Finite Group Databasebeta

← #31900556

Subgroups — #31900556

Order 18176.

400 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 97 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#22596703C211yesmaximal, characteristic
2#22589317C211yesmaximal, characteristic
3#22586905C212yesmaximal
4#22586199C212yesmaximal
5C71×#1276C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7C2×#14556120—21no—
8#17986023C2211yescharacteristic
9#17985897—24no—
10#17985869—21no—
11#17985865C2212yes—
12#17985759—21no—
13#17985759—21no—
14#17985737—24no—
15#431×C71C2212yes—
16C43×C71C2211yescharacteristic
17C2×#11765849—21no—
18#14556128Q812yes—
19#14556120—21no—
20#14556120—42no—
21#14556120—42no—
22#14556107—22no—
23C4×Dic142—22no—
24#14556087—21no—
25C2×C4⋊Dic71—24no—
26C2×C4⋊Dic71—22no—
27C2×Dic71⋊C4—24no—
28C4⋊(C71⋊C8)—22no—
29C71⋊#95—22no—
30C71⋊#95D414yes—
31C2×C42×C71—22no—
32C2×C42×C71D411yescharacteristic
33C2×C42×C71D411yescharacteristic
34C2×C42×C71C2311yescharacteristic
35C4⋊C8×C71—22no—
36C22⋊C8×C71—22no—
37C4×C8×C71—22no—
38C2×Dic142—22no—
39C22×Dic71—24no—
40#11765849—41no—
41#11765849—41no—
42C2×C71⋊C8—41no—
43C4⋊Dic71—22no—
44C4⋊Dic71—22no—
45C4⋊Dic71—48no—
46C4⋊Dic71—42no—
47C4⋊Dic71—42no—
48Dic71⋊C4—48no—
49Dic71⋊C4—42no—
50C22⋊Dic71—41no—
51C4×Dic71—41no—
52C22×C4×C71—21no—
53C22×C4×C71—21no—
54C22×C4×C71C2×D412yes—
55C22×C4×C71C2×D411yescharacteristic
56C2×C8×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×C71C22⋊Q812yes—
72C2×C4×C71C22⋊Q811yescharacteristic
73C2×C4×C71#12412yes—
74C2×C4×C71#12411yescharacteristic
75C2×C4×C71C2×SD1611yescharacteristic
76C2×C4×C71C2×Q1611yescharacteristic
77C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
78C2×C4×C71#13711yescharacteristic
79C2×Dic71—48no—
80C2×Dic71—42no—
81C2×Dic71—44no—
82Dic142—82no—
83C8×C71—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#47511yescharacteristic
94C4×C71#47711yescharacteristic
95C4×C71#48711yescharacteristic
96C4×C71#48911yescharacteristic
97Dic71—84no—
98Dic71—82no—
99#10365—711nomaximal, Sylow
100C2×C71—21no—
101C2×C71—21no—
102C2×C71#202411yescharacteristic
103C2×C71#203112yes—
104#2859—711no—
105#1994—711no—
106#1518—712no—
107#1286—712no—
108#1276D7111yescharacteristic
109C71#1036511yescharacteristic, complemented, Sylow
110#533—711no—
111C2×#125—1421no—
112#431C2×D7112yes—
113#423—711no—
114#389—1424no—
115#384—1421no—
116#382—712no—
117C43C2×D7111yescharacteristic
118#339—1421no—
119#339—1421no—
120#328—1424no—
121C2×M16—1421no—
122C4⋊Q8—712no—
123#125—2842no—
124#125—2842no—
125#125—1421no—
126C22⋊Q8—1422no—
127C4×Q8—1422no—
128#117—1421no—
129C2×C4⋊C4—1424no—
130C2×C4⋊C4—1424no—
131C2×C4⋊C4—1422no—
132C2×C42—22no—
133C2×C42D28411yescharacteristic
134C2×C42D28411yescharacteristic
135C2×C42C22×D7111yescharacteristic
136C4⋊C8—22no—
137C4⋊C8—1422no—
138C22⋊C8—22no—
139C4×C8—22no—
140#95—714no—
141#95—1422no—
142C2×Q8—1422no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—1424no—
146C22×C4D4×D7112yes—
147C22×C4C2×D28411yescharacteristic
148M16—2841no—
149M16—2841no—
150C2×C8—42no—
151C2×C8—2841no—
152C4⋊C4—2848no—
153C4⋊C4—2848no—
154C4⋊C4—2842no—
155C4⋊C4—2842no—
156C4⋊C4—2842no—
157C4⋊C4—1422no—
158C4⋊C4—1422no—
159C22⋊C4—2841no—
160C42—21no—
161C42—21no—
162C42—44no—
163C42—2841no—
164C42#1176585412yes—
165C42Dic28412yes—
166C42Q8×D7111yescharacteristic
167C42#1176586411yescharacteristic
168C42#1176586512yes—
169C23#1455609811yescharacteristic
170Q8—5682no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—2848no—
177C2×C4—2844no—
178C2×C4—2842no—
179C2×C4#1455610612yes—
180C2×C4#1455610811yescharacteristic
181C2×C4#1455611412yes—
182C2×C4#1455611611yescharacteristic
183C2×C4C2×#1176585411yescharacteristic
184C2×C4C2×Dic28411yescharacteristic
185C2×C4#1455615611yescharacteristic
186C2×C4#1455616111yescharacteristic
187C8—82no—
188C8—5681no—
189C22—22no—
190C22#1798591211yescharacteristic
191C22#1798615211yescharacteristic
192C22#1798615711yescharacteristic
193C4—24no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—5684no—
198C4—5682no—
199C4#1798628511yescharacteristic
200C4#1798629511yescharacteristic
201C4#1798635011yescharacteristic
202C4#1798635911yescharacteristic
203C2—21no—
204C2—21no—
205C2#2258948211yescharacteristic
206C2#2258951312yes—
subgroupG/H
centerC22#17985912
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1276C2
socleC22×C71#393
prime radicalC23×C71#120
hypercenter#1276D71
hypocenterC71#10365
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

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
#225967032
#225893172
#225869052
#225861992
C71×#12762
#1036571