Finite Group Databasebeta

← #37509443

Subgroups — #37509443

Order 22912.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#26038950C211yesmaximal, characteristic
2C2×#19153041C212yesmaximal, complemented
3#26038403C211yesmaximal, characteristic, complemented
4#26038199C212yesmaximal, complemented
5C22×C42×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2211yescharacteristic
7C2×#15668243C2212yes—
8#19153096—21no—
9#19153092C412yescomplemented
10#19153087C412yescomplemented
11#19153082—21no—
12C4×C22⋊D179—22no—
13#19153072C2211yescharacteristic
14#19153041—24no—
15C22⋊(C179⋊C8)—22no—
16C23×C4×C179—21no—
17C23×C4×C179C2211yescharacteristic
18C2×C42×C179—24no—
19C2×C42×C179—21no—
20C2×C42×C179C2212yes—
21#15668259—22no—
22#15668259—22no—
23C2×Dic358C2×C411yescharacteristic, complemented
24C2×D716C2×C411yescharacteristic, complemented
25C2×C22⋊D179—22no—
26C2×C22⋊D179—21no—
27C2×C4×D179—21no—
28#15668243—22no—
29#15668243—22no—
30C2×C179⋊C8—22no—
31C4⋊Dic179C2×C412yescomplemented
32Dic179⋊C4—22no—
33C22⋊Dic179—22no—
34C22⋊Dic179—21no—
35(C2×D179)⋊C4—22no—
36C4×Dic179—21no—
37C24×C179D411yescharacteristic
38C22×C4×C179—24no—
39C22×C4×C179—28no—
40C22×C4×C179—22no—
41C22×C4×C179—22no—
42C22×C4×C179—22no—
43C22×C4×C179C2×C412yes—
44C22×C4×C179D411yescharacteristic
45C22×C4×C179D412yes—
46C22×C4×C179C2311yescharacteristic
47C42×C179—24no—
48C42×C179—22no—
49C42×C179D414yescomplemented
50C22×D179—21no—
51Dic358—22no—
52D716—22no—
53C22⋊D179—44no—
54C22⋊D179—41no—
55C22⋊D179—41no—
56C2×Dic179—22no—
57C2×Dic179—21no—
58C4×D179—42no—
59C179⋊C8—42no—
60C23×C179—24no—
61C23×C179—21no—
62C23×C179—21no—
63C23×C179C22⋊C412yes—
64C23×C179C2×D411yescharacteristic
65C2×C4×C179—216no—
66C2×C4×C179—28no—
67C2×C4×C179—24no—
68C2×C4×C179—21no—
69C2×C4×C179—21no—
70C2×C4×C179—24no—
71C2×C4×C179—24no—
72C2×C4×C179C22⋊C412yes—
73C2×C4×C179C22×C411yescharacteristic
74C2×C4×C179C2×D412yes—
75C2×C4×C179C2×D411yescharacteristic
76C2×C4×C179#4112yes—
77C22×C179—24no—
78C22×C179—24no—
79C22×C179—22no—
80C22×C179—22no—
81C22×C179—22no—
82C22×C179#10414yescomplemented
83C22×C179C2×C22⋊C411yescharacteristic
84C22×C179#12011yescharacteristic
85C22×C179#12311yescharacteristic
86C2×D179—41no—
87C2×D179—42no—
88C4×C179—28no—
89C4×C179—22no—
90C4×C179C4×D412yes—
91C4×C179#12112yes—
92Dic179—41no—
93Dic179—42no—
94C2×C179—21no—
95C2×C179—24no—
96C2×C179—21no—
97C2×C179#38611yescharacteristic
98C2×C179C2×#10412yes—
99D179—81no—
100C179#189611yescharacteristic, complemented, Sylow
101#1896—1791nomaximal, Sylow
102#525—1791no—
103C22×C42D17911yescharacteristic, complemented
104C2×#104—1792no—
105#407—1791no—
106#339—1792no—
107C2×#41—1791no—
108C23×C4—21no—
109C23×C4C2×D17911yescharacteristic
110C2×M16—1792no—
111#123—3581no—
112C22⋊Q8—1792no—
113#121—1792no—
114#120—3581no—
115C4×D4—3582no—
116#117—1791no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D17912yes—
120#104—3584no—
121C22⋊C8—3582no—
122C24C22⋊D17911yescharacteristic
123#41—3582no—
124#41—3582no—
125C2×Q8—1791no—
126C2×D4—3582no—
127C2×D4—3581no—
128C2×D4—1791no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—3581no—
135C22×C4C4×D17912yes—
136C22×C4C22⋊D17911yescharacteristic
137C22×C4D71612yes—
138C22×C4C22×D17911yescharacteristic
139M16—3582no—
140M16—3582no—
141C2×C8—3582no—
142C4⋊C4—3582no—
143C4⋊C4—1792no—
144C22⋊C4—3582no—
145C22⋊C4—3581no—
146C22⋊C4—3582no—
147C42—24no—
148C42—22no—
149C42—3581no—
150C42C22⋊D17914yescomplemented
151C23—24no—
152C23—21no—
153C23—21no—
154C23—3581no—
155C23(C2×D179)⋊C412yes—
156C23C2×C22⋊D17911yescharacteristic
157Q8—3582no—
158D4—3582no—
159D4—7164no—
160D4—7161no—
161D4—7161no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—3582no—
170C2×C4—3581no—
171C2×C4—7162no—
172C2×C4(C2×D179)⋊C412yes—
173C2×C4C2×C4×D17911yescharacteristic
174C2×C4C2×C22⋊D17912yes—
175C2×C4C2×D71611yescharacteristic
176C2×C4#1566825912yes—
177C8—7162no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—7161no—
184C22—7162no—
185C22#1915304114yescomplemented
186C22C2×(C2×D179)⋊C411yescharacteristic
187C22#1915308211yescharacteristic
188C22#1915309611yescharacteristic
189C4—28no—
190C4—22no—
191C4—7161no—
192C4—7162no—
193C4C4×C22⋊D17912yes—
194C4#1915308712yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—14321no—
199C2#2603831811yescharacteristic
200C2C2×#1915304112yes—
subgroupG/H
centerC2×C4C2×D716
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
FittingC22×C42×C179C2
socleC22×C179C2×C22⋊C4
prime radical#26038950C2
cosocleC22×C4×C179C23
hypercenterC22×C42D179
hypocenterC179#1896
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#19153041C2semidirect
3. #26038403C2semidirect
4. #26038199C2semidirect
5. C22×C42×C179C2semidirect
9. #19153092C4semidirect
10. #19153087C4semidirect
23. C2×Dic358C2×C4semidirect
24. C2×D716C2×C4semidirect
31. C4⋊Dic179C2×C4semidirect
49. C42×C179D4semidirect
82. C22×C179#104semidirect
100. C179#1896semidirect
103. C22×C42D179semidirect
150. C42C22⋊D179semidirect
185. C22#19153041semidirect

Maximal subgroups

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

subgroupindex
#260389502
C2×#191530412
#260384032
#260381992
C22×C42×C1792
#1896179