Finite Group Databasebeta

← #37509525

Subgroups — #37509525

Order 22912.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#26039206C211yesmaximal, characteristic, complemented
2#26039003C211yesmaximal, characteristic
3#26038443C212yesmaximal, complemented
4#26038195C212yesmaximal, complemented
5#410×C179C211yesmaximal, characteristic, complemented
6C22×D716C2211yescharacteristic
7#19153120—24no—
8C2×#15668243—22no—
9C2×#15668243C2212yes—
10#19153109C412yescomplemented
11#19153102C412yescomplemented
12#19153087—22no—
13#19153081—22no—
14#19153072C2211yescharacteristic
15#19153033—24no—
16C2×#41×C179C2211yescharacteristic
17#117×C179C2212yes—
18#99×C179—22no—
19C23×D179—21no—
20C2×D716—22no—
21C2×D716—21no—
22C2×D716—22no—
23C2×D716C2×C412yes—
24C2×C22⋊D179—24no—
25#15668243—24no—
26#15668243—22no—
27#15668243—22no—
28C2×C179⋊C8—24no—
29C2×C179⋊C8D414yescomplemented
30C4⋊Dic179—21no—
31C22⋊Dic179—21no—
32(C2×D179)⋊C4—24no—
33C4×Dic179C2×C412yes—
34#41×C179—22no—
35#41×C179—22no—
36C2×Q8×C179C2×C411yescharacteristic
37C2×D4×C179C2×C411yescharacteristic
38C2×D4×C179D412yes—
39C22×C4×C179D412yes—
40C22×C4×C179C2311yescharacteristic
41C4⋊C4×C179—22no—
42C22⋊C4×C179—22no—
43C42×C179—22no—
44C22×D179—22no—
45C22×D179—42no—
46C22×D179—82no—
47D716—24no—
48D716—41no—
49D716—41no—
50C22⋊D179—84no—
51C2×Dic179—22no—
52C179⋊C8—24no—
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×C179—42no—
61C2×C4×C179C22⋊C412yes—
62C2×C4×C179C22×C411yescharacteristic
63C2×C4×C179C2×D412yes—
64C2×C4×C179C2×D411yescharacteristic
65C2×C4×C179#4112yes—
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×C179—82no—
76C4×C179C4×D412yes—
77C4×C179#12011yescharacteristic
78C4×C179#12311yescharacteristic
79Dic179—42no—
80C2×C179—21no—
81C2×C179—22no—
82C2×C179—42no—
83C2×C179#38611yescharacteristic
84D179—82no—
85C179#191011yescharacteristic, complemented, Sylow
86#1910—1791nomaximal, Sylow
87#568—1791no—
88#535—1791no—
89#418—1792no—
90#410D17911yescharacteristic, complemented
91H(3,ℤ/4)—1792no—
92C2×#41C2×D17911yescharacteristic
93C22×D4—1791no—
94#131—3584no—
95C2×M16—3582no—
96C2×M16—1792no—
97#127—1792no—
98#124—1792no—
99#121—3582no—
100#120—3582no—
101#117—1791no—
102#117C2×D17912yes—
103#102—3584no—
104#99—22no—
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×D4C22⋊D17912yes—
116C22×C4D71612yes—
117C22×C4C22×D17911yescharacteristic
118M16—3584no—
119M16—3582no—
120M16—3582no—
121C2×C8—3584no—
122C2×C8—1794no—
123C4⋊C4—22no—
124C4⋊C4—3581no—
125C22⋊C4—22no—
126C22⋊C4—3581no—
127C22⋊C4—3584no—
128C42—22no—
129C42—1792no—
130C23—3582no—
131C23—14322no—
132C23—7162no—
133C23(C2×D179)⋊C412yes—
134C23C2×C22⋊D17911yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—3584no—
139D4—14324no—
140D4—7161no—
141D4—7161no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—3582no—
146C2×C4(C2×D179)⋊C412yes—
147C2×C4C2×C4×D17911yescharacteristic
148C2×C4D4×D17912yes—
149C2×C4C2×D71611yescharacteristic
150C2×C4#1566825812yes—
151C8—3584no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—14324no—
156C22—7164no—
157C22—7162no—
158C22C2×(C2×D179)⋊C411yescharacteristic
159C22#1915308512yes—
160C4—22no—
161C4—82no—
162C4—7162no—
163C4(C22⋊D179)⋊C412yes—
164C4#1915308311yescharacteristic
165C4#1915309711yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—14322no—
170C2#2603831611yescharacteristic
subgroupG/H
centerC2#26038316
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#410×C179C2
socleC2×C179#386
prime radical#26039003C2
cosocleC22×C4×C179C23
hypercenter#410D179
hypocenterC179#1910
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #26039206C2semidirect
3. #26038443C2semidirect
4. #26038195C2semidirect
5. #410×C179C2semidirect
10. #19153109C4semidirect
11. #19153102C4semidirect
29. C2×C179⋊C8D4semidirect
85. C179#1910semidirect
90. #410D179semidirect

Maximal subgroups

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

subgroupindex
#260392062
#260390032
#260384432
#260381952
#410×C1792
#1910179