Finite Group Databasebeta

← #44398693

Subgroups — #44398693

Order 29056.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#28409658C211yesmaximal, characteristic
2#28409190C212yesmaximal, complemented
3#28408944C211yesmaximal, characteristic
4#28408770C212yesmaximal, complemented
5#323×C227C211yesmaximal, characteristic
6C22×Dic454C2211yescharacteristic
7C2×Dic908—22no—
8#20300267—21no—
9#20300267C2212yes—
10#20300265—22no—
11#20300265—22no—
12C2×C22⋊Dic227—21no—
13C2×C22⋊Dic227—21no—
14#20300213—22no—
15C227⋊#95—21no—
16C227⋊#95C2211yescharacteristic
17C2×C22⋊C4×C227C2211yescharacteristic
18C22⋊C8×C227C2212yes—
19C2×Dic454—21no—
20C2×Dic454—22no—
21C2×Dic454—42no—
22C2×Dic454D412yes—
23C22×Dic227—21no—
24C22×Dic227—21no—
25C22×Dic227—21no—
26C22×Dic227—21no—
27Dic908—44no—
28C4⋊Dic227—21no—
29C4⋊Dic227D412yes—
30Dic227⋊C4—22no—
31Dic227⋊C4—22no—
32C22⋊Dic227—22no—
33C22⋊Dic227—22no—
34C22⋊Dic227—42no—
35C24×C227D411yescharacteristic
36C22×C4×C227D411yescharacteristic
37C22×C4×C227C2311yescharacteristic
38C2×C8×C227—22no—
39C22⋊C4×C227—22no—
40Dic454—24no—
41Dic454—42no—
42Dic454—42no—
43Dic454—81no—
44C2×Dic227—22no—
45C2×Dic227—22no—
46C2×Dic227—22no—
47C2×Dic227—41no—
48C2×Dic227—41no—
49C2×Dic227—41no—
50C2×Dic227—41no—
51C2×Dic227—41no—
52C2×Dic227—81no—
53C2×Dic227—81no—
54C23×C227—21no—
55C23×C227—21no—
56C23×C227—41no—
57C23×C227—41no—
58C23×C227Q1612yescomplemented
59C23×C227C2×D411yescharacteristic
60C2×C4×C227—21no—
61C2×C4×C227—41no—
62C2×C4×C227—41no—
63C2×C4×C227C2×D412yes—
64C8×C227—42no—
65C22×C227—21no—
66C22×C227—21no—
67C22×C227—22no—
68C22×C227—42no—
69C22×C227—41no—
70C22×C227—41no—
71C22×C227—81no—
72C22×C227#12011yescharacteristic
73C22×C227C2×Q1611yescharacteristic
74C22×C227#13711yescharacteristic
75C4×C227—22no—
76C4×C227—41no—
77Dic227—42no—
78Dic227—42no—
79Dic227—42no—
80Dic227—81no—
81C2×C227—21no—
82C2×C227—21no—
83C2×C227—42no—
84C2×C227#45111yescharacteristic
85C2×C227#45511yescharacteristic
86C2×C227B(4,𝔽2)11yescharacteristic
87C227#160211yescharacteristic, complemented, Sylow
88#1602—2271nomaximal, Sylow
89#543—2271no—
90#451—2272no—
91#393—2271no—
92#328—2272no—
93#323D22711yescharacteristic
94C22×Q8—2271no—
95C2×Q16—4542no—
96C22⋊Q8—4542no—
97C22⋊Q8—4542no—
98C22⋊Q8—4541no—
99C22⋊Q8—2272no—
100C2×C22⋊C4—4541no—
101C2×C22⋊C4—4541no—
102C2×C22⋊C4C2×D22711yescharacteristic
103#103—4542no—
104C22⋊C8C2×D22712yes—
105#95—4541no—
106#95—2271no—
107C24D90811yescharacteristic
108C2×Q8—9082no—
109C2×Q8—4541no—
110C2×Q8—4542no—
111C2×Q8—2272no—
112C22×C4—4541no—
113C22×C4—4541no—
114C22×C4—4541no—
115C22×C4—4541no—
116C22×C4D90811yescharacteristic
117C22×C4C22×D22711yescharacteristic
118Q16—9084no—
119C2×C8—22no—
120C4⋊C4—4542no—
121C4⋊C4—4541no—
122C4⋊C4—4542no—
123C4⋊C4—2272no—
124C22⋊C4—22no—
125C22⋊C4—9082no—
126C22⋊C4—4542no—
127C22⋊C4—4542no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic90812yescomplemented
133C23C2×D90811yescharacteristic
134Q8—18161no—
135Q8—9082no—
136Q8—9082no—
137Q8—4544no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—18161no—
142C2×C4—18161no—
143C2×C4—9081no—
144C2×C4—9081no—
145C2×C4—9081no—
146C2×C4—9081no—
147C2×C4—9081no—
148C2×C4—4542no—
149C2×C4—4542no—
150C2×C4—4542no—
151C2×C4D4×D22712yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#2030025811yescharacteristic
161C22C2×Dic90811yescharacteristic
162C22#2030032111yescharacteristic
163C4—22no—
164C4—41no—
165C4—18161no—
166C4—9082no—
167C4—9082no—
168C4—9082no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2840919011yescharacteristic
173C2#2840921211yescharacteristic
174C2#2840922011yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#323×C227C2
socleC22×C227#120
prime radicalC24×C227D4
hypercenter#323D227
hypocenterC227#1602
LCSG ▷ C22×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #28409190C2semidirect
4. #28408770C2semidirect
58. C23×C227Q16semidirect
87. C227#1602semidirect
132. C23Dic908semidirect

Maximal subgroups

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

subgroupindex
#284096582
#284091902
#284089442
#284087702
#323×C2272
#1602227