Finite Group Databasebeta

← #37511020

Subgroups — #37511020

Order 22912.

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

normal subgroups only

This group has 216 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#26039189C211yesmaximal, characteristic, complemented
2#26038622C212yesmaximal, complemented
3#26038237C212yesmaximal, complemented
4#26038236C211yesmaximal, characteristic, complemented
5#454×C179C211yesmaximal, characteristic
6#19153143—22no—
7#19153111C2212yes—
8#19153106—22no—
9#19153105—21no—
10#19153094—21no—
11#19153090—22no—
12#19153040—22no—
13#19153032C2212yes—
142−1+4×C179C2211yescharacteristic
15AGL(1,ℤ/8)×C179—21no—
16#124×C179C2211yescharacteristic
17#104×C179—21no—
18#101×C179C2211yescharacteristic
19C2×Dic358D412yes—
20C22×Dic179—21no—
21#15668249—42no—
22#15668247—42no—
23#15668243—22no—
24C4⋊Dic179—21no—
25C4⋊Dic179—21no—
26Dic179⋊C4—22no—
27Dic179⋊C4—22no—
28Dic179⋊C4—22no—
29C22⋊Dic179—22no—
30C22⋊Dic179—21no—
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
43Dic358—42no—
44C2×Dic179—22no—
45C2×Dic179—22no—
46C2×Dic179—81no—
47C179⋊C8—42no—
48C23×C179—21no—
49Q8×C179—21no—
50Q8×C179—22no—
51Q8×C179—41no—
52D4×C179—21no—
53D4×C179—22no—
54D4×C179—41no—
55D4×C179—41no—
56C2×C4×C179—21no—
57C2×C4×C179—41no—
58C2×C4×C179—42no—
59C2×C4×C179C2×D412yes—
60C2×C4×C179C2×D411yescharacteristic
61C8×C179—41no—
62C22×C179—42no—
63C22×C179—41no—
64C22×C179—41no—
65C22×C179#12011yescharacteristic
66C4×C179—21no—
67C4×C179—22no—
68C4×C179—41no—
69C4×C179—41no—
70Dic179—42no—
71Dic179—42no—
72C2×C179—21no—
73C2×C179—81no—
74C2×C179—81no—
75C2×C179B(4,𝔽2)11yescharacteristic
76C179#220311yescharacteristic, complemented, Sylow
77#2203—1791nomaximal, Sylow
78#563—1791no—
79#456—1792no—
80#454D17911yescharacteristic
81#356—1792no—
82#355—1791no—
832−1+4C2×D17911yescharacteristic
84#137—3582no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1792no—
87#126—3582no—
88#125—3581no—
89#124C2×D17911yescharacteristic
90#123—3581no—
91C22⋊Q8—3582no—
92#104—21no—
93#104—3582no—
94#101—1792no—
95#101C2×D17911yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1792no—
101C2×Q8C22×D17911yescharacteristic
102C2×D4C22⋊D17911yescharacteristic
103C22×C4—3581no—
104Q16—7162no—
105SD16—41no—
106SD16—7162no—
107D8—41no—
108M16—21no—
109M16—3582no—
110C4⋊C4—3582no—
111C4⋊C4—3581no—
112C4⋊C4—3581no—
113C4⋊C4—3582no—
114C4⋊C4—3582no—
115C22⋊C4—22no—
116C22⋊C4—3582no—
117C22⋊C4—3581no—
118C42—1792no—
119C42C22⋊D17911yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—7162no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—3582no—
133C2×C4—3582no—
134C2×C4—14321no—
135C2×C4D4×D17912yes—
136C2×C4C2×C22⋊D17911yescharacteristic
137C8—41no—
138C8—7162no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#1915308111yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—7162no—
148C4—7162no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting, prime radical#454×C179C2
socleC2×C179B(4,𝔽2)
hypercenter#454D179
hypocenterC179#2203
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. #26039189C2semidirect
2. #26038622C2semidirect
3. #26038237C2semidirect
4. #26038236C2semidirect
76. C179#2203semidirect

Maximal subgroups

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

subgroupindex
#260391892
#260386222
#260382372
#260382362
#454×C1792
#2203179