Finite Group Databasebeta

← #44398249

Subgroups — #44398249

Order 29056.

346 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×#20300262C211yesmaximal, characteristic
2#28408996C211yesmaximal, characteristic, complemented
3#28408765C214yesmaximal, complemented
4C4×C22⋊C4×C227C211yesmaximal, characteristic, complemented
5C22×D908C411yescharacteristic, complemented
6C22×C22⋊D227—21no—
7C2×#16813868—21no—
8#20300262—21no—
9#20300262—21no—
10#20300262C412yescomplemented
11#20300262C2212yes—
12C2×C4⋊Dic227C411yescharacteristic, complemented
13C2×(C2×D227)⋊C4—21no—
14C22⋊(C227⋊C8)—21no—
15C22⋊(C227⋊C8)C2212yescomplemented
16C23×C4×C227C2211yescharacteristic
17C2×C22⋊C4×C227—21no—
18C2×C42×C227—21no—
19#95×C227C2212yescomplemented
20C23×D227—21no—
21C2×D908—21no—
22C2×D908—42no—
23C2×D908C2×C412yes—
24C2×C22⋊D227—22no—
25C2×C22⋊D227—22no—
26C2×C22⋊D227—44no—
27C22×Dic227—21no—
28#16813868—42no—
29C2×C227⋊C8—22no—
30C4⋊Dic227—21no—
31C4⋊Dic227C2×C412yes—
32(C2×D227)⋊C4—22no—
33(C2×D227)⋊C4—22no—
34C24×C227D411yescharacteristic
35C22×C4×C227—22no—
36C22×C4×C227—42no—
37C22×C4×C227C2×C411yescharacteristic
38C22×C4×C227C2×C411yescharacteristic
39C22×C4×C227D411yescharacteristic
40C22×C4×C227D411yescharacteristic
41C22×C4×C227D411yescharacteristic
42C22×C4×C227C2311yescharacteristic
43C22⋊C4×C227—24no—
44C42×C227—42no—
45C22×D227—22no—
46C22×D227—42no—
47C22×D227—42no—
48C22×D227—42no—
49D908—42no—
50D908—42no—
51C22⋊D227—44no—
52C22⋊D227—44no—
53C2×Dic227—22no—
54C2×Dic227—42no—
55C227⋊C8—42no—
56C23×C227—22no—
57C23×C227—22no—
58C23×C227—41no—
59C23×C227C22⋊C411yescharacteristic
60C23×C227C22⋊C411yescharacteristic
61C23×C227C2×D411yescharacteristic
62C2×C4×C227—22no—
63C2×C4×C227—22no—
64C2×C4×C227—22no—
65C2×C4×C227—24no—
66C2×C4×C227—41no—
67C2×C4×C227—41no—
68C2×C4×C227—41no—
69C2×C4×C227—42no—
70C2×C4×C227—42no—
71C2×C4×C227C22⋊C411yescharacteristic
72C2×C4×C227C22⋊C411yescharacteristic
73C2×C4×C227C22×C411yescharacteristic
74C2×C4×C227C2×D411yescharacteristic
75C22×C227—21no—
76C22×C227—21no—
77C22×C227—21no—
78C22×C227—21no—
79C22×C227—21no—
80C22×C227—21no—
81C22×C227—21no—
82C22×C227—44no—
83C22×C227#10412yes—
84C22×C227C2×C22⋊C411yescharacteristic
85C22×C227AGL(1,ℤ/8)12yes—
86C2×D227—42no—
87C2×D227—42no—
88C2×D227—42no—
89C2×D227—84no—
90C4×C227—21no—
91C4×C227—21no—
92C4×C227—21no—
93C4×C227—44no—
94C4×C227#9912yes—
95Dic227—42no—
96C2×C227—22no—
97C2×C227—22no—
98C2×C227—41no—
99C2×C227C2×#9911yescharacteristic
100C2×C227#41811yescharacteristic
101C2×C227C2×#10411yescharacteristic
102D227—82no—
103C227#150711yescharacteristic, complemented, Sylow
104#1507—2271nomaximal, Sylow
105C2×#121—2271no—
106#407—2271no—
107C4×C22⋊C4D22711yescharacteristic, complemented
108#327—2274no—
109C22×D4—4541no—
110C22×D4—2271no—
111C23×C4C2×D22711yescharacteristic
112C2×M16—4541no—
113#121—4541no—
114#121—4541no—
115#121—2272no—
116#121—2272no—
117C2×C4⋊C4—2271no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—4541no—
120C2×C42—21no—
121C22⋊C8—4541no—
122C22⋊C8—2272no—
123#95C2×D22712yescomplemented
124C24—4541no—
125C24C22⋊D22711yescharacteristic
126C2×D4—9082no—
127C2×D4—9084no—
128C2×D4—4542no—
129C2×D4—4542no—
130C2×D4—4541no—
131C2×D4—2272no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—4541no—
135C22×C4C4×D22711yescharacteristic
136C22×C4C4×D22711yescharacteristic
137C22×C4C22⋊D22711yescharacteristic
138C22×C4D90811yescharacteristic
139C22×C4D90811yescharacteristic
140C22×C4C22×D22711yescharacteristic
141M16—9082no—
142C2×C8—4542no—
143C4⋊C4—4541no—
144C4⋊C4—2272no—
145C22⋊C4—24no—
146C22⋊C4—4542no—
147C22⋊C4—4542no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—9082no—
153C23—9082no—
154C23—9082no—
155C23—4542no—
156C23(C2×D227)⋊C411yescharacteristic
157C23(C2×D227)⋊C411yescharacteristic
158C23C2×C22⋊D22711yescharacteristic
159D4—9082no—
160D4—9084no—
161D4—9084no—
162D4—9082no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—9082no—
173C2×C4—4542no—
174C2×C4(C2×D227)⋊C411yescharacteristic
175C2×C4(C2×D227)⋊C411yescharacteristic
176C2×C4C2×C4×D22711yescharacteristic
177C2×C4C2×D90811yescharacteristic
178C8—9082no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—18164no—
188C22—9082no—
189C22—9082no—
190C22—9082no—
191C22#2030021612yes—
192C22C2×(C2×D227)⋊C411yescharacteristic
193C22#2030031312yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—44no—
198C4—9082no—
199C4#2030020212yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—18162no—
204C2C2×#2030020211yescharacteristic
205C2#2840903611yescharacteristic
206C2C2×#2030021611yescharacteristic
subgroupG/H
centerC22C2×(C2×D227)⋊C4
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
FittingC4×C22⋊C4×C227C2
socleC22×C227C2×C22⋊C4
prime radicalC2×#20300262C2
cosocleC22×C4×C227C23
hypercenterC4×C22⋊C4D227
hypocenterC227#1507
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #28408996C2semidirect
3. #28408765C2semidirect
4. C4×C22⋊C4×C227C2semidirect
5. C22×D908C4semidirect
10. #20300262C4semidirect
12. C2×C4⋊Dic227C4semidirect
15. C22⋊(C227⋊C8)C22semidirect
19. #95×C227C22semidirect
103. C227#1507semidirect
107. C4×C22⋊C4D227semidirect
123. #95C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002622
#284089962
#284087652
C4×C22⋊C4×C2272
#1507227