Finite Group Databasebeta

← #31911859

Subgroups — #31911859

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#22590513C211yesmaximal, characteristic
2#22590418C211yesmaximal, characteristic
3#22590183C211yesmaximal, characteristic
4#22586408C211yesmaximal, characteristic, complemented
5#22586401C211yesmaximal, characteristic, complemented
6#22586379C211yesmaximal, characteristic, complemented
7C71×#2245C211yesmaximal, characteristic, complemented
8#17986440—21no—
9#17986415—21no—
10#17986117C2211yescharacteristic
11#17985821—21no—
12#17985820C2211yescharacteristic
13#17985820C2211yescharacteristic
14#17985820C2211yescharacteristic
15#17985815—21no—
16#17985808—21no—
17#17985782—21no—
18#499×C71C2211yescharacteristic
19#494×C71C2211yescharacteristic
20#358×C71—21no—
21#358×C71—21no—
22#346×C71C2211yescharacteristic
23#14556150D411yescharacteristic
24#14556134—21no—
25#14556074—41no—
26#14556071—41no—
27#14556068—21no—
28#14556068—21no—
29#14556068—21no—
30#14556067—41no—
31C16×D71—41no—
32#14556064D411yescharacteristic
33#14556056—21no—
34C2×Q16×C71—21no—
35C2×Q16×C71—21no—
36C4⋊Q8×C71—21no—
37#125×C71—21no—
38C2×C16×C71D411yescharacteristic
39C2×C16×C71D411yescharacteristic
40#107×C71—21no—
41#106×C71—21no—
42C4×C8×C71C2311yescharacteristic
43#11765865—21no—
44Dic284SD1611yescharacteristic
45#11765854—21no—
46D568SD1611yescharacteristic
47#11765849—21no—
48#11765848—41no—
49C8×D71—41no—
50C71⋊C16—41no—
51C71⋊C16—41no—
52C71⋊C16—41no—
53C2×Q8×C71—41no—
54Q16×C71—41no—
55Q16×C71—41no—
56Q16×C71—41no—
57Q16×C71—41no—
58C2×C8×C71C2×D411yescharacteristic
59C2×C8×C71C2×D411yescharacteristic
60C4⋊C4×C71—41no—
61C4⋊C4×C71—81no—
62C4⋊C4×C71—81no—
63C42×C71#4111yescharacteristic
64C16×C71—21no—
65C16×C71—21no—
66Q8×C71—81no—
67Q8×C71—81no—
68C2×C4×C71—21no—
69C2×C4×C71—81no—
70C2×C4×C71—81no—
71C2×C4×C71#12111yescharacteristic
72C22⋊D71—41no—
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—161no—
86C4×C71#46311yescharacteristic
87C4×C71#46511yescharacteristic
88Dic71—41no—
89#12466—711nomaximal, Sylow
90C2×C71—21no—
91C2×C71#167711yescharacteristic
92D71—81no—
93#2245D7111yescharacteristic, complemented
94#2231—711no—
95#2214—711no—
96#2171—711no—
97#1369—711no—
98#1366—711no—
99#1358—711no—
100C71#1246611yescharacteristic, complemented, Sylow
101#510—1421no—
102#504—1421no—
103#499C2×D7111yescharacteristic
104#494C2×D7111yescharacteristic
105#443—711no—
106#365—1421no—
107#364—711no—
108#364—711no—
109#364—711no—
110#362—1421no—
111#360—1421no—
112#358—21no—
113#358—21no—
114#350—1421no—
115#346C2×D7111yescharacteristic
116#135—711no—
117C2×Q16—21no—
118C2×Q16—21no—
119#131—1421no—
120C4⋊Q8—21no—
121#125—21no—
122Q32—2841no—
123SD32—2841no—
124M32—2841no—
125M32—1421no—
126M32—1421no—
127M32—1421no—
128C2×C16—2841no—
129C2×C16C22⋊D7111yescharacteristic
130C2×C16C22⋊D7111yescharacteristic
131#108—711no—
132#107—21no—
133#106—21no—
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×C8C2×C22⋊D7111yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C42#1176586411yescharacteristic
154C16—21no—
155C16—21no—
156C16—2841no—
157C16—2841no—
158C16—2841no—
159Q8—81no—
160Q8—81no—
161Q8—1421no—
162D4—2841no—
163D4—1421no—
164C2×C4—21no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—2841no—
168C2×C4#1455609911yescharacteristic
169C8—2841no—
170C8SD16×D7111yescharacteristic
171C8#1455614811yescharacteristic
172C8#1455614911yescharacteristic
173C8#1455615311yescharacteristic
174C22—2841no—
175C22#1798626611yescharacteristic
176C4—41no—
177C4—161no—
178C4—161no—
179C4—2841no—
180C4#1798621111yescharacteristic
181C4#1798622511yescharacteristic
182C2—21no—
183C2—5681no—
184C2#2258774611yescharacteristic
subgroupG/H
centerC2#22587746
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#2245C2
socleC2×C71#1677
prime radical#14556150D4
hypercenter#2245D71
hypocenterC71#12466
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2245

Complements

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

NH ≅ G/Ntype
4. #22586408C2semidirect
5. #22586401C2semidirect
6. #22586379C2semidirect
7. C71×#2245C2semidirect
93. #2245D71semidirect
100. C71#12466semidirect

Maximal subgroups

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

subgroupindex
#225905132
#225904182
#225901832
#225864082
#225864012
#225863792
C71×#22452
#1246671