Finite Group Databasebeta

← #37507971

Subgroups — #37507971

Order 22912.

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

normal subgroups only

This group has 282 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#26039020C211yesmaximal, characteristic
2#26038580C212yesmaximal, complemented
3#26038359C211yesmaximal, characteristic, complemented
4#26038174C212yesmaximal, complemented
5#324×C179C211yesmaximal, characteristic, complemented
6C2×#15668259—21no—
7C22×Dic358C2211yescharacteristic
8C2×#15668248—22no—
9#19153093—22no—
10#19153088—22no—
11#19153087—21no—
12#19153087C2212yes—
13C2×Dic179⋊C4—21no—
14#19153038—22no—
15C179⋊#95—21no—
16C179⋊#95C2211yescharacteristic
17C2×C4⋊C4×C179C2211yescharacteristic
18C22⋊C8×C179C2212yes—
19#15668259—42no—
20#15668259—42no—
21C2×Dic358—21no—
22C2×Dic358—22no—
23C2×Dic358—42no—
24C2×D716—21no—
25C2×D716—21no—
26C2×D716D412yes—
27C2×C22⋊D179—22no—
28C22×Dic179—21no—
29C22×Dic179—21no—
30C2×C4×D179—22no—
31#15668248—44no—
32C4⋊Dic179—21no—
33C4⋊Dic179D412yes—
34Dic179⋊C4—42no—
35(C2×D179)⋊C4—22no—
36(C2×D179)⋊C4—22no—
37C22×C4×C179D411yescharacteristic
38C22×C4×C179D411yescharacteristic
39C22×C4×C179C2311yescharacteristic
40C2×C8×C179—22no—
41C4⋊C4×C179—22no—
42C22×D179—22no—
43Dic358—24no—
44Dic358—42no—
45Dic358—42no—
46D716—42no—
47D716—42no—
48D716—81no—
49C22⋊D179—42no—
50C22⋊D179—42no—
51C2×Dic179—22no—
52C2×Dic179—22no—
53C2×Dic179—22no—
54C2×Dic179—41no—
55C2×Dic179—41no—
56C2×Dic179—41no—
57C2×Dic179—41no—
58C4×D179—44no—
59C23×C179C2×D411yescharacteristic
60C2×C4×C179—21no—
61C2×C4×C179—21no—
62C2×C4×C179—21no—
63C2×C4×C179—41no—
64C2×C4×C179—41no—
65C2×C4×C179SD1612yescomplemented
66C2×C4×C179C2×D412yes—
67C8×C179—42no—
68C22×C179—21no—
69C22×C179—21no—
70C22×C179#12011yescharacteristic
71C22×C179C2×SD1611yescharacteristic
72C22×C179AGL(1,ℤ/8)11yescharacteristic
73C2×D179—42no—
74C2×D179—42no—
75C2×D179—42no—
76C4×C179—22no—
77C4×C179—22no—
78C4×C179—41no—
79Dic179—42no—
80Dic179—42no—
81Dic179—42no—
82C2×C179—21no—
83C2×C179—21no—
84C2×C179#45011yescharacteristic
85C2×C179#45611yescharacteristic
86C2×C179B(4,𝔽2)11yescharacteristic
87D179—82no—
88C179#160411yescharacteristic, complemented, Sylow
89#1604—1791nomaximal, Sylow
90#537—1791no—
91#448—1792no—
92#395—1791no—
93#327—1792no—
94#324D17911yescharacteristic, complemented
95C2×#41—3581no—
96C22×Q8—1791no—
97C2×SD16—3582no—
98C22⋊Q8—3582no—
99#121—3582no—
100#121—3581no—
101#121—1792no—
102C2×C4⋊C4—3581no—
103C2×C4⋊C4C2×D17911yescharacteristic
104#103—3582no—
105C22⋊C8C2×D17912yes—
106#95—3581no—
107#95—1791no—
108#41—7162no—
109#41—7162no—
110C2×Q8—3581no—
111C2×Q8—3582no—
112C2×Q8—7162no—
113C2×D4—3581no—
114C2×D4—3581no—
115C2×D4—3582no—
116C2×D4—1792no—
117C22×C4—3582no—
118C22×C4—3581no—
119C22×C4—3581no—
120C22×C4D71611yescharacteristic
121C22×C4D71611yescharacteristic
122C22×C4C22×D17911yescharacteristic
123SD16—7164no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—3581no—
127C4⋊C4—1792no—
128C4⋊C4—7162no—
129C22⋊C4—3582no—
130C22⋊C4—3582no—
131C23—3582no—
132C23C2×D71611yescharacteristic
133Q8—3584no—
134Q8—7162no—
135Q8—7162no—
136D4—14321no—
137D4—7162no—
138D4—7162no—
139D4—7162no—
140D4—7162no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3582no—
147C2×C4—3582no—
148C2×C4—3582no—
149C2×C4—7164no—
150C2×C4—7161no—
151C2×C4—7161no—
152C2×C4—7161no—
153C2×C4—7161no—
154C2×C4#1566824812yescomplemented
155C2×C4D4×D17912yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—7162no—
160C22—7162no—
161C22—7162no—
162C22#1915308311yescharacteristic
163C22C2×#1566824811yescharacteristic
164C22#1915314111yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—7162no—
169C4—7162no—
170C4—7162no—
171C2—21no—
172C2—21no—
173C2—14322no—
174C2#2603859211yescharacteristic
175C2#2603862411yescharacteristic
176C2#2603862711yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#324×C179C2
socleC22×C179#120
prime radical#26039020C2
hypercenter#324D179
hypocenterC179#1604
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
2. #26038580C2semidirect
3. #26038359C2semidirect
4. #26038174C2semidirect
5. #324×C179C2semidirect
65. C2×C4×C179SD16semidirect
88. C179#1604semidirect
94. #324D179semidirect
154. C2×C4#15668248semidirect

Maximal subgroups

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

subgroupindex
#260390202
#260385802
#260383592
#260381742
#324×C1792
#1604179