Finite Group Databasebeta

← #31901548

Subgroups — #31901548

Order 18176.

344 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 81 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#22588459C211yesmaximal, characteristic
2#22587040C212yesmaximal
3#22586852C212yesmaximal
4#22586210C211yesmaximal, characteristic
5C71×#3249C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×C4⋊(C71⋊C8)C2211yescharacteristic
8#17985845C2211yescharacteristic
9#17985753—22no—
10#17985745—22no—
11#17985745—22no—
12#17985736—24no—
13#533×C71C2212yes—
14#477×C71—22no—
15#467×C71—22no—
16#462×C71—22no—
17C2×C4⋊C8×C71C2211yescharacteristic
18C22×C71⋊C8—21no—
19C2×C4×Dic71—21no—
20C4⋊(C71⋊C8)—22no—
21C4⋊(C71⋊C8)—22no—
22C22⋊(C71⋊C8)—22no—
23#14556039—22no—
24C71⋊#95—21no—
25C71⋊#95D412yes—
26C2×Q16×C71—22no—
27C22×C8×C71D412yes—
28C4⋊Q8×C71D412yes—
29C4⋊Q8×C71D412yes—
30#125×C71—22no—
31C22⋊Q8×C71—22no—
32C22⋊Q8×C71—22no—
33C4×Q8×C71—22no—
34C4×Q8×C71—22no—
35#117×C71—22no—
36C2×C42×C71C2311yescharacteristic
37#107×C71—22no—
38C4⋊C8×C71—22no—
39#103×C71—24no—
40C22×Dic71—22no—
41C2×C71⋊C8—42no—
42C2×C71⋊C8—41no—
43C2×C71⋊C8—42no—
44C4×Dic71—41no—
45C4×Dic71—41no—
46C2×Q8×C71—22no—
47C2×Q8×C71—22no—
48C22×C4×C71C2×D411yescharacteristic
49C22×C4×C71C2×D411yescharacteristic
50C22×C4×C71#4111yescharacteristic
51Q16×C71—82no—
52C2×C8×C71—22no—
53C2×C8×C71—42no—
54C4⋊C4×C71—22no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—42no—
59C4⋊C4×C71—42no—
60C22⋊C4×C71—42no—
61C42×C71—42no—
62C42×C71C2×D412yes—
63C42×C71#4112yes—
64C23×C71#12111yescharacteristic
65Q8×C71—42no—
66Q8×C71—42no—
67Q8×C71—82no—
68C2×C4×C71—22no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—41no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71#12011yescharacteristic
75C2×C4×C71#12112yes—
76C2×C4×C71#12312yes—
77C2×C4×C71#12411yescharacteristic
78C2×Dic71—42no—
79C2×Dic71—42no—
80C2×Dic71—42no—
81C8×C71—42no—
82C71⋊C8—82no—
83C71⋊C8—81no—
84C22×C71—21no—
85C22×C71—21no—
86C22×C71#39411yescharacteristic
87C22×C71#47312yes—
88C4×C71—22no—
89C4×C71—21no—
90C4×C71—22no—
91C4×C71—82no—
92C4×C71—82no—
93C4×C71#45411yescharacteristic
94C4×C71#45511yescharacteristic
95Dic71—82no—
96#10517—711nomaximal, Sylow
97C2×C71—22no—
98C2×C71#204011yescharacteristic
99C2×C71#204711yescharacteristic
100C2×C71#205311yescharacteristic
101#3249D7111yescharacteristic
102#1832—711no—
103#1552—712no—
104#1506—712no—
105#1290—711no—
106C71#1051711yescharacteristic, complemented, Sylow
107#533C2×D7112yes—
108#477—22no—
109#467—22no—
110#462—22no—
111#432—712no—
112C2×C4⋊C8—711no—
113C2×C4⋊C8C2×D7111yescharacteristic
114#376—711no—
115#336—1422no—
116#332—1422no—
117#332—1422no—
118#328—1424no—
119C2×Q16—22no—
120C22×C8—1421no—
121C22×C8C22⋊D7112yes—
122C4⋊Q8C22⋊D7112yes—
123C4⋊Q8C22⋊D7112yes—
124#125—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129#117—22no—
130C2×C42—1421no—
131C2×C42C22×D7111yescharacteristic
132#107—22no—
133C4⋊C8—22no—
134C4⋊C8—1422no—
135C4⋊C8—1422no—
136#103—24no—
137C22⋊C8—1422no—
138#97—1422no—
139#95—712no—
140#95—1421no—
141C2×Q8—22no—
142C2×Q8—22no—
143C22×C4—1422no—
144C22×C4D4×D7111yescharacteristic
145C22×C4C2×C22⋊D7111yescharacteristic
146C22×C4#1176586411yescharacteristic
147Q16—82no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—2842no—
151C2×C8—2841no—
152C2×C8—2842no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C22⋊C4—42no—
160C42—42no—
161C42—2841no—
162C42—2841no—
163C42C2×C22⋊D7112yes—
164C42#1176586412yes—
165C23#1455609911yescharacteristic
166Q8—42no—
167Q8—42no—
168Q8—82no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—2842no—
176C2×C4—2842no—
177C2×C4—2842no—
178C2×C4#1455609711yescharacteristic
179C2×C4#1455610012yes—
180C2×C4#1455610912yes—
181C2×C4#1455611511yescharacteristic
182C8—42no—
183C8—5682no—
184C8—5681no—
185C22—21no—
186C22—21no—
187C22#1798591311yescharacteristic
188C22#1798627112yes—
189C4—21no—
190C4—22no—
191C4—22no—
192C4—82no—
193C4—82no—
194C4—5682no—
195C4#1798617211yescharacteristic
196C4#1798617711yescharacteristic
197C2—22no—
198C2#2258956611yescharacteristic
199C2#2258960511yescharacteristic
200C2#2258964311yescharacteristic
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3249C2
socleC22×C71#394
prime radicalC23×C71#121
hypercenter#3249D71
hypocenterC71#10517
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
106. C71#10517semidirect

Maximal subgroups

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

subgroupindex
#225884592
#225870402
#225868522
#225862102
C71×#32492
#1051771