Finite Group Databasebeta

← #31894992

Subgroups — #31894992

Order 18176.

422 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 95 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#22598720C211yesmaximal, characteristic
2#22591521C211yesmaximal, characteristic
3#22586892C212yesmaximal
4#22586206C212yesmaximal
5C71×#1558C211yesmaximal, characteristic
6#17986556C2211yescharacteristic
7#17986367—22no—
8#17986349C414yescomplemented
9#17986008—22no—
10C4×#11765849C2211yescharacteristic
11#17985906—22no—
12#17985885—24no—
13#17985867C2212yes—
14#17985737—24no—
15C2×C4⋊Q8×C71C2211yescharacteristic
16#432×C71C2212yes—
17#333×C71—22no—
18C2×#11765849—21no—
19#14556128C2×C412yes—
20#14556120—21no—
21#14556107—22no—
22C4×Dic142—22no—
23#14556087—21no—
24C2×C4⋊Dic71—22no—
25C2×C4×Dic71—22no—
26#14556052—28no—
27#14556039—21no—
28C4×C71⋊C8C2×C412yes—
29C71⋊#95—24no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C71⋊#95D414yes—
33C22×Q8×C71D412yes—
34C4⋊Q8×C71—22no—
35C4⋊Q8×C71C2×C412yes—
36C4⋊Q8×C71D412yes—
37C2×C4⋊C4×C71—22no—
38C2×C42×C71C2311yescharacteristic
39C4⋊C8×C71—22no—
40C22⋊C8×C71—22no—
41#97×C71—22no—
42C2×Dic142—22no—
43C22×Dic71—24no—
44#11765849—42no—
45C2×C71⋊C8—22no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—82no—
49Dic71⋊C4—42no—
50C22⋊Dic71—41no—
51C4×Dic71—41no—
52C4×Dic71—82no—
53C2×Q8×C71—22no—
54C2×Q8×C71—42no—
55C2×Q8×C71—42no—
56C2×Q8×C71C22⋊C414yes—
57C22×C4×C71—22no—
58C22×C4×C71C2×D412yes—
59C22×C4×C71C2×D411yescharacteristic
60C2×C8×C71—42no—
61C4⋊C4×C71—24no—
62C4⋊C4×C71—24no—
63C42×C71C22×C411yescharacteristic
64C42×C71C2×D411yescharacteristic
65C42×C71#4112yes—
66C23×C71#12011yescharacteristic
67Q8×C71—44no—
68Q8×C71—82no—
69C2×C4×C71—22no—
70C2×C4×C71—24no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71C2×C22⋊C411yescharacteristic
75C2×C4×C71C4×D412yes—
76C2×C4×C71#12112yes—
77C2×C4×C71#12311yescharacteristic
78C2×Dic71—48no—
79C2×Dic71—42no—
80C2×Dic71—44no—
81Dic142—82no—
82C8×C71—82no—
83C71⋊C8—42no—
84C22×C71—22no—
85C22×C71#38611yescharacteristic
86C22×C71#45512yes—
87C4×C71—21no—
88C4×C71—21no—
89C4×C71—22no—
90C4×C71—22no—
91C4×C71—44no—
92Dic71—84no—
93Dic71—82no—
94#9426—711nomaximal, Sylow
95C2×C71—22no—
96C2×C71#188511yescharacteristic
97C2×C71#190512yes—
98#3139—711no—
99#2381—711no—
100#1558D7111yescharacteristic
101#1515—712no—
102#1288—712no—
103C71#942611yescharacteristic, complemented, Sylow
104#533—711no—
105C2×C4⋊Q8C2×D7111yescharacteristic
106#491—1422no—
107#487—714no—
108#432C2×D7112yes—
109#419—1422no—
110C4×M16—711no—
111#391—1422no—
112#387—1424no—
113#383—712no—
114#333—22no—
115#328—1424no—
116C22×Q8D28412yes—
117C2×M16—1421no—
118C4⋊Q8—22no—
119C4⋊Q8—712no—
120C4⋊Q8C4×D7112yes—
121C4⋊Q8C22⋊D7112yes—
122#125—1421no—
123C22⋊Q8—1422no—
124C4×Q8—1422no—
125#117—1421no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—1422no—
128C2×C42—1422no—
129C2×C42C22×D7111yescharacteristic
130C4⋊C8—22no—
131#103—1428no—
132C22⋊C8—22no—
133#97—22no—
134#97—1421no—
135C4×C8—712no—
136#95—714no—
137#95—1424no—
138#95—1422no—
139#95—1422no—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—1422no—
144C2×Q8(C2×D71)⋊C414yes—
145C22×C4—22no—
146C22×C4—1424no—
147C22×C4D4×D7112yes—
148C22×C4C2×D28411yescharacteristic
149M16—2842no—
150C2×C8—42no—
151C2×C8—1422no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—2842no—
155C4⋊C4—5682no—
156C4⋊C4—1422no—
157C4⋊C4—1422no—
158C22⋊C4—2841no—
159C42—2841no—
160C42—5682no—
161C42C2×C4×D7111yescharacteristic
162C42C2×C22⋊D7111yescharacteristic
163C42#1176586412yes—
164C23#1455609811yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—5682no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—2848no—
174C2×C4—2842no—
175C2×C4—2844no—
176C2×C4C2×(C2×D71)⋊C411yescharacteristic
177C2×C4(C22⋊D71)⋊C412yes—
178C2×C4#1455610012yes—
179C2×C4#1455611211yescharacteristic
180C8—82no—
181C8—2842no—
182C22—22no—
183C22#1798587611yescharacteristic
184C22#1798617912yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—5684no—
191C4—5682no—
192C2—22no—
193C2#2258872611yescharacteristic
194C2#2258883212yes—
subgroupG/H
centerC22#17985876
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1558C2
socleC22×C71#386
prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenter#1558D71
hypocenterC71#9426
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
8. #17986349C4semidirect
103. C71#9426semidirect

Maximal subgroups

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

subgroupindex
#225987202
#225915212
#225868922
#225862062
C71×#15582
#942671