Finite Group Databasebeta

← #37511016

Subgroups — #37511016

Order 22912.

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

normal subgroups only

This group has 248 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#26038612C212yesmaximal, complemented
3#26038236C211yesmaximal, characteristic, complemented
4#26038235C212yesmaximal, complemented
5#454×C179C211yesmaximal, characteristic, complemented
6#19153138—22no—
7#19153102C2212yes—
8#19153101—21no—
9#19153099—22no—
10#19153083—22no—
11#19153081—21no—
12#19153040—22no—
13#19153032C2212yescomplemented
142−1+4×C179C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C179—21no—
16#124×C179C2211yescharacteristic
17#104×C179—21no—
18#101×C179C2211yescharacteristic, complemented
19C23×D179—21no—
20C2×Dic358—21no—
21C2×D716—21no—
22C2×D716D412yes—
23C2×C22⋊D179—22no—
24#15668246—42no—
25#15668244—42no—
26#15668243—22no—
27C22⋊Dic179—21no—
28(C2×D179)⋊C4—22no—
29(C2×D179)⋊C4—22no—
30(C2×D179)⋊C4—22no—
31C4×Dic179D412yes—
32#41×C179—21no—
33#41×C179—22no—
34#41×C179—41no—
35C2×Q8×C179—41no—
36C2×Q8×C179C2311yescharacteristic
37C2×D4×C179D411yescharacteristic
38SD16×C179—41no—
39D8×C179—41no—
40M16×C179—21no—
41C22⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43C22×D179—22no—
44C22×D179—82no—
45C22×D179—81no—
46Dic358—81no—
47D716—42no—
48D716—81no—
49C22⋊D179—82no—
50C2×Dic179—22no—
51C179⋊C8—42no—
52C23×C179—21no—
53Q8×C179—21no—
54Q8×C179—22no—
55Q8×C179—41no—
56D4×C179—21no—
57D4×C179—22no—
58D4×C179—41no—
59D4×C179—41no—
60C2×C4×C179—21no—
61C2×C4×C179—41no—
62C2×C4×C179—42no—
63C2×C4×C179C2×D412yes—
64C2×C4×C179C2×D411yescharacteristic
65C8×C179—41no—
66C22×C179—42no—
67C22×C179—41no—
68C22×C179—41no—
69C22×C179#12011yescharacteristic
70C2×D179—42no—
71C2×D179—42no—
72C2×D179—42no—
73C2×D179—84no—
74C4×C179—21no—
75C4×C179—22no—
76C4×C179—41no—
77C4×C179—41no—
78Dic179—42no—
79C2×C179—21no—
80C2×C179—81no—
81C2×C179—81no—
82C2×C179B(4,𝔽2)11yescharacteristic
83D179—82no—
84C179#220211yescharacteristic, complemented, Sylow
85#2202—1791nomaximal, Sylow
86H(3,𝔽4)—1791no—
87#454—1792no—
88#454D17911yescharacteristic, complemented
89#355—1791no—
90#355—1792no—
912−1+4C2×D17911yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—3582no—
94#124—3582no—
95#124—3581no—
96#124—1792no—
97#124C2×D17911yescharacteristic
98#120—3582no—
99#120—3581no—
100#104—21no—
101#104—3582no—
102#101—1792no—
103#101C2×D17911yescharacteristic, complemented
104C24—3581no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—3581no—
110C2×Q8C22×D17911yescharacteristic
111C2×D4—3581no—
112C2×D4—3582no—
113C2×D4—1792no—
114C2×D4C22⋊D17911yescharacteristic
115SD16—41no—
116SD16—7162no—
117D8—41no—
118D8—7162no—
119M16—21no—
120M16—3582no—
121C22⋊C4—22no—
122C22⋊C4—3582no—
123C22⋊C4—3581no—
124C22⋊C4—3582no—
125C22⋊C4—3582no—
126C42—1792no—
127C42C22⋊D17911yescharacteristic
128C23—21no—
129C23—3582no—
130C23—14322no—
131C23—14321no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—14321no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—14321no—
141D4—14322no—
142D4—7162no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—3582no—
147C2×C4D4×D17912yes—
148C2×C4C2×C22⋊D17911yescharacteristic
149C8—41no—
150C8—7162no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—14324no—
155C22—7162no—
156C22—7162no—
157C22—7162no—
158C22#1915308111yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—7162no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—14322no—
168C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#454×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#454D179
hypocenterC179#2202
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #26039181C2semidirect
2. #26038612C2semidirect
3. #26038236C2semidirect
4. #26038235C2semidirect
5. #454×C179C2semidirect
13. #19153032C22semidirect
14. 2−1+4×C179C22semidirect
18. #101×C179C22semidirect
84. C179#2202semidirect
88. #454D179semidirect
91. 2−1+4C2×D179semidirect
103. #101C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260391812
#260386122
#260382362
#260382352
#454×C1792
#2202179