Finite Group Databasebeta

← #44401702

Subgroups — #44401702

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#28409834C211yesmaximal, characteristic
2C2×#20300231C211yesmaximal, characteristic, complemented
3#28409304C211yesmaximal, characteristic, complemented
4#28408846C211yesmaximal, characteristic, complemented
5#28408839C211yesmaximal, characteristic, complemented
6#28408817C211yesmaximal, characteristic, complemented
7#510×C227C211yesmaximal, characteristic, complemented
8#20300334—21no—
9#20300319—21no—
10#20300315—22no—
11#20300313—21no—
12#20300311—21no—
13C2×D1816C2211yescharacteristic
14D8×D227—21no—
15#20300294C2211yescharacteristic
16#20300231—22no—
17C2×C227⋊C16C2211yescharacteristic, complemented
18#20300224C2211yescharacteristic, complemented
19#20300207—21no—
20#135×C227C2211yescharacteristic
21C2×Q16×C227C2211yescharacteristic, complemented
22Q32×C227—21no—
23SD32×C227—21no—
24M32×C227C2211yescharacteristic, complemented
25#16813885—22no—
26#16813884—41no—
27#16813884D411yescharacteristic
28C2×D908—21no—
29C2×D908—41no—
30D4×D227—22no—
31D4×D227—42no—
32#16813874—41no—
33#16813871—22no—
34#16813871—41no—
35D1816—21no—
36D1816D412yes—
37#16813869—22no—
38#16813868—21no—
39#16813868D411yescharacteristic
40#16813867—21no—
41C8×D227—21no—
42C227⋊C16—21no—
43#41×C227—21no—
44C2×Q8×C227—21no—
45Q16×C227—21no—
46Q16×C227—21no—
47Q16×C227D411yescharacteristic
48SD16×C227—21no—
49D8×C227D411yescharacteristic
50C2×C8×C227C2311yescharacteristic
51C16×C227—21no—
52C22×D227—42no—
53C22×D227—41no—
54Dic454D811yescharacteristic
55D908—22no—
56D908—42no—
57D908—41no—
58D908D811yescharacteristic
59C22⋊D227—21no—
60C22⋊D227—42no—
61C4×D227—21no—
62C4×D227—42no—
63C227⋊C8—21no—
64C227⋊C8—41no—
65Q8×C227—21no—
66Q8×C227—41no—
67Q8×C227—41no—
68D4×C227—21no—
69D4×C227—41no—
70C2×C4×C227—41no—
71C2×C4×C227—41no—
72C2×C4×C227C2×D411yescharacteristic
73C8×C227C2×D411yescharacteristic
74C8×C227C2×D411yescharacteristic
75C22×C227—41no—
76C22×C227AGL(1,ℤ/8)11yescharacteristic
77C2×D227—21no—
78C2×D227—42no—
79C2×D227—162no—
80C2×D227—161no—
81C4×C227—41no—
82C4×C227—81no—
83C4×C227#12011yescharacteristic
84C4×C227C2×D811yescharacteristic
85Dic227—21no—
86C2×C227—21no—
87C2×C227—81no—
88C2×C227#44711yescharacteristic
89D227—41no—
90D227—82no—
91C227#219411yescharacteristic, complemented, Sylow
92#2194—2271nomaximal, Sylow
93#576—2271no—
94#510D22711yescharacteristic, complemented
95C2×SD32—2271no—
96#473—2271no—
97#361—2271no—
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#135C2×D22711yescharacteristic
106C2×Q16C2×D22711yescharacteristic, complemented
107C2×D8—4541no—
108C2×D8—2271no—
109#131—2271no—
110Q32—21no—
111SD32—21no—
112SD32—4542no—
113M32C2×D22711yescharacteristic, complemented
114C2×C16—2271no—
115#108—2271no—
116#101—4541no—
117#41—21no—
118#41—9081no—
119#41—4542no—
120#41—2271no—
121C2×Q8—21no—
122C2×D4—9081no—
123C2×D4—9082no—
124C2×D4—4542no—
125C2×D4—4541no—
126Q16—21no—
127Q16—21no—
128Q16—9081no—
129Q16C22⋊D22711yescharacteristic
130SD16—21no—
131SD16—9081no—
132SD16—4542no—
133D8—4542no—
134D8—4541no—
135D8—2272no—
136D8C22⋊D22711yescharacteristic
137M16—4541no—
138M16—4541no—
139M16—2271no—
140C2×C8—4541no—
141C2×C8C22×D22711yescharacteristic
142C16—21no—
143C16—4541no—
144C23—9082no—
145C23—9081no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—2271no—
150D4—21no—
151D4—41no—
152D4—9082no—
153D4—9082no—
154D4—9081no—
155D4—4541no—
156D4—4542no—
157D4—2271no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—9082no—
161C2×C4—4541no—
162C2×C4D4×D22711yescharacteristic
163C8—9081no—
164C8—4541no—
165C8D4×D22711yescharacteristic
166C8C2×C22⋊D22711yescharacteristic
167C22—41no—
168C22—36321no—
169C22—36322no—
170C22—9082no—
171C22—4541no—
172C22#2030031211yescharacteristic
173C4—41no—
174C4—81no—
175C4—4541no—
176C4#2030025611yescharacteristic
177C4D8×D22711yescharacteristic
178C2—21no—
179C2—81no—
180C2—18162no—
181C2—9081no—
182C2#2840916411yescharacteristic
subgroupG/H
centerC2#28409164
commutator, cosocleC2×C8×C227C23
FrattiniC2×C8C22×D227
Fitting#510×C227C2
socleC2×C227#447
prime radical#28409834C2
hypercenter#510D227
hypocenterC227#2194
LCSG ▷ C2×C8×C227 ▷ C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#20300231C2semidirect
3. #28409304C2semidirect
4. #28408846C2semidirect
5. #28408839C2semidirect
6. #28408817C2semidirect
7. #510×C227C2semidirect
17. C2×C227⋊C16C22semidirect
18. #20300224C22semidirect
21. C2×Q16×C227C22semidirect
24. M32×C227C22semidirect
91. C227#2194semidirect
94. #510D227semidirect
106. C2×Q16C2×D227semidirect
113. M32C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098342
C2×#203002312
#284093042
#284088462
#284088392
#284088172
#510×C2272
#2194227