Finite Group Databasebeta

← #31900552

Subgroups — #31900552

Order 18176.

400 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 101 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#22589317C211yesmaximal, characteristic
2#22586903C212yesmaximal
3#22586198C212yesmaximal
4#22586173C211yesmaximal, characteristic
5C71×#2859C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986052C2212yes—
8#17986023C2211yescharacteristic
9#17985759—21no—
10#17985759—21no—
11#17985735—24no—
12C2×#125×C71—21no—
13#389×C71—24no—
14#384×C71—21no—
15#382×C71C2212yes—
16C43×C71C2211yescharacteristic
17C2×#11765849—21no—
18#14556128Q812yes—
19#14556120—21no—
20#14556107—22no—
21C4×Dic142—22no—
22#14556087—21no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C22⋊(C71⋊C8)—22no—
26C4×C71⋊C8—22no—
27#125×C71—42no—
28#125×C71—42no—
29C2×C4⋊C4×C71—24no—
30C2×C4⋊C4×C71—24no—
31C2×C4⋊C4×C71—22no—
32C2×C42×C71—22no—
33C2×C42×C71D411yescharacteristic
34C2×C42×C71D411yescharacteristic
35C2×C42×C71C2311yescharacteristic
36#95×C71—22no—
37#95×C71D414yes—
38C2×Dic142—22no—
39#11765849—41no—
40#11765849—41no—
41C2×C71⋊C8—42no—
42C2×C71⋊C8—41no—
43C4⋊Dic71—22no—
44C4⋊Dic71—22no—
45Dic71⋊C4—42no—
46C22⋊Dic71—41no—
47C4×Dic71—41no—
48C22×C4×C71—21no—
49C22×C4×C71—21no—
50C22×C4×C71—24no—
51C22×C4×C71C2×D412yes—
52C22×C4×C71C2×D411yescharacteristic
53C4⋊C4×C71—48no—
54C4⋊C4×C71—48no—
55C4⋊C4×C71—42no—
56C4⋊C4×C71—42no—
57C42×C71—21no—
58C42×C71—21no—
59C42×C71—44no—
60C42×C71SD1612yes—
61C42×C71Q1612yes—
62C42×C71C2×Q811yescharacteristic
63C42×C71#4111yescharacteristic
64C42×C71#4112yes—
65C23×C71#12011yescharacteristic
66C2×C4×C71—24no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—42no—
70C2×C4×C71—42no—
71C2×C4×C71—48no—
72C2×C4×C71—44no—
73C2×C4×C71C22⋊Q812yes—
74C2×C4×C71C22⋊Q811yescharacteristic
75C2×C4×C71#12412yes—
76C2×C4×C71#12411yescharacteristic
77C2×C4×C71C2×SD1611yescharacteristic
78C2×C4×C71C2×Q1611yescharacteristic
79C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C71#13711yescharacteristic
81C2×Dic71—42no—
82Dic142—82no—
83C71⋊C8—82no—
84C71⋊C8—81no—
85C22×C71—22no—
86C22×C71#39311yescharacteristic
87C22×C71#45011yescharacteristic
88C22×C71#45111yescharacteristic
89C4×C71—24no—
90C4×C71—22no—
91C4×C71—22no—
92C4×C71—42no—
93C4×C71—84no—
94C4×C71#47511yescharacteristic
95C4×C71#47711yescharacteristic
96C4×C71#48711yescharacteristic
97C4×C71#48911yescharacteristic
98Dic71—82no—
99#10365—711nomaximal, Sylow
100C2×C71—21no—
101C2×C71—21no—
102C2×C71#202411yescharacteristic
103C2×C71#203112yes—
104#2859D7111yescharacteristic
105#1994—711no—
106#1518—712no—
107#1286—712no—
108#1276—711no—
109C71#1036511yescharacteristic, complemented, Sylow
110#533—711no—
111C2×#125—21no—
112#431—712no—
113#423—711no—
114#389—24no—
115#384—21no—
116#382C2×D7112yes—
117C43C2×D7111yescharacteristic
118#339—1421no—
119#339—1421no—
120#328—1424no—
121C2×M16—1421no—
122C4⋊Q8—712no—
123#125—42no—
124#125—42no—
125#125—1421no—
126C22⋊Q8—1422no—
127C4×Q8—1422no—
128#117—1421no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C42—22no—
133C2×C42C22⋊D7111yescharacteristic
134C2×C42C22⋊D7111yescharacteristic
135C2×C42C22×D7111yescharacteristic
136C4⋊C8—1422no—
137C4⋊C8—1422no—
138C22⋊C8—1422no—
139C4×C8—1422no—
140#95—22no—
141#95C22⋊D7114yes—
142C2×Q8—1422no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4C2×C22⋊D7112yes—
147C22×C4C2×C22⋊D7111yescharacteristic
148M16—2841no—
149M16—2841no—
150C2×C8—2842no—
151C2×C8—2841no—
152C4⋊C4—48no—
153C4⋊C4—48no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—2842no—
157C4⋊C4—1422no—
158C4⋊C4—1422no—
159C22⋊C4—2841no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—2841no—
164C42#1176585312yes—
165C42#1176585512yes—
166C42Q8×D7111yescharacteristic
167C42#1176586611yescharacteristic
168C42#1176586612yes—
169C23#1455609711yescharacteristic
170Q8—5682no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—48no—
177C2×C4—44no—
178C2×C4—2842no—
179C2×C4#1455610412yes—
180C2×C4#1455610411yescharacteristic
181C2×C4#1455611712yes—
182C2×C4#1455611711yescharacteristic
183C2×C4C2×#1176585311yescharacteristic
184C2×C4C2×#1176585511yescharacteristic
185C2×C4#1455615411yescharacteristic
186C2×C4#1455615911yescharacteristic
187C8—5682no—
188C8—5681no—
189C22—22no—
190C22#1798591011yescharacteristic
191C22#1798615011yescharacteristic
192C22#1798615511yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—84no—
198C4—5682no—
199C4#1798628111yescharacteristic
200C4#1798629111yescharacteristic
201C4#1798635111yescharacteristic
202C4#1798636111yescharacteristic
203C2—21no—
204C2—21no—
205C2#2258947811yescharacteristic
206C2#2258950912yes—
subgroupG/H
centerC22#17985910
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2859C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#2859D71
hypocenterC71#10365
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2859

Complements

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

NH ≅ G/Ntype
109. C71#10365semidirect

Maximal subgroups

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

subgroupindex
#225893172
#225869032
#225861982
#225861732
C71×#28592
#1036571