Finite Group Databasebeta

← #44400180

Subgroups — #44400180

Order 29056.

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

normal subgroups only

This group has 358 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#28409543C211yesmaximal, characteristic
2C2×#20300247C211yesmaximal, characteristic, complemented
3C2×#20300214C212yesmaximal, complemented
4#28408793C212yesmaximal, complemented
5#407×C227C211yesmaximal, characteristic, complemented
6C2×#16813884C2211yescharacteristic
7#20300271—21no—
8#20300267C412yescomplemented
9#20300262C412yescomplemented
10#20300257—21no—
11C4×C22⋊D227—22no—
12#20300247—24no—
13#20300247C2211yescharacteristic
14C2×C4⋊Dic227—21no—
15C2×C22⋊Dic227—21no—
16C2×C4×Dic227C2212yes—
17#20300214—24no—
18C23×C4×C227C2211yescharacteristic
19C2×M16×C227C2212yes—
20C22⋊C8×C227—22no—
21#16813884—22no—
22#16813884—22no—
23C2×Dic454C2×C411yescharacteristic
24C2×D908C2×C411yescharacteristic
25C2×C22⋊D227—22no—
26C2×C22⋊D227—21no—
27C22×Dic227—22no—
28C2×C4×D227—21no—
29C4⋊Dic227—24no—
30C4⋊Dic227C2×C412yes—
31Dic227⋊C4—22no—
32C22⋊Dic227—24no—
33C22⋊Dic227—22no—
34C22⋊Dic227—21no—
35(C2×D227)⋊C4—22no—
36C4×Dic227—22no—
37C4×Dic227—21no—
38C4×Dic227D414yescomplemented
39C24×C227D411yescharacteristic
40C22×C4×C227—22no—
41C22×C4×C227—22no—
42C22×C4×C227C2×C412yes—
43C22×C4×C227D412yes—
44C22×C4×C227D411yescharacteristic
45C22×C4×C227C2311yescharacteristic
46M16×C227—22no—
47M16×C227—22no—
48C2×C8×C227—22no—
49C22×D227—21no—
50Dic454—22no—
51D908—22no—
52C22⋊D227—44no—
53C22⋊D227—41no—
54C22⋊D227—41no—
55C2×Dic227—22no—
56C2×Dic227—21no—
57C2×Dic227—24no—
58C2×Dic227—44no—
59C4×D227—42no—
60C23×C227—21no—
61C23×C227—21no—
62C23×C227—42no—
63C23×C227C22⋊C412yes—
64C23×C227C2×D411yescharacteristic
65C2×C4×C227—21no—
66C2×C4×C227—21no—
67C2×C4×C227—24no—
68C2×C4×C227—24no—
69C2×C4×C227C22⋊C412yes—
70C2×C4×C227C22×C411yescharacteristic
71C2×C4×C227C2×D412yes—
72C2×C4×C227C2×D411yescharacteristic
73C2×C4×C227#4112yes—
74C8×C227—42no—
75C22×C227—22no—
76C22×C227—22no—
77C22×C227—22no—
78C22×C227—22no—
79C22×C227—42no—
80C22×C227—42no—
81C22×C227C2×C22⋊C411yescharacteristic
82C22×C227#12011yescharacteristic
83C22×C227#12311yescharacteristic
84C2×D227—41no—
85C2×D227—42no—
86C4×C227—22no—
87C4×C227C4×D412yes—
88C4×C227#12112yes—
89Dic227—44no—
90Dic227—41no—
91Dic227—42no—
92C2×C227—21no—
93C2×C227—21no—
94C2×C227—42no—
95C2×C227#38611yescharacteristic
96C2×C227#42112yes—
97D227—81no—
98C227#189711yescharacteristic, complemented, Sylow
99#1897—2271nomaximal, Sylow
100#525—2271no—
101C2×#117—2271no—
102C2×#104—2272no—
103#407D22711yescharacteristic, complemented
104#339—2272no—
105C2×#41—2271no—
106C23×C4C2×D22711yescharacteristic
107C2×M16C2×D22712yes—
108#123—4541no—
109C22⋊Q8—2272no—
110#121—2272no—
111#120—4541no—
112C4×D4—4542no—
113#117—4544no—
114#117—2271no—
115C2×C4⋊C4—4541no—
116C2×C22⋊C4—4541no—
117C2×C42—2272no—
118#104—4544no—
119C22⋊C8—22no—
120C24D90811yescharacteristic
121#41—4542no—
122#41—4542no—
123C2×Q8—2271no—
124C2×D4—4542no—
125C2×D4—4541no—
126C2×D4—2271no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—4542no—
130C22×C4—4541no—
131C22×C4C4×D22712yes—
132C22×C4C22⋊D22712yes—
133C22×C4D90811yescharacteristic
134C22×C4C22×D22711yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—4544no—
139C4⋊C4—4542no—
140C4⋊C4—2272no—
141C22⋊C4—4544no—
142C22⋊C4—4542no—
143C22⋊C4—4541no—
144C22⋊C4—4542no—
145C42—4542no—
146C42—4541no—
147C42—2274no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—4541no—
152C23(C2×D227)⋊C412yes—
153C23C2×D90811yescharacteristic
154Q8—4542no—
155D4—9084no—
156D4—9081no—
157D4—9081no—
158D4—4542no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—9084no—
164C2×C4—9082no—
165C2×C4—4542no—
166C2×C4—4541no—
167C2×C4—4544no—
168C2×C4(C2×D227)⋊C412yes—
169C2×C4C2×C4×D22711yescharacteristic
170C2×C4D4×D22712yes—
171C2×C4C2×C22⋊D22711yescharacteristic
172C2×C4#1681388312yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—9081no—
181C22—9082no—
182C22C2×(C2×D227)⋊C411yescharacteristic
183C22#2030025811yescharacteristic
184C22#2030027211yescharacteristic
185C4—22no—
186C4—9084no—
187C4—9081no—
188C4—9082no—
189C4(C22⋊D227)⋊C412yes—
190C4#2030026012yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—18161no—
195C2#2840890911yescharacteristic
196C2#2840904812yes—
subgroupG/H
centerC2×C4C2×C22⋊D227
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#407×C227C2
socleC22×C227C2×C22⋊C4
prime radical#28409543C2
cosocleC22×C4×C227C23
hypercenter#407D227
hypocenterC227#1897
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#20300247C2semidirect
3. C2×#20300214C2semidirect
4. #28408793C2semidirect
5. #407×C227C2semidirect
8. #20300267C4semidirect
9. #20300262C4semidirect
38. C4×Dic227D4semidirect
98. C227#1897semidirect
103. #407D227semidirect

Maximal subgroups

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

subgroupindex
#284095432
C2×#203002472
C2×#203002142
#284087932
#407×C2272
#1897227