Finite Group Databasebeta

← #37509537

Subgroups — #37509537

Order 22912.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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#26039013C211yesmaximal, characteristic
2#26038457C212yesmaximal
3C2×#19153032C211yesmaximal, characteristic, complemented
4#26038195C212yesmaximal, complemented
5#519×C179C211yesmaximal, characteristic, complemented
6C2×#15668259C2211yescharacteristic
7C2×#15668243C2212yes—
8#19153111C412yescomplemented
9#19153102C412yescomplemented
10#19153096—22no—
11#19153089—22no—
12#19153072C2211yescharacteristic
13#19153041—24no—
14#19153032—22no—
15C22×Q8×C179C2211yescharacteristic
16C4×Q8×C179—24no—
17#117×C179—22no—
18#117×C179C2212yes—
19#15668259—22no—
20#15668259—22no—
21C2×Dic358C2×C411yescharacteristic
22C2×D716C2×C411yescharacteristic
23C2×C22⋊D179—21no—
24C2×C4×D179—21no—
25#15668243—22no—
26#15668243—22no—
27C2×C179⋊C8—22no—
28C4⋊Dic179—21no—
29Dic179⋊C4—24no—
30C22⋊Dic179—21no—
31(C2×D179)⋊C4—24no—
32C4×Dic179C2×C412yes—
33C2×Q8×C179—22no—
34C2×Q8×C179—22no—
35C2×Q8×C179C2×C412yes—
36C2×Q8×C179D412yes—
37C22×C4×C179D412yes—
38C22×C4×C179C2311yescharacteristic
39C4⋊C4×C179—24no—
40C4⋊C4×C179—22no—
41C22⋊C4×C179—22no—
42C42×C179—24no—
43C42×C179D414yes—
44C22×D179—21no—
45Dic358—22no—
46D716—22no—
47C22⋊D179—41no—
48C22⋊D179—41no—
49C2×Dic179—22no—
50C2×Dic179—21no—
51C4×D179—42no—
52C179⋊C8—42no—
53C23×C179C2×D411yescharacteristic
54Q8×C179—24no—
55Q8×C179—24no—
56C2×C4×C179—22no—
57C2×C4×C179—22no—
58C2×C4×C179—24no—
59C2×C4×C179C22⋊C414yes—
60C2×C4×C179C22×C411yescharacteristic
61C2×C4×C179C2×D412yes—
62C2×C4×C179C2×D411yescharacteristic
63C2×C4×C179#4112yes—
64C22×C179—41no—
65C22×C179C2×C22⋊C411yescharacteristic
66C22×C179#12011yescharacteristic
67C22×C179#12311yescharacteristic
68C2×D179—42no—
69C2×D179—41no—
70C4×C179—24no—
71C4×C179—44no—
72C4×C179C4×D412yes—
73C4×C179#12112yes—
74Dic179—42no—
75Dic179—41no—
76C2×C179—21no—
77C2×C179—21no—
78C2×C179—21no—
79C2×C179#38611yescharacteristic
80D179—81no—
81C179#191211yescharacteristic, complemented, Sylow
82#1912—1791nomaximal, Sylow
83#536—1791no—
84#519D17911yescharacteristic, complemented
85#421—1792no—
86C2×#101—1791no—
87H(3,ℤ/4)—1792no—
88C2×#41—1791no—
89C22×Q8C2×D17911yescharacteristic
90C2×M16—1792no—
91C4⋊Q8—1792no—
92#124—1792no—
93#123—3582no—
94C22⋊Q8—3582no—
95C4×Q8—24no—
96#117—22no—
97#117—1791no—
98#117C2×D17912yes—
99#104—3584no—
100#101—3582no—
101#41—3582no—
102#41—3582no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—1791no—
106C2×Q8C4×D17912yes—
107C2×Q8D71612yes—
108C2×D4—3581no—
109C2×D4—1791no—
110C22×C4—3581no—
111C22×C4C22⋊D17912yes—
112C22×C4C22×D17911yescharacteristic
113M16—3582no—
114M16—3582no—
115C2×C8—3582no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—3584no—
119C4⋊C4—3581no—
120C22⋊C4—22no—
121C22⋊C4—3581no—
122C22⋊C4—3584no—
123C42—24no—
124C42—1792no—
125C42C22⋊D17914yes—
126C23—3581no—
127C23C2×C22⋊D17911yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—3582no—
131D4—3582no—
132D4—7161no—
133D4—7161no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—3582no—
138C2×C4—3581no—
139C2×C4—7162no—
140C2×C4(C2×D179)⋊C414yes—
141C2×C4C2×C4×D17911yescharacteristic
142C2×C4C2×C22⋊D17912yes—
143C2×C4C2×D71611yescharacteristic
144C2×C4#1566825912yes—
145C8—7162no—
146C22—41no—
147C22—7162no—
148C22—7161no—
149C22C2×(C2×D179)⋊C411yescharacteristic
150C22#1915308211yescharacteristic
151C22#1915309611yescharacteristic
152C4—24no—
153C4—44no—
154C4—7162no—
155C4—7161no—
156C4C4×C22⋊D17912yes—
157C4#1915308712yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—14321no—
162C2#2603831811yescharacteristic
subgroupG/H
centerC2#26038318
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#519×C179C2
socleC2×C179#386
prime radicalC2×#15668259C22
cosocleC22×C4×C179C23
hypercenter#519D179
hypocenterC179#1912
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. C2×#19153032C2semidirect
4. #26038195C2semidirect
5. #519×C179C2semidirect
8. #19153111C4semidirect
9. #19153102C4semidirect
81. C179#1912semidirect
84. #519D179semidirect

Maximal subgroups

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

subgroupindex
#260390132
#260384572
C2×#191530322
#260381952
#519×C1792
#1912179