Finite Group Databasebeta

← #37511021

Subgroups — #37511021

Order 22912.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#26038621C212yesmaximal, complemented
2#26038611C211yesmaximal, characteristic
3#26038238C212yesmaximal, complemented
4#26038236C211yesmaximal, characteristic, complemented
5#563×C179C211yesmaximal, characteristic, complemented
6#19153161C2211yescharacteristic
7#19153144—22no—
8#19153139—21no—
9#19153041—21no—
10#19153041—22no—
11#19153032C2211yescharacteristic
12#19153032C2212yes—
13C4⋊Q8×C179C2212yes—
14#126×C179—22no—
15#125×C179—21no—
16#124×C179C2211yescharacteristic
17#123×C179—21no—
18C22⋊Q8×C179—22no—
19#15668260—41no—
20#15668259—21no—
21#15668259—22no—
22Q8×D179—41no—
23#15668249—42no—
24#15668247—41no—
25#15668246—42no—
26#15668244—41no—
27#15668243—21no—
28#15668243—22no—
29C2×Q8×C179D412yes—
30C2×Q8×C179C2311yescharacteristic
31C2×D4×C179D411yescharacteristic
32C22×C4×C179—21no—
33C4⋊C4×C179—21no—
34C4⋊C4×C179—22no—
35C4⋊C4×C179—21no—
36C4⋊C4×C179—22no—
37C4⋊C4×C179—22no—
38C22⋊C4×C179—22no—
39C22⋊C4×C179—21no—
40C22⋊C4×C179—22no—
41C42×C179D411yescharacteristic
42C42×C179D412yes—
43Dic358—21no—
44Dic358—22no—
45D716—21no—
46D716—22no—
47C22⋊D179—41no—
48C4×D179—41no—
49C4×D179—42no—
50C179⋊C8—41no—
51C179⋊C8—42no—
52C23×C179—21no—
53Q8×C179—41no—
54Q8×C179—42no—
55D4×C179—41no—
56C2×C4×C179—22no—
57C2×C4×C179—21no—
58C2×C4×C179—22no—
59C2×C4×C179—81no—
60C2×C4×C179C2×D411yescharacteristic
61C2×C4×C179C2×D412yes—
62C22×C179—42no—
63C22×C179—41no—
64C22×C179#12011yescharacteristic
65C2×D179—41no—
66C4×C179—21no—
67C4×C179—22no—
68C4×C179—41no—
69C4×C179—42no—
70C4×C179—42no—
71Dic179—41no—
72C2×C179—21no—
73C2×C179—81no—
74C2×C179B(4,𝔽2)11yescharacteristic
75D179—81no—
76C179#220311yescharacteristic, complemented, Sylow
77#2203—1791nomaximal, Sylow
78#563D17911yescharacteristic, complemented
79#456—1792no—
80#454—1791no—
81#356—1792no—
82#355—1791no—
832−1+4—1791no—
84#137—3582no—
85AGL(1,ℤ/8)—3581no—
86C4⋊Q8C2×D17912yes—
87#126—22no—
88#125—21no—
89#124C2×D17911yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—3581no—
93#104—3582no—
94#101—1792no—
95#101—1791no—
96#41—3581no—
97#41—3582no—
98#41—7161no—
99C2×Q8—7161no—
100C2×Q8C22⋊D17912yes—
101C2×Q8C22×D17911yescharacteristic
102C2×D4C22⋊D17911yescharacteristic
103C22×C4—21no—
104Q16—7162no—
105SD16—7161no—
106SD16—7162no—
107D8—7161no—
108M16—3581no—
109M16—3582no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D17911yescharacteristic
119C42C22⋊D17912yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—3581no—
124Q8—3582no—
125D4—41no—
126D4—3581no—
127D4—3582no—
128D4—7161no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—7161no—
134C2×C4—7162no—
135C2×C4C2×C22⋊D17912yes—
136C2×C4C2×C22⋊D17911yescharacteristic
137C8—7161no—
138C8—7162no—
139C22—42no—
140C22—41no—
141C22—7161no—
142C22#1915308211yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—7161no—
149C2—21no—
150C2—81no—
151C2—14321no—
152C2#2603862611yescharacteristic
subgroupG/H
centerC2#26038626
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#563×C179C2
socleC2×C179B(4,𝔽2)
prime radical#26038611C2
hypercenter#563D179
hypocenterC179#2203
LCSG ▷ C2×Q8×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #26038621C2semidirect
3. #26038238C2semidirect
4. #26038236C2semidirect
5. #563×C179C2semidirect
76. C179#2203semidirect
78. #563D179semidirect

Maximal subgroups

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

subgroupindex
#260386212
#260386112
#260382382
#260382362
#563×C1792
#2203179