Finite Group Databasebeta

← #37508010

Subgroups — #37508010

Order 22912.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#26039070C211yesmaximal, characteristic
2#26038770C212yesmaximal
3#26038369C211yesmaximal, characteristic
4#26038177C212yesmaximal
5#324×C179C211yesmaximal, characteristic
6#19153112—22no—
7#19153104—22no—
8#19153092—21no—
9#19153092C2212yes—
10#19153072—21no—
11C2×C4⋊Dic179C2211yescharacteristic
12C2×Dic179⋊C4—21no—
13#19153045—22no—
14#19153038—22no—
15C179⋊#95—21no—
16C179⋊#95C2211yescharacteristic
17C2×C4⋊C4×C179C2211yescharacteristic
18C22⋊C8×C179C2212yes—
19C2×Dic358—21no—
20C2×Dic358D412yes—
21C22×Dic179—21no—
22C22×Dic179—21no—
23C4⋊Dic179—21no—
24C4⋊Dic179—21no—
25C4⋊Dic179—21no—
26C4⋊Dic179—22no—
27C4⋊Dic179D412yes—
28Dic179⋊C4—22no—
29Dic179⋊C4—22no—
30Dic179⋊C4—42no—
31C22⋊Dic179—22no—
32C4×Dic179—22no—
33C22×C4×C179D411yescharacteristic
34C22×C4×C179D411yescharacteristic
35C22×C4×C179C2311yescharacteristic
36C2×C8×C179—22no—
37C4⋊C4×C179—22no—
38Dic358—42no—
39Dic358—81no—
40C2×Dic179—22no—
41C2×Dic179—22no—
42C2×Dic179—22no—
43C2×Dic179—41no—
44C2×Dic179—41no—
45C2×Dic179—41no—
46C2×Dic179—41no—
47C2×Dic179—41no—
48C23×C179C2×D411yescharacteristic
49C2×C4×C179—21no—
50C2×C4×C179—21no—
51C2×C4×C179—21no—
52C2×C4×C179—41no—
53C2×C4×C179—41no—
54C2×C4×C179SD1612yes—
55C2×C4×C179C2×D412yes—
56C8×C179—42no—
57C22×C179—21no—
58C22×C179—21no—
59C22×C179#12011yescharacteristic
60C22×C179C2×SD1611yescharacteristic
61C22×C179#13711yescharacteristic
62C4×C179—22no—
63C4×C179—22no—
64C4×C179—41no—
65Dic179—42no—
66Dic179—42no—
67Dic179—42no—
68Dic179—81no—
69C2×C179—21no—
70C2×C179—21no—
71C2×C179#44811yescharacteristic
72C2×C179#45611yescharacteristic
73C2×C179#45811yescharacteristic
74C179#161111yescharacteristic, complemented, Sylow
75#1611—1791nomaximal, Sylow
76#544—1791no—
77#483—1792no—
78#398—1791no—
79#328—1792no—
80#324D17911yescharacteristic
81C4⋊Q8—3582no—
82#125—3582no—
83C22⋊Q8—3581no—
84C22⋊Q8—1792no—
85#117—3581no—
86C2×C4⋊C4—3581no—
87C2×C4⋊C4—1791no—
88C2×C4⋊C4C2×D17911yescharacteristic
89#106—3582no—
90#103—3582no—
91C22⋊C8C2×D17912yes—
92#95—3581no—
93#95—1791no—
94C2×Q8—3581no—
95C2×Q8—1792no—
96C22×C4—3581no—
97C22×C4—3581no—
98C22×C4D71611yescharacteristic
99C22×C4D71611yescharacteristic
100C22×C4C22×D17911yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—3582no—
104C4⋊C4—3581no—
105C4⋊C4—3581no—
106C4⋊C4—3581no—
107C4⋊C4—3582no—
108C4⋊C4—3582no—
109C4⋊C4—1792no—
110C4⋊C4—7162no—
111C22⋊C4—3582no—
112C42—3582no—
113C23C2×D71611yescharacteristic
114Q8—14321no—
115Q8—7162no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—3582no—
122C2×C4—3582no—
123C2×C4—3582no—
124C2×C4—7161no—
125C2×C4—7161no—
126C2×C4—7161no—
127C2×C4—7161no—
128C2×C4—7161no—
129C2×C4#1566824812yes—
130C2×C4D4×D17912yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#1915308311yescharacteristic
135C22C2×#1566824811yescharacteristic
136C22#1915314611yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—14321no—
141C4—7162no—
142C4—7162no—
143C4—7162no—
144C2—21no—
145C2—21no—
146C2#2603858011yescharacteristic
147C2#2603862411yescharacteristic
148C2#2603863011yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#324×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#324D179
hypocenterC179#1611
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
74. C179#1611semidirect

Maximal subgroups

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

subgroupindex
#260390702
#260387702
#260383692
#260381772
#324×C1792
#1611179