Finite Group Databasebeta

← #37507959

Subgroups — #37507959

Order 22912.

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

normal subgroups only

This group has 256 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#26039065C211yesmaximal, characteristic
2#26038597C212yesmaximal, complemented
3#26038351C211yesmaximal, characteristic
4#26038177C212yesmaximal, complemented
5#323×C179C211yesmaximal, characteristic
6C22×Dic358C2211yescharacteristic
7C2×Dic716—22no—
8#19153092—21no—
9#19153092C2212yes—
10#19153090—22no—
11#19153090—22no—
12C2×C22⋊Dic179—21no—
13C2×C22⋊Dic179—21no—
14#19153038—22no—
15C179⋊#95—21no—
16C179⋊#95C2211yescharacteristic
17C2×C22⋊C4×C179C2211yescharacteristic
18C22⋊C8×C179C2212yes—
19C2×Dic358—21no—
20C2×Dic358—22no—
21C2×Dic358—42no—
22C2×Dic358D412yes—
23C22×Dic179—21no—
24C22×Dic179—21no—
25C22×Dic179—21no—
26C22×Dic179—21no—
27Dic716—44no—
28C4⋊Dic179—21no—
29C4⋊Dic179D412yes—
30Dic179⋊C4—22no—
31Dic179⋊C4—22no—
32C22⋊Dic179—22no—
33C22⋊Dic179—22no—
34C22⋊Dic179—42no—
35C24×C179D411yescharacteristic
36C22×C4×C179D411yescharacteristic
37C22×C4×C179C2311yescharacteristic
38C2×C8×C179—22no—
39C22⋊C4×C179—22no—
40Dic358—24no—
41Dic358—42no—
42Dic358—42no—
43Dic358—81no—
44C2×Dic179—22no—
45C2×Dic179—22no—
46C2×Dic179—22no—
47C2×Dic179—41no—
48C2×Dic179—41no—
49C2×Dic179—41no—
50C2×Dic179—41no—
51C2×Dic179—41no—
52C2×Dic179—81no—
53C2×Dic179—81no—
54C23×C179—21no—
55C23×C179—21no—
56C23×C179—41no—
57C23×C179—41no—
58C23×C179Q1612yescomplemented
59C23×C179C2×D411yescharacteristic
60C2×C4×C179—21no—
61C2×C4×C179—41no—
62C2×C4×C179—41no—
63C2×C4×C179C2×D412yes—
64C8×C179—42no—
65C22×C179—21no—
66C22×C179—21no—
67C22×C179—22no—
68C22×C179—42no—
69C22×C179—41no—
70C22×C179—41no—
71C22×C179—81no—
72C22×C179#12011yescharacteristic
73C22×C179C2×Q1611yescharacteristic
74C22×C179#13711yescharacteristic
75C4×C179—22no—
76C4×C179—41no—
77Dic179—42no—
78Dic179—42no—
79Dic179—42no—
80Dic179—81no—
81C2×C179—21no—
82C2×C179—21no—
83C2×C179—42no—
84C2×C179#45111yescharacteristic
85C2×C179#45511yescharacteristic
86C2×C179B(4,𝔽2)11yescharacteristic
87C179#160211yescharacteristic, complemented, Sylow
88#1602—1791nomaximal, Sylow
89#543—1791no—
90#451—1792no—
91#393—1791no—
92#328—1792no—
93#323D17911yescharacteristic
94C22×Q8—1791no—
95C2×Q16—3582no—
96C22⋊Q8—3582no—
97C22⋊Q8—3582no—
98C22⋊Q8—3581no—
99C22⋊Q8—1792no—
100C2×C22⋊C4—3581no—
101C2×C22⋊C4—3581no—
102C2×C22⋊C4C2×D17911yescharacteristic
103#103—3582no—
104C22⋊C8C2×D17912yes—
105#95—3581no—
106#95—1791no—
107C24D71611yescharacteristic
108C2×Q8—3581no—
109C2×Q8—3582no—
110C2×Q8—1792no—
111C2×Q8—7162no—
112C22×C4—3581no—
113C22×C4—3581no—
114C22×C4—3581no—
115C22×C4—3581no—
116C22×C4D71611yescharacteristic
117C22×C4C22×D17911yescharacteristic
118Q16—7164no—
119C2×C8—22no—
120C4⋊C4—3582no—
121C4⋊C4—3581no—
122C4⋊C4—3582no—
123C4⋊C4—1792no—
124C22⋊C4—22no—
125C22⋊C4—3582no—
126C22⋊C4—3582no—
127C22⋊C4—7162no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic71612yescomplemented
133C23C2×D71611yescharacteristic
134Q8—3584no—
135Q8—14321no—
136Q8—7162no—
137Q8—7162no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—3582no—
142C2×C4—3582no—
143C2×C4—3582no—
144C2×C4—14321no—
145C2×C4—14321no—
146C2×C4—7161no—
147C2×C4—7161no—
148C2×C4—7161no—
149C2×C4—7161no—
150C2×C4—7161no—
151C2×C4D4×D17912yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1915308311yescharacteristic
161C22C2×Dic71611yescharacteristic
162C22#1915314611yescharacteristic
163C4—22no—
164C4—41no—
165C4—14321no—
166C4—7162no—
167C4—7162no—
168C4—7162no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2603859711yescharacteristic
173C2#2603861911yescharacteristic
174C2#2603862711yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#323×C179C2
socleC22×C179#120
prime radicalC24×C179D4
hypercenter#323D179
hypocenterC179#1602
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #26038597C2semidirect
4. #26038177C2semidirect
58. C23×C179Q16semidirect
87. C179#1602semidirect
132. C23Dic716semidirect

Maximal subgroups

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

subgroupindex
#260390652
#260385972
#260383512
#260381772
#323×C1792
#1602179