Finite Group Databasebeta

← #37507523

Subgroups — #37507523

Order 22912.

242 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 73 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#26038973C211yesmaximal, characteristic
2#26038496C211yesmaximal, characteristic, complemented
3#26038175C212yesmaximal, complemented
4#26038172C212yesmaximal
5C4×C4⋊C4×C179C211yesmaximal, characteristic, complemented
6#19153107—21no—
7#19153105C411yescharacteristic, complemented
8#19153101C411yescharacteristic, complemented
9#19153096—21no—
10#19153092C2211yescharacteristic
11#19153087C2211yescharacteristic
12C4×Dic358C411yescharacteristic, complemented
13C4×D716C411yescharacteristic, complemented
14C4×C22⋊D179—21no—
15#19153070—21no—
16C4⋊(C179⋊C8)—21no—
17C22⋊(C179⋊C8)C2212yes—
18#19153024—21no—
19C2×C4⋊C4×C179—21no—
20C2×C42×C179—21no—
21C2×C42×C179C2211yescharacteristic
22#95×C179C2212yes—
23C2×Dic358C2×C411yescharacteristic
24C2×D716C2×C411yescharacteristic
25C2×C22⋊D179—21no—
26C2×C4×D179—21no—
27C2×C179⋊C8—22no—
28C4⋊Dic179C2×C411yescharacteristic
29C4⋊Dic179C2×C411yescharacteristic
30Dic179⋊C4—21no—
31Dic179⋊C4—21no—
32Dic179⋊C4—21no—
33Dic179⋊C4—21no—
34C22⋊Dic179—21no—
35(C2×D179)⋊C4—21no—
36(C2×D179)⋊C4—21no—
37(C2×D179)⋊C4—21no—
38(C2×D179)⋊C4—21no—
39C4×Dic179—21no—
40C22×C4×C179—22no—
41C22×C4×C179D411yescharacteristic
42C22×C4×C179D411yescharacteristic
43C22×C4×C179C2311yescharacteristic
44C4⋊C4×C179—24no—
45C42×C179—42no—
46C42×C179C2×C411yescharacteristic
47C42×C179C2×C411yescharacteristic
48C42×C179D411yescharacteristic
49C42×C179D411yescharacteristic
50C22×D179—21no—
51Dic358—41no—
52D716—41no—
53C22⋊D179—41no—
54C22⋊D179—41no—
55C2×Dic179—21no—
56C2×Dic179—21no—
57C2×Dic179—21no—
58C4×D179—42no—
59C179⋊C8—42no—
60C23×C179C2×D411yescharacteristic
61C2×C4×C179—21no—
62C2×C4×C179—21no—
63C2×C4×C179—22no—
64C2×C4×C179—24no—
65C2×C4×C179—41no—
66C2×C4×C179—42no—
67C2×C4×C179—42no—
68C2×C4×C179C22⋊C411yescharacteristic
69C2×C4×C179C22⋊C411yescharacteristic
70C2×C4×C179C22⋊C411yescharacteristic
71C2×C4×C179C22⋊C411yescharacteristic
72C2×C4×C179C22×C411yescharacteristic
73C2×C4×C179C2×D411yescharacteristic
74C22×C179—21no—
75C22×C179—21no—
76C22×C179C2×C22⋊C411yescharacteristic
77C22×C179AGL(1,ℤ/8)11yescharacteristic
78C22×C179#13711yescharacteristic
79C2×D179—41no—
80C2×D179—41no—
81C2×D179—41no—
82C4×C179—21no—
83C4×C179—21no—
84C4×C179—21no—
85C4×C179—21no—
86C4×C179—21no—
87C4×C179—44no—
88C4×C179#10412yes—
89Dic179—41no—
90Dic179—41no—
91Dic179—41no—
92C2×C179—21no—
93C2×C179—21no—
94C2×C179#41011yescharacteristic
95C2×C179#41711yescharacteristic
96C2×C179C2×#10411yescharacteristic
97D179—81no—
98C179#150911yescharacteristic, complemented, Sylow
99#1509—1791nomaximal, Sylow
100#529—1791no—
101#432—1791no—
102C4×C4⋊C4D17911yescharacteristic, complemented
103#328—1792no—
104#327—1792no—
105#126—3581no—
106#125—1791no—
107#124—1791no—
108#123—3581no—
109C22⋊Q8—1791no—
110#121—1791no—
111C4×Q8—1791no—
112C4×D4—3581no—
113C4×D4—1791no—
114#117—3581no—
115C2×C4⋊C4—21no—
116C2×C42—21no—
117C2×C42C2×D17911yescharacteristic
118C4⋊C8—3581no—
119C22⋊C8—1792no—
120#97—3581no—
121#95C2×D17912yes—
122C2×Q8—1791no—
123C2×D4—3581no—
124C2×D4—1791no—
125C22×C4—22no—
126C22×C4—3581no—
127C22×C4C22⋊D17911yescharacteristic
128C22×C4C22⋊D17911yescharacteristic
129C22×C4C22×D17911yescharacteristic
130C2×C8—3582no—
131C4⋊C4—24no—
132C4⋊C4—3581no—
133C4⋊C4—3581no—
134C4⋊C4—3581no—
135C4⋊C4—3581no—
136C4⋊C4—1791no—
137C4⋊C4—1791no—
138C22⋊C4—3581no—
139C22⋊C4—3581no—
140C22⋊C4—3581no—
141C22⋊C4—3581no—
142C22⋊C4—3581no—
143C42—42no—
144C42—3581no—
145C42C4×D17911yescharacteristic
146C42C4×D17911yescharacteristic
147C42D71611yescharacteristic
148C42D71611yescharacteristic
149C23—3581no—
150C23C2×C22⋊D17911yescharacteristic
151Q8—7161no—
152D4—7161no—
153D4—7161no—
154D4—7161no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—3581no—
163C2×C4—3581no—
164C2×C4—3581no—
165C2×C4—7162no—
166C2×C4(C2×D179)⋊C411yescharacteristic
167C2×C4(C2×D179)⋊C411yescharacteristic
168C2×C4(C2×D179)⋊C411yescharacteristic
169C2×C4(C2×D179)⋊C411yescharacteristic
170C2×C4C2×C4×D17911yescharacteristic
171C2×C4C2×D71611yescharacteristic
172C8—7162no—
173C22—21no—
174C22—21no—
175C22—7161no—
176C22—7161no—
177C22—7161no—
178C22C2×(C2×D179)⋊C411yescharacteristic
179C22#1915313911yescharacteristic
180C22#1915314411yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—44no—
187C4—7161no—
188C4—7161no—
189C4—7161no—
190C4#1915304112yes—
191C2—21no—
192C2—21no—
193C2—14321no—
194C2#2603841211yescharacteristic
195C2#2603843811yescharacteristic
196C2C2×#1915304111yescharacteristic
subgroupG/H
centerC22C2×(C2×D179)⋊C4
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
FittingC4×C4⋊C4×C179C2
socleC22×C179C2×C22⋊C4
prime radical#19153087C22
cosocleC22×C4×C179C23
hypercenterC4×C4⋊C4D179
hypocenterC179#1509
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #26038496C2semidirect
3. #26038175C2semidirect
5. C4×C4⋊C4×C179C2semidirect
7. #19153105C4semidirect
8. #19153101C4semidirect
12. C4×Dic358C4semidirect
13. C4×D716C4semidirect
98. C179#1509semidirect
102. C4×C4⋊C4D179semidirect

Maximal subgroups

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

subgroupindex
#260389732
#260384962
#260381752
#260381722
C4×C4⋊C4×C1792
#1509179