Finite Group Databasebeta

← #37511000

Subgroups — #37511000

Order 22912.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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#26039181C211yesmaximal, characteristic, complemented
2#26038626C212yesmaximal, complemented
3#26038234C211yesmaximal, characteristic, complemented
4#26038227C212yesmaximal, complemented
5#455×C179C211yesmaximal, characteristic, complemented
6#19153102—21no—
7#19153101—21no—
8#19153099—22no—
9#19153083—22no—
10#19153081—22no—
11#19153081C2212yes—
12#19153028—22no—
13#19153028C2212yes—
14H(5,𝔽2)×C179C2211yescharacteristic
15#137×C179—21no—
16#124×C179C2211yescharacteristic
17#104×C179—21no—
18#100×C179C2211yescharacteristic
19C23×D179D412yescomplemented
20C2×Dic358—21no—
21C2×D716—21no—
22C2×D716—21no—
23C2×C22⋊D179—22no—
24C2×C22⋊D179—42no—
25C22⋊Dic179—22no—
26C22⋊Dic179D412yescomplemented
27(C2×D179)⋊C4—22no—
28(C2×D179)⋊C4—22no—
29(C2×D179)⋊C4—22no—
30(C2×D179)⋊C4—42no—
31C4×Dic179—21no—
32#41×C179—21no—
33#41×C179—41no—
34C2×Q8×C179D411yescharacteristic, complemented
35C2×D4×C179—22no—
36C2×D4×C179—41no—
37C2×D4×C179C2311yescharacteristic
38Q16×C179—41no—
39SD16×C179—41no—
40M16×C179—21no—
41C22⋊C4×C179—22no—
42C42×C179D411yescharacteristic, complemented
43C22×D179—22no—
44C22×D179—42no—
45C22×D179—82no—
46Dic358—81no—
47D716—81no—
48D716—81no—
49C22⋊D179—82no—
50C22⋊D179—84no—
51C2×Dic179—22no—
52C2×Dic179—42no—
53C23×C179—22no—
54C23×C179C2×D412yes—
55Q8×C179—21no—
56Q8×C179—41no—
57D4×C179—21no—
58D4×C179—41no—
59D4×C179—42no—
60D4×C179—41no—
61C2×C4×C179—22no—
62C2×C4×C179—21no—
63C2×C4×C179—21no—
64C2×C4×C179—41no—
65C2×C4×C179C2×D411yescharacteristic
66C8×C179—41no—
67C22×C179—22no—
68C22×C179—42no—
69C22×C179—41no—
70C22×C179—82no—
71C22×C179#12011yescharacteristic
72C2×D179—42no—
73C2×D179—42no—
74C2×D179—44no—
75C2×D179—42no—
76C2×D179—82no—
77C4×C179—21no—
78C4×C179—41no—
79C4×C179—41no—
80C4×C179—41no—
81Dic179—42no—
82Dic179—82no—
83C2×C179—21no—
84C2×C179—42no—
85C2×C179—81no—
86C2×C179B(4,𝔽2)11yescharacteristic
87D179—82no—
88C179#219911yescharacteristic, complemented, Sylow
89#2199—1791nomaximal, Sylow
90H(3,𝔽4)—1791no—
91B(4,𝔽2)—1792no—
92#455D17911yescharacteristic, complemented
93#354—1791no—
94AΣL(1,𝔽16)—1792no—
95H(5,𝔽2)C2×D17911yescharacteristic
96#137—21no—
97#124—3582no—
98#124—3581no—
99#124—3581no—
100#124C2×D17911yescharacteristic
101#120—3582no—
102#120—3582no—
103#120—1792no—
104#104—21no—
105#100C2×D17911yescharacteristic
106#99—3582no—
107#99—1792no—
108C24—1792no—
109#41—21no—
110#41—41no—
111C2×Q8—3581no—
112C2×Q8C22⋊D17911yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—3581no—
116C2×D4—3581no—
117C2×D4—3582no—
118C2×D4—7162no—
119C2×D4C22×D17911yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—3582no—
125C22⋊C4—3582no—
126C22⋊C4—3582no—
127C22⋊C4—3582no—
128C22⋊C4—1792no—
129C22⋊C4—7162no—
130C42—3581no—
131C42C22⋊D17911yescharacteristic, complemented
132C23—22no—
133C23—3582no—
134C23—14322no—
135C23—7162no—
136C23D4×D17912yes—
137Q8—21no—
138Q8—41no—
139Q8—14321no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—14321no—
145D4—14321no—
146D4—14324no—
147D4—14322no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—3582no—
153C2×C4—7162no—
154C2×C4C2×C22⋊D17911yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—14322no—
161C22—7162no—
162C22—7162no—
163C22—7164no—
164C22—7162no—
165C22#1915308111yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—14322no—
171C4—7162no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—14322no—
176C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#455×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#455D179
hypocenterC179#2199
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #26039181C2semidirect
2. #26038626C2semidirect
3. #26038234C2semidirect
4. #26038227C2semidirect
5. #455×C179C2semidirect
19. C23×D179D4semidirect
26. C22⋊Dic179D4semidirect
34. C2×Q8×C179D4semidirect
42. C42×C179D4semidirect
88. C179#2199semidirect
92. #455D179semidirect
112. C2×Q8C22⋊D179semidirect
131. C42C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
#260391812
#260386262
#260382342
#260382272
#455×C1792
#2199179