Finite Group Databasebeta

← #36369695

Subgroups — #36369695

Order 22144.

412 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#19149741C214yesmaximal, complemented
2#26012450C211yesmaximal, characteristic, complemented
3#26012435C211yesmaximal, characteristic, complemented
4#127×D173C211yesmaximal, characteristic
5C2×D173⋊C8C2212yes—
6C2×C4⋊#10621924C2212yes—
7#19149741—28no—
8#19149734—22no—
9(C2×#10621924)⋊C4—22no—
10C2×D4×D173—22no—
11C2×D4×D173C2212yes—
12#19149676C411yescharacteristic, complemented
13#19149675—22no—
14#19149675C412yescomplemented
15C42×D173C2211yescharacteristic
16#127×C173C411yescharacteristic, complemented
17C22×#10621924—22no—
18D173⋊C8—22no—
19D173⋊C8D414yescomplemented
20C2×#13063443—22no—
21C4⋊#10621924—24no—
22C23×D173—22no—
23C2×D692—22no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—24no—
26C2×C22⋊D173—22no—
27D4×D173—24no—
28D4×D173—24no—
29D4×D173—48no—
30C2×C4×D173D412yes—
31C2×C4×D173C2311yescharacteristic
32C4×Dic173C2×C411yescharacteristic
33C4×Dic173D412yes—
34C2×D4×C173—22no—
35C2×D4×C173C2×C412yes—
36C42×C173C2×C411yescharacteristic
37C2×#10621924—42no—
38C2×#10621924—44no—
39#13063443—24no—
40C22×D173—22no—
41C22×D173—42no—
42C22×D173—42no—
43C22×D173—48no—
44C22×D173C2×D411yescharacteristic
45D692—24no—
46D692—44no—
47C22⋊D173—48no—
48C22⋊D173—44no—
49C2×Dic173C22⋊C412yes—
50C2×Dic173C2×D411yescharacteristic
51C4×D173—24no—
52C4×D173C2×D412yes—
53C4×D173#4112yes—
54C23×C173—22no—
55D4×C173—24no—
56D4×C173—44no—
57C2×C4×C173C22⋊C412yes—
58C2×C4×C173C22×C411yescharacteristic
59C22×C173—42no—
60C22×C173—44no—
61C22×C173C2×C22⋊C411yescharacteristic
62C2×D173—42no—
63C2×D173—44no—
64C2×D173—84no—
65C2×D173—84no—
66C2×D173#12411yescharacteristic
67C2×D173#12711yescharacteristic
68C2×D173AGL(1,ℤ/8)14yes—
69#10621924—82no—
70C4×C173—22no—
71C4×C173C4×D412yes—
72Dic173—22no—
73Dic173#12112yes—
74C2×C173—82no—
75C2×C173#39011yescharacteristic
76C2×C173#41812yes—
77D173—82no—
78D173#49012yes—
79D173#49612yes—
80C173#196111yescharacteristic, complemented, Sylow
81#1961—1731nomaximal, Sylow
82C2×#127—1731no—
83C2×#102—1734no—
84C2×#97—1731no—
85#382—1731no—
86C22×D4—3462no—
87C22×D4—1732no—
88C22×C8—1732no—
89#127—3462no—
90#127—1732no—
91#127—1731no—
92#127#1062192411yescharacteristic, complemented
93C2×C4⋊C4—1732no—
94C2×C42—1731no—
95#102—3468no—
96#97—3462no—
97#95—3462no—
98C24—3462no—
99C2×D4—22no—
100C2×D4—3464no—
101C2×D4—3464no—
102C2×D4—3462no—
103C2×D4—3464no—
104C2×D4—3462no—
105C2×D4—1732no—
106C2×D4—6928no—
107C2×D4C2×#1062192412yes—
108C22×C4—3462no—
109C22×C4—1732no—
110C22×C4—1731no—
111C2×C8—3462no—
112C2×C8—3462no—
113C2×C8—1734no—
114C4⋊C4—3464no—
115C42—1732no—
116C42—1731no—
117C42C2×#1062192411yescharacteristic
118C23—22no—
119C23—3462no—
120C23—1731no—
121C23—6922no—
122C23—6922no—
123C23—6928no—
124D4—24no—
125D4—44no—
126D4—3464no—
127D4—6924no—
128D4—6928no—
129D4—6924no—
130C2×C4—3464no—
131C2×C4—1732no—
132C2×C4—1731no—
133C2×C4—1732no—
134C2×C4—1732no—
135C2×C4—6922no—
136C2×C4—6924no—
137C2×C4C22⋊#1062192412yes—
138C2×C4C22×#1062192411yescharacteristic
139C8—3464no—
140C22—42no—
141C22—44no—
142C22—13844no—
143C22—13844no—
144C22—1731no—
145C22—1731no—
146C22—1734no—
147C22—6922no—
148C22—6924no—
149C22C2×C22⋊#1062192411yescharacteristic
150C4—22no—
151C4—3462no—
152C4—13842no—
153C4—1732no—
154C4D4×#1062192412yes—
155C2—82no—
156C2—13842no—
157C2—1732no—
158C2—1732no—
159C2#2601244311yescharacteristic
160C2#2601248312yes—
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC22×C173C2×C22⋊C4
prime radical#127×D173C2
cosocleC2×C4×D173C23
hypercenter#127#10621924
hypocenterC173#1961
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#19149741C2semidirect
2. #26012450C2semidirect
3. #26012435C2semidirect
12. #19149676C4semidirect
14. #19149675C4semidirect
16. #127×C173C4semidirect
19. D173⋊C8D4semidirect
80. C173#1961semidirect
92. #127#10621924semidirect

Maximal subgroups

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

subgroupindex
C2×#191497412
#260124502
#260124352
#127×D1732
#1961173