Finite Group Databasebeta

← #37509532

Subgroups — #37509532

Order 22912.

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

normal subgroups only

This group has 362 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#26039006C211yesmaximal, characteristic
2#26038457C212yesmaximal, complemented
3C2×#19153030C211yesmaximal, characteristic, complemented
4#26038195C212yesmaximal, complemented
5#518×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2211yescharacteristic
7C2×#15668243C2212yes—
8#19153109C412yescomplemented
9#19153100C412yescomplemented
10#19153084—22no—
11#19153082—22no—
12#19153072C2211yescharacteristic
13#19153041—24no—
14#19153030—22no—
15C22×D4×C179C2211yescharacteristic
16C4×D4×C179—24no—
17#117×C179C2212yes—
18C2×C22⋊C4×C179—22no—
19#15668259—22no—
20#15668259—22no—
21C2×Dic358C2×C411yescharacteristic, complemented
22C2×D716C2×C411yescharacteristic, complemented
23C2×C22⋊D179—24no—
24C2×C22⋊D179—21no—
25C2×C4×D179—21no—
26#15668243—22no—
27#15668243—22no—
28C2×C179⋊C8—22no—
29C4⋊Dic179—21no—
30C22⋊Dic179—24no—
31C22⋊Dic179—21no—
32C4×Dic179C2×C412yescomplemented
33C24×C179D412yes—
34C2×D4×C179—22no—
35C2×D4×C179—22no—
36C2×D4×C179C2×C412yes—
37C2×D4×C179D412yes—
38C22×C4×C179—24no—
39C22×C4×C179C2311yescharacteristic
40C4⋊C4×C179—22no—
41C22⋊C4×C179—24no—
42C22⋊C4×C179—22no—
43C42×C179D414yescomplemented
44C22×D179—21no—
45Dic358—22no—
46D716—22no—
47C22⋊D179—41no—
48C22⋊D179—41no—
49C22⋊D179—84no—
50C2×Dic179—22no—
51C2×Dic179—21no—
52C4×D179—42no—
53C179⋊C8—42no—
54C23×C179—22no—
55C23×C179—22no—
56C23×C179—82no—
57C23×C179C22⋊C414yes—
58C23×C179C2×D411yescharacteristic
59D4×C179—24no—
60D4×C179—24no—
61C2×C4×C179—24no—
62C2×C4×C179—84no—
63C2×C4×C179C22×C411yescharacteristic
64C2×C4×C179C2×D412yes—
65C2×C4×C179C2×D411yescharacteristic
66C2×C4×C179#4112yes—
67C22×C179—24no—
68C22×C179—41no—
69C22×C179—44no—
70C22×C179—44no—
71C22×C179—82no—
72C22×C179C2×C22⋊C411yescharacteristic
73C22×C179#12011yescharacteristic
74C22×C179#12311yescharacteristic
75C2×D179—42no—
76C2×D179—41no—
77C4×C179—44no—
78C4×C179C4×D412yes—
79C4×C179#12112yes—
80Dic179—42no—
81Dic179—41no—
82C2×C179—21no—
83C2×C179—21no—
84C2×C179—21no—
85C2×C179—44no—
86C2×C179#38611yescharacteristic
87D179—81no—
88C179#191111yescharacteristic, complemented, Sylow
89#1911—1791nomaximal, Sylow
90#535—1791no—
91#518D17911yescharacteristic, complemented
92#421—1792no—
93C2×#100—1791no—
94H(3,ℤ/4)—1792no—
95C2×#41—1791no—
96C22×D4C2×D17911yescharacteristic
97C2×M16—1792no—
98#127—1792no—
99#124—1792no—
100#121—3582no—
101#120—3582no—
102C4×D4—24no—
103#117—1791no—
104#117C2×D17912yes—
105C2×C22⋊C4—22no—
106#104—3584no—
107#100—3582no—
108C24C22⋊D17912yes—
109#41—3582no—
110#41—3582no—
111C2×Q8—1791no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—3584no—
115C2×D4—3581no—
116C2×D4—1791no—
117C2×D4C4×D17912yes—
118C2×D4D71612yes—
119C22×C4—24no—
120C22×C4—3581no—
121C22×C4C22×D17911yescharacteristic
122M16—3582no—
123M16—3582no—
124C2×C8—3582no—
125C4⋊C4—22no—
126C4⋊C4—3581no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—3584no—
130C22⋊C4—3581no—
131C42—1792no—
132C42C22⋊D17914yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—3581no—
137C23(C2×D179)⋊C414yes—
138C23C2×C22⋊D17911yescharacteristic
139Q8—3582no—
140D4—24no—
141D4—24no—
142D4—3582no—
143D4—14324no—
144D4—7161no—
145D4—7161no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—3582no—
149C2×C4—3581no—
150C2×C4—7162no—
151C2×C4C2×C4×D17911yescharacteristic
152C2×C4C2×C22⋊D17912yes—
153C2×C4C2×D71611yescharacteristic
154C2×C4#1566825912yes—
155C8—7162no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—7162no—
162C22—7161no—
163C22C2×(C2×D179)⋊C411yescharacteristic
164C22#1915308211yescharacteristic
165C22#1915309611yescharacteristic
166C4—44no—
167C4—7162no—
168C4—7161no—
169C4C4×C22⋊D17912yes—
170C4#1915308712yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—14321no—
176C2#2603831811yescharacteristic
subgroupG/H
centerC2#26038318
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#518×C179C2
socleC2×C179#386
prime radical#26039006C2
cosocleC22×C4×C179C23
hypercenter#518D179
hypocenterC179#1911
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #26038457C2semidirect
3. C2×#19153030C2semidirect
4. #26038195C2semidirect
5. #518×C179C2semidirect
8. #19153109C4semidirect
9. #19153100C4semidirect
21. C2×Dic358C2×C4semidirect
22. C2×D716C2×C4semidirect
32. C4×Dic179C2×C4semidirect
43. C42×C179D4semidirect
88. C179#1911semidirect
91. #518D179semidirect
132. C42C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260390062
#260384572
C2×#191530302
#260381952
#518×C1792
#1911179