Finite Group Databasebeta

← #31911849

Subgroups — #31911849

Order 18176.

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

normal subgroups only

This group has 172 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#22590616C211yesmaximal, characteristic, complemented
2#22590517C211yesmaximal, characteristic
3#22590513C211yesmaximal, characteristic
4#22590176C211yesmaximal, characteristic
5#22586415C211yesmaximal, characteristic, complemented
6#22586402C211yesmaximal, characteristic, complemented
7C71×#1366C211yesmaximal, characteristic, complemented
8#17986417—21no—
9#17986401C2211yescharacteristic
10#17986117C2211yescharacteristic
11#17985826—21no—
12#17985824—21no—
13#17985822—21no—
14#17985821—21no—
15#17985820C2211yescharacteristic
16#17985810—21no—
17#17985808—21no—
18#17985784—21no—
19C4×C71⋊C16C2211yescharacteristic
20#499×C71C2211yescharacteristic
21#364×C71C2211yescharacteristic
22#364×C71C2211yescharacteristic
23#14556150D411yescharacteristic
24#14556134—21no—
25#14556120—21no—
26#14556068—21no—
27#14556067—41no—
28C2×C71⋊C16D411yescharacteristic
29C2×C71⋊C16D411yescharacteristic
30C16×D71—41no—
31#14556064D411yescharacteristic
32#14556062—21no—
33#14556060—21no—
34#14556056—21no—
35#125×C71—21no—
36M32×C71—21no—
37M32×C71—21no—
38#107×C71—21no—
39#106×C71—21no—
40C4×C8×C71C2311yescharacteristic
41#11765865—21no—
42Dic284Q1611yescharacteristic
43#11765854—21no—
44D568Q1611yescharacteristic
45#11765849—21no—
46#11765848—41no—
47C8×D71—41no—
48C4⋊Dic71—41no—
49Dic71⋊C4—81no—
50C71⋊C16—21no—
51C71⋊C16—41no—
52C71⋊C16SD1611yescharacteristic
53C71⋊C16SD1611yescharacteristic
54C2×C8×C71C2×D411yescharacteristic
55C2×C8×C71C2×D411yescharacteristic
56C4⋊C4×C71—41no—
57C4⋊C4×C71—81no—
58C42×C71#4111yescharacteristic
59C16×C71—41no—
60C16×C71—41no—
61C2×C4×C71—21no—
62C2×C4×C71—81no—
63C2×C4×C71#12111yescharacteristic
64C22⋊D71—41no—
65C2×Dic71—81no—
66D284—21no—
67C4×D71—41no—
68Dic142—21no—
69C8×C71C2×SD1611yescharacteristic
70C8×C71C2×Q1611yescharacteristic
71C8×C71#13511yescharacteristic
72C8×C71#13711yescharacteristic
73C71⋊C8—41no—
74C22×C71#47211yescharacteristic
75C2×D71—41no—
76C4×C71—41no—
77C4×C71—161no—
78C4×C71#46111yescharacteristic
79C4×C71#46711yescharacteristic
80Dic71—41no—
81Dic71—161no—
82#12464—711nomaximal, Sylow
83C2×C71—21no—
84C2×C71#167511yescharacteristic
85D71—81no—
86#2253—711no—
87#2231—711no—
88#2231—711no—
89#2170—711no—
90#1373—711no—
91#1366—711no—
92#1366D7111yescharacteristic, complemented
93C71#1246411yescharacteristic, complemented, Sylow
94#504—1421no—
95#499—711no—
96#499C2×D7111yescharacteristic
97#443—711no—
98#367—1421no—
99#366—1421no—
100#365—1421no—
101#365—1421no—
102#364—711no—
103#364C2×D7111yescharacteristic
104#364C2×D7111yescharacteristic
105#360—1421no—
106#360—1421no—
107#350—1421no—
108C4×C16—711no—
109#135—711no—
110#131—1421no—
111#125—21no—
112#125—1421no—
113M32—21no—
114M32—21no—
115M32—2841no—
116M32—1421no—
117C2×C16—2841no—
118C2×C16—711no—
119C2×C16—711no—
120#108—711no—
121#107—21no—
122#107—1421no—
123#106—21no—
124#106—1421no—
125#104—1421no—
126C4×C8C22×D7111yescharacteristic
127#41—1421no—
128Q16—711no—
129SD16—1421no—
130D8—711no—
131M16—2841no—
132M16—1421no—
133C2×C8—2841no—
134C2×C8D4×D7111yescharacteristic
135C2×C8D4×D7111yescharacteristic
136C4⋊C4—41no—
137C4⋊C4—81no—
138C4⋊C4—2841no—
139C4⋊C4—5681no—
140C42#1176586511yescharacteristic
141C16—41no—
142C16—41no—
143C16—2841no—
144C16—711no—
145C16—711no—
146C16—1421no—
147Q8—1421no—
148D4—2841no—
149D4—1421no—
150C2×C4—21no—
151C2×C4—81no—
152C2×C4—2841no—
153C2×C4—5681no—
154C2×C4#1455610111yescharacteristic
155C8—2841no—
156C8SD16×D7111yescharacteristic
157C8Q16×D7111yescharacteristic
158C8#1455614911yescharacteristic
159C8#1455616011yescharacteristic
160C22—2841no—
161C22#1798626811yescharacteristic
162C4—41no—
163C4—161no—
164C4—2841no—
165C4—11361no—
166C4#1798620311yescharacteristic
167C4#1798623911yescharacteristic
168C2—21no—
169C2—5681no—
170C2#2258773811yescharacteristic
subgroupG/H
centerC2#22587738
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1366C2
socleC2×C71#1675
prime radical#14556150D4
hypercenter#1366D71
hypocenterC71#12464
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
1. #22590616C2semidirect
5. #22586415C2semidirect
6. #22586402C2semidirect
7. C71×#1366C2semidirect
92. #1366D71semidirect
93. C71#12464semidirect

Maximal subgroups

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

subgroupindex
#225906162
#225905172
#225905132
#225901762
#225864152
#225864022
C71×#13662
#1246471