Finite Group Databasebeta

← #37507976

Subgroups — #37507976

Order 22912.

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

normal subgroups only

This group has 238 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#26038597C212yesmaximal
3#26038359C211yesmaximal, characteristic
4#26038177C212yesmaximal
5#324×C179C211yesmaximal, characteristic
6C22×Dic358C2211yescharacteristic
7C2×Dic716—22no—
8#19153112—22no—
9#19153104—22no—
10#19153092—21no—
11#19153092C2212yes—
12#19153072—21no—
13C2×Dic179⋊C4—21no—
14#19153038—22no—
15C179⋊#95—21no—
16C179⋊#95C2211yescharacteristic
17C2×C4⋊C4×C179C2211yescharacteristic
18C22⋊C8×C179C2212yes—
19C2×Dic358—21no—
20C2×Dic358—22no—
21C2×Dic358—42no—
22C2×Dic358D412yes—
23C22×Dic179—21no—
24C22×Dic179—21no—
25Dic716—44no—
26C4⋊Dic179—21no—
27C4⋊Dic179—21no—
28C4⋊Dic179—21no—
29C4⋊Dic179D412yes—
30Dic179⋊C4—22no—
31Dic179⋊C4—22no—
32Dic179⋊C4—42no—
33C22⋊Dic179—22no—
34C4×Dic179—22no—
35C22×C4×C179D411yescharacteristic
36C22×C4×C179D411yescharacteristic
37C22×C4×C179C2311yescharacteristic
38C2×C8×C179—22no—
39C4⋊C4×C179—22no—
40Dic358—24no—
41Dic358—42no—
42Dic358—42no—
43Dic358—81no—
44C2×Dic179—22no—
45C2×Dic179—22no—
46C2×Dic179—22no—
47C2×Dic179—22no—
48C2×Dic179—41no—
49C2×Dic179—41no—
50C2×Dic179—41no—
51C2×Dic179—41no—
52C23×C179C2×D411yescharacteristic
53C2×C4×C179—21no—
54C2×C4×C179—21no—
55C2×C4×C179—21no—
56C2×C4×C179—41no—
57C2×C4×C179—41no—
58C2×C4×C179Q1612yescomplemented
59C2×C4×C179C2×D412yes—
60C8×C179—42no—
61C22×C179—21no—
62C22×C179—21no—
63C22×C179#12011yescharacteristic
64C22×C179C2×Q1611yescharacteristic
65C22×C179#13711yescharacteristic
66C4×C179—22no—
67C4×C179—22no—
68C4×C179—41no—
69Dic179—42no—
70Dic179—42no—
71Dic179—42no—
72Dic179—42no—
73C2×C179—21no—
74C2×C179—21no—
75C2×C179#45111yescharacteristic
76C2×C179#45611yescharacteristic
77C2×C179B(4,𝔽2)11yescharacteristic
78C179#160511yescharacteristic, complemented, Sylow
79#1605—1791nomaximal, Sylow
80#544—1791no—
81#451—1792no—
82#395—1791no—
83#328—1792no—
84#324D17911yescharacteristic
85C22×Q8—1791no—
86C2×Q16—3582no—
87C4⋊Q8—3582no—
88#125—3582no—
89C22⋊Q8—3581no—
90C22⋊Q8—1792no—
91#117—3581no—
92C2×C4⋊C4—3581no—
93C2×C4⋊C4C2×D17911yescharacteristic
94#103—3582no—
95C22⋊C8C2×D17912yes—
96#95—3581no—
97#95—1791no—
98C2×Q8—3581no—
99C2×Q8—3582no—
100C2×Q8—1792no—
101C2×Q8—7162no—
102C22×C4—3581no—
103C22×C4—3581no—
104C22×C4D71611yescharacteristic
105C22×C4D71611yescharacteristic
106C22×C4C22×D17911yescharacteristic
107Q16—7164no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—3582no—
111C4⋊C4—3581no—
112C4⋊C4—3581no—
113C4⋊C4—3581no—
114C4⋊C4—3582no—
115C4⋊C4—1792no—
116C4⋊C4—7162no—
117C22⋊C4—3582no—
118C42—3582no—
119C23C2×D71611yescharacteristic
120Q8—3584no—
121Q8—14321no—
122Q8—7162no—
123Q8—7162no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—3582no—
130C2×C4—3582no—
131C2×C4—3582no—
132C2×C4—3582no—
133C2×C4—7161no—
134C2×C4—7161no—
135C2×C4—7161no—
136C2×C4—7161no—
137C2×C4Dic71612yescomplemented
138C2×C4D4×D17912yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#1915308311yescharacteristic
143C22C2×Dic71611yescharacteristic
144C22#1915314611yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—7162no—
149C4—7162no—
150C4—7162no—
151C4—7162no—
152C2—21no—
153C2—21no—
154C2#2603859711yescharacteristic
155C2#2603862411yescharacteristic
156C2#2603862711yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#324×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#324D179
hypocenterC179#1605
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
58. C2×C4×C179Q16semidirect
78. C179#1605semidirect
137. C2×C4Dic716semidirect

Maximal subgroups

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

subgroupindex
#260390702
#260385972
#260383592
#260381772
#324×C1792
#1605179