Finite Group Databasebeta

← #36352895

Subgroups — #36352895

Order 22144.

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

normal subgroups only

This group has 184 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
1#26012245C211yesmaximal, characteristic
2#26011859C211yesmaximal, characteristic, complemented
3#26011853C211yesmaximal, characteristic
4#26011233C211yesmaximal, characteristic, complemented
5#26011232C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic
7#501×C173C211yesmaximal, characteristic, complemented
8#19149678—21no—
9#19149674—21no—
10#19149674—21no—
11#19149671C2211yescharacteristic
12#19149667C2211yescharacteristic
13#19149664—21no—
14#19149659—21no—
15#19149657—21no—
16#19149603—21no—
17#19149603—21no—
18#19149602—21no—
19#19149599—21no—
20#19149590C2211yescharacteristic
21#19149590C2211yescharacteristic
22C4⋊Q8×C173C2211yescharacteristic
23#125×C173C2211yescharacteristic
24#107×C173—21no—
25#106×C173—21no—
26#97×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346D411yescharacteristic
29C2×D692D411yescharacteristic
30C2×C4×D173—21no—
31C2×#13063436—21no—
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173D411yescharacteristic
35C4⋊Dic173D411yescharacteristic
36#14583805—21no—
37#14583805—21no—
38#14583805—21no—
39#14583803—21no—
40#14583803—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—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
54C22×D173—21no—
55Dic346—41no—
56Dic346—81no—
57D692—41no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C4×D173—81no—
62#13063436—41no—
63#13063436—41no—
64Q8×C173—42no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C2×C4×C173C2×D411yescharacteristic
71C8×C173—22no—
72C22×C173#12011yescharacteristic
73C2×D173—41no—
74C2×D173—41no—
75C2×D173—41no—
76C4×C173—21no—
77C4×C173—21no—
78C4×C173—21no—
79C4×C173—41no—
80C4×C173—41no—
81C4×C173—81no—
82Dic173—41no—
83Dic173—41no—
84Dic173—41no—
85C2×C173#45411yescharacteristic
86C2×C173#45511yescharacteristic
87C2×C173#45611yescharacteristic
88D173—81no—
89C173#166411yescharacteristic, complemented, Sylow
90#1664—1731nomaximal, Sylow
91#563—1731no—
92#501D17311yescharacteristic, complemented
93#491—1731no—
94#489—1731no—
95#330—1731no—
96#330—1731no—
97#329—1731no—
98C4⋊Q8—3461no—
99C4⋊Q8C2×D17311yescharacteristic
100#126—3461no—
101#126—3461no—
102#125—1731no—
103#125C2×D17311yescharacteristic
104#124—1731no—
105#123—3461no—
106C22⋊Q8—3461no—
107C22⋊Q8—3461no—
108#107—21no—
109#106—21no—
110#103—3461no—
111#103—3461no—
112#103—3461no—
113#102—3461no—
114#97—1731no—
115#97—1731no—
116#97C2×D17311yescharacteristic
117C2×Q8—21no—
118C2×Q8—3461no—
119C2×Q8—1731no—
120C2×D4—1731no—
121C22×C4—3461no—
122C2×C8—21no—
123C2×C8—3461no—
124C2×C8—3461no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4—1731no—
134C4⋊C4—1731no—
135C4⋊C4C22⋊D17311yescharacteristic
136C4⋊C4C22⋊D17311yescharacteristic
137C22⋊C4—3461no—
138C22⋊C4—3461no—
139C22⋊C4—3461no—
140C22⋊C4—3461no—
141C22⋊C4—3461no—
142C42—3461no—
143C42C22×D17311yescharacteristic
144C23—3461no—
145Q8—42no—
146Q8—13841no—
147Q8—6921no—
148D4—6921no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—3461no—
155C2×C4—13841no—
156C2×C4D4×D17311yescharacteristic
157C2×C4D4×D17311yescharacteristic
158C2×C4C2×C22⋊D17311yescharacteristic
159C8—22no—
160C8—6921no—
161C8—6921no—
162C22—6921no—
163C22—6921no—
164C22—6921no—
165C22#1914964711yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—6921no—
173C4—6921no—
174C4—6921no—
175C2—13841no—
176C2#2601166211yescharacteristic
177C2#2601166711yescharacteristic
178C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#501×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#501D173
hypocenterC173#1664
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #26011859C2semidirect
4. #26011233C2semidirect
5. #26011232C2semidirect
7. #501×C173C2semidirect
89. C173#1664semidirect
92. #501D173semidirect

Maximal subgroups

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

subgroupindex
#260122452
#260118592
#260118532
#260112332
#260112322
#260112302
#501×C1732
#1664173