Finite Group Databasebeta

← #37508208

Subgroups — #37508208

Order 22912.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039150C211yesmaximal, characteristic
2#26038789C211yesmaximal, characteristic, complemented
3#26038725C211yesmaximal, characteristic, complemented
4#26038645C211yesmaximal, characteristic
5#26038169C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#333×C179C211yesmaximal, characteristic, complemented
8C2×#15668248—21no—
9#19153113C2211yescharacteristic
10#19153104—21no—
11#19153093—21no—
12#19153089—22no—
13C4×Dic358C2211yescharacteristic
14C4×D716C2211yescharacteristic
15#19153075—21no—
16C4⋊C4×D179—21no—
17#19153045—21no—
18#19153038—21no—
19#19153038—21no—
20#19153037—22no—
21C4×C179⋊C8C2211yescharacteristic
22C4⋊Q8×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4⋊C8×C179C2211yescharacteristic
25C2×Dic358—21no—
26C2×Dic358D411yescharacteristic
27C2×D716D411yescharacteristic
28C2×C4×D179—22no—
29C2×C4×D179—21no—
30#15668248—42no—
31C2×C179⋊C8—21no—
32C4⋊Dic179—22no—
33C4⋊Dic179—21no—
34C4⋊Dic179—21no—
35C4⋊Dic179D411yescharacteristic
36C4⋊Dic179D411yescharacteristic
37Dic179⋊C4—22no—
38Dic179⋊C4—21no—
39(C2×D179)⋊C4—22no—
40(C2×D179)⋊C4—21no—
41C4×Dic179—21no—
42C2×Q8×C179D412yes—
43C2×C8×C179—21no—
44C2×C8×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C42×C179C2311yescharacteristic
48C22×D179—21no—
49Dic358—21no—
50Dic358—21no—
51Dic358—41no—
52Dic358—41no—
53D716—21no—
54D716SD1612yes—
55C2×Dic179—21no—
56C2×Dic179—21no—
57C2×Dic179—21no—
58C2×Dic179—41no—
59C4×D179—41no—
60C4×D179—41no—
61C4×D179—82no—
62C179⋊C8—41no—
63Q8×C179—42no—
64C2×C4×C179—22no—
65C2×C4×C179C2×D411yescharacteristic
66C2×C4×C179C2×D411yescharacteristic
67C2×C4×C179C2×D411yescharacteristic
68C8×C179—41no—
69C8×C179—41no—
70C22×C179#12011yescharacteristic
71C2×D179—22no—
72C2×D179—41no—
73C2×D179—41no—
74C4×C179—21no—
75C4×C179—42no—
76C4×C179C2×SD1611yescharacteristic
77C4×C179#13511yescharacteristic
78C4×C179#13711yescharacteristic
79C4×C179#13711yescharacteristic
80Dic179—41no—
81Dic179—41no—
82Dic179—41no—
83Dic179—81no—
84C2×C179#44811yescharacteristic
85C2×C179#45211yescharacteristic
86C2×C179#45511yescharacteristic
87D179—42no—
88C179#164311yescharacteristic, complemented, Sylow
89#1643—1791nomaximal, Sylow
90#554—1791no—
91#487—1791no—
92#475—1791no—
93#461—1791no—
94#333D17911yescharacteristic, complemented
95Q8⋊C8—1791no—
96D4⋊C8—1791no—
97C2×SD16—3581no—
98C4⋊Q8—1791no—
99C4⋊Q8C2×D17911yescharacteristic
100#125—3581no—
101C22⋊Q8—3581no—
102C22⋊Q8—3582no—
103C4×Q8—1791no—
104C4×D4—3581no—
105C4×D4—1791no—
106C2×C4⋊C4—3581no—
107#106—3581no—
108C4⋊C8C2×D17911yescharacteristic
109C4⋊C8C2×D17911yescharacteristic
110#103—3581no—
111#103—3582no—
112#103—3581no—
113C4×C8—1791no—
114C2×Q8—3581no—
115C2×Q8—1791no—
116C2×Q8D71612yes—
117C2×D4—1791no—
118C22×C4—3582no—
119C22×C4—3581no—
120SD16—7162no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—3581no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3582no—
127C4⋊C4—3582no—
128C4⋊C4—3581no—
129C4⋊C4—3581no—
130C4⋊C4—3581no—
131C4⋊C4—1791no—
132C4⋊C4—1791no—
133C22⋊C4—3582no—
134C22⋊C4—3581no—
135C42—3581no—
136C42C22×D17911yescharacteristic
137C23—3581no—
138Q8—42no—
139Q8—3581no—
140Q8—3581no—
141Q8—7161no—
142Q8—7161no—
143D4—3581no—
144D4—1792no—
145C2×C4—22no—
146C2×C4—3581no—
147C2×C4—3581no—
148C2×C4—3581no—
149C2×C4—14322no—
150C2×C4—7161no—
151C2×C4—7161no—
152C2×C4—7161no—
153C2×C4D4×D17911yescharacteristic
154C2×C4D4×D17911yescharacteristic
155C2×C4C2×D71611yescharacteristic
156C8—41no—
157C8—41no—
158C8—7161no—
159C22—3582no—
160C22—7161no—
161C22—7161no—
162C22#1915308311yescharacteristic
163C4—21no—
164C4—42no—
165C4—14321no—
166C4—7161no—
167C4—7161no—
168C4—7161no—
169C4SD16×D17911yescharacteristic
170C4#1915313411yescharacteristic
171C4#1915314511yescharacteristic
172C4#1915314511yescharacteristic
173C2—7162no—
174C2#2603857911yescharacteristic
175C2#2603860411yescharacteristic
176C2#2603861911yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#333×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#333D179
hypocenterC179#1643
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #26038789C2semidirect
3. #26038725C2semidirect
5. #26038169C2semidirect
7. #333×C179C2semidirect
88. C179#1643semidirect
94. #333D179semidirect

Maximal subgroups

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

subgroupindex
#260391502
#260387892
#260387252
#260386452
#260381692
#260381662
#333×C1792
#1643179