Finite Group Databasebeta

← #37509527

Subgroups — #37509527

Order 22912.

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

normal subgroups only

This group has 344 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#26039003C211yesmaximal, characteristic
2#26038445C212yesmaximal, complemented
3#26038413C211yesmaximal, characteristic, complemented
4#26038196C212yesmaximal, complemented
5#568×C179C211yesmaximal, characteristic, complemented
6C22×D716C2211yescharacteristic
7#19153109C412yescomplemented
8#19153102C412yescomplemented
9#19153087—22no—
10#19153081—22no—
11#19153072C2212yes—
12#19153072C2211yescharacteristic
13#19153035—24no—
14#19153028—22no—
15C2×#41×C179C2211yescharacteristic
16#131×C179—24no—
17C2×M16×C179—22no—
18C2×M16×C179C2212yes—
19C23×D179—21no—
20C2×D716—22no—
21C2×D716—21no—
22C2×D716—22no—
23C2×D716C2×C412yes—
24C2×C22⋊D179—24no—
25C4⋊Dic179—21no—
26C4⋊Dic179—22no—
27C22⋊Dic179—21no—
28C22⋊Dic179—22no—
29(C2×D179)⋊C4—24no—
30C4×Dic179—22no—
31C4×Dic179C2×C412yes—
32#41×C179—22no—
33#41×C179—22no—
34C2×Q8×C179C2×C411yescharacteristic
35C2×D4×C179C2×C411yescharacteristic
36C2×D4×C179D412yes—
37C22×C4×C179D412yes—
38C22×C4×C179C2311yescharacteristic
39M16×C179—24no—
40M16×C179—22no—
41M16×C179—22no—
42C2×C8×C179—24no—
43C2×C8×C179D414yescomplemented
44C22×D179—22no—
45C22×D179—42no—
46C22×D179—82no—
47D716—24no—
48D716—41no—
49D716—41no—
50C22⋊D179—84no—
51C2×Dic179—22no—
52C2×Dic179—42no—
53C23×C179C22⋊C412yes—
54C23×C179C2×D411yescharacteristic
55Q8×C179—22no—
56D4×C179—24no—
57D4×C179—22no—
58C2×C4×C179—22no—
59C2×C4×C179—22no—
60C2×C4×C179C22⋊C412yes—
61C2×C4×C179C22×C411yescharacteristic
62C2×C4×C179C2×D412yes—
63C2×C4×C179C2×D411yescharacteristic
64C2×C4×C179#4112yes—
65C8×C179—24no—
66C22×C179—22no—
67C22×C179—41no—
68C22×C179—42no—
69C22×C179C2×C22⋊C411yescharacteristic
70C22×C179#12112yes—
71C2×D179—44no—
72C2×D179—42no—
73C2×D179—84no—
74C4×C179—22no—
75C4×C179C4×D412yes—
76C4×C179#12011yescharacteristic
77C4×C179#12311yescharacteristic
78Dic179—42no—
79Dic179—82no—
80C2×C179—21no—
81C2×C179—22no—
82C2×C179—42no—
83C2×C179#38611yescharacteristic
84D179—82no—
85C179#191011yescharacteristic, complemented, Sylow
86#1910—1791nomaximal, Sylow
87#568D17911yescharacteristic, complemented
88#535—1791no—
89#418—1792no—
90#410—1791no—
91H(3,ℤ/4)—1792no—
92C2×#41C2×D17911yescharacteristic
93C22×D4—1791no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D17912yes—
97#127—1792no—
98#124—1792no—
99#121—3582no—
100#120—3582no—
101#117—1792no—
102#117—1791no—
103#102—3584no—
104#99—3582no—
105C24—3581no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D17911yescharacteristic
109C2×D4—3584no—
110C2×D4—3582no—
111C2×D4—3581no—
112C2×D4—3582no—
113C2×D4—1792no—
114C2×D4C4×D17911yescharacteristic
115C2×D4D71612yes—
116C22×C4C22⋊D17912yes—
117C22×C4C22×D17911yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D17914yescomplemented
123C4⋊C4—3581no—
124C4⋊C4—3582no—
125C22⋊C4—3581no—
126C22⋊C4—3584no—
127C22⋊C4—3582no—
128C42—3582no—
129C42—1792no—
130C23—3582no—
131C23—14322no—
132C23—7162no—
133C23(C2×D179)⋊C412yes—
134C23C2×D71611yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—3584no—
139D4—14324no—
140D4—7161no—
141D4—7161no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—3582no—
145C2×C4—7162no—
146C2×C4(C2×D179)⋊C412yes—
147C2×C4C2×C4×D17911yescharacteristic
148C2×C4C2×C22⋊D17912yes—
149C2×C4C2×C22⋊D17911yescharacteristic
150C2×C4#1566825912yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—14324no—
156C22—7164no—
157C22—7162no—
158C22C2×(C2×D179)⋊C411yescharacteristic
159C22#1915308712yes—
160C4—22no—
161C4—14322no—
162C4—7162no—
163C4C4×C22⋊D17912yes—
164C4#1915308211yescharacteristic
165C4#1915309611yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—14322no—
170C2#2603831811yescharacteristic
subgroupG/H
centerC2#26038318
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#568×C179C2
socleC2×C179#386
prime radical#26039003C2
cosocleC22×C4×C179C23
hypercenter#568D179
hypocenterC179#1910
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26038445C2semidirect
3. #26038413C2semidirect
4. #26038196C2semidirect
5. #568×C179C2semidirect
7. #19153109C4semidirect
8. #19153102C4semidirect
43. C2×C8×C179D4semidirect
85. C179#1910semidirect
87. #568D179semidirect
122. C2×C8C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260390032
#260384452
#260384132
#260381962
#568×C1792
#1910179