—
94Dic173—42no—
95Dic173—82no—
96C2×C173—22no—
97C2×C173—21no—
98C2×C173—22no—
99C2×C173—21no—
100C2×C173#38611yescharacteristic
101C2×C173#41012yes—
102D173—81no—
103C173#189911yescharacteristic, complemented, Sylow
104#1899—1731nomaximal, Sylow
105C2×#123—1731no—
106C2×#117D17311yescharacteristic, complemented
107C2×#99—1732no—
108#381—1731no—
109#342—1732no—
110C22×D4—1731no—
111C23×C4C2×D17311yescharacteristic
112#123—3461no—
113#123—3461no—
114#123—3462no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—1732no—
118C2×C22⋊C4—1732no—
119C2×C22⋊C4—1731no—
120C2×C22⋊C4—1732no—
121C2×C22⋊C4C2×D17312yes—
122C2×C42—21no—
123#99—3464no—
124#95—3462no—
125C24—1731no—
126C24C22×D17311yescharacteristic
127C2×D4—3462no—
128C2×D4—3461no—
129C2×D4—3461no—
130C2×D4—3462no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—3462no—
135C22×C4—1731no—
136C22×C4—1732no—
137C22×C4C4×D17312yes—
138C22×C4C22⋊D17311yescharacteristic
139C22×C4C22⋊D17311yescharacteristic
140C22×C4D69212yes—
141C4⋊C4—24no—
142C4⋊C4—3462no—
143C4⋊C4—3462no—
144C22⋊C4—22no—
145C22⋊C4—3461no—
146C22⋊C4—3461no—
147C22⋊C4—3462no—
148C22⋊C4—3462no—
149C22⋊C4—3462no—
150C22⋊C4—3462no—
151C22⋊C4C22⋊D17314yes—
152C42—24no—
153C23—42no—
154C23—3462no—
155C23—3461no—
156C23—6922no—
157C23C2×C4×D17311yescharacteristic
158C23C2×C22⋊D17311yescharacteristic
159C23C2×C22⋊D17312yes—
160C23C2×D69211yescharacteristic
161C23#1458382612yes—
162D4—6921no—
163D4—6922no—
164D4—6921no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—44no—
173C2×C4—3462no—
174C2×C4—3461no—
175C2×C4—3462no—
176C2×C4—6924no—
177C2×C4—6922no—
178C2×C4—6922no—
179C2×C4#1458380312yes—
180C2×C4#1458380312yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—13842no—
187C22—6922no—
188C22—6921no—
189C22C2×#1458380311yescharacteristic
190C22C4×C22⋊D17312yes—
191C22#1914964811yescharacteristic
192C22#1914965312yes—
193C22#1914966211yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—13842no—
198C4—6922no—
199C4—6921no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—13841no—
205C2#2601137011yescharacteristic
206C2#2601146412yes—
subgroupG/H
centerC22C2×#14583803
commutatorC23×C173C22×C4
FrattiniC24C22×D173
FittingC2×#117×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC22×C22⋊D173C22
cosocleC24×C173C23
hypercenterC2×#117D173
hypocenterC173#1899
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #26011354C2semidirect
4. #26011257C2semidirect
5. C2×#117×C173C2semidirect
10. C2×#14583805C4semidirect
13. C2×#14583803C4semidirect
103. C173#1899semidirect
106. C2×#117D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496622
C2×#191495932
#260113542
#260112572
C2×#117×C1732
#1899173