Finite Group Databasebeta

← #37509389

Subgroups — #37509389

Order 22912.

360 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 97 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#26038914C211yesmaximal, characteristic
2#26038448C212yesmaximal, complemented
3C2×#19153030C211yesmaximal, characteristic
4#26038199C212yesmaximal, complemented
5C2×#124×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7D4×Dic179—24no—
8#19153072C2212yes—
9C2×C22⋊Dic179—22no—
10#19153037—24no—
11#19153030—22no—
12C22×Q8×C179C2211yescharacteristic
13C22×D4×C179C2211yescharacteristic
14#124×C179—22no—
15#124×C179C414yescomplemented
16C2×C22⋊C4×C179—22no—
17C2×C42×C179C2211yescharacteristic
18C22×Dic179—24no—
19#15668243—22no—
20#15668243—22no—
21C2×C179⋊C8—22no—
22C4⋊Dic179D414yes—
23C22⋊Dic179—24no—
24C22⋊Dic179—22no—
25C4×Dic179—22no—
26C24×C179D412yes—
27C2×Q8×C179—21no—
28C2×Q8×C179—22no—
29C2×Q8×C179—22no—
30C2×Q8×C179C2×C412yescomplemented
31C2×D4×C179—22no—
32C2×D4×C179—22no—
33C2×D4×C179C2×C412yes—
34C2×D4×C179D412yes—
35C22×C4×C179—21no—
36C22×C4×C179—21no—
37C22×C4×C179C2311yescharacteristic
38C22⋊C4×C179—28no—
39C42×C179—21no—
40C42×C179C2×C412yescomplemented
41C2×Dic179—24no—
42C2×Dic179—84no—
43C179⋊C8—42no—
44C23×C179—22no—
45C23×C179—22no—
46C23×C179—44no—
47C23×C179C22⋊C414yes—
48C23×C179C2×D411yescharacteristic
49Q8×C179—24no—
50Q8×C179—42no—
51D4×C179—24no—
52D4×C179—24no—
53C2×C4×C179—22no—
54C2×C4×C179—22no—
55C2×C4×C179—42no—
56C2×C4×C179—41no—
57C2×C4×C179—41no—
58C2×C4×C179C22×C411yescharacteristic
59C2×C4×C179C2×D412yes—
60C2×C4×C179C2×D411yescharacteristic
61C2×C4×C179#4112yes—
62C22×C179—22no—
63C22×C179—24no—
64C22×C179—44no—
65C22×C179—44no—
66C22×C179—82no—
67C22×C179C2×C22⋊C411yescharacteristic
68C22×C179#12011yescharacteristic
69C22×C179#12311yescharacteristic
70C4×C179—42no—
71C4×C179—42no—
72C4×C179C4×D412yes—
73C4×C179#12112yes—
74Dic179—44no—
75C2×C179—21no—
76C2×C179—21no—
77C2×C179—44no—
78C2×C179#38611yescharacteristic
79C2×C179#45512yes—
80C179#188511yescharacteristic, complemented, Sylow
81#1885—1791nomaximal, Sylow
82C2×#124D17911yescharacteristic
83#518—1791no—
84#419—1792no—
85C2×#100—1791no—
86#339—1792no—
87C22×Q8C2×D17911yescharacteristic
88C22×D4C2×D17911yescharacteristic
89C2×M16—1792no—
90#124—22no—
91#124Dic17914yescomplemented
92C4×D4—3584no—
93#117—1792no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—3582no—
96C2×C42C2×D17911yescharacteristic
97#103—3584no—
98#100—3582no—
99C24C22⋊D17912yes—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8C2×Dic17912yescomplemented
104C2×D4—22no—
105C2×D4—22no—
106C2×D4C2×Dic17912yes—
107C2×D4C22⋊D17912yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—3584no—
111C22×C4C22×D17911yescharacteristic
112M16—3582no—
113M16—3582no—
114C2×C8—3582no—
115C4⋊C4—1794no—
116C22⋊C4—28no—
117C22⋊C4—3584no—
118C22⋊C4—3582no—
119C42—21no—
120C42—3582no—
121C42C2×Dic17912yescomplemented
122C23—22no—
123C23—22no—
124C23—44no—
125C23C22⋊Dic17914yes—
126C23C2×C22⋊D17911yescharacteristic
127Q8—24no—
128Q8—42no—
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—3584no—
137C2×C4—14324no—
138C2×C4C22×Dic17911yescharacteristic
139C2×C4D4×D17912yes—
140C2×C4C2×C22⋊D17911yescharacteristic
141C2×C4#1566825812yes—
142C8—7162no—
143C22—22no—
144C22—24no—
145C22—44no—
146C22—44no—
147C22—82no—
148C22C2×C22⋊Dic17911yescharacteristic
149C22#1915308111yescharacteristic
150C22#1915309411yescharacteristic
151C4—42no—
152C4—42no—
153C4—7164no—
154C4D4×Dic17912yes—
155C4#1915308512yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2#2603831911yescharacteristic
160C2#2603861612yes—
subgroupG/H
centerC22#19153081
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
FittingC2×#124×C179C2
socleC22×C179#120
prime radicalC22×D4×C179C22
cosocleC22×C4×C179C23
hypercenterC2×#124D179
hypocenterC179#1885
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #26038448C2semidirect
4. #26038199C2semidirect
15. #124×C179C4semidirect
30. C2×Q8×C179C2×C4semidirect
40. C42×C179C2×C4semidirect
80. C179#1885semidirect
91. #124Dic179semidirect
103. C2×Q8C2×Dic179semidirect
121. C42C2×Dic179semidirect

Maximal subgroups

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

subgroupindex
#260389142
#260384482
C2×#191530302
#260381992
C2×#124×C1792
#1885179