Finite Group Databasebeta

← #36352890

Subgroups — #36352890

Order 22144.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C173⋊#564C211yesmaximal, characteristic
2#26011897C211yesmaximal, characteristic
3#26011859C211yesmaximal, characteristic
4#26011233C211yesmaximal, characteristic
5#26011232C211yesmaximal, characteristic
6#26011232C211yesmaximal, characteristic
7#501×C173C211yesmaximal, characteristic
8#19149671C2211yescharacteristic
9#19149671C2211yescharacteristic
10#19149670—21no—
11#19149670—21no—
12#19149669—21no—
13#19149669—21no—
14#19149669—21no—
15#19149669—21no—
16#19149612—21no—
17#19149610—21no—
18#19149603—21no—
19#19149603—21no—
20#19149590C2211yescharacteristic
21#19149590C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#125×C173C2211yescharacteristic
24#107×C173—21no—
25#106×C173—21no—
26#97×C173C2211yescharacteristic
27C2×#13063436—21no—
28C2×#13063436—21no—
29C4⋊Dic173—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173D411yescharacteristic
32C4⋊Dic173D411yescharacteristic
33C4⋊Dic173D411yescharacteristic
34C4⋊Dic173D411yescharacteristic
35#14583805—21no—
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583805—21no—
42#14583805—21no—
43C4×Dic173—21no—
44C4×Dic173—21no—
45C2×Q8×C173—21no—
46C2×C8×C173—21no—
47C4⋊C4×C173—21no—
48C4⋊C4×C173—21no—
49C4⋊C4×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173D411yescharacteristic
52C4⋊C4×C173D411yescharacteristic
53C42×C173C2311yescharacteristic
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58#13063436—22no—
59#13063436—41no—
60Q8×C173—42no—
61C2×C4×C173—21no—
62C2×C4×C173—21no—
63C2×C4×C173—41no—
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C2×C4×C173C2×D411yescharacteristic
67C8×C173—22no—
68C22×C173#12011yescharacteristic
69C4×C173—21no—
70C4×C173—21no—
71C4×C173—21no—
72C4×C173—41no—
73C4×C173—41no—
74C4×C173—81no—
75Dic173—41no—
76Dic173—41no—
77Dic173—41no—
78Dic173—41no—
79C2×C173#45411yescharacteristic
80C2×C173#45611yescharacteristic
81C2×C173#45611yescharacteristic
82C173#166311yescharacteristic, complemented, Sylow
83#1663—1731nomaximal, Sylow
84#564—1731no—
85#501—1731no—
86#501D17311yescharacteristic
87#491—1731no—
88#330—1731no—
89#330—1731no—
90#330—1731no—
91C4⋊Q8C2×D17311yescharacteristic
92#125—3461no—
93#125—3461no—
94#125—3461no—
95#125—3461no—
96#125—3461no—
97#125—3461no—
98#125—1731no—
99#125—1731no—
100#125C2×D17311yescharacteristic
101#107—21no—
102#107—3461no—
103#106—21no—
104#106—3461no—
105#103—3461no—
106#103—3461no—
107#97—1731no—
108#97—1731no—
109#97C2×D17311yescharacteristic
110C2×Q8—21no—
111C2×C8—21no—
112C2×C8—3461no—
113C2×C8—3461no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—3461no—
119C4⋊C4—3461no—
120C4⋊C4—3461no—
121C4⋊C4—3461no—
122C4⋊C4—3461no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—3461no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—1731no—
129C4⋊C4—1731no—
130C4⋊C4—1731no—
131C4⋊C4—1731no—
132C4⋊C4C22⋊D17311yescharacteristic
133C4⋊C4C22⋊D17311yescharacteristic
134C42—3461no—
135C42—3461no—
136C42C22×D17311yescharacteristic
137Q8—42no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—3461no—
142C2×C4—3461no—
143C2×C4—3461no—
144C2×C4—3461no—
145C2×C4D4×D17311yescharacteristic
146C2×C4D4×D17311yescharacteristic
147C2×C4C2×C22⋊D17311yescharacteristic
148C8—22no—
149C8—3462no—
150C8—6921no—
151C22#1914964711yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—41no—
156C4—41no—
157C4—81no—
158C4—6921no—
159C4—6921no—
160C4—6921no—
161C4—6921no—
162C2#2601166211yescharacteristic
163C2#2601167211yescharacteristic
164C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#501×C173C2
socle, prime radicalC22×C173#120
hypercenter#501D173
hypocenterC173#1663
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
82. C173#1663semidirect

Maximal subgroups

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

subgroupindex
C173⋊#5642
#260118972
#260118592
#260112332
#260112322
#501×C1732
#1663173