Finite Group Databasebeta

← #37511018

Subgroups — #37511018

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#26038614C212yesmaximal, complemented
3#26038611C211yesmaximal, characteristic, complemented
4#26038235C212yesmaximal, complemented
5#355×C179C211yesmaximal, characteristic, complemented
6#19153161C2211yescharacteristic, complemented
7#19153141—22no—
8#19153139—21no—
9#19153102C2212yes—
10#19153101—21no—
11#19153099—22no—
12#19153083—22no—
13#19153081—21no—
14#19153041—21no—
15#19153039—22no—
16#19153032C2211yescharacteristic, complemented
17#124×C179C2211yescharacteristic
18#101×C179C2212yescomplemented
19C23×D179—21no—
20#15668260—41no—
21#15668259—22no—
22#15668259—21no—
23C2×Dic358—21no—
24Q8×D179—41no—
25C2×D716—21no—
26C2×D716D412yes—
27C2×C22⋊D179—22no—
28#15668248—42no—
29#15668247—41no—
30D1432—42no—
31#15668244—41no—
32#15668243—21no—
33C22⋊Dic179—21no—
34(C2×D179)⋊C4—22no—
35(C2×D179)⋊C4—22no—
36(C2×D179)⋊C4—22no—
37C4×Dic179D412yes—
38C2×Q8×C179C2311yescharacteristic
39C2×D4×C179D411yescharacteristic
40M16×C179—22no—
41C22⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43C22×D179—22no—
44C22×D179—82no—
45C22×D179—81no—
46Dic358—21no—
47Dic358—22no—
48Dic358—81no—
49D716—21no—
50D716—22no—
51D716—42no—
52D716—81no—
53C22⋊D179—41no—
54C22⋊D179—82no—
55C2×Dic179—22no—
56C4×D179—42no—
57C4×D179—41no—
58C179⋊C8—41no—
59C23×C179—21no—
60Q8×C179—41no—
61D4×C179—41no—
62C2×C4×C179—21no—
63C2×C4×C179C2×D412yes—
64C2×C4×C179C2×D411yescharacteristic
65C8×C179—42no—
66C22×C179—42no—
67C22×C179—41no—
68C22×C179#12011yescharacteristic
69C2×D179—42no—
70C2×D179—42no—
71C2×D179—41no—
72C2×D179—42no—
73C2×D179—84no—
74C4×C179—22no—
75C4×C179—21no—
76C4×C179—41no—
77Dic179—42no—
78Dic179—41no—
79C2×C179—21no—
80C2×C179—81no—
81C2×C179B(4,𝔽2)11yescharacteristic
82D179—81no—
83D179—82no—
84C179#220211yescharacteristic, complemented, Sylow
85#2202—1791nomaximal, Sylow
86H(3,𝔽4)—1791no—
87#454—1791no—
88#454—1792no—
89#355—1792no—
90#355D17911yescharacteristic, complemented
912−1+4—1791no—
92AGL(1,ℤ/8)—3582no—
93AGL(1,ℤ/8)—3581no—
94#124—3582no—
95#124—3581no—
96#124—1792no—
97#124C2×D17911yescharacteristic
98#120—3582no—
99#120—3581no—
100#104—3582no—
101#104—3581no—
102#101—1791no—
103#101C2×D17912yescomplemented
104C24—3581no—
105#41—3582no—
106#41—3581no—
107#41—7161no—
108C2×Q8—3581no—
109C2×Q8—7161no—
110C2×Q8C22×D17911yescharacteristic
111C2×D4—3581no—
112C2×D4—3582no—
113C2×D4—1792no—
114C2×D4D71611yescharacteristic
115SD16—7162no—
116SD16—7161no—
117D8—7162no—
118D8—7161no—
119M16—22no—
120M16—3581no—
121C22⋊C4—22no—
122C22⋊C4—3582no—
123C22⋊C4—3581no—
124C22⋊C4—3582no—
125C22⋊C4—3582no—
126C42—1792no—
127C42D71611yescharacteristic
128C23—21no—
129C23—3582no—
130C23—14322no—
131C23—14321no—
132Q8—41no—
133Q8—3582no—
134Q8—3581no—
135Q8—14321no—
136D4—41no—
137D4—3582no—
138D4—3581no—
139D4—14321no—
140D4—14322no—
141D4—7162no—
142D4—7161no—
143C2×C4—21no—
144C2×C4—3582no—
145C2×C4—7162no—
146C2×C4—7161no—
147C2×C4D4×D17912yes—
148C2×C4C2×D71611yescharacteristic
149C8—42no—
150C8—7161no—
151C22—42no—
152C22—41no—
153C22—14324no—
154C22—7162no—
155C22—7162no—
156C22—7161no—
157C22—7162no—
158C22#1915308311yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—7162no—
163C4—7161no—
164C2—21no—
165C2—81no—
166C2—14322no—
167C2—14321no—
168C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#355×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#355D179
hypocenterC179#2202
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #26039181C2semidirect
2. #26038614C2semidirect
3. #26038611C2semidirect
4. #26038235C2semidirect
5. #355×C179C2semidirect
6. #19153161C22semidirect
16. #19153032C22semidirect
18. #101×C179C22semidirect
84. C179#2202semidirect
90. #355D179semidirect
103. #101C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260391812
#260386142
#260386112
#260382352
#355×C1792
#2202179