Finite Group Databasebeta

← #36353981

Subgroups — #36353981

Order 22144.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#26011998C211yesmaximal, characteristic
2#26011475C211yesmaximal, characteristic, complemented
3#26011454C212yesmaximal, complemented
4#26011262C212yesmaximal, complemented
5#568×C173C211yesmaximal, characteristic, complemented
6C22×C4×D173C2211yescharacteristic
7C2×#14583810C2212yes—
8#19149660C412yes—
9#19149655—21no—
10#19149650—21no—
11#19149647C412yes—
12C4×C22⋊D173—22no—
13#19149638C2211yescharacteristic
14#19149598—21no—
15#19149596—21no—
16#19149591—24no—
17C2×#41×C173C2211yescharacteristic
18#131×C173—24no—
19C2×M16×C173—22no—
20C2×M16×C173C2212yes—
21C23×D173C2×C411yescharacteristic
22C2×C22⋊D173—22no—
23C22×Dic173C2×C411yescharacteristic
24C2×C4×D173—22no—
25C2×C4×D173—21no—
26C2×C4×D173—21no—
27C2×C4×D173—22no—
28#14583810—22no—
29#14583810—22no—
30C2×#13063436—22no—
31C4⋊Dic173—21no—
32#14583805—22no—
33C22⋊Dic173—22no—
34C22⋊Dic173C2×C412yes—
35#14583803—22no—
36C4×Dic173—21no—
37#41×C173—22no—
38#41×C173—22no—
39C2×Q8×C173D411yescharacteristic
40C2×D4×C173C2×C412yes—
41C2×D4×C173D411yescharacteristic
42C22×C4×C173D412yes—
43C22×C4×C173C2311yescharacteristic
44M16×C173—24no—
45M16×C173—22no—
46M16×C173—22no—
47C2×C8×C173—24no—
48C2×C8×C173D414yescomplemented
49C22×D173—21no—
50C22×D173—22no—
51C22×D173—42no—
52C22⋊D173—44no—
53C2×Dic173—21no—
54C2×Dic173—22no—
55C2×Dic173—22no—
56C4×D173—41no—
57C4×D173—42no—
58C4×D173—41no—
59#13063436—42no—
60C23×C173C22⋊C412yes—
61C23×C173C22×C411yescharacteristic
62Q8×C173—22no—
63D4×C173—24no—
64D4×C173—22no—
65C2×C4×C173—22no—
66C2×C4×C173—22no—
67C2×C4×C173C22⋊C412yes—
68C2×C4×C173C2×D412yes—
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C2×C4×C173#4112yes—
72C8×C173—24no—
73C22×C173—22no—
74C22×C173—41no—
75C22×C173—42no—
76C22×C173C2×C22⋊C411yescharacteristic
77C22×C173C4×D412yes—
78C2×D173—22no—
79C2×D173—24no—
80C2×D173—41no—
81C2×D173—42no—
82C2×D173—41no—
83C4×C173—22no—
84C4×C173#12011yescharacteristic
85C4×C173#12112yes—
86C4×C173#12311yescharacteristic
87Dic173—42no—
88Dic173—41no—
89C2×C173—21no—
90C2×C173—22no—
91C2×C173—42no—
92C2×C173#38611yescharacteristic
93D173—42no—
94C173#189111yescharacteristic, complemented, Sylow
95#1891—1731nomaximal, Sylow
96#568D17311yescharacteristic, complemented
97#525—1731no—
98#413—1731no—
99#407—1732no—
100#344—1732no—
101C2×#41C2×D17311yescharacteristic
102C23×C4—1731no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1732no—
106C2×M16C2×D17312yes—
107#123—1732no—
108C22⋊Q8—3461no—
109#121—3461no—
110#120—1732no—
111C4×D4—3462no—
112#117—1731no—
113#101—3461no—
114#100—3461no—
115C22⋊C8—3464no—
116C24—1731no—
117#41—22no—
118#41—22no—
119C2×Q8D69211yescharacteristic
120C2×D4—3462no—
121C2×D4C4×D17312yes—
122C2×D4D69211yescharacteristic
123C22×C4—3462no—
124C22×C4—3461no—
125C22×C4—3461no—
126C22×C4—3462no—
127C22×C4—1731no—
128C22×C4C22⋊D17312yes—
129C22×C4C22×D17311yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—3462no—
134M16—3462no—
135C2×C8—24no—
136C2×C8—3462no—
137C2×C8C22⋊D17314yescomplemented
138C4⋊C4—3461no—
139C4⋊C4—3462no—
140C22⋊C4—3462no—
141C22⋊C4—3462no—
142C22⋊C4—1732no—
143C42—3461no—
144C23—3461no—
145C23—3462no—
146C23—6922no—
147C23#1458380312yes—
148C23C2×C4×D17311yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—6924no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—3462no—
156C2×C4—3461no—
157C2×C4—3462no—
158C2×C4—6921no—
159C2×C4—6922no—
160C2×C4—6921no—
161C2×C4#1458380312yes—
162C2×C4C2×C22⋊D17312yes—
163C2×C4C2×C22⋊D17311yescharacteristic
164C2×C4C2×D69211yescharacteristic
165C2×C4#1458382612yes—
166C8—24no—
167C8—6922no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—3462no—
172C22—3464no—
173C22—6921no—
174C22—6922no—
175C22—6921no—
176C22C2×#1458380311yescharacteristic
177C22C4×C22⋊D17312yes—
178C4—22no—
179C4—6922no—
180C4—6921no—
181C4#1914964811yescharacteristic
182C4#1914965312yes—
183C4#1914966211yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—6922no—
188C2#2601137011yescharacteristic
subgroupG/H
centerC4#19149648
commutatorC23×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#568×C173C2
socleC2×C173#386
prime radical#26011998C2
cosocleC22×C4×C173C23
hypercenter#568D173
hypocenterC173#1891
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26011475C2semidirect
3. #26011454C2semidirect
4. #26011262C2semidirect
5. #568×C173C2semidirect
48. C2×C8×C173D4semidirect
94. C173#1891semidirect
96. #568D173semidirect
137. C2×C8C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260119982
#260114752
#260114542
#260112622
#568×C1732
#1891173