Finite Group Databasebeta

← #44401704

Subgroups — #44401704

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#28409473C211yesmaximal, characteristic, complemented
3#28409304C211yesmaximal, characteristic, complemented
4#28408845C211yesmaximal, characteristic, complemented
5#28408840C211yesmaximal, characteristic, complemented
6#28408815C211yesmaximal, characteristic, complemented
7C2×SD32×C227C211yesmaximal, characteristic, complemented
8#20300333—21no—
9#20300319—21no—
10#20300314—21no—
11#20300312—22no—
12#20300310C2211yescharacteristic
13#20300307—21no—
14D8×D227—21no—
15#20300294C2211yescharacteristic
16#20300234—21no—
17#20300231—21no—
18#20300228C2211yescharacteristic, complemented
19#20300224C2211yescharacteristic, complemented
20#20300207—21no—
21C2×Q16×C227C2211yescharacteristic, complemented
22C2×D8×C227C2211yescharacteristic
23SD32×C227—22no—
24C2×C16×C227C2211yescharacteristic, complemented
25#16813884—21no—
26#16813884D411yescharacteristic
27#16813883—22no—
28#16813883—41no—
29C2×C22⋊D227—42no—
30D4×D227—22no—
31D4×D227—41no—
32Dic908D411yescharacteristic
33#16813874—41no—
34#16813873—21no—
35#16813872—22no—
36#16813871—41no—
37D1816D411yescharacteristic
38#16813869—22no—
39#16813868—21no—
40#16813868D411yescharacteristic
41#16813867—21no—
42C8×D227—21no—
43C227⋊C16—21no—
44C2×Q8×C227—21no—
45C2×D4×C227—21no—
46Q16×C227—21no—
47Q16×C227—21no—
48D8×C227—21no—
49D8×C227D412yes—
50C2×C8×C227C2311yescharacteristic
51C16×C227—21no—
52C22×D227—42no—
53Dic454—21no—
54Dic454D811yescharacteristic
55D908—21no—
56D908D811yescharacteristic
57C22⋊D227—21no—
58C22⋊D227—42no—
59C22⋊D227—42no—
60C22⋊D227—41no—
61C2×Dic227—42no—
62C4×D227—21no—
63C4×D227—41no—
64C227⋊C8—21no—
65C227⋊C8—41no—
66C23×C227—41no—
67Q8×C227—41no—
68Q8×C227—41no—
69D4×C227—22no—
70D4×C227—41no—
71C2×C4×C227—41no—
72C2×C4×C227C2×D411yescharacteristic
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—81no—
82C4×C227#12011yescharacteristic
83C4×C227C2×D811yescharacteristic
84Dic227—21no—
85Dic227—41no—
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×SD32D22711yescharacteristic, complemented
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#135—2271no—
106C2×Q16C2×D22711yescharacteristic, complemented
107C2×D8—4541no—
108C2×D8C2×D22711yescharacteristic
109#131—2271no—
110Q32—4541no—
111SD32—22no—
112SD32—4541no—
113M32—2271no—
114C2×C16C2×D22711yescharacteristic, complemented
115#108—2271no—
116#101—4541no—
117#41—9081no—
118#41—4542no—
119#41—4541no—
120#41—2271no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—9081no—
124C2×D4—9082no—
125C2×D4—4542no—
126Q16—21no—
127Q16—21no—
128Q16—9081no—
129Q16—2271no—
130SD16—9081no—
131SD16—4542no—
132SD16—4541no—
133D8—21no—
134D8—4542no—
135D8—2271no—
136D8C22⋊D22712yes—
137M16—4541no—
138M16—4541no—
139M16—2271no—
140C2×C8—4541no—
141C2×C8C22×D22711yescharacteristic
142C16—21no—
143C16—4541no—
144C23—41no—
145C23—9082no—
146Q8—41no—
147Q8—41no—
148Q8—4541no—
149Q8—2271no—
150D4—22no—
151D4—41no—
152D4—9082no—
153D4—9081no—
154D4—9082no—
155D4—4541no—
156D4—4541no—
157D4—2271no—
158C2×C4—41no—
159C2×C4—9082no—
160C2×C4—9081no—
161C2×C4—4541no—
162C2×C4D4×D22711yescharacteristic
163C8—9081no—
164C8—4541no—
165C8D4×D22711yescharacteristic
166C8C2×C22⋊D22711yescharacteristic
167C22—42no—
168C22—161no—
169C22—36322no—
170C22—9081no—
171C22—4541no—
172C22#2030031211yescharacteristic
173C4—81no—
174C4—9081no—
175C4—4541no—
176C4#2030025611yescharacteristic
177C4D8×D22711yescharacteristic
178C2—21no—
179C2—82no—
180C2—18161no—
181C2—9081no—
182C2#2840916411yescharacteristic
subgroupG/H
centerC2#28409164
commutator, cosocleC2×C8×C227C23
FrattiniC2×C8C22×D227
FittingC2×SD32×C227C2
socleC2×C227#447
prime radical#28409833C2
hypercenterC2×SD32D227
hypocenterC227#2194
LCSG ▷ C2×C8×C227 ▷ C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #28409473C2semidirect
3. #28409304C2semidirect
4. #28408845C2semidirect
5. #28408840C2semidirect
6. #28408815C2semidirect
7. C2×SD32×C227C2semidirect
18. #20300228C22semidirect
19. #20300224C22semidirect
21. C2×Q16×C227C22semidirect
24. C2×C16×C227C22semidirect
91. C227#2194semidirect
95. C2×SD32D227semidirect
106. C2×Q16C2×D227semidirect
114. C2×C16C2×D227semidirect

Maximal subgroups

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

subgroupindex
#284098332
#284094732
#284093042
#284088452
#284088402
#284088152
C2×SD32×C2272
#2194227