Finite Group Databasebeta

← #37507939

Subgroups — #37507939

Order 22912.

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

normal subgroups only

This group has 266 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, complemented
2#26038590C212yesmaximal
3#26038175C212yesmaximal, complemented
4#26038165C211yesmaximal, characteristic, complemented
5#394×C179C211yesmaximal, characteristic
6C2×#15668247—22no—
7#19153112—22no—
8#19153104—22no—
9#19153092—21no—
10#19153092C2212yes—
11#19153072—21no—
12#19153034—22no—
13C22⋊(C179⋊C8)C2212yes—
14C22×D4×C179C2211yescharacteristic
15C2×C4⋊C4×C179C2211yescharacteristic
16C2×C22⋊C4×C179—21no—
17C2×C22⋊C4×C179—21no—
18#95×C179C2211yescharacteristic
19C2×Dic358—21no—
20C2×Dic358D412yes—
21#15668247—44no—
22C2×C179⋊C8—22no—
23C4⋊Dic179—21no—
24C4⋊Dic179—21no—
25C4⋊Dic179—21no—
26C4⋊Dic179D412yes—
27Dic179⋊C4—22no—
28Dic179⋊C4—22no—
29C22⋊Dic179—22no—
30C4×Dic179—22no—
31C24×C179—21no—
32C2×D4×C179—22no—
33C2×D4×C179—42no—
34C22×C4×C179—21no—
35C22×C4×C179D411yescharacteristic
36C22×C4×C179D411yescharacteristic
37C22×C4×C179C2311yescharacteristic
38C4⋊C4×C179—22no—
39C22⋊C4×C179—42no—
40C22⋊C4×C179—42no—
41Dic358—42no—
42Dic358—81no—
43C2×Dic179—22no—
44C2×Dic179—22no—
45C179⋊C8—42no—
46C23×C179—22no—
47C23×C179—41no—
48C23×C179—41no—
49C23×C179—41no—
50C23×C179—41no—
51C23×C179—41no—
52C23×C179—41no—
53C23×C179C2×D411yescharacteristic
54D4×C179—24no—
55D4×C179—42no—
56C2×C4×C179—21no—
57C2×C4×C179—21no—
58C2×C4×C179—21no—
59C2×C4×C179—41no—
60C2×C4×C179—41no—
61C2×C4×C179—41no—
62C2×C4×C179—41no—
63C2×C4×C179—41no—
64C2×C4×C179SD1612yescomplemented
65C2×C4×C179C2×D412yes—
66C22×C179—21no—
67C22×C179—21no—
68C22×C179—42no—
69C22×C179—42no—
70C22×C179—42no—
71C22×C179—81no—
72C22×C179—81no—
73C22×C179—81no—
74C22×C179—81no—
75C22×C179#12011yescharacteristic
76C22×C179C2×SD1611yescharacteristic
77C22×C179#13711yescharacteristic
78C4×C179—22no—
79C4×C179—22no—
80C4×C179—41no—
81C4×C179—81no—
82Dic179—42no—
83Dic179—42no—
84C2×C179—21no—
85C2×C179—21no—
86C2×C179—82no—
87C2×C179#44811yescharacteristic
88C2×C179#45411yescharacteristic
89C2×C179B(4,𝔽2)11yescharacteristic
90C179#159911yescharacteristic, complemented, Sylow
91#1599—1791nomaximal, Sylow
92#544—1791no—
93#450—1792no—
94#394D17911yescharacteristic
95#328—1792no—
96#324—1791no—
97C22×D4C2×D17911yescharacteristic
98C2×SD16—3582no—
99C4⋊Q8—3582no—
100#125—3582no—
101C22⋊Q8—3581no—
102C22⋊Q8—1792no—
103#117—3581no—
104C2×C4⋊C4C2×D17911yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#102—3582no—
108C22⋊C8—1792no—
109#95C2×D17911yescharacteristic
110C24—21no—
111C2×Q8—3581no—
112C2×Q8—1792no—
113C2×D4—22no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4C22⋊D17911yescharacteristic
117C22×C4C22⋊D17911yescharacteristic
118C22×C4C22×D17911yescharacteristic
119SD16—7164no—
120C2×C8—3582no—
121C4⋊C4—22no—
122C4⋊C4—3582no—
123C4⋊C4—3581no—
124C4⋊C4—3581no—
125C4⋊C4—3581no—
126C4⋊C4—3582no—
127C4⋊C4—1792no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—3582no—
131C42—3582no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C2×C22⋊D17911yescharacteristic
140Q8—14321no—
141Q8—7162no—
142D4—24no—
143D4—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3582no—
153C2×C4—3582no—
154C2×C4#1566824712yescomplemented
155C2×C4D4×D17912yes—
156C8—7162no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1915308111yescharacteristic
167C22C2×#1566824711yescharacteristic
168C22#1915314311yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—81no—
173C4—7162no—
174C4—7162no—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#2603857711yescharacteristic
179C2#2603861111yescharacteristic
180C2#2603862511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#394×C179C2
socleC22×C179#120
prime radicalC22×D4×C179C22
hypercenter#394D179
hypocenterC179#1599
LCSG ▷ C22×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #26039070C2semidirect
3. #26038175C2semidirect
4. #26038165C2semidirect
64. C2×C4×C179SD16semidirect
90. C179#1599semidirect
154. C2×C4#15668247semidirect

Maximal subgroups

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

subgroupindex
#260390702
#260385902
#260381752
#260381652
#394×C1792
#1599179