Finite Group Databasebeta

← #37510666

Subgroups — #37510666

Order 22912.

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

normal subgroups only

This group has 286 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#99×D179C212yesmaximal, complemented
3#26038233C212yesmaximal, complemented
4#26038232C211yesmaximal, characteristic, complemented
5#353×C179C211yesmaximal, characteristic, complemented
6C2×#15668258C411yescharacteristic, complemented
7C2×D4×D179C2211yescharacteristic
8#19153109C411yescharacteristic, complemented
9#19153101C411yescharacteristic, complemented
10#19153100C2211yescharacteristic
11#19153085—22no—
12#19153081—22no—
13#19153070C411yescharacteristic, complemented
14C22⋊C4×D179—22no—
15#19153028C2212yes—
16#19153027—22no—
17#127×C179C2211yescharacteristic
18#99×C179C2212yes—
19C23×D179D412yes—
20#15668258—42no—
21C2×Dic358C2×C411yescharacteristic, complemented
22C2×D716C2×C411yescharacteristic
23C2×C22⋊D179—21no—
24C2×C22⋊D179—22no—
25C2×C22⋊D179D412yes—
26D4×D179—42no—
27C22×Dic179—21no—
28C2×C4×D179—42no—
29C2×C4×D179C2×C411yescharacteristic
30Dic179⋊C4—21no—
31C22⋊Dic179—22no—
32C22⋊Dic179—22no—
33(C2×D179)⋊C4—21no—
34(C2×D179)⋊C4—22no—
35(C2×D179)⋊C4—42no—
36C4×Dic179C2×C411yescharacteristic, complemented
37C2×D4×C179—22no—
38C2×D4×C179C2×C411yescharacteristic, complemented
39C2×D4×C179C2311yescharacteristic
40C22⋊C4×C179—22no—
41C42×C179C2×C411yescharacteristic, complemented
42C22×D179—22no—
43C22×D179—22no—
44C22×D179—82no—
45C22×D179C22⋊C412yes—
46C22×D179C22⋊C411yescharacteristic
47Dic358—41no—
48D716—41no—
49C22⋊D179—42no—
50C22⋊D179—42no—
51C22⋊D179—82no—
52C2×Dic179—21no—
53C2×Dic179—21no—
54C2×Dic179—42no—
55C2×Dic179—42no—
56C2×Dic179C22⋊C411yescharacteristic
57C4×D179—22no—
58C4×D179—84no—
59C23×C179—21no—
60C23×C179C2×D412yes—
61D4×C179—41no—
62D4×C179—41no—
63D4×C179—82no—
64C2×C4×C179—21no—
65C2×C4×C179—42no—
66C2×C4×C179C22×C411yescharacteristic
67C22×C179—22no—
68C22×C179—42no—
69C22×C179—42no—
70C22×C179—41no—
71C22×C179C2×C22⋊C411yescharacteristic
72C2×D179—22no—
73C2×D179—41no—
74C2×D179—44no—
75C2×D179—42no—
76C2×D179—82no—
77C2×D179#9912yes—
78C4×C179—21no—
79C4×C179—41no—
80C4×C179—82no—
81Dic179—21no—
82Dic179—41no—
83Dic179—41no—
84Dic179—82no—
85C2×C179—21no—
86C2×C179—42no—
87C2×C179—81no—
88C2×C179C2×#9911yescharacteristic
89D179—22no—
90D179—42no—
91C179#212711yescharacteristic, complemented, Sylow
92#2127—1791nomaximal, Sylow
93#535—1791no—
94C2×#99—1792no—
95#354—1792no—
96#353—1791no—
97#353D17911yescharacteristic, complemented
98C2×#41—1791no—
99C22×D4—1791no—
100#127—1791no—
101#127C2×D17911yescharacteristic
102#124—1791no—
103#124—1791no—
104#121—3582no—
105#120—3582no—
106#117—1791no—
107C2×C22⋊C4—3582no—
108#99—3582no—
109#99—1792no—
110#99C2×D17912yes—
111C24—1792no—
112#41—7162no—
113C2×Q8—1791no—
114C2×D4—22no—
115C2×D4—3582no—
116C2×D4—3581no—
117C2×D4—1792no—
118C2×D4—1791no—
119C2×D4—7162no—
120C2×D4C4×D17911yescharacteristic, complemented
121C2×D4C22×D17911yescharacteristic
122C22×C4—3581no—
123C22×C4—1791no—
124C22×C4—7162no—
125C4⋊C4—3581no—
126C22⋊C4—22no—
127C22⋊C4—3581no—
128C22⋊C4—3582no—
129C22⋊C4—3582no—
130C22⋊C4—3582no—
131C22⋊C4—7162no—
132C42—1791no—
133C42C4×D17911yescharacteristic, complemented
134C23—21no—
135C23—3582no—
136C23—3582no—
137C23—14322no—
138C23—1792no—
139C23—1791no—
140C23D4×D17912yes—
141Q8—7161no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—14322no—
146D4—7161no—
147D4—7162no—
148D4—7162no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—3582no—
152C2×C4—3581no—
153C2×C4—3581no—
154C2×C4—14324no—
155C2×C4—1791no—
156C2×C4—7162no—
157C2×C4—7162no—
158C2×C4C2×C4×D17911yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—3582no—
164C22—14322no—
165C22—1792no—
166C22—7161no—
167C22—7162no—
168C22—7164no—
169C22C22⋊C4×D17911yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—3581no—
174C4—14322no—
175C4—7161no—
176C4—7161no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—3582no—
181C2—7162no—
182C2#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#2127
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. #99×D179C2semidirect
3. #26038233C2semidirect
4. #26038232C2semidirect
5. #353×C179C2semidirect
6. C2×#15668258C4semidirect
8. #19153109C4semidirect
9. #19153101C4semidirect
13. #19153070C4semidirect
21. C2×Dic358C2×C4semidirect
36. C4×Dic179C2×C4semidirect
38. C2×D4×C179C2×C4semidirect
41. C42×C179C2×C4semidirect
91. C179#2127semidirect
97. #353D179semidirect
120. C2×D4C4×D179semidirect
133. C42C4×D179semidirect

Maximal subgroups

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

subgroupindex
#260390022
#99×D1792
#260382332
#260382322
#353×C1792
#2127179