Finite Group Databasebeta

← #44401708

Subgroups — #44401708

Order 29056.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#28409833C211yesmaximal, characteristic
2#28409475C211yesmaximal, characteristic, complemented
3C2×#20300232C211yesmaximal, characteristic, complemented
4#28409308C211yesmaximal, characteristic, complemented
5#28408838C211yesmaximal, characteristic, complemented
6#28408817C211yesmaximal, characteristic, complemented
7#361×C227C211yesmaximal, characteristic, complemented
8#20300333—21no—
9#20300321—21no—
10#20300314—21no—
11#20300312—22no—
12#20300310C2211yescharacteristic
13#20300307—21no—
14C2×Dic908C2211yescharacteristic, complemented
15D8×D227—21no—
16#20300294C2211yescharacteristic
17Dic1816—21no—
18#20300233—21no—
19#20300232—22no—
20C2×C227⋊C16C2211yescharacteristic, complemented
21#20300206—21no—
22C2×D8×C227C2211yescharacteristic
23M32×C227C2211yescharacteristic, complemented
24#108×C227C2211yescharacteristic, complemented
25#16813884—21no—
26#16813884D411yescharacteristic
27#16813883—22no—
28#16813883—41no—
29C2×Dic454—21no—
30C2×C22⋊D227—42no—
31D4×D227—22no—
32D4×D227—41no—
33Dic908—21no—
34Dic908—21no—
35Dic908—41no—
36Dic908D411yescharacteristic
37#16813873—21no—
38#16813873—41no—
39#16813872—22no—
40D1816D411yescharacteristic
41#16813869—22no—
42#16813868D411yescharacteristic
43#16813867—21no—
44C8×D227—21no—
45C227⋊C16—21no—
46C2×D4×C227—21no—
47D8×C227—21no—
48D8×C227D412yes—
49M16×C227—21no—
50C2×C8×C227C2311yescharacteristic
51C16×C227—21no—
52C22×D227—42no—
53Dic454—21no—
54Dic454—41no—
55Dic454—41no—
56Dic454D811yescharacteristic
57D908—21no—
58D908D811yescharacteristic
59C22⋊D227—21no—
60C22⋊D227—42no—
61C22⋊D227—42no—
62C22⋊D227—41no—
63C2×Dic227—42no—
64C2×Dic227—41no—
65C4×D227—21no—
66C4×D227—41no—
67C227⋊C8—21no—
68C23×C227—41no—
69D4×C227—22no—
70D4×C227—41no—
71C2×C4×C227C2×D411yescharacteristic
72C8×C227—41no—
73C8×C227C2×D411yescharacteristic
74C8×C227C2×D411yescharacteristic
75C22×C227—42no—
76C22×C227—161no—
77C22×C227AGL(1,ℤ/8)11yescharacteristic
78C2×D227—21no—
79C2×D227—41no—
80C2×D227—162no—
81C4×C227#12011yescharacteristic
82C4×C227C2×D811yescharacteristic
83Dic227—21no—
84Dic227—41no—
85Dic227—81no—
86C2×C227—21no—
87C2×C227—82no—
88C2×C227#44711yescharacteristic
89D227—41no—
90D227—81no—
91C227#219411yescharacteristic, complemented, Sylow
92#2194—2271nomaximal, Sylow
93#576—2271no—
94#510—2271no—
95C2×SD32—2271no—
96#473—2271no—
97#361D22711yescharacteristic, complemented
98#359—2271no—
99#350—2271no—
100H(5,𝔽2)—4541no—
101#137—4541no—
102AGL(1,ℤ/8)—4542no—
103AGL(1,ℤ/8)—4541no—
104#135—4541no—
105#135—2271no—
106C2×Q16—2271no—
107C2×D8—4541no—
108C2×D8C2×D22711yescharacteristic
109#131—2271no—
110Q32—4541no—
111SD32—4541no—
112SD32—4542no—
113M32C2×D22711yescharacteristic, complemented
114C2×C16—2271no—
115#108C2×D22711yescharacteristic, complemented
116#101—4541no—
117#41—9081no—
118#41—4542no—
119#41—4541no—
120#41—2271no—
121C2×Q8—4541no—
122C2×D4—21no—
123C2×D4—9081no—
124C2×D4—9082no—
125C2×D4—4542no—
126Q16—9081no—
127Q16—4541no—
128Q16—4541no—
129Q16—2271no—
130SD16—9081no—
131SD16—4542no—
132SD16—4541no—
133D8—21no—
134D8—4542no—
135D8—2271no—
136D8D90812yes—
137M16—21no—
138M16—4541no—
139M16—2271no—
140C2×C8—4541no—
141C2×C8C22×D22711yescharacteristic
142C16—21no—
143C16—4541no—
144C23—41no—
145C23—9082no—
146Q8—9081no—
147Q8—9081no—
148Q8—4541no—
149Q8—2271no—
150D4—22no—
151D4—41no—
152D4—9082no—
153D4—9081no—
154D4—9082no—
155D4—4541no—
156D4—4541no—
157D4—2271no—
158C2×C4—9081no—
159C2×C4—9082no—
160C2×C4—9081no—
161C2×C4—4541no—
162C2×C4D4×D22711yescharacteristic
163C8—41no—
164C8—4541no—
165C8D4×D22711yescharacteristic
166C8C2×D90811yescharacteristic
167C22—42no—
168C22—161no—
169C22—36322no—
170C22—9081no—
171C22—4541no—
172C22#2030031511yescharacteristic
173C4—18161no—
174C4—9081no—
175C4—4541no—
176C4#2030025811yescharacteristic
177C4D8×D22711yescharacteristic
178C2—21no—
179C2—82no—
180C2—18161no—
181C2—9081no—
182C2#2840916711yescharacteristic
subgroupG/H
centerC2#28409167
commutator, cosocleC2×C8×C227C23
FrattiniC2×C8C22×D227
Fitting#361×C227C2
socleC2×C227#447
prime radical#28409833C2
hypercenter#361D227
hypocenterC227#2194
LCSG ▷ C2×C8×C227 ▷ C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #28409475C2semidirect
3. C2×#20300232C2semidirect
4. #28409308C2semidirect
5. #28408838C2semidirect
6. #28408817C2semidirect
7. #361×C227C2semidirect
14. C2×Dic908C22semidirect
20. C2×C227⋊C16C22semidirect
23. M32×C227C22semidirect
24. #108×C227C22semidirect
91. C227#2194semidirect
97. #361D227semidirect
113. M32C2×D227semidirect
115. #108C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098332
#284094752
C2×#203002322
#284093082
#284088382
#284088172
#361×C2272
#2194227