Finite Group Databasebeta

← #37511023

Subgroups — #37511023

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#26038624C212yesmaximal, complemented
3#26038611C211yesmaximal, characteristic
4#26038237C212yesmaximal, complemented
5#355×C179C211yesmaximal, characteristic, complemented
6#19153161C2211yescharacteristic
7#19153146—22no—
8#19153139—21no—
9#19153111C2212yes—
10#19153106—22no—
11#19153105—21no—
12#19153094—21no—
13#19153090—22no—
14#19153041—21no—
15#19153039—22no—
16#19153032C2211yescharacteristic
17#124×C179C2211yescharacteristic
18#101×C179C2212yes—
19#15668260—41no—
20#15668259—22no—
21#15668259—21no—
22C2×Dic358D412yes—
23Q8×D179—41no—
24C22×Dic179—21no—
25Dic716—42no—
26#15668248—42no—
27#15668247—41no—
28#15668244—41no—
29#15668243—21no—
30C4⋊Dic179—21no—
31C4⋊Dic179—21no—
32Dic179⋊C4—22no—
33Dic179⋊C4—22no—
34Dic179⋊C4—22no—
35C22⋊Dic179—22no—
36C22⋊Dic179—21no—
37C4×Dic179D412yes—
38C2×Q8×C179C2311yescharacteristic
39C2×D4×C179D411yescharacteristic
40M16×C179—22no—
41C22⋊C4×C179—22no—
42C42×C179D411yescharacteristic
43Dic358—21no—
44Dic358—22no—
45Dic358—42no—
46D716—21no—
47D716—22no—
48C22⋊D179—41no—
49C2×Dic179—22no—
50C2×Dic179—22no—
51C2×Dic179—81no—
52C4×D179—42no—
53C4×D179—41no—
54C179⋊C8—41no—
55C23×C179—21no—
56Q8×C179—41no—
57D4×C179—41no—
58C2×C4×C179—21no—
59C2×C4×C179C2×D412yes—
60C2×C4×C179C2×D411yescharacteristic
61C8×C179—42no—
62C22×C179—42no—
63C22×C179—41no—
64C22×C179#12011yescharacteristic
65C2×D179—41no—
66C4×C179—22no—
67C4×C179—21no—
68C4×C179—41no—
69Dic179—42no—
70Dic179—41no—
71Dic179—42no—
72C2×C179—21no—
73C2×C179—81no—
74C2×C179B(4,𝔽2)11yescharacteristic
75D179—81no—
76C179#220311yescharacteristic, complemented, Sylow
77#2203—1791nomaximal, Sylow
78#563—1791no—
79#456—1792no—
80#454—1791no—
81#356—1792no—
82#355D17911yescharacteristic, complemented
832−1+4—1791no—
84#137—3582no—
85AGL(1,ℤ/8)—3581no—
86C4⋊Q8—1792no—
87#126—3582no—
88#125—3581no—
89#124C2×D17911yescharacteristic
90#123—3581no—
91C22⋊Q8—3582no—
92#104—3582no—
93#104—3581no—
94#101—1791no—
95#101C2×D17912yes—
96#41—3582no—
97#41—3581no—
98#41—7161no—
99C2×Q8—1792no—
100C2×Q8—7161no—
101C2×Q8C22×D17911yescharacteristic
102C2×D4D71611yescharacteristic
103C22×C4—3581no—
104Q16—7162no—
105SD16—7162no—
106SD16—7161no—
107D8—7161no—
108M16—22no—
109M16—3581no—
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—
119C42D71611yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—3581no—
123Q8—3582no—
124Q8—7162no—
125D4—41no—
126D4—3581no—
127D4—3582no—
128D4—7161no—
129C2×C4—21no—
130C2×C4—3582no—
131C2×C4—3582no—
132C2×C4—14321no—
133C2×C4—7162no—
134C2×C4—7161no—
135C2×C4D4×D17912yes—
136C2×C4C2×D71611yescharacteristic
137C8—42no—
138C8—7161no—
139C22—42no—
140C22—41no—
141C22—7161no—
142C22#1915308311yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—7162no—
147C4—7162no—
148C4—7161no—
149C2—21no—
150C2—81no—
151C2—14321no—
152C2#2603862711yescharacteristic
subgroupG/H
centerC2#26038627
commutator, cosocleC2×Q8×C179C23
FrattiniC2×Q8C22×D179
Fitting#355×C179C2
socleC2×C179B(4,𝔽2)
prime radical#26038611C2
hypercenter#355D179
hypocenterC179#2203
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. #26039189C2semidirect
2. #26038624C2semidirect
4. #26038237C2semidirect
5. #355×C179C2semidirect
76. C179#2203semidirect
82. #355D179semidirect

Maximal subgroups

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

subgroupindex
#260391892
#260386242
#260386112
#260382372
#355×C1792
#2203179