Finite Group Databasebeta

← #36352618

Subgroups — #36352618

Order 22144.

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

normal subgroups only

This group has 336 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#26012126C211yesmaximal, characteristic, complemented
2#26011867C211yesmaximal, characteristic, complemented
3#26011687C212yesmaximal, complemented
4#26011218C212yesmaximal, complemented
5#331×C173C211yesmaximal, characteristic, complemented
6C2×D4×D173—22no—
7C2×D1384—22no—
8C2×#14583811—22no—
9#19149676C2211yescharacteristic, complemented
10#19149650—22no—
11#19149647—22no—
12C4×D692C2212yes—
13#19149601—22no—
14C4×#13063436C2211yescharacteristic, complemented
15#127×C173C2211yescharacteristic
16C4⋊C8×C173C2212yescomplemented
17C23×D173—22no—
18C2×D692—22no—
19C2×D692—21no—
20C2×D692D412yes—
21C2×C22⋊D173—24no—
22D4×D173—44no—
23D4×D173—44no—
24C2×C4×D173—22no—
25D1384—44no—
26#14583811—44no—
27C2×#13063436—21no—
28C4⋊Dic173D412yes—
29C22⋊Dic173—22no—
30#14583803—22no—
31C2×D4×C173—22no—
32C2×D4×C173D412yes—
33C2×C8×C173—22no—
34C42×C173C2311yescharacteristic
35C22×D173—22no—
36C22×D173—48no—
37C22×D173—41no—
38C22×D173—82no—
39D692—22no—
40D692—42no—
41D692—42no—
42D692—42no—
43D692D814yescomplemented
44C22⋊D173—48no—
45C2×Dic173—22no—
46C4×D173—42no—
47C4×D173—42no—
48#13063436—22no—
49C23×C173—22no—
50D4×C173—42no—
51D4×C173—42no—
52D4×C173—42no—
53C2×C4×C173C2×D412yes—
54C2×C4×C173C2×D411yescharacteristic
55C8×C173—42no—
56C22×C173—44no—
57C22×C173—42no—
58C22×C173#12011yescharacteristic
59C2×D173—24no—
60C2×D173—42no—
61C2×D173—42no—
62C2×D173—88no—
63C2×D173—82no—
64C2×D173—81no—
65C4×C173—21no—
66C4×C173C2×D812yes—
67C4×C173AGL(1,ℤ/8)12yes—
68Dic173—42no—
69C2×C173—82no—
70C2×C173#44712yes—
71C2×C173#45311yescharacteristic
72D173—44no—
73D173—161no—
74C173#161911yescharacteristic, complemented, Sylow
75#1619—1731nomaximal, Sylow
76D42—1731no—
77#493—1731no—
78#459—1732no—
79#331D17311yescharacteristic, complemented
80D4⋊C8—1732no—
81C22×D4—3462no—
82C2×D8—3462no—
83C2×D8—3462no—
84#127—1731no—
85#127C2×D17311yescharacteristic
86#121—3462no—
87#120—3462no—
88C4×D4—1732no—
89C4⋊C8C2×D17312yescomplemented
90#102—3462no—
91C4×C8—1731no—
92C24—3462no—
93C2×D4—22no—
94C2×D4—3464no—
95C2×D4—3462no—
96C2×D4—3461no—
97C2×D4—1732no—
98C2×D4—6924no—
99C2×D4—6924no—
100C2×D4D69212yes—
101C22×C4—3462no—
102D8—6924no—
103D8—6924no—
104C2×C8—22no—
105C2×C8—3461no—
106C4⋊C4—1732no—
107C22⋊C4—3462no—
108C22⋊C4—3462no—
109C42C22×D17311yescharacteristic
110C23—22no—
111C23—3462no—
112C23—13842no—
113C23—6928no—
114C23—6921no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—3462no—
119D4—1734no—
120D4—6922no—
121D4—6928no—
122D4—6922no—
123D4—6922no—
124C2×C4—3462no—
125C2×C4—6922no—
126C2×C4—6922no—
127C2×C4D4×D17312yes—
128C2×C4C2×D69211yescharacteristic
129C8—42no—
130C8—3462no—
131C22—42no—
132C22—44no—
133C22—3464no—
134C22—13848no—
135C22—13841no—
136C22—13842no—
137C22—6922no—
138C22—6922no—
139C22#1914964911yescharacteristic
140C4—21no—
141C4—6922no—
142C4D8×D17312yes—
143C4#1914970612yes—
144C2—82no—
145C2—6924no—
146C2—27681no—
147C2#2601162612yes—
148C2#2601166111yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#331×C173C2
socleC22×C173#120
hypercenter#331D173
hypocenterC173#1619
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #26012126C2semidirect
2. #26011867C2semidirect
3. #26011687C2semidirect
4. #26011218C2semidirect
5. #331×C173C2semidirect
9. #19149676C22semidirect
14. C4×#13063436C22semidirect
16. C4⋊C8×C173C22semidirect
43. D692D8semidirect
74. C173#1619semidirect
79. #331D173semidirect
89. C4⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121262
#260118672
#260116872
#260112182
#331×C1732
#1619173