Finite Group Databasebeta

← #36352899

Subgroups — #36352899

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#26011901C211yesmaximal, characteristic, complemented
3#26011860C211yesmaximal, characteristic, complemented
4#26011851C211yesmaximal, characteristic
5#26011232C211yesmaximal, characteristic, complemented
6#26011230C211yesmaximal, characteristic
7#330×C173C211yesmaximal, characteristic, complemented
8#19149679C2211yescharacteristic
9#19149678—21no—
10#19149674—21no—
11#19149674—21no—
12#19149671C2211yescharacteristic
13#19149667C2211yescharacteristic
14#19149664—21no—
15#19149659—21no—
16#19149657—21no—
17#19149612—21no—
18#19149610—21no—
19#19149604—21no—
20#19149604—21no—
21#19149604—21no—
22#19149601—21no—
23#19149590C2211yescharacteristic
24#125×C173C2211yescharacteristic
25#97×C173C2211yescharacteristic
26#97×C173C2211yescharacteristic
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346D411yescharacteristic
30C2×D692D411yescharacteristic
31C2×C4×D173—21no—
32C2×#13063436—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173—21no—
35C4⋊Dic173—21no—
36C4⋊Dic173D411yescharacteristic
37C4⋊Dic173D411yescharacteristic
38#14583805—21no—
39#14583805—21no—
40#14583805—21no—
41#14583803—21no—
42#14583803—21no—
43#14583803—21no—
44#14583803—21no—
45#14583803—21no—
46C4×Dic173—21no—
47C2×C8×C173—21no—
48C2×C8×C173—21no—
49C4⋊C4×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173D411yescharacteristic
52C4⋊C4×C173D411yescharacteristic
53C42×C173C2311yescharacteristic
54C22×D173—21no—
55Dic346—41no—
56Dic346—42no—
57Dic346—81no—
58D692—41no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—41no—
63C4×D173—81no—
64#13063436—22no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173C2×D411yescharacteristic
68C2×C4×C173C2×D411yescharacteristic
69C2×C4×C173C2×D411yescharacteristic
70C8×C173—41no—
71C8×C173—41no—
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—
81Dic173—41no—
82Dic173—41no—
83Dic173—41no—
84Dic173—81no—
85C2×C173#45411yescharacteristic
86C2×C173#45511yescharacteristic
87C2×C173#45611yescharacteristic
88D173—81no—
89C173#166411yescharacteristic, complemented, Sylow
90#1664—1731nomaximal, Sylow
91#563—1731no—
92#501—1731no—
93#491—1731no—
94#489—1731no—
95#330—1731no—
96#330D17311yescharacteristic, complemented
97#329—1731no—
98C4⋊Q8—3461no—
99C4⋊Q8—1731no—
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—3461no—
109#106—3461no—
110#103—3461no—
111#103—3461no—
112#103—3461no—
113#102—3461no—
114#97—1731no—
115#97C2×D17311yescharacteristic
116#97C2×D17311yescharacteristic
117C2×Q8—3461no—
118C2×Q8—3461no—
119C2×Q8—1731no—
120C2×D4—1731no—
121C22×C4—3461no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—3461no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—3461no—
130C4⋊C4—3461no—
131C4⋊C4—3461no—
132C4⋊C4—3461no—
133C4⋊C4—1731no—
134C4⋊C4—1731no—
135C4⋊C4D69211yescharacteristic
136C4⋊C4D69211yescharacteristic
137C22⋊C4—3461no—
138C22⋊C4—3461no—
139C22⋊C4—3461no—
140C22⋊C4—3461no—
141C22⋊C4—3461no—
142C42—3461no—
143C42C22×D17311yescharacteristic
144C23—3461no—
145Q8—13841no—
146Q8—6922no—
147Q8—6921no—
148D4—6921no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—3461no—
154C2×C4—13841no—
155C2×C4—6921no—
156C2×C4D4×D17311yescharacteristic
157C2×C4D4×D17311yescharacteristic
158C2×C4C2×D69211yescharacteristic
159C8—41no—
160C8—41no—
161C8—3462no—
162C22—6921no—
163C22—6921no—
164C22—6921no—
165C22#1914964911yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—13841no—
172C4—6921no—
173C4—6921no—
174C4—6921no—
175C2—13841no—
176C2#2601166511yescharacteristic
177C2#2601167011yescharacteristic
178C2#2601167611yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#330×C173C2
socleC22×C173#120
prime radicalC2×D692D4
hypercenter#330D173
hypocenterC173#1664
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #26011901C2semidirect
3. #26011860C2semidirect
5. #26011232C2semidirect
7. #330×C173C2semidirect
89. C173#1664semidirect
96. #330D173semidirect

Maximal subgroups

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

subgroupindex
#260122452
#260119012
#260118602
#260118512
#260112322
#260112302
#330×C1732
#1664173