Finite Group Databasebeta

← #31911861

Subgroups — #31911861

Order 18176.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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#22590585C211yesmaximal, characteristic, complemented
2#22590517C211yesmaximal, characteristic
3#22590418C211yesmaximal, characteristic
4#22590181C211yesmaximal, characteristic
5#22586407C211yesmaximal, characteristic, complemented
6#22586402C211yesmaximal, characteristic, complemented
7C71×#1358C211yesmaximal, characteristic, complemented
8#17986440—21no—
9#17986417—21no—
10#17986401C2211yescharacteristic
11#17986117C2211yescharacteristic
12#17985822—21no—
13#17985820C2211yescharacteristic
14#17985815—21no—
15#17985810—21no—
16#17985803—21no—
17#17985803—21no—
18#17985784—21no—
19#17985775C2211yescharacteristic
20#494×C71C2211yescharacteristic
21#364×C71C2211yescharacteristic
22#364×C71C2211yescharacteristic
23#14556150D411yescharacteristic
24#14556134—21no—
25#14556120—21no—
26#14556074—41no—
27#14556071—41no—
28#14556068—21no—
29#14556067—41no—
30C2×C71⋊C16D411yescharacteristic
31C2×C71⋊C16D411yescharacteristic
32C16×D71—41no—
33#14556064D411yescharacteristic
34#14556062—21no—
35#14556060—21no—
36#14556056—21no—
37C2×Q16×C71—21no—
38C2×Q16×C71—21no—
39C4⋊Q8×C71—21no—
40M32×C71—21no—
41M32×C71—21no—
42C4×C8×C71C2311yescharacteristic
43#11765865—21no—
44Dic284SD1611yescharacteristic
45#11765854—21no—
46D568SD1611yescharacteristic
47#11765849—21no—
48#11765848—41no—
49C8×D71—41no—
50C4⋊Dic71—41no—
51Dic71⋊C4—81no—
52C71⋊C16—21no—
53C71⋊C16—21no—
54C71⋊C16—41no—
55C2×Q8×C71—41no—
56Q16×C71—41no—
57Q16×C71—41no—
58Q16×C71—41no—
59Q16×C71—41no—
60C2×C8×C71C2×D411yescharacteristic
61C2×C8×C71C2×D411yescharacteristic
62C4⋊C4×C71—81no—
63C42×C71#4111yescharacteristic
64C16×C71—41no—
65C16×C71—41no—
66Q8×C71—81no—
67Q8×C71—81no—
68C2×C4×C71—21no—
69C2×C4×C71—81no—
70C2×C4×C71#12111yescharacteristic
71C22⋊D71—41no—
72C2×Dic71—81no—
73D284—21no—
74C4×D71—41no—
75Dic142—21no—
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—161no—
85C4×C71#46311yescharacteristic
86C4×C71#46511yescharacteristic
87Dic71—41no—
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#1369—711no—
98#1366—711no—
99#1358D7111yescharacteristic, complemented
100C71#1246611yescharacteristic, complemented, Sylow
101#510—1421no—
102#504—1421no—
103#499—711no—
104#494C2×D7111yescharacteristic
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#346—711no—
116#135—711no—
117C2×Q16—21no—
118C2×Q16—21no—
119#131—1421no—
120C4⋊Q8—21no—
121#125—1421no—
122Q32—2841no—
123SD32—2841no—
124M32—21no—
125M32—21no—
126M32—2841no—
127M32—1421no—
128C2×C16—2841no—
129C2×C16—711no—
130C2×C16—711no—
131#108—711no—
132#107—1421no—
133#106—1421no—
134#104—1421no—
135C4×C8C22×D7111yescharacteristic
136#41—1421no—
137C2×Q8—41no—
138Q16—41no—
139Q16—41no—
140Q16—41no—
141Q16—41no—
142Q16—711no—
143SD16—1421no—
144D8—711no—
145M16—2841no—
146M16—1421no—
147C2×C8—2841no—
148C2×C8D4×D7111yescharacteristic
149C2×C8D4×D7111yescharacteristic
150C4⋊C4—81no—
151C4⋊C4—2841no—
152C4⋊C4—5681no—
153C42#1176586511yescharacteristic
154C16—41no—
155C16—41no—
156C16—2841no—
157C16—1421no—
158C16—1421no—
159Q8—81no—
160Q8—81no—
161Q8—1421no—
162D4—2841no—
163D4—1421no—
164C2×C4—21no—
165C2×C4—81no—
166C2×C4—2841no—
167C2×C4—5681no—
168C2×C4#1455610111yescharacteristic
169C8—2841no—
170C8SD16×D7111yescharacteristic
171C8#1455614711yescharacteristic
172C8#1455614911yescharacteristic
173C8#1455615211yescharacteristic
174C22—2841no—
175C22#1798626811yescharacteristic
176C4—41no—
177C4—161no—
178C4—2841no—
179C4—11361no—
180C4#1798621311yescharacteristic
181C4#1798622711yescharacteristic
182C2—21no—
183C2—5681no—
184C2#2258774811yescharacteristic
subgroupG/H
centerC2#22587748
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1358C2
socleC2×C71#1677
prime radical#14556150D4
hypercenter#1358D71
hypocenterC71#12466
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1358

Complements

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

NH ≅ G/Ntype
1. #22590585C2semidirect
5. #22586407C2semidirect
6. #22586402C2semidirect
7. C71×#1358C2semidirect
99. #1358D71semidirect
100. C71#12466semidirect

Maximal subgroups

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

subgroupindex
#225905852
#225905172
#225904182
#225901812
#225864072
#225864022
C71×#13582
#1246671