Finite Group Databasebeta

← #36354011

Subgroups — #36354011

Order 22144.

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

normal subgroups only

This group has 358 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
1#26012002C211yesmaximal, characteristic
2C2×#19149607C212yesmaximal, complemented
3#26011455C211yesmaximal, characteristic, complemented
4#26011251C212yesmaximal, complemented
5C2×#117×C173C211yesmaximal, characteristic, complemented
6C2×#14583826C2211yescharacteristic
7C2×#14583810C2212yes—
8#19149662—21no—
9#19149658C412yescomplemented
10#19149653C412yescomplemented
11#19149648—21no—
12C4×C22⋊D173—22no—
13#19149638C2211yescharacteristic
14#19149607—24no—
15C22⋊#13063436—22no—
16C23×C4×C173C2211yescharacteristic
17#117×C173—24no—
18C2×C4⋊C4×C173—21no—
19C2×C22⋊C4×C173—21no—
20C2×C42×C173C2212yes—
21#14583826—22no—
22#14583826—22no—
23C2×Dic346C2×C411yescharacteristic
24C2×D692C2×C411yescharacteristic
25C2×C22⋊D173—22no—
26C2×C22⋊D173—21no—
27C2×C4×D173—21no—
28#14583810—22no—
29#14583810—22no—
30C2×#13063436—22no—
31C4⋊Dic173C2×C412yes—
32#14583805—22no—
33C22⋊Dic173—22no—
34C22⋊Dic173—21no—
35#14583803—22no—
36C4×Dic173—21no—
37C24×C173D411yescharacteristic
38C22×C4×C173—22no—
39C22×C4×C173—22no—
40C22×C4×C173—22no—
41C22×C4×C173C2×C412yes—
42C22×C4×C173D411yescharacteristic
43C22×C4×C173D412yes—
44C22×C4×C173C2311yescharacteristic
45C4⋊C4×C173—24no—
46C22⋊C4×C173—24no—
47C42×C173—22no—
48C42×C173D414yescomplemented
49C22×D173—21no—
50Dic346—22no—
51D692—22no—
52C22⋊D173—44no—
53C22⋊D173—41no—
54C22⋊D173—41no—
55C2×Dic173—21no—
56C2×Dic173—22no—
57C4×D173—42no—
58#13063436—42no—
59C23×C173—21no—
60C23×C173—21no—
61C23×C173—42no—
62C23×C173C22⋊C412yes—
63C23×C173C2×D411yescharacteristic
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—24no—
67C2×C4×C173—24no—
68C2×C4×C173—24no—
69C2×C4×C173—44no—
70C2×C4×C173C22⋊C412yes—
71C2×C4×C173C22×C411yescharacteristic
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173#4112yes—
75C22×C173—22no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—22no—
79C22×C173—42no—
80C22×C173—42no—
81C22×C173C2×C22⋊C411yescharacteristic
82C22×C173#12011yescharacteristic
83C22×C173#12311yescharacteristic
84C2×D173—41no—
85C2×D173—42no—
86C4×C173—22no—
87C4×C173—44no—
88C4×C173C4×D412yes—
89C4×C173#12112yes—
90Dic173—41no—
91Dic173—42no—
92C2×C173—21no—
93C2×C173—21no—
94C2×C173—42no—
95C2×C173#38611yescharacteristic
96C2×C173#42112yes—
97D173—81no—
98C173#189711yescharacteristic, complemented, Sylow
99#1897—1731nomaximal, Sylow
100#525—1731no—
101C2×#117D17311yescharacteristic, complemented
102C2×#104—1732no—
103#407—1731no—
104#339—1732no—
105C2×#41—1731no—
106C23×C4C2×D17311yescharacteristic
107C2×M16—1732no—
108#123—3461no—
109C22⋊Q8—1732no—
110#121—1732no—
111#120—3461no—
112C4×D4—3462no—
113#117—24no—
114#117—1731no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D17312yes—
118#104—3464no—
119C22⋊C8—3462no—
120C24C22⋊D17311yescharacteristic
121#41—3462no—
122#41—3462no—
123C2×Q8—1731no—
124C2×D4—3461no—
125C2×D4—3462no—
126C2×D4—1731no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—3461no—
131C22×C4C4×D17312yes—
132C22×C4C22⋊D17311yescharacteristic
133C22×C4D69212yes—
134C22×C4C22×D17311yescharacteristic
135M16—3462no—
136M16—3462no—
137C2×C8—3462no—
138C4⋊C4—24no—
139C4⋊C4—3462no—
140C4⋊C4—1732no—
141C22⋊C4—24no—
142C22⋊C4—3461no—
143C22⋊C4—3462no—
144C22⋊C4—3462no—
145C42—22no—
146C42—3461no—
147C42C22⋊D17314yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—3461no—
152C23#1458380312yes—
153C23C2×C22⋊D17311yescharacteristic
154Q8—3462no—
155D4—3462no—
156D4—6924no—
157D4—6921no—
158D4—6921no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—3462no—
166C2×C4—3461no—
167C2×C4—6922no—
168C2×C4#1458380312yes—
169C2×C4C2×C4×D17311yescharacteristic
170C2×C4C2×C22⋊D17312yes—
171C2×C4C2×D69211yescharacteristic
172C2×C4#1458382612yes—
173C8—6922no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—6922no—
181C22—6921no—
182C22C2×#1458380311yescharacteristic
183C22#1914964811yescharacteristic
184C22#1914966211yescharacteristic
185C4—22no—
186C4—44no—
187C4—6922no—
188C4—6921no—
189C4C4×C22⋊D17312yes—
190C4#1914965312yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—13841no—
195C2#2601137011yescharacteristic
196C2#2601150912yes—
subgroupG/H
centerC2×C4C2×D692
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC2×#117×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012002C2
cosocleC22×C4×C173C23
hypercenterC2×#117D173
hypocenterC173#1897
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19149607C2semidirect
3. #26011455C2semidirect
4. #26011251C2semidirect
5. C2×#117×C173C2semidirect
9. #19149658C4semidirect
10. #19149653C4semidirect
48. C42×C173D4semidirect
98. C173#1897semidirect
101. C2×#117D173semidirect
147. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
C2×#191496072
#260114552
#260112512
C2×#117×C1732
#1897173