Finite Group Databasebeta

← #37508131

Subgroups — #37508131

Order 22912.

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

normal subgroups only

This group has 214 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#26039150C211yesmaximal, characteristic
2#26038731C211yesmaximal, characteristic, complemented
3#26038726C211yesmaximal, characteristic
4#26038188C211yesmaximal, characteristic, complemented
5#26038171C211yesmaximal, characteristic, complemented
6#26038168C211yesmaximal, characteristic
7#494×C179C211yesmaximal, characteristic, complemented
8#19153104—21no—
9#19153093—21no—
10#19153089—22no—
11C4×Dic358C2211yescharacteristic
12C4×D716C2211yescharacteristic
13#19153075—21no—
14C4⋊C4×D179—21no—
15#19153046—21no—
16#19153044—21no—
17C4⋊(C179⋊C8)C2211yescharacteristic
18C4⋊(C179⋊C8)C2211yescharacteristic
19#19153036—21no—
20#19153033—21no—
21C2×Q16×C179—22no—
22C4⋊Q8×C179C2211yescharacteristic
23C4⋊Q8×C179C2211yescharacteristic
24C4×C8×C179C2211yescharacteristic
25C2×Dic358D411yescharacteristic
26C2×D716D411yescharacteristic
27C2×C4×D179—22no—
28C2×C4×D179—21no—
29C2×C179⋊C8—21no—
30C2×C179⋊C8—21no—
31C4⋊Dic179—22no—
32C4⋊Dic179D411yescharacteristic
33C4⋊Dic179D411yescharacteristic
34Dic179⋊C4—22no—
35Dic179⋊C4—21no—
36(C2×D179)⋊C4—22no—
37(C2×D179)⋊C4—21no—
38C4×Dic179—21no—
39C2×Q8×C179—21no—
40C2×Q8×C179D412yes—
41Q16×C179—44no—
42C2×C8×C179—21no—
43C4⋊C4×C179—21no—
44C4⋊C4×C179—21no—
45C4⋊C4×C179—21no—
46C4⋊C4×C179—21no—
47C42×C179C2311yescharacteristic
48C22×D179—21no—
49Dic358—21no—
50Dic358—21no—
51D716—21no—
52D716Q1612yescomplemented
53C2×Dic179—21no—
54C2×Dic179—21no—
55C2×Dic179—21no—
56C4×D179—41no—
57C4×D179—41no—
58C4×D179—82no—
59C179⋊C8—41no—
60C179⋊C8—41no—
61Q8×C179—42no—
62Q8×C179—42no—
63C2×C4×C179—22no—
64C2×C4×C179—41no—
65C2×C4×C179C2×D411yescharacteristic
66C2×C4×C179C2×D411yescharacteristic
67C2×C4×C179C2×D411yescharacteristic
68C8×C179—22no—
69C22×C179#12011yescharacteristic
70C2×D179—22no—
71C2×D179—41no—
72C2×D179—41no—
73C4×C179—21no—
74C4×C179—42no—
75C4×C179—81no—
76C4×C179C2×Q1611yescharacteristic
77C4×C179#13511yescharacteristic
78C4×C179#13711yescharacteristic
79C4×C179#13711yescharacteristic
80Dic179—41no—
81Dic179—41no—
82Dic179—41no—
83C2×C179#45111yescharacteristic
84C2×C179#45211yescharacteristic
85C2×C179#45311yescharacteristic
86D179—42no—
87C179#163211yescharacteristic, complemented, Sylow
88#1632—1791nomaximal, Sylow
89#554—1791no—
90#494D17911yescharacteristic, complemented
91#477—1791no—
92#476—1791no—
93#333—1791no—
94Q8⋊C8—1791no—
95D4⋊C8—1791no—
96C2×Q16—22no—
97C4⋊Q8C2×D17911yescharacteristic
98C4⋊Q8C2×D17911yescharacteristic
99#125—3581no—
100C22⋊Q8—3581no—
101C22⋊Q8—3582no—
102C4×Q8—1791no—
103C4×D4—3581no—
104C4×D4—1791no—
105C2×C4⋊C4—3581no—
106#107—3581no—
107#106—3581no—
108C4⋊C8—1791no—
109C4⋊C8—1791no—
110#103—3581no—
111#102—3581no—
112C4×C8C2×D17911yescharacteristic
113C2×Q8—21no—
114C2×Q8—1791no—
115C2×Q8C22⋊D17912yes—
116C2×D4—1791no—
117C22×C4—3582no—
118C22×C4—3581no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—3581no—
122C2×C8—3581no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3582no—
128C4⋊C4—3582no—
129C4⋊C4—3581no—
130C4⋊C4—1791no—
131C4⋊C4—1791no—
132C22⋊C4—3582no—
133C22⋊C4—3581no—
134C42—3581no—
135C42C22×D17911yescharacteristic
136C23—3581no—
137Q8—42no—
138Q8—42no—
139Q8—3581no—
140Q8—3581no—
141D4—3581no—
142D4—1792no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—3581no—
146C2×C4—3581no—
147C2×C4—3581no—
148C2×C4—14322no—
149C2×C4—7161no—
150C2×C4—7161no—
151C2×C4D4×D17911yescharacteristic
152C2×C4D4×D17911yescharacteristic
153C2×C4C2×C22⋊D17911yescharacteristic
154C8—22no—
155C8—7161no—
156C8—7161no—
157C22—3582no—
158C22—7161no—
159C22—7161no—
160C22#1915308111yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—7161no—
165C4—7161no—
166C4—7161no—
167C4Q16×D17911yescharacteristic
168C4#1915313611yescharacteristic
169C4#1915314511yescharacteristic
170C4#1915314511yescharacteristic
171C2—7162no—
172C2#2603859311yescharacteristic
173C2#2603860011yescharacteristic
174C2#2603860611yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#494×C179C2
socleC22×C179#120
prime radicalC2×D716D4
hypercenter#494D179
hypocenterC179#1632
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #26038731C2semidirect
4. #26038188C2semidirect
5. #26038171C2semidirect
7. #494×C179C2semidirect
52. D716Q16semidirect
87. C179#1632semidirect
90. #494D179semidirect

Maximal subgroups

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

subgroupindex
#260391502
#260387312
#260387262
#260381882
#260381712
#260381682
#494×C1792
#1632179