Finite Group Databasebeta

← #44400837

Subgroups — #44400837

Order 29056.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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#28409830C212yesmaximal, complemented
2#28409000C211yesmaximal, characteristic, complemented
3#28408999C212yesmaximal, complemented
4C227⋊#338C211yesmaximal, characteristic, complemented
5#575×C227C211yesmaximal, characteristic, complemented
6C2×#16813884C2211yescharacteristic
7#20300319—22no—
8#20300318—22no—
9#20300314—22no—
10#20300313—22no—
11#20300308—24no—
12C2×#16813868C2212yes—
13C22×C227⋊C8C2211yescharacteristic
14C22⋊(C227⋊C8)—24no—
15(C2×D227)⋊C8—22no—
16C2×#41×C227C2212yes—
17#137×C227—22no—
18AGL(1,ℤ/8)×C227—22no—
19#135×C227—22no—
20C2×M16×C227C2211yescharacteristic
21#16813884—22no—
22#16813884—22no—
23C2×Dic454D411yescharacteristic
24C2×D908D411yescharacteristic
25C2×C22⋊D227—21no—
26C2×C4×D227—21no—
27#16813874—24no—
28#16813872—24no—
29#16813871—24no—
30#16813869—24no—
31#16813868—22no—
32#16813868—22no—
33C2×C227⋊C8—21no—
34C2×C227⋊C8—22no—
35C2×C227⋊C8D414yes—
36#41×C227—22no—
37#41×C227—22no—
38#41×C227—22no—
39#41×C227—22no—
40C2×Q8×C227D412yes—
41C2×D4×C227—22no—
42C2×D4×C227D412yes—
43C22×C4×C227—22no—
44C22×C4×C227C2311yescharacteristic
45Q16×C227—22no—
46SD16×C227—24no—
47D8×C227—22no—
48M16×C227—22no—
49C2×C8×C227D412yes—
50C22×D227—21no—
51Dic454—22no—
52D908—22no—
53C22⋊D227—42no—
54C2×Dic227—21no—
55C4×D227—41no—
56C4×D227—41no—
57C227⋊C8—24no—
58C227⋊C8—41no—
59C23×C227—22no—
60C23×C227#4111yescharacteristic
61Q8×C227—22no—
62Q8×C227—22no—
63D4×C227—22no—
64D4×C227—22no—
65D4×C227—42no—
66D4×C227—42no—
67C2×C4×C227—22no—
68C2×C4×C227—42no—
69C2×C4×C227—42no—
70C2×C4×C227C2×D411yescharacteristic
71C2×C4×C227C2×D412yes—
72C2×C4×C227C2×D412yes—
73C2×C4×C227C2×D411yescharacteristic
74C8×C227—22no—
75C22×C227—41no—
76C22×C227—44no—
77C22×C227—42no—
78C22×C227#12111yescharacteristic
79C22×C227#12112yes—
80C2×D227—42no—
81C2×D227—41no—
82C4×C227—42no—
83C4×C227#12012yes—
84C4×C227#12011yescharacteristic
85C4×C227#12711yescharacteristic
86Dic227—41no—
87C2×C227—21no—
88C2×C227—22no—
89C2×C227—82no—
90C2×C227#39211yescharacteristic
91D227—81no—
92C227#201711yescharacteristic, complemented, Sylow
93#2017—2271nomaximal, Sylow
94#575—2272no—
95#575D22711yescharacteristic, complemented
96#408—2272no—
97#408—2271no—
98#338—2271no—
99C2×#41—2271no—
100C2×#41C2×D22712yes—
101#137—22no—
102#137—4542no—
103#137—4542no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—4542no—
106AGL(1,ℤ/8)—4542no—
107#135—22no—
108#135—4544no—
109C2×M16—2272no—
110C2×M16C2×D22711yescharacteristic
111C22×C8—2271no—
112C22⋊C8—4544no—
113C22⋊C8—4542no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—4542no—
119#41—4542no—
120C2×Q8—2271no—
121C2×Q8C22⋊D22712yes—
122C2×D4—22no—
123C2×D4—4541no—
124C2×D4—2271no—
125C2×D4C22⋊D22712yes—
126C22×C4—22no—
127C22×C4—4541no—
128C22×C4C22×D22711yescharacteristic
129Q16—22no—
130Q16—4544no—
131SD16—24no—
132SD16—4544no—
133SD16—4544no—
134D8—22no—
135D8—4544no—
136M16—22no—
137M16—4542no—
138M16—4542no—
139C2×C8—4542no—
140C2×C8—4541no—
141C2×C8—2274no—
142C2×C8C22⋊D22712yes—
143C23—22no—
144C23—4541no—
145C23#1681388311yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—4542no—
149D4—22no—
150D4—22no—
151D4—42no—
152D4—42no—
153D4—9082no—
154D4—4542no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—9081no—
159C2×C4—9081no—
160C2×C4—4541no—
161C2×C4D4×D22711yescharacteristic
162C2×C4D4×D22712yes—
163C2×C4C2×C22⋊D22712yes—
164C2×C4C2×C22⋊D22711yescharacteristic
165C8—22no—
166C8—9081no—
167C8—4544no—
168C22—41no—
169C22—44no—
170C22—42no—
171C22—9082no—
172C22—9081no—
173C22#2030025911yescharacteristic
174C22#2030026012yes—
175C4—42no—
176C4—9081no—
177C4#2030025612yes—
178C4#2030025711yescharacteristic
179C4#2030028411yescharacteristic
180C2—22no—
181C2—21no—
182C2—82no—
183C2—18161no—
184C2#2840894011yescharacteristic
subgroupG/H
centerC4#20300284
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#575×C227C2
socleC2×C227#392
hypercenter#575D227
hypocenterC227#2017
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #28409830C2semidirect
2. #28409000C2semidirect
3. #28408999C2semidirect
4. C227⋊#338C2semidirect
5. #575×C227C2semidirect
92. C227#2017semidirect
95. #575D227semidirect

Maximal subgroups

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

subgroupindex
#284098302
#284090002
#284089992
C227⋊#3382
#575×C2272
#2017227