Finite Group Databasebeta

← #37508174

Subgroups — #37508174

Order 22912.

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

normal subgroups only

This group has 194 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#26039159C211yesmaximal, characteristic
2#26038790C211yesmaximal, characteristic, complemented
3#26038735C211yesmaximal, characteristic, complemented
4#26038729C211yesmaximal, characteristic
5#26038184C211yesmaximal, characteristic, complemented
6#26038171C211yesmaximal, characteristic, complemented
7D4⋊C8×C179C211yesmaximal, characteristic
8#19153113C2211yescharacteristic
9#19153113C2211yescharacteristic
10#19153106—22no—
11#19153100—21no—
12#19153092—21no—
13C4×Dic358C2211yescharacteristic
14#19153078—21no—
15#19153072—21no—
16#19153047—21no—
17#19153044—21no—
18C4⋊(C179⋊C8)C2211yescharacteristic
19#19153038—21no—
20#19153038—22no—
21#19153034—21no—
22C4×D4×C179C2211yescharacteristic
23C4⋊C8×C179C2211yescharacteristic
24C4×C8×C179C2211yescharacteristic
25C2×Dic358—21no—
26C2×Dic358—21no—
27C2×Dic358D411yescharacteristic
28C2×C179⋊C8—21no—
29C4⋊Dic179—21no—
30C4⋊Dic179—21no—
31C4⋊Dic179—21no—
32C4⋊Dic179D412yes—
33C4⋊Dic179D411yescharacteristic
34Dic179⋊C4—22no—
35Dic179⋊C4—21no—
36C22⋊Dic179—22no—
37C22⋊Dic179—22no—
38C4×Dic179—22no—
39C4×Dic179—21no—
40C2×D4×C179D411yescharacteristic
41C22×C4×C179—21no—
42C2×C8×C179—21no—
43C2×C8×C179—21no—
44C4⋊C4×C179D411yescharacteristic
45C22⋊C4×C179—21no—
46C42×C179C2311yescharacteristic
47Dic358—21no—
48Dic358—21no—
49Dic358—41no—
50Dic358—41no—
51Dic358—42no—
52C2×Dic179—21no—
53C2×Dic179—21no—
54C2×Dic179—22no—
55C2×Dic179—41no—
56C179⋊C8—41no—
57C23×C179—21no—
58D4×C179—21no—
59D4×C179—21no—
60C2×C4×C179—21no—
61C2×C4×C179—41no—
62C2×C4×C179—41no—
63C2×C4×C179C2×D411yescharacteristic
64C2×C4×C179C2×D411yescharacteristic
65C2×C4×C179C2×D411yescharacteristic
66C8×C179—41no—
67C8×C179—41no—
68C22×C179—41no—
69C22×C179—41no—
70C22×C179—41no—
71C22×C179#12011yescharacteristic
72C4×C179—21no—
73C4×C179—41no—
74C4×C179#13511yescharacteristic
75C4×C179#13511yescharacteristic
76C4×C179#13711yescharacteristic
77C4×C179#13711yescharacteristic
78Dic179—41no—
79Dic179—42no—
80Dic179—41no—
81Dic179—81no—
82C2×C179—81no—
83C2×C179#45211yescharacteristic
84C2×C179#45211yescharacteristic
85C2×C179#45411yescharacteristic
86C179#163811yescharacteristic, complemented, Sylow
87#1638—1791nomaximal, Sylow
88#556—1791no—
89#487—1791no—
90#477—1791no—
91#476—1791no—
92#332—1791no—
93Q8⋊C8—1791no—
94D4⋊C8D17911yescharacteristic
95C4⋊Q8—1791no—
96C4⋊Q8—1791no—
97#126—3582no—
98#124—3581no—
99C22⋊Q8—3581no—
100C4×Q8—3581no—
101C4×Q8—1791no—
102C4×D4C2×D17911yescharacteristic
103#117—3581no—
104#107—3581no—
105#106—3581no—
106C4⋊C8—1791no—
107C4⋊C8C2×D17911yescharacteristic
108#103—3581no—
109#103—3582no—
110#102—3581no—
111C4×C8C2×D17911yescharacteristic
112C2×Q8—3581no—
113C2×Q8—3581no—
114C2×Q8—1791no—
115C2×D4D71611yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3581no—
120C4⋊C4—3582no—
121C4⋊C4—3581no—
122C4⋊C4—3581no—
123C4⋊C4—3581no—
124C4⋊C4—3581no—
125C4⋊C4—1792no—
126C4⋊C4—1791no—
127C4⋊C4D71611yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3582no—
130C22⋊C4—3582no—
131C42—3582no—
132C42—3581no—
133C42C22×D17911yescharacteristic
134C23—21no—
135Q8—3581no—
136Q8—3581no—
137Q8—7162no—
138Q8—7161no—
139Q8—7161no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—3581no—
146C2×C4—3581no—
147C2×C4—3582no—
148C2×C4—7161no—
149C2×C4D4×D17911yescharacteristic
150C2×C4D4×D17911yescharacteristic
151C2×C4C2×D71611yescharacteristic
152C8—41no—
153C8—41no—
154C8—7161no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1915308311yescharacteristic
159C4—21no—
160C4—41no—
161C4—14321no—
162C4—7161no—
163C4—7162no—
164C4—7161no—
165C4#1915313211yescharacteristic
166C4#1915313511yescharacteristic
167C4#1915314211yescharacteristic
168C4#1915314611yescharacteristic
169C2—81no—
170C2#2603860211yescharacteristic
171C2#2603860311yescharacteristic
172C2#2603861311yescharacteristic
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
FittingD4⋊C8×C179C2
socleC22×C179#120
prime radicalC2×D4×C179D4
hypercenterD4⋊C8D179
hypocenterC179#1638
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #26038790C2semidirect
3. #26038735C2semidirect
5. #26038184C2semidirect
6. #26038171C2semidirect
86. C179#1638semidirect

Maximal subgroups

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

subgroupindex
#260391592
#260387902
#260387352
#260387292
#260381842
#260381712
D4⋊C8×C1792
#1638179