Finite Group Databasebeta

← #31911848

Subgroups — #31911848

Order 18176.

172 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 59 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#22590616C211yesmaximal, characteristic, complemented
2#22590516C211yesmaximal, characteristic
3#22590177C211yesmaximal, characteristic
4#22586415C211yesmaximal, characteristic, complemented
5#22586401C211yesmaximal, characteristic, complemented
6#22586400C211yesmaximal, characteristic, complemented
7C71×#2231C211yesmaximal, characteristic
8#17986416—21no—
9#17986401C2211yescharacteristic
10#17985826—21no—
11#17985824—21no—
12#17985823—21no—
13#17985820C2211yescharacteristic
14#17985820C2211yescharacteristic
15#17985809—21no—
16#17985783—21no—
17C4×C71⋊C16C2211yescharacteristic
18#499×C71C2211yescharacteristic
19#443×C71C2211yescharacteristic
20#365×C71—21no—
21#364×C71C2211yescharacteristic
22#360×C71—21no—
23#14556120—21no—
24#14556068—21no—
25#14556068—21no—
26#14556068—41no—
27C2×C71⋊C16—41no—
28C2×C71⋊C16D411yescharacteristic
29C2×C71⋊C16D411yescharacteristic
30#14556062—21no—
31#14556060—21no—
32#135×C71D411yescharacteristic
33#131×C71—21no—
34#125×C71—21no—
35M32×C71—21no—
36#108×C71D411yescharacteristic
37#107×C71—21no—
38#106×C71—21no—
39#104×C71—21no—
40C4×C8×C71C2311yescharacteristic
41C4⋊Dic71—41no—
42Dic71⋊C4—81no—
43C71⋊C16—21no—
44C71⋊C16—41no—
45C71⋊C16—41no—
46C71⋊C16SD1611yescharacteristic
47C71⋊C16SD1611yescharacteristic
48#41×C71—21no—
49Q16×C71Q1611yescharacteristic
50SD16×C71—21no—
51D8×C71Q1611yescharacteristic
52M16×C71—21no—
53M16×C71—41no—
54C2×C8×C71—41no—
55C2×C8×C71C2×D411yescharacteristic
56C2×C8×C71C2×D411yescharacteristic
57C4⋊C4×C71—41no—
58C4⋊C4×C71—81no—
59C42×C71#4111yescharacteristic
60C16×C71—41no—
61Q8×C71—21no—
62D4×C71—21no—
63D4×C71—41no—
64C2×C4×C71—21no—
65C2×C4×C71—41no—
66C2×C4×C71—81no—
67C2×C4×C71#12111yescharacteristic
68C2×Dic71—81no—
69C8×C71—41no—
70C8×C71C2×SD1611yescharacteristic
71C8×C71C2×Q1611yescharacteristic
72C8×C71#13511yescharacteristic
73C8×C71#13711yescharacteristic
74C22×C71—41no—
75C22×C71#47211yescharacteristic
76C4×C71—41no—
77C4×C71—41no—
78C4×C71—161no—
79C4×C71#46111yescharacteristic
80C4×C71#46711yescharacteristic
81Dic71—161no—
82#12464—711nomaximal, Sylow
83C2×C71—21no—
84C2×C71—81no—
85C2×C71#167511yescharacteristic
86#2253—711no—
87#2231—711no—
88#2231D7111yescharacteristic
89#2170—711no—
90#1373—711no—
91#1366—711no—
92#1366—711no—
93C71#1246411yescharacteristic, complemented, Sylow
94#504—1421no—
95#499—711no—
96#499C2×D7111yescharacteristic
97#443C2×D7111yescharacteristic
98#367—1421no—
99#366—1421no—
100#365—21no—
101#365—1421no—
102#364—711no—
103#364—711no—
104#364C2×D7111yescharacteristic
105#360—21no—
106#360—1421no—
107#350—1421no—
108C4×C16—711no—
109#135C22⋊D7111yescharacteristic
110#131—21no—
111#125—21no—
112#125—1421no—
113M32—21no—
114M32—2841no—
115M32—1421no—
116M32—1421no—
117C2×C16—2841no—
118C2×C16—711no—
119C2×C16—711no—
120#108C22⋊D7111yescharacteristic
121#107—21no—
122#107—1421no—
123#106—21no—
124#106—1421no—
125#104—21no—
126C4×C8C22×D7111yescharacteristic
127#41—21no—
128Q16#1176585511yescharacteristic
129SD16—21no—
130D8#1176585511yescharacteristic
131M16—21no—
132M16—41no—
133C2×C8—41no—
134C2×C8D4×D7111yescharacteristic
135C2×C8C2×C22⋊D7111yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—81no—
138C4⋊C4—2841no—
139C4⋊C4—5681no—
140C42#1176586411yescharacteristic
141C16—41no—
142C16—2841no—
143C16—2841no—
144C16—711no—
145C16—711no—
146C16—1421no—
147Q8—21no—
148D4—21no—
149D4—41no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—81no—
153C2×C4—5681no—
154C2×C4#1455610011yescharacteristic
155C8—41no—
156C8SD16×D7111yescharacteristic
157C8C2×#1176585511yescharacteristic
158C8#1455614811yescharacteristic
159C8#1455615711yescharacteristic
160C22—41no—
161C22#1798626711yescharacteristic
162C4—41no—
163C4—41no—
164C4—161no—
165C4—11361no—
166C4#1798620211yescharacteristic
167C4#1798623811yescharacteristic
168C2—21no—
169C2—81no—
170C2#2258773711yescharacteristic
subgroupG/H
centerC2#22587737
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2231C2
socleC2×C71#1675
prime radical#135×C71D4
hypercenter#2231D71
hypocenterC71#12464
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2231

Complements

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

NH ≅ G/Ntype
1. #22590616C2semidirect
4. #22586415C2semidirect
5. #22586401C2semidirect
6. #22586400C2semidirect
93. C71#12464semidirect

Maximal subgroups

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

subgroupindex
#225906162
#225905162
#225901772
#225864152
#225864012
#225864002
C71×#22312
#1246471