Finite Group Databasebeta

← #44400147

Subgroups — #44400147

Order 29056.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20300269C211yesmaximal, characteristic
2C2×#20300203C211yesmaximal, characteristic
3#28408893C212yesmaximal, complemented
4#28408799C212yesmaximal, complemented
5#518×C227C211yesmaximal, characteristic
6C23×Dic227C2211yescharacteristic
7#20300269—22no—
8#20300269C414yescomplemented
9C2×Dic227⋊C4—21no—
10C2×C22⋊Dic227—21no—
11C2×C22⋊Dic227C2212yes—
12C2×C22⋊Dic227C2211yescharacteristic
13#20300203—22no—
14C227⋊#95—24no—
15C22×D4×C227C2211yescharacteristic
16C4×D4×C227—24no—
17#117×C227—21no—
18C2×C22⋊C4×C227—21no—
19C2×C22⋊C4×C227C2212yes—
20C22×Dic227—22no—
21C22×Dic227—21no—
22C22×Dic227—21no—
23C22×Dic227—21no—
24C22×Dic227—22no—
25C22×Dic227—22no—
26C22×Dic227C2×C412yescomplemented
27Dic227⋊C4—24no—
28C22⋊Dic227—24no—
29C22⋊Dic227—21no—
30C22⋊Dic227—22no—
31C22⋊Dic227—22no—
32C22⋊Dic227C2×C412yescomplemented
33C24×C227D411yescharacteristic
34C24×C227C2311yescharacteristic
35C2×D4×C227—22no—
36C2×D4×C227—22no—
37C2×D4×C227C2×C412yes—
38C2×D4×C227D412yes—
39C22×C4×C227—22no—
40C22×C4×C227D411yescharacteristic
41C22×C4×C227D414yes—
42C4⋊C4×C227—22no—
43C22⋊C4×C227—22no—
44C22⋊C4×C227—22no—
45C22⋊C4×C227—22no—
46C42×C227—22no—
47C2×Dic227—22no—
48C2×Dic227—22no—
49C2×Dic227—24no—
50C2×Dic227—42no—
51C2×Dic227—41no—
52C2×Dic227—42no—
53C2×Dic227—41no—
54C2×Dic227—44no—
55C2×Dic227—42no—
56C23×C227—21no—
57C23×C227—21no—
58C23×C227—42no—
59C23×C227—42no—
60C23×C227C22⋊C412yes—
61C23×C227C22×C411yescharacteristic
62C23×C227C2×D411yescharacteristic
63C23×C227C2×D412yes—
64C23×C227C2×D411yescharacteristic
65C23×C227#4112yes—
66D4×C227—24no—
67D4×C227—24no—
68C2×C4×C227—21no—
69C2×C4×C227—21no—
70C2×C4×C227—24no—
71C2×C4×C227—44no—
72C2×C4×C227—82no—
73C2×C4×C227C22⋊C412yes—
74C22×C227—24no—
75C22×C227—22no—
76C22×C227—22no—
77C22×C227—41no—
78C22×C227—41no—
79C22×C227—42no—
80C22×C227—42no—
81C22×C227—42no—
82C22×C227—81no—
83C22×C227C2×C22⋊C411yescharacteristic
84C22×C227C4×D412yes—
85C22×C227#12011yescharacteristic
86C22×C227#12112yes—
87C22×C227#12311yescharacteristic
88C4×C227—22no—
89C4×C227—44no—
90Dic227—42no—
91Dic227—42no—
92Dic227—82no—
93C2×C227—21no—
94C2×C227—22no—
95C2×C227—21no—
96C2×C227—22no—
97C2×C227—42no—
98C2×C227#38611yescharacteristic
99C2×C227#45812yes—
100C227#189011yescharacteristic, complemented, Sylow
101#1890—2271nomaximal, Sylow
102C2×#123—2271no—
103#518D22711yescharacteristic
104C2×#99—2271no—
105#381—2272no—
106#342—2272no—
107C22×D4C2×D22711yescharacteristic
108C23×C4—2271no—
109#123—4542no—
110#123—2274no—
111C4×D4—24no—
112#117—21no—
113C2×C4⋊C4—4541no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—4541no—
116C2×C22⋊C4—2272no—
117C2×C22⋊C4—2271no—
118C2×C22⋊C4C2×D22712yes—
119#99—4542no—
120#95—4544no—
121C24C22⋊D22711yescharacteristic
122C24C22×D22711yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C4×D22712yes—
126C2×D4D90812yes—
127C22×C4—22no—
128C22×C4—4542no—
129C22×C4—4542no—
130C22×C4—4541no—
131C22×C4—4541no—
132C22×C4—4541no—
133C22×C4—4542no—
134C22×C4—2272no—
135C22×C4C22⋊D22711yescharacteristic
136C22×C4C22⋊D22714yes—
137C4⋊C4—22no—
138C4⋊C4—4544no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—4542no—
143C22⋊C4—4542no—
144C22⋊C4—4544no—
145C22⋊C4—4541no—
146C22⋊C4—2272no—
147C42—22no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23(C2×D227)⋊C412yes—
153C23C2×C4×D22711yescharacteristic
154C23C2×C22⋊D22711yescharacteristic
155C23C2×C22⋊D22712yes—
156C23C2×D90811yescharacteristic
157C23#1681388412yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—82no—
165C2×C4—9084no—
166C2×C4—9082no—
167C2×C4—9082no—
168C2×C4—9081no—
169C2×C4—9082no—
170C2×C4—9081no—
171C2×C4—4542no—
172C2×C4—4542no—
173C2×C4—4544no—
174C2×C4(C2×D227)⋊C412yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×(C2×D227)⋊C411yescharacteristic
185C22C4×C22⋊D22712yes—
186C22#2030025711yescharacteristic
187C22#2030026212yes—
188C22#2030027111yescharacteristic
189C4—22no—
190C4—44no—
191C4—18162no—
192C4—9082no—
193C4—9082no—
194C2—22no—
195C2—21no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#2840891111yescharacteristic
200C2#2840922212yes—
subgroupG/H
centerC22#20300257
commutatorC23×C227C22×C4
FrattiniC24C22×D227
Fitting#518×C227C2
socleC22×C227#120
prime radicalC22×D4×C227C22
cosocleC24×C227C23
hypercenter#518D227
hypocenterC227#1890
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #28408893C2semidirect
4. #28408799C2semidirect
8. #20300269C4semidirect
26. C22×Dic227C2×C4semidirect
32. C22⋊Dic227C2×C4semidirect
100. C227#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#203002692
C2×#203002032
#284088932
#284087992
#518×C2272
#1890227