Finite Group Databasebeta

← #37508255

Subgroups — #37508255

Order 22912.

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

normal subgroups only

This group has 200 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#26039162C211yesmaximal, characteristic
2#26038842C211yesmaximal, characteristic, complemented
3#26038641C211yesmaximal, characteristic
4#26038184C211yesmaximal, characteristic, complemented
5Q8⋊(C179⋊C8)C211yesmaximal, characteristic, complemented
6#26038166C211yesmaximal, characteristic
7#475×C179C211yesmaximal, characteristic, complemented
8C2×#15668246—21no—
9#19153113C2211yescharacteristic
10#19153108—22no—
11#19153101—21no—
12#19153089—21no—
13C4×Dic358—21no—
14C4×D716C2211yescharacteristic
15#19153071—21no—
16#19153046—22no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18#19153037—21no—
19C4×C179⋊C8C2211yescharacteristic
20C4⋊Q8×C179C2211yescharacteristic
21C4×Q8×C179C2211yescharacteristic
22#106×C179—21no—
23C4⋊C8×C179C2211yescharacteristic
24#103×C179—21no—
25C2×Dic358—21no—
26C2×D716D411yescharacteristic
27C2×C4×D179—21no—
28#15668246—42no—
29C2×C179⋊C8—21no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—21no—
32C4⋊Dic179D411yescharacteristic
33C4⋊Dic179D412yes—
34Dic179⋊C4—22no—
35(C2×D179)⋊C4—22no—
36(C2×D179)⋊C4—22no—
37(C2×D179)⋊C4—21no—
38C4×Dic179—22no—
39C2×Q8×C179—21no—
40C2×Q8×C179D411yescharacteristic
41C2×C8×C179—21no—
42C4⋊C4×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179D411yescharacteristic
46C42×C179—21no—
47C42×C179C2311yescharacteristic
48C22×D179—21no—
49Dic358—41no—
50Dic358—41no—
51D716—21no—
52D716—21no—
53C2×Dic179—21no—
54C2×Dic179—22no—
55C4×D179—41no—
56C4×D179—41no—
57C179⋊C8—22no—
58C179⋊C8—41no—
59Q8×C179—21no—
60Q8×C179—21no—
61Q8×C179—41no—
62Q8×C179—41no—
63C2×C4×C179—21no—
64C2×C4×C179—21no—
65C2×C4×C179—41no—
66C2×C4×C179C2×D411yescharacteristic
67C2×C4×C179C2×D411yescharacteristic
68C2×C4×C179C2×D411yescharacteristic
69C8×C179—41no—
70C22×C179#12011yescharacteristic
71C2×D179—41no—
72C2×D179—41no—
73C2×D179—41no—
74C4×C179—21no—
75C4×C179—41no—
76C4×C179—41no—
77C4×C179—81no—
78C4×C179#13511yescharacteristic
79C4×C179#13511yescharacteristic
80C4×C179AGL(1,ℤ/8)11yescharacteristic
81C4×C179#13711yescharacteristic
82Dic179—42no—
83Dic179—41no—
84C2×C179#44911yescharacteristic
85C2×C179#45211yescharacteristic
86C2×C179#45611yescharacteristic
87D179—81no—
88C179#165011yescharacteristic, complemented, Sylow
89#1650—1791nomaximal, Sylow
90#556—1791no—
91#500—1791no—
92#475D17911yescharacteristic, complemented
93#461—1791no—
94#332—1791no—
95Q8⋊C8—1791no—
96D4⋊C8—1791no—
97C2×SD16—3581no—
98C4⋊Q8—1791no—
99C4⋊Q8C2×D17911yescharacteristic
100#126—3582no—
101#124—3581no—
102C22⋊Q8—3581no—
103C4×Q8—3581no—
104C4×Q8C2×D17911yescharacteristic
105C4×D4—1791no—
106#117—3581no—
107#107—3582no—
108#106—21no—
109C4⋊C8—1791no—
110C4⋊C8C2×D17911yescharacteristic
111#103—21no—
112#103—3581no—
113C4×C8—1791no—
114C2×Q8—21no—
115C2×Q8—3581no—
116C2×Q8C22⋊D17911yescharacteristic
117C2×D4—1791no—
118C22×C4—3581no—
119SD16—7162no—
120C2×C8—21no—
121C2×C8—3581no—
122C2×C8—3581no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3582no—
127C4⋊C4—3581no—
128C4⋊C4—1791no—
129C4⋊C4—1792no—
130C4⋊C4C22⋊D17911yescharacteristic
131C22⋊C4—3582no—
132C22⋊C4—3582no—
133C22⋊C4—3581no—
134C42—21no—
135C42—3582no—
136C42C22×D17911yescharacteristic
137C23—3581no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—7161no—
143Q8—7161no—
144D4—3581no—
145D4—3581no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3581no—
150C2×C4—3582no—
151C2×C4—7161no—
152C2×C4—7161no—
153C2×C4D4×D17911yescharacteristic
154C2×C4D4×D17911yescharacteristic
155C2×C4C2×C22⋊D17911yescharacteristic
156C8—41no—
157C8—3582no—
158C8—7161no—
159C22—7161no—
160C22—7161no—
161C22—7161no—
162C22#1915308111yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—7162no—
168C4—7161no—
169C4#1915313311yescharacteristic
170C4#1915313411yescharacteristic
171C4#1915314011yescharacteristic
172C4#1915314411yescharacteristic
173C2—14321no—
174C2#2603858311yescharacteristic
175C2#2603859911yescharacteristic
176C2#2603862011yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#475×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#475D179
hypocenterC179#1650
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #26038842C2semidirect
4. #26038184C2semidirect
5. Q8⋊(C179⋊C8)C2semidirect
7. #475×C179C2semidirect
88. C179#1650semidirect
92. #475D179semidirect

Maximal subgroups

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

subgroupindex
#260391622
#260388422
#260386412
#260381842
Q8⋊(C179⋊C8)2
#260381662
#475×C1792
#1650179