Finite Group Databasebeta

← #36352807

Subgroups — #36352807

Order 22144.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012219C211yesmaximal, characteristic
2#26011896C211yesmaximal, characteristic
3#26011777C211yesmaximal, characteristic
4#26011701C211yesmaximal, characteristic
5#26011236C211yesmaximal, characteristic
6#26011223C211yesmaximal, characteristic
7Q8⋊C8×C173C211yesmaximal, characteristic
8C2×#14583816—21no—
9#19149679C2211yescharacteristic
10#19149679C2211yescharacteristic
11#19149677—21no—
12#19149671—21no—
13#19149669—22no—
14C4×Dic346—21no—
15C4×Dic346C2211yescharacteristic
16#19149644—21no—
17#19149613—22no—
18#19149611—21no—
19C4⋊#13063436C2211yescharacteristic
20#19149604—21no—
21#19149602—21no—
22C4×Q8×C173C2211yescharacteristic
23C4⋊C8×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic
25C2×Dic346—21no—
26C2×Dic346—21no—
27C2×Dic346D411yescharacteristic
28#14583816—42no—
29C2×#13063436—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173—21no—
32C4⋊Dic173—21no—
33C4⋊Dic173D411yescharacteristic
34C4⋊Dic173D412yes—
35#14583805—22no—
36#14583805—22no—
37#14583805—22no—
38#14583805—21no—
39C4×Dic173—22no—
40C4×Dic173—21no—
41C2×Q8×C173D411yescharacteristic
42C2×C8×C173—21no—
43C2×C8×C173—21no—
44C4⋊C4×C173—21no—
45C4⋊C4×C173D411yescharacteristic
46C42×C173—21no—
47C42×C173C2311yescharacteristic
48Dic346—21no—
49Dic346—41no—
50Dic346—41no—
51Dic346—41no—
52Dic346—41no—
53Dic346Q1612yescomplemented
54C2×Dic173—21no—
55C2×Dic173—22no—
56C2×Dic173—21no—
57C2×Dic173—41no—
58#13063436—41no—
59Q8×C173—21no—
60Q8×C173—21no—
61C2×C4×C173—21no—
62C2×C4×C173—21no—
63C2×C4×C173C2×D411yescharacteristic
64C2×C4×C173C2×D411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C8×C173—22no—
67C8×C173—41no—
68C22×C173#12011yescharacteristic
69C4×C173—21no—
70C4×C173—41no—
71C4×C173—41no—
72C4×C173C2×Q1611yescharacteristic
73C4×C173#13511yescharacteristic
74C4×C173AGL(1,ℤ/8)11yescharacteristic
75C4×C173#13711yescharacteristic
76Dic173—22no—
77Dic173—42no—
78Dic173—41no—
79Dic173—81no—
80C2×C173#44911yescharacteristic
81C2×C173#45111yescharacteristic
82C2×C173#45611yescharacteristic
83C173#164811yescharacteristic, complemented, Sylow
84#1648—1731nomaximal, Sylow
85#557—1731no—
86#500—1731no—
87#475—1731no—
88#462—1731no—
89#332—1731no—
90Q8⋊C8—1731no—
91Q8⋊C8D17311yescharacteristic
92C2×Q16—3461no—
93C4⋊Q8—3461no—
94C4⋊Q8—1731no—
95C4⋊Q8—1731no—
96#125—3461no—
97#125—3462no—
98C4×Q8—3461no—
99C4×Q8—3461no—
100C4×Q8—1731no—
101C4×Q8C2×D17311yescharacteristic
102#107—3462no—
103#106—3461no—
104C4⋊C8—1731no—
105C4⋊C8C2×D17311yescharacteristic
106#103—3461no—
107#103—3461no—
108C4×C8C2×D17311yescharacteristic
109C2×Q8—3461no—
110C2×Q8—3461no—
111C2×Q8—1731no—
112C2×Q8D69211yescharacteristic
113Q16—6922no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—3461no—
117C4⋊C4—21no—
118C4⋊C4—3462no—
119C4⋊C4—3462no—
120C4⋊C4—3462no—
121C4⋊C4—3461no—
122C4⋊C4—3461no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—1731no—
126C4⋊C4—1732no—
127C4⋊C4D69211yescharacteristic
128C42—21no—
129C42—3462no—
130C42—3461no—
131C42C22×D17311yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—3461no—
135Q8—1732no—
136Q8—6921no—
137Q8—6921no—
138Q8—6921no—
139Q8—6921no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—3461no—
143C2×C4—3461no—
144C2×C4—3462no—
145C2×C4—6921no—
146C2×C4D4×D17311yescharacteristic
147C2×C4D4×D17311yescharacteristic
148C2×C4C2×D69211yescharacteristic
149C8—22no—
150C8—41no—
151C8—6921no—
152C22#1914964911yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—3462no—
157C4—13841no—
158C4—6921no—
159C4—6922no—
160C4Q16×D17311yescharacteristic
161C4#1914970111yescharacteristic
162C4#1914970711yescharacteristic
163C4#1914970811yescharacteristic
164C2#2601163811yescharacteristic
165C2#2601164811yescharacteristic
166C2#2601167511yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
FittingQ8⋊C8×C173C2
socle, prime radicalC22×C173#120
hypercenterQ8⋊C8D173
hypocenterC173#1648
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
53. Dic346Q16semidirect
83. C173#1648semidirect

Maximal subgroups

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

subgroupindex
#260122192
#260118962
#260117772
#260117012
#260112362
#260112232
Q8⋊C8×C1732
#1648173