Finite Group Databasebeta

← #31895076

Subgroups — #31895076

Order 18176.

434 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 99 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#22598889C211yesmaximal, characteristic
2#22586967C211yesmaximal, characteristic
3#22586851C212yesmaximal
4#22586233C212yesmaximal
5C71×#2371C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986409C414yescomplemented
8#17986398—22no—
9#17986056C2212yes—
10#17986039—22no—
11C4×#11765849C2211yescharacteristic
12#17985741—22no—
13#17985735—22no—
14#17985735—22no—
15C2×#125×C71C2211yescharacteristic
16#387×C71—22no—
17#387×C71—22no—
18C4×C4⋊C4×C71—22no—
19#376×C71C2212yes—
20C2×#11765849—21no—
21#14556128C2×C412yes—
22#14556120—21no—
23#14556107—22no—
24C4×Dic142—22no—
25#14556087—21no—
26#14556061—24no—
27#14556059—24no—
28C4⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30#14556039—22no—
31#14556039C2×C412yes—
32C4×C71⋊C8—21no—
33#125×C71—22no—
34#125×C71C2×C412yes—
35#125×C71D412yes—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71D412yes—
39C2×C42×C71—22no—
40C2×C42×C71—22no—
41C2×C42×C71C2311yescharacteristic
42#95×C71—22no—
43#95×C71—22no—
44#95×C71—22no—
45#95×C71D412yes—
46#95×C71D412yes—
47C2×Dic142—22no—
48#11765849—24no—
49C2×C71⋊C8—22no—
50C2×C71⋊C8—42no—
51C4⋊Dic71—22no—
52C4⋊Dic71—22no—
53Dic71⋊C4—42no—
54C22⋊Dic71—41no—
55C4×Dic71—41no—
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71C2×D411yescharacteristic
60C22×C4×C71C2×D412yes—
61C4⋊C4×C71—24no—
62C4⋊C4×C71—24no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71—44no—
65C4⋊C4×C71C22⋊C414yes—
66C42×C71—44no—
67C42×C71—44no—
68C42×C71C22×C411yescharacteristic
69C42×C71C2×D411yescharacteristic
70C42×C71#4112yes—
71C23×C71#12011yescharacteristic
72C2×C4×C71—22no—
73C2×C4×C71—24no—
74C2×C4×C71—24no—
75C2×C4×C71—42no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71C2×C22⋊C411yescharacteristic
84C2×C4×C71C4×D412yes—
85C2×C4×C71#12112yes—
86C2×C4×C71#12311yescharacteristic
87C2×Dic71—42no—
88Dic142—82no—
89C71⋊C8—24no—
90C71⋊C8—82no—
91C22×C71—22no—
92C22×C71#38611yescharacteristic
93C22×C71#45612yes—
94C4×C71—21no—
95C4×C71—21no—
96C4×C71—22no—
97C4×C71—22no—
98C4×C71—44no—
99C4×C71—44no—
100C4×C71—82no—
101Dic71—82no—
102#9440—711nomaximal, Sylow
103C2×C71—22no—
104C2×C71#188611yescharacteristic
105C2×C71#190612yes—
106#3166—711no—
107#2371D7111yescharacteristic
108#1534—711no—
109#1506—712no—
110#1300—712no—
111C71#944011yescharacteristic, complemented, Sylow
112#533—711no—
113C2×#125C2×D7111yescharacteristic
114#501—714no—
115#499—1422no—
116#432—712no—
117#428—1422no—
118C4×M16—711no—
119#387—22no—
120#387—22no—
121C4×C4⋊C4—22no—
122#376C2×D7112yes—
123#330—1422no—
124#328—1422no—
125#328—1422no—
126C2×M16—1421no—
127C4⋊Q8—712no—
128#125—22no—
129#125—1421no—
130#125C4×D7112yes—
131#125D28412yes—
132C22⋊Q8—1422no—
133C4×Q8—1422no—
134#117—1421no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4C22⋊D7112yes—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C22×D7111yescharacteristic
141#107—1424no—
142#106—1424no—
143C4⋊C8—1422no—
144C22⋊C8—1422no—
145#97—712no—
146#97—1422no—
147C4×C8—1421no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95C22⋊D7112yes—
152#95C22⋊D7112yes—
153C2×Q8—1422no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4C2×C22⋊D7111yescharacteristic
158C22×C4C2×C22⋊D7112yes—
159M16—1424no—
160C2×C8—2842no—
161C2×C8—1422no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—44no—
166C4⋊C4—2842no—
167C4⋊C4—1422no—
168C4⋊C4—1422no—
169C4⋊C4(C2×D71)⋊C414yes—
170C22⋊C4—2841no—
171C42—44no—
172C42—44no—
173C42—2841no—
174C42C2×C4×D7111yescharacteristic
175C42C2×D28411yescharacteristic
176C42#1176586512yes—
177C23#1455609711yescharacteristic
178Q8—5682no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—2842no—
191C2×C4C2×(C2×D71)⋊C411yescharacteristic
192C2×C4C4×C22⋊D7112yes—
193C2×C4#1455610212yes—
194C2×C4#1455611111yescharacteristic
195C8—5682no—
196C8—1424no—
197C22—22no—
198C22#1798587811yescharacteristic
199C22#1798618212yes—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—44no—
206C4—82no—
207C4—5682no—
208C2—22no—
209C2#2258873311yescharacteristic
210C2#2258884112yes—
subgroupG/H
centerC22#17985878
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2371C2
socleC22×C71#386
prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenter#2371D71
hypocenterC71#9440
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
7. #17986409C4semidirect
111. C71#9440semidirect

Maximal subgroups

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

subgroupindex
#225988892
#225869672
#225868512
#225862332
C71×#23712
#944071