Finite Group Databasebeta

← #44401747

Subgroups — #44401747

Order 29056.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#28409221C212yesmaximal
2#28409209C211yesmaximal, characteristic
3#28408827C211yesmaximal, characteristic, complemented
4#28408822C212yesmaximal, complemented
5#563×C227C211yesmaximal, characteristic, complemented
6#20300333C2211yescharacteristic
7#20300319—21no—
8#20300271—22no—
9#20300216—21no—
10#20300205C2211yescharacteristic
11#20300203C2212yes—
12#20300202—22no—
13C4⋊Q8×C227—21no—
14#126×C227—22no—
15#125×C227—21no—
16#124×C227C2211yescharacteristic
17#123×C227C2212yes—
18C22⋊Q8×C227—22no—
19#16813884—21no—
20#16813883—41no—
21C2×C22⋊D227—22no—
22D4×D227—41no—
23#16813874—41no—
24#16813871—41no—
25#16813868—21no—
26Dic227⋊C4—42no—
27C22⋊Dic227—22no—
28(C2×D227)⋊C4—42no—
29C2×Q8×C227—21no—
30C2×Q8×C227D411yescharacteristic
31C2×D4×C227C2311yescharacteristic
32C22×C4×C227D412yes—
33C4⋊C4×C227—21no—
34C4⋊C4×C227—22no—
35C4⋊C4×C227—22no—
36C4⋊C4×C227—21no—
37C4⋊C4×C227—22no—
38C22⋊C4×C227—22no—
39C22⋊C4×C227—22no—
40C22⋊C4×C227D412yes—
41C42×C227—21no—
42C42×C227D411yescharacteristic
43C22×D227—22no—
44Dic454—21no—
45D908—21no—
46C22⋊D227—42no—
47C22⋊D227—41no—
48C2×Dic227—22no—
49C2×Dic227—42no—
50C4×D227—41no—
51C227⋊C8—41no—
52C23×C227C2×D412yes—
53Q8×C227—41no—
54Q8×C227—81no—
55D4×C227—41no—
56C2×C4×C227—21no—
57C2×C4×C227—22no—
58C2×C4×C227—21no—
59C2×C4×C227—22no—
60C2×C4×C227—42no—
61C2×C4×C227C2×D411yescharacteristic
62C22×C227—22no—
63C22×C227—42no—
64C22×C227#12011yescharacteristic
65C2×D227—41no—
66C2×D227—82no—
67C4×C227—21no—
68C4×C227—42no—
69C4×C227—41no—
70C4×C227—42no—
71C4×C227—41no—
72Dic227—41no—
73Dic227—82no—
74C2×C227—21no—
75C2×C227—42no—
76C2×C227B(4,𝔽2)11yescharacteristic
77D227—81no—
78C227#220111yescharacteristic, complemented, Sylow
79#2201—2271nomaximal, Sylow
80#563D22711yescharacteristic, complemented
81#458—2272no—
82#455—2271no—
83#354—2271no—
84#352—2272no—
85H(5,𝔽2)—2271no—
86#137—4541no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D22711yescharacteristic
91#123—4542no—
92#123C2×D22712yes—
93C22⋊Q8—22no—
94#104—4541no—
95#100—2271no—
96#99—4542no—
97#99—2272no—
98#41—9081no—
99#41—4541no—
100C2×Q8—21no—
101C2×Q8C22⋊D22711yescharacteristic
102C2×D4—9081no—
103C2×D4—4542no—
104C2×D4C22×D22711yescharacteristic
105C22×C4C22⋊D22712yes—
106Q16—9081no—
107SD16—9081no—
108M16—4541no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—9082no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—9082no—
118C22⋊C4—4542no—
119C22⋊C4C22⋊D22712yes—
120C42—21no—
121C42C22⋊D22711yescharacteristic
122C23—4542no—
123C23C2×C22⋊D22712yes—
124Q8—41no—
125Q8—81no—
126Q8—4541no—
127D4—41no—
128D4—9082no—
129D4—9081no—
130D4—4541no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—9081no—
137C2×C4—9082no—
138C2×C4—4542no—
139C2×C4C2×C22⋊D22711yescharacteristic
140C8—9081no—
141C22—22no—
142C22—42no—
143C22—18162no—
144C22—9081no—
145C22#2030025711yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—18162no—
152C4—9081no—
153C2—21no—
154C2—42no—
155C2—18161no—
156C2#2840921911yescharacteristic
subgroupG/H
centerC2#28409219
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
Fitting#563×C227C2
socleC2×C227B(4,𝔽2)
prime radical#20300333C22
hypercenter#563D227
hypocenterC227#2201
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #28408827C2semidirect
4. #28408822C2semidirect
5. #563×C227C2semidirect
78. C227#2201semidirect
80. #563D227semidirect

Maximal subgroups

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

subgroupindex
#284092212
#284092092
#284088272
#284088222
#563×C2272
#2201227