Finite Group Databasebeta

← #37511003

Subgroups — #37511003

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#26038627C212yesmaximal, complemented
3#26038616C211yesmaximal, characteristic, complemented
4#26038225C212yesmaximal, complemented
5#354×C179C211yesmaximal, characteristic, complemented
6#19153158C2211yescharacteristic
7#19153144—21no—
8#19153102—21no—
9#19153101—21no—
10#19153099—22no—
11#19153083—22no—
12#19153083—22no—
13#19153081C2212yes—
14#19153041—21no—
15#19153030C2211yescharacteristic
16#19153027—22no—
17#124×C179C2211yescharacteristic
18#99×C179C2212yes—
19C23×D179D412yescomplemented
20#15668259—21no—
21#15668258—41no—
22C2×Dic358—21no—
23C2×D716—21no—
24C2×D716—21no—
25C2×D716—42no—
26C2×C22⋊D179—22no—
27C2×C22⋊D179—22no—
28D4×D179—41no—
29#15668249—41no—
30#15668246—41no—
31#15668243—21no—
32C22⋊Dic179D412yescomplemented
33(C2×D179)⋊C4—22no—
34(C2×D179)⋊C4—22no—
35(C2×D179)⋊C4—22no—
36(C2×D179)⋊C4—42no—
37C4×Dic179—21no—
38C2×Q8×C179D411yescharacteristic, complemented
39C2×D4×C179C2311yescharacteristic
40C22⋊C4×C179—22no—
41C22⋊C4×C179—22no—
42C42×C179D411yescharacteristic, complemented
43C22×D179—22no—
44C22×D179—22no—
45C22×D179—42no—
46C22×D179—82no—
47Dic358—21no—
48Dic358—81no—
49D716—21no—
50D716—81no—
51D716—81no—
52D716—84no—
53C22⋊D179—42no—
54C22⋊D179—41no—
55C22⋊D179—82no—
56C2×Dic179—22no—
57C2×Dic179—22no—
58C4×D179—41no—
59C179⋊C8—41no—
60C23×C179C2×D412yes—
61Q8×C179—41no—
62D4×C179—41no—
63C2×C4×C179—21no—
64C2×C4×C179—21no—
65C2×C4×C179—42no—
66C2×C4×C179C2×D411yescharacteristic
67C22×C179—22no—
68C22×C179—42no—
69C22×C179#12011yescharacteristic
70C2×D179—42no—
71C2×D179—42no—
72C2×D179—44no—
73C2×D179—41no—
74C2×D179—42no—
75C2×D179—82no—
76C2×D179—82no—
77C4×C179—21no—
78C4×C179—41no—
79C4×C179—41no—
80C4×C179—82no—
81Dic179—42no—
82Dic179—41no—
83C2×C179—21no—
84C2×C179—42no—
85C2×C179B(4,𝔽2)11yescharacteristic
86D179—82no—
87D179—81no—
88C179#219911yescharacteristic, complemented, Sylow
89#2199—1791nomaximal, Sylow
90H(3,𝔽4)—1791no—
91B(4,𝔽2)—1792no—
92#455—1791no—
93#354D17911yescharacteristic, complemented
94AΣL(1,𝔽16)—1792no—
95H(5,𝔽2)—1791no—
96#137—3581no—
97#124—3582no—
98#124—3581no—
99#124—3581no—
100#124C2×D17911yescharacteristic
101#120—3582no—
102#120—3582no—
103#120—1792no—
104#104—3581no—
105#100—1791no—
106#99—3582no—
107#99C2×D17912yes—
108C24—1792no—
109#41—3581no—
110#41—7161no—
111C2×Q8—3581no—
112C2×Q8D71611yescharacteristic, complemented
113C2×D4—3581no—
114C2×D4—3581no—
115C2×D4—3582no—
116C2×D4—3582no—
117C2×D4—7162no—
118C2×D4—7161no—
119C2×D4C22×D17911yescharacteristic
120Q16—7161no—
121SD16—7161no—
122M16—3581no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—3582no—
126C22⋊C4—3582no—
127C22⋊C4—3582no—
128C22⋊C4—1792no—
129C22⋊C4—7162no—
130C42—3581no—
131C42D71611yescharacteristic, complemented
132C23—3582no—
133C23—3582no—
134C23—14322no—
135C23—7162no—
136C23D4×D17912yes—
137Q8—41no—
138Q8—3581no—
139Q8—14321no—
140D4—41no—
141D4—3581no—
142D4—14321no—
143D4—14321no—
144D4—14324no—
145D4—14322no—
146D4—7162no—
147D4—7161no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—3582no—
152C2×C4—3582no—
153C2×C4—7161no—
154C2×C4C2×D71611yescharacteristic
155C8—7161no—
156C22—22no—
157C22—42no—
158C22—14322no—
159C22—14322no—
160C22—7162no—
161C22—7162no—
162C22—7164no—
163C22—7161no—
164C22—7162no—
165C22#1915308311yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—82no—
170C4—7162no—
171C4—7161no—
172C2—21no—
173C2—42no—
174C2—14321no—
175C2—14322no—
176C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×D4×C179C23
FrattiniC2×D4C22×D179
Fitting#354×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#354D179
hypocenterC179#2199
LCSG ▷ C2×D4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #26039181C2semidirect
2. #26038627C2semidirect
3. #26038616C2semidirect
4. #26038225C2semidirect
5. #354×C179C2semidirect
19. C23×D179D4semidirect
32. C22⋊Dic179D4semidirect
38. C2×Q8×C179D4semidirect
42. C42×C179D4semidirect
88. C179#2199semidirect
93. #354D179semidirect
112. C2×Q8D716semidirect
131. C42D716semidirect

Maximal subgroups

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

subgroupindex
#260391812
#260386272
#260386162
#260382252
#354×C1792
#2199179