Finite Group Databasebeta

← #37511029

Subgroups — #37511029

Order 22912.

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

normal subgroups only

This group has 224 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#26039169C211yesmaximal, characteristic, complemented
2#26038622C212yesmaximal
3#26038238C211yesmaximal, characteristic, complemented
4#26038237C212yesmaximal, complemented
5#456×C179C211yesmaximal, characteristic
6#19153143—22no—
7#19153112—22no—
8#19153111—21no—
9#19153111C2212yes—
10C4×Dic358—21no—
11#19153078—22no—
12#19153040—22no—
13#19153032C2212yes—
142−1+4×C179C2211yescharacteristic
15#137×C179—21no—
16C4⋊Q8×C179C2211yescharacteristic
17#104×C179—21no—
18#101×C179C2211yescharacteristic
19C2×Dic358—21no—
20C2×Dic358—21no—
21C2×Dic358D412yes—
22#15668249—42no—
23#15668247—42no—
24#15668243—22no—
25C4⋊Dic179—21no—
26Dic179⋊C4—22no—
27Dic179⋊C4—22no—
28Dic179⋊C4—22no—
29C4×Dic179—21no—
30C4×Dic179—22no—
31C4×Dic179D412yes—
32#41×C179—21no—
33#41×C179—22no—
34#41×C179—41no—
35C2×Q8×C179—41no—
36C2×Q8×C179D411yescharacteristic
37C2×Q8×C179C2311yescharacteristic
38Q16×C179—41no—
39SD16×C179—41no—
40M16×C179—21no—
41C4⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43Dic358—24no—
44Dic358—42no—
45Dic358—81no—
46C2×Dic179—22no—
47C2×Dic179—22no—
48C179⋊C8—42no—
49Q8×C179—21no—
50Q8×C179—22no—
51Q8×C179—41no—
52Q8×C179—41no—
53D4×C179—21no—
54D4×C179—22no—
55D4×C179—41no—
56C2×C4×C179—21no—
57C2×C4×C179—21no—
58C2×C4×C179—41no—
59C2×C4×C179—42no—
60C2×C4×C179C2×D412yes—
61C2×C4×C179C2×D411yescharacteristic
62C8×C179—41no—
63C22×C179—41no—
64C22×C179#12011yescharacteristic
65C4×C179—22no—
66C4×C179—21no—
67C4×C179—22no—
68C4×C179—41no—
69C4×C179—41no—
70Dic179—24no—
71Dic179—42no—
72C2×C179—21no—
73C2×C179—81no—
74C2×C179B(4,𝔽2)11yescharacteristic
75C179#220511yescharacteristic, complemented, Sylow
76#2205—1791nomaximal, Sylow
77Q82—1791no—
78#456—1792no—
79#456D17911yescharacteristic
80#356—1791no—
81#356—1792no—
822−1+4C2×D17911yescharacteristic
83#137—21no—
84#137—3582no—
85C4⋊Q8—3582no—
86C4⋊Q8—3581no—
87C4⋊Q8—1792no—
88C4⋊Q8C2×D17911yescharacteristic
89C4×Q8—3582no—
90C4×Q8—3581no—
91#104—21no—
92#104—3582no—
93#101—1792no—
94#101C2×D17911yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—3581no—
100C2×Q8—3581no—
101C2×Q8—1792no—
102C2×Q8C22⋊D17911yescharacteristic
103C2×Q8C22×D17911yescharacteristic
104Q16—41no—
105Q16—7162no—
106SD16—41no—
107SD16—7162no—
108M16—21no—
109M16—3582no—
110C4⋊C4—22no—
111C4⋊C4—3582no—
112C4⋊C4—3581no—
113C4⋊C4—3582no—
114C4⋊C4—3582no—
115C42—3582no—
116C42—3581no—
117C42—1792no—
118C42C22⋊D17911yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—3584no—
124Q8—14321no—
125Q8—7162no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—3582no—
134C2×C4—3582no—
135C2×C4D4×D17912yes—
136C2×C4C2×C22⋊D17911yescharacteristic
137C8—41no—
138C8—7162no—
139C22—41no—
140C22#1915308111yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—3584no—
147C4—7162no—
148C2—21no—
149C2—81no—
150C2#2603862511yescharacteristic
subgroupG/H
centerC2#26038625
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#456×C179C2
socleC2×C179B(4,𝔽2)
prime radical2−1+4×C179C22
hypercenter#456D179
hypocenterC179#2205
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #26039169C2semidirect
3. #26038238C2semidirect
4. #26038237C2semidirect
75. C179#2205semidirect

Maximal subgroups

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

subgroupindex
#260391692
#260386222
#260382382
#260382372
#456×C1792
#2205179