Finite Group Databasebeta

← #37510670

Subgroups — #37510670

Order 22912.

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

normal subgroups only

This group has 270 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#26039002C211yesmaximal, characteristic
2#26038411C212yesmaximal, complemented
3#26038232C211yesmaximal, characteristic, complemented
4#26038231C212yesmaximal, complemented
5#353×C179C211yesmaximal, characteristic, complemented
6C2×#15668258C2211yescharacteristic
7C2×D4×D179C411yescharacteristic, complemented
8#19153109C2211yescharacteristic
9#19153101C411yescharacteristic, complemented
10#19153100C411yescharacteristic, complemented
11#19153085—22no—
12#19153081—22no—
13#19153070C411yescharacteristic, complemented
14#19153069—22no—
15#19153028C2212yescomplemented
16#19153027—22no—
17#127×C179C2211yescharacteristic
18#99×C179C2212yescomplemented
19C23×D179—21no—
20#15668258—42no—
21C2×Dic358C2×C411yescharacteristic
22C2×D716C2×C411yescharacteristic
23C2×C22⋊D179—21no—
24C2×C22⋊D179—22no—
25C2×C22⋊D179D412yes—
26D4×D179—42no—
27C22×Dic179D412yes—
28C2×C4×D179C2×C411yescharacteristic
29Dic179⋊C4—21no—
30Dic179⋊C4—42no—
31C22⋊Dic179—22no—
32C22⋊Dic179—22no—
33(C2×D179)⋊C4—21no—
34(C2×D179)⋊C4—22no—
35C4×Dic179—42no—
36C4×Dic179C2×C411yescharacteristic
37C2×D4×C179—22no—
38C2×D4×C179C2×C411yescharacteristic
39C2×D4×C179C2311yescharacteristic
40C22⋊C4×C179—22no—
41C42×C179C2×C411yescharacteristic
42C22×D179—21no—
43C22×D179—42no—
44C22×D179—82no—
45C22×D179C22⋊C411yescharacteristic
46Dic358—41no—
47D716—41no—
48C22⋊D179—42no—
49C22⋊D179—42no—
50C22⋊D179—82no—
51C2×Dic179—22no—
52C2×Dic179—22no—
53C2×Dic179—21no—
54C2×Dic179—42no—
55C2×Dic179C22⋊C412yes—
56C2×Dic179C22⋊C411yescharacteristic
57C4×D179—22no—
58C23×C179—21no—
59C23×C179C2×D412yes—
60D4×C179—41no—
61D4×C179—41no—
62D4×C179—82no—
63C2×C4×C179—21no—
64C2×C4×C179—42no—
65C2×C4×C179C22×C411yescharacteristic
66C22×C179—22no—
67C22×C179—42no—
68C22×C179—42no—
69C22×C179—41no—
70C22×C179C2×C22⋊C411yescharacteristic
71C2×D179—21no—
72C2×D179—41no—
73C2×D179—42no—
74C2×D179—41no—
75C2×D179—84no—
76C4×C179—21no—
77C4×C179—41no—
78C4×C179—82no—
79Dic179—22no—
80Dic179—41no—
81Dic179—82no—
82Dic179#9912yes—
83C2×C179—21no—
84C2×C179—42no—
85C2×C179—81no—
86C2×C179C2×#9911yescharacteristic
87D179—41no—
88D179—81no—
89C179#212811yescharacteristic, complemented, Sylow
90#2128—1791nomaximal, Sylow
91#535—1791no—
92#410—1792no—
93#353—1791no—
94#353—1792no—
95#353D17911yescharacteristic, complemented
96C2×#41—1791no—
97C22×D4—1791no—
98#127—1791no—
99#127C2×D17911yescharacteristic
100#124—1791no—
101#124—1791no—
102#121—3582no—
103#120—3582no—
104#117—3582no—
105#117—1791no—
106#99—3582no—
107#99—1792no—
108#99C2×D17912yescomplemented
109C24—3581no—
110#41—7162no—
111C2×Q8—1791no—
112C2×D4—22no—
113C2×D4—3582no—
114C2×D4—3581no—
115C2×D4—1792no—
116C2×D4—1791no—
117C2×D4—7162no—
118C2×D4C4×D17911yescharacteristic
119C2×D4C22×D17911yescharacteristic
120C22×C4—1792no—
121C22×C4—1791no—
122C4⋊C4—3581no—
123C4⋊C4—7162no—
124C22⋊C4—22no—
125C22⋊C4—3581no—
126C22⋊C4—3582no—
127C22⋊C4—3582no—
128C22⋊C4—3582no—
129C42—1791no—
130C42—7162no—
131C42C4×D17911yescharacteristic
132C23—21no—
133C23—3581no—
134C23—14322no—
135C23—1791no—
136C23—7162no—
137C23D4×D17912yes—
138Q8—7161no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—14322no—
143D4—7161no—
144D4—7162no—
145D4—7162no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—3582no—
149C2×C4—3582no—
150C2×C4—3582no—
151C2×C4—3581no—
152C2×C4—1792no—
153C2×C4—1791no—
154C2×C4—7162no—
155C2×C4C2×C4×D17911yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—3581no—
161C22—14324no—
162C22—7162no—
163C22—7161no—
164C22—7161no—
165C22C22⋊C4×D17911yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—3582no—
170C4—14322no—
171C4—1792no—
172C4—7161no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—14321no—
177C2—7161no—
178C2#99×D17911yescharacteristic
subgroupG/H
centerC2#99×D179
commutatorC2×C4×C179C22×C4
FrattiniC2×D4C22×D179
Fitting#353×C179C2
socleC2×C179C2×#99
prime radical#26039002C2
cosocleC2×D4×C179C23
hypercenter#353D179
hypocenterC179#2128
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #26038411C2semidirect
3. #26038232C2semidirect
4. #26038231C2semidirect
5. #353×C179C2semidirect
7. C2×D4×D179C4semidirect
9. #19153101C4semidirect
10. #19153100C4semidirect
13. #19153070C4semidirect
15. #19153028C22semidirect
18. #99×C179C22semidirect
89. C179#2128semidirect
95. #353D179semidirect
108. #99C2×D179semidirect

Maximal subgroups

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

subgroupindex
#260390022
#260384112
#260382322
#260382312
#353×C1792
#2128179