Finite Group Databasebeta

← #31911863

Subgroups — #31911863

Order 18176.

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

normal subgroups only

This group has 186 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#22590586C211yesmaximal, characteristic, complemented
2#22590517C211yesmaximal, characteristic
3#22590422C211yesmaximal, characteristic
4#22590183C211yesmaximal, characteristic
5#22586400C211yesmaximal, characteristic, complemented
6#22586378C211yesmaximal, characteristic, complemented
7C71×#1369C211yesmaximal, characteristic, complemented
8#17986442—21no—
9#17986415—21no—
10#17986401C2211yescharacteristic
11#17986379C2211yescharacteristic
12#17986117C2211yescharacteristic
13#17985822—21no—
14#17985820C2211yescharacteristic
15#17985816—21no—
16#17985810—21no—
17#17985804—21no—
18#17985804—21no—
19#17985782—21no—
20#364×C71C2211yescharacteristic
21#364×C71C2211yescharacteristic
22#346×C71C2211yescharacteristic
23#14556150D411yescharacteristic
24C2×Dic284—21no—
25C2×Dic284—21no—
26#14556134—21no—
27#14556128—21no—
28#14556120—21no—
29Dic568—41no—
30#14556073—41no—
31#14556068—21no—
32#14556067—41no—
33C16×D71—41no—
34#14556064D411yescharacteristic
35#14556062—21no—
36#14556060—21no—
37#14556056—21no—
38M32×C71—21no—
39M32×C71—21no—
40C2×C16×C71D411yescharacteristic
41C2×C16×C71D411yescharacteristic
42C4×C8×C71C2311yescharacteristic
43#11765865—21no—
44C2×Dic142—41no—
45Dic284—41no—
46Dic284—41no—
47Dic284—41no—
48Dic284—41no—
49Dic284SD1611yescharacteristic
50#11765854—21no—
51D568SD1611yescharacteristic
52#11765849—21no—
53#11765848—41no—
54C8×D71—41no—
55C4⋊Dic71—41no—
56C4⋊Dic71—81no—
57Dic71⋊C4—81no—
58C71⋊C16—41no—
59C2×C8×C71C2×D411yescharacteristic
60C2×C8×C71C2×D411yescharacteristic
61C42×C71#4111yescharacteristic
62C16×C71—21no—
63C16×C71—21no—
64C16×C71—41no—
65C16×C71—41no—
66C2×C4×C71—21no—
67C2×C4×C71#12111yescharacteristic
68C22⋊D71—41no—
69C2×Dic71—81no—
70C2×Dic71—81no—
71D284—21no—
72C4×D71—41no—
73Dic142—21no—
74Dic142—81no—
75Dic142—81no—
76C8×C71C2×SD1611yescharacteristic
77C8×C71#13511yescharacteristic
78C8×C71#13511yescharacteristic
79C8×C71AGL(1,ℤ/8)11yescharacteristic
80C71⋊C8—41no—
81C22×C71#47211yescharacteristic
82C2×D71—41no—
83C4×C71—41no—
84C4×C71#46311yescharacteristic
85C4×C71#46511yescharacteristic
86Dic71—41no—
87Dic71—161no—
88Dic71—161no—
89#12466—711nomaximal, Sylow
90C2×C71—21no—
91C2×C71#167711yescharacteristic
92D71—81no—
93#2245—711no—
94#2231—711no—
95#2214—711no—
96#2171—711no—
97#1369D7111yescharacteristic, complemented
98#1366—711no—
99#1358—711no—
100C71#1246611yescharacteristic, complemented, Sylow
101#510—1421no—
102#504—1421no—
103#499—711no—
104#494—711no—
105#443—711no—
106#365—1421no—
107#364—711no—
108#364C2×D7111yescharacteristic
109#364C2×D7111yescharacteristic
110#362—1421no—
111#360—1421no—
112#358—1421no—
113#358—1421no—
114#350—1421no—
115#346C2×D7111yescharacteristic
116#135—711no—
117C2×Q16—1421no—
118C2×Q16—1421no—
119#131—1421no—
120C4⋊Q8—1421no—
121#125—1421no—
122Q32—2841no—
123SD32—2841no—
124M32—21no—
125M32—21no—
126M32—2841no—
127M32—1421no—
128C2×C16—2841no—
129C2×C16D28411yescharacteristic
130C2×C16D28411yescharacteristic
131#108—711no—
132#107—1421no—
133#106—1421no—
134#104—1421no—
135C4×C8C22×D7111yescharacteristic
136#41—1421no—
137C2×Q8—2841no—
138Q16—2841no—
139Q16—2841no—
140Q16—2841no—
141Q16—2841no—
142Q16—711no—
143SD16—1421no—
144D8—711no—
145M16—2841no—
146M16—1421no—
147C2×C8—2841no—
148C2×C8D4×D7111yescharacteristic
149C2×C8C2×D28411yescharacteristic
150C4⋊C4—2841no—
151C4⋊C4—5681no—
152C4⋊C4—5681no—
153C42#1176586611yescharacteristic
154C16—21no—
155C16—21no—
156C16—41no—
157C16—41no—
158C16—2841no—
159Q8—5681no—
160Q8—5681no—
161Q8—1421no—
162D4—2841no—
163D4—1421no—
164C2×C4—21no—
165C2×C4—2841no—
166C2×C4—5681no—
167C2×C4—5681no—
168C2×C4#1455610311yescharacteristic
169C8—2841no—
170C8SD16×D7111yescharacteristic
171C8#1455614911yescharacteristic
172C8#1455615011yescharacteristic
173C8#1455615611yescharacteristic
174C22—2841no—
175C22#1798627011yescharacteristic
176C4—41no—
177C4—2841no—
178C4—11361no—
179C4—11361no—
180C4#1798621511yescharacteristic
181C4#1798622911yescharacteristic
182C2—21no—
183C2—5681no—
184C2#2258775011yescharacteristic
subgroupG/H
centerC2#22587750
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1369C2
socleC2×C71#1677
prime radical#14556150D4
hypercenter#1369D71
hypocenterC71#12466
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1369

Complements

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

NH ≅ G/Ntype
1. #22590586C2semidirect
5. #22586400C2semidirect
6. #22586378C2semidirect
7. C71×#1369C2semidirect
97. #1369D71semidirect
100. C71#12466semidirect

Maximal subgroups

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

subgroupindex
#225905862
#225905172
#225904222
#225901832
#225864002
#225863782
C71×#13692
#1246671