Finite Group Databasebeta

← #44400281

Subgroups — #44400281

Order 29056.

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

normal subgroups only

This group has 340 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#28409596C211yesmaximal, characteristic
2#28409016C211yesmaximal, characteristic, complemented
3C2×#20300204C212yesmaximal, complemented
4#28408804C212yesmaximal, complemented
5#568×C227C211yesmaximal, characteristic, complemented
6C22×D908C2211yescharacteristic
7C2×#16813868C2212yes—
8#20300284—21no—
9#20300277—21no—
10#20300262—22no—
11#20300256C414yes—
12#20300247C2211yescharacteristic
13#20300207—21no—
14#20300205—21no—
15#20300204—24no—
16C2×#41×C227C2211yescharacteristic
17#131×C227—24no—
18C2×M16×C227—22no—
19C2×M16×C227C2212yes—
20C23×D227C2×C412yes—
21C2×D908—22no—
22C2×D908—21no—
23C2×D908—21no—
24C2×D908—22no—
25C2×C22⋊D227—24no—
26#16813868—22no—
27#16813868—22no—
28C2×C227⋊C8—22no—
29C4⋊Dic227—21no—
30C22⋊Dic227C2×C412yes—
31(C2×D227)⋊C4—24no—
32C4×Dic227—21no—
33#41×C227—22no—
34#41×C227—22no—
35C2×Q8×C227D411yescharacteristic
36C2×D4×C227C2×C412yes—
37C2×D4×C227D411yescharacteristic
38C22×C4×C227D412yes—
39C22×C4×C227C2311yescharacteristic
40M16×C227—24no—
41M16×C227—22no—
42M16×C227D414yescomplemented
43C2×C8×C227—24no—
44C2×C8×C227—22no—
45C22×D227—22no—
46C22×D227—24no—
47C22×D227—82no—
48D908—41no—
49D908—42no—
50D908—41no—
51C22⋊D227—84no—
52C2×Dic227—22no—
53C227⋊C8—42no—
54C23×C227C22⋊C412yes—
55C23×C227C22×C411yescharacteristic
56Q8×C227—22no—
57D4×C227—24no—
58D4×C227—22no—
59C2×C4×C227—22no—
60C2×C4×C227—22no—
61C2×C4×C227C22⋊C412yes—
62C2×C4×C227C2×D412yes—
63C2×C4×C227C2×D411yescharacteristic
64C2×C4×C227C2×D411yescharacteristic
65C2×C4×C227#4112yes—
66C8×C227—24no—
67C22×C227—22no—
68C22×C227—41no—
69C22×C227—42no—
70C22×C227C2×C22⋊C411yescharacteristic
71C22×C227C4×D412yes—
72C2×D227—44no—
73C2×D227—42no—
74C2×D227—84no—
75C4×C227—22no—
76C4×C227#12011yescharacteristic
77C4×C227#12112yes—
78C4×C227#12311yescharacteristic
79Dic227—42no—
80C2×C227—21no—
81C2×C227—22no—
82C2×C227—42no—
83C2×C227#38611yescharacteristic
84D227—82no—
85C227#191411yescharacteristic, complemented, Sylow
86#1914—2271nomaximal, Sylow
87#568D22711yescharacteristic, complemented
88#535—2271no—
89#413—2271no—
90C2×#100—2272no—
91#344—2272no—
92C2×#41C2×D22711yescharacteristic
93C22×D4—2271no—
94#131—24no—
95C2×M16—22no—
96C2×M16—2272no—
97C2×M16C2×D22712yes—
98#127—4541no—
99#124—4541no—
100#121—4542no—
101#120—2274no—
102#117—2271no—
103#101—4541no—
104#100—4541no—
105#100—4544no—
106C24—2272no—
107#41—22no—
108#41—22no—
109C2×Q8D90811yescharacteristic
110C2×D4—4544no—
111C2×D4—4542no—
112C2×D4—4541no—
113C2×D4—4541no—
114C2×D4—4542no—
115C2×D4C4×D22712yes—
116C2×D4D90811yescharacteristic
117C22×C4C22⋊D22712yes—
118C22×C4C22×D22711yescharacteristic
119M16—24no—
120M16—22no—
121M16—4542no—
122M16—4542no—
123M16C22⋊D22714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—4542no—
127C4⋊C4—4541no—
128C22⋊C4—4544no—
129C22⋊C4—2272no—
130C42—4541no—
131C23—18162no—
132C23—4544no—
133C23—4542no—
134C23(C2×D227)⋊C412yes—
135C23C2×C4×D22711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—18164no—
140D4—9081no—
141D4—9082no—
142D4—9081no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—4542no—
146C2×C4(C2×D227)⋊C412yes—
147C2×C4C2×C22⋊D22712yes—
148C2×C4C2×C22⋊D22711yescharacteristic
149C2×C4C2×D90811yescharacteristic
150C2×C4#1681388412yes—
151C8—24no—
152C8—9082no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—18164no—
157C22—9084no—
158C22—9082no—
159C22C2×(C2×D227)⋊C411yescharacteristic
160C22C4×C22⋊D22712yes—
161C4—22no—
162C4—9082no—
163C4#2030025711yescharacteristic
164C4#2030026212yes—
165C4#2030027111yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—18162no—
170C2#2840891111yescharacteristic
subgroupG/H
centerC2#28408911
commutatorC23×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#568×C227C2
socleC2×C227#386
prime radical#28409596C2
cosocleC22×C4×C227C23
hypercenter#568D227
hypocenterC227#1914
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #28409016C2semidirect
3. C2×#20300204C2semidirect
4. #28408804C2semidirect
5. #568×C227C2semidirect
42. M16×C227D4semidirect
85. C227#1914semidirect
87. #568D227semidirect
123. M16C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
#284095962
#284090162
C2×#203002042
#284088042
#568×C2272
#1914227