Finite Group Databasebeta

← #37508213

Subgroups — #37508213

Order 22912.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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#26039146C211yesmaximal, characteristic
2#26038791C211yesmaximal, characteristic, complemented
3#26038646C211yesmaximal, characteristic, complemented
4#26038187C211yesmaximal, characteristic, complemented
5#26038169C211yesmaximal, characteristic, complemented
6D4⋊(C179⋊C8)C211yesmaximal, characteristic
7#474×C179C211yesmaximal, characteristic
8C2×#15668249—21no—
9#19153113C2211yescharacteristic
10#19153104—21no—
11#19153092—21no—
12#19153090—22no—
13C4×Dic358C2211yescharacteristic
14D4×Dic179—21no—
15C2×C4⋊Dic179—21no—
16C4⋊(C179⋊C8)C2211yescharacteristic
17#19153037—21no—
18#19153036—22no—
19C4×C179⋊C8C2211yescharacteristic
20C4⋊Q8×C179C2211yescharacteristic
21C4×D4×C179C2211yescharacteristic
22#107×C179—21no—
23C4⋊C8×C179C2211yescharacteristic
24#102×C179—21no—
25C2×Dic358D411yescharacteristic
26C2×Dic358D412yes—
27C22×Dic179—22no—
28#15668249—42no—
29C2×C179⋊C8—21no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—22no—
32C4⋊Dic179—21no—
33C4⋊Dic179—21no—
34C4⋊Dic179D411yescharacteristic
35Dic179⋊C4—22no—
36Dic179⋊C4—21no—
37C22⋊Dic179—22no—
38C4×Dic179—21no—
39C2×Q8×C179—21no—
40C2×D4×C179D411yescharacteristic
41C22×C4×C179—21no—
42C2×C8×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179D411yescharacteristic
46C22⋊C4×C179—21no—
47C42×C179C2311yescharacteristic
48Dic358—21no—
49Dic358—21no—
50Dic358—42no—
51C2×Dic179—21no—
52C2×Dic179—21no—
53C2×Dic179—22no—
54C2×Dic179—82no—
55C179⋊C8—41no—
56C179⋊C8—41no—
57C23×C179—21no—
58Q8×C179—41no—
59Q8×C179—41no—
60D4×C179—21no—
61D4×C179Q1612yescomplemented
62C2×C4×C179—21no—
63C2×C4×C179—41no—
64C2×C4×C179—41no—
65C2×C4×C179—41no—
66C2×C4×C179C2×D411yescharacteristic
67C2×C4×C179C2×D411yescharacteristic
68C2×C4×C179C2×D411yescharacteristic
69C8×C179—41no—
70C22×C179—22no—
71C22×C179—41no—
72C22×C179—41no—
73C22×C179#12011yescharacteristic
74C4×C179—21no—
75C4×C179—41no—
76C4×C179—81no—
77C4×C179C2×Q1611yescharacteristic
78C4×C179#13511yescharacteristic
79C4×C179AGL(1,ℤ/8)11yescharacteristic
80C4×C179#13711yescharacteristic
81Dic179—41no—
82Dic179—41no—
83Dic179—42no—
84C2×C179—42no—
85C2×C179#44911yescharacteristic
86C2×C179#45111yescharacteristic
87C2×C179#45511yescharacteristic
88C179#164411yescharacteristic, complemented, Sylow
89#1644—1791nomaximal, Sylow
90#554—1791no—
91#487—1791no—
92#474D17911yescharacteristic
93#462—1791no—
94#333—1791no—
95Q8⋊C8—1791no—
96D4⋊C8—1791no—
97C2×Q16—3581no—
98C4⋊Q8—1791no—
99C4⋊Q8C2×D17911yescharacteristic
100#125—3581no—
101C22⋊Q8—3582no—
102C22⋊Q8—3581no—
103C4×Q8—1791no—
104C4×D4—3581no—
105C4×D4C2×D17911yescharacteristic
106C2×C4⋊C4—3581no—
107#107—21no—
108C4⋊C8—1791no—
109C4⋊C8C2×D17911yescharacteristic
110#103—3582no—
111#103—3581no—
112#102—21no—
113C4×C8—1791no—
114C2×Q8—21no—
115C2×Q8—1791no—
116C2×Q8—1792no—
117C2×D4C22⋊D17911yescharacteristic
118C22×C4—21no—
119C22×C4—3582no—
120Q16—7162no—
121C2×C8—21no—
122C2×C8—3581no—
123C2×C8—3581no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3582no—
127C4⋊C4—3582no—
128C4⋊C4—3581no—
129C4⋊C4—3581no—
130C4⋊C4—3581no—
131C4⋊C4—1791no—
132C4⋊C4C22⋊D17911yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—3582no—
135C42—3581no—
136C42C22×D17911yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—3581no—
141Q8—3581no—
142Q8—7162no—
143D4—21no—
144D4#1566824912yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3581no—
150C2×C4—3581no—
151C2×C4—3582no—
152C2×C4—14322no—
153C2×C4D4×D17911yescharacteristic
154C2×C4D4×D17911yescharacteristic
155C2×C4C2×C22⋊D17911yescharacteristic
156C8—41no—
157C8—7161no—
158C8—7161no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#1915308111yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—7161no—
167C4—7161no—
168C4—7162no—
169C4C2×#1566824911yescharacteristic
170C4#1915313211yescharacteristic
171C4#1915313711yescharacteristic
172C4#1915314311yescharacteristic
173C2—42no—
174C2#2603858111yescharacteristic
175C2#2603859411yescharacteristic
176C2#2603861511yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#474×C179C2
socleC22×C179#120
prime radicalC2×D4×C179D4
hypercenter#474D179
hypocenterC179#1644
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #26038791C2semidirect
3. #26038646C2semidirect
4. #26038187C2semidirect
5. #26038169C2semidirect
61. D4×C179Q16semidirect
88. C179#1644semidirect
144. D4#15668249semidirect

Maximal subgroups

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

subgroupindex
#260391462
#260387912
#260386462
#260381872
#260381692
D4⋊(C179⋊C8)2
#474×C1792
#1644179