Finite Group Databasebeta

← #31901543

Subgroups — #31901543

Order 18176.

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

normal subgroups only

This group has 364 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#22588493C211yesmaximal, characteristic
2#22587040C212yesmaximal
3#22586852C212yesmaximal
4#22586210C211yesmaximal, characteristic
5C71×#3249C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×C4⋊(C71⋊C8)C2211yescharacteristic
8C2×Dic71⋊C8—22no—
9#17985845C2211yescharacteristic
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—
22Dic71⋊C8—44no—
23C22⋊(C71⋊C8)—22no—
24#14556039—22no—
25C71⋊#95—21no—
26C71⋊#95D412yes—
27C2×Q16×C71—22no—
28C22×C8×C71D412yes—
29C4⋊Q8×C71D412yes—
30C4⋊Q8×C71D412yes—
31#125×C71—22no—
32C22⋊Q8×C71—22no—
33C22⋊Q8×C71—22no—
34C4×Q8×C71—22no—
35C4×Q8×C71—22no—
36#117×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#107×C71—22no—
39C4⋊C8×C71—22no—
40#103×C71—24no—
41C22×Dic71—22no—
42C2×C71⋊C8—42no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—42no—
45C4×Dic71—22no—
46C4×Dic71—22no—
47C2×Q8×C71—22no—
48C2×Q8×C71—22no—
49C22×C4×C71C2×D411yescharacteristic
50C22×C4×C71C2×D411yescharacteristic
51C22×C4×C71#4111yescharacteristic
52Q16×C71—82no—
53C2×C8×C71—22no—
54C2×C8×C71—42no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—42no—
60C4⋊C4×C71—42no—
61C22⋊C4×C71—42no—
62C42×C71—42no—
63C42×C71C2×D412yes—
64C42×C71#4112yes—
65C23×C71#12111yescharacteristic
66Q8×C71—42no—
67Q8×C71—42no—
68Q8×C71—82no—
69C2×C4×C71—22no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—41no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71#12011yescharacteristic
76C2×C4×C71#12112yes—
77C2×C4×C71#12312yes—
78C2×C4×C71#12411yescharacteristic
79C2×Dic71—24no—
80C2×Dic71—42no—
81C2×Dic71—42no—
82C8×C71—42no—
83C71⋊C8—82no—
84C71⋊C8—81no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71#39411yescharacteristic
88C22×C71#47312yes—
89C4×C71—22no—
90C4×C71—21no—
91C4×C71—22no—
92C4×C71—82no—
93C4×C71—82no—
94C4×C71#45311yescharacteristic
95C4×C71#45611yescharacteristic
96Dic71—44no—
97#10516—711nomaximal, Sylow
98C2×C71—22no—
99C2×C71#203911yescharacteristic
100C2×C71#204711yescharacteristic
101C2×C71#205311yescharacteristic
102#3249D7111yescharacteristic
103#1843—711no—
104#1552—712no—
105#1506—712no—
106#1290—711no—
107C71#1051611yescharacteristic, complemented, Sylow
108#533C2×D7112yes—
109#477—22no—
110#467—22no—
111#462—22no—
112#432—712no—
113C2×C4⋊C8—711no—
114C2×C4⋊C8—1422no—
115C2×C4⋊C8C2×D7111yescharacteristic
116#376—711no—
117#332—1422no—
118#332—1422no—
119#328—1424no—
120C2×Q16—22no—
121C22×C8—1421no—
122C22×C8C22⋊D7112yes—
123C4⋊Q8C22⋊D7112yes—
124C4⋊Q8C22⋊D7112yes—
125#125—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130#117—22no—
131C2×C42—1421no—
132C2×C42C22×D7111yescharacteristic
133#107—22no—
134C4⋊C8—22no—
135C4⋊C8—2844no—
136C4⋊C8—1422no—
137C4⋊C8—1422no—
138#103—24no—
139C22⋊C8—1422no—
140#97—1422no—
141#95—712no—
142#95—1421no—
143C2×Q8—22no—
144C2×Q8—22no—
145C22×C4—1422no—
146C22×C4D4×D7111yescharacteristic
147C22×C4C2×C22⋊D7111yescharacteristic
148C22×C4#1176586411yescharacteristic
149Q16—82no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—2842no—
153C2×C8—2841no—
154C2×C8—2842no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C22⋊C4—42no—
162C42—42no—
163C42—1422no—
164C42—1422no—
165C42C2×C22⋊D7112yes—
166C42#1176586412yes—
167C23#1455609911yescharacteristic
168Q8—42no—
169Q8—42no—
170Q8—82no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—2842no—
178C2×C4—2842no—
179C2×C4—1424no—
180C2×C4#1455609711yescharacteristic
181C2×C4#1455610012yes—
182C2×C4#1455610912yes—
183C2×C4#1455611511yescharacteristic
184C8—42no—
185C8—5682no—
186C8—5681no—
187C22—21no—
188C22—21no—
189C22#1798591311yescharacteristic
190C22#1798627112yes—
191C4—21no—
192C4—22no—
193C4—22no—
194C4—82no—
195C4—82no—
196C4—2844no—
197C4#1798616711yescharacteristic
198C4#1798618211yescharacteristic
199C2—22no—
200C2#2258956111yescharacteristic
201C2#2258960511yescharacteristic
202C2#2258964311yescharacteristic
subgroupG/H
centerC22#17985913
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#3249C2
socleC22×C71#394
prime radicalC23×C71#121
hypercenter#3249D71
hypocenterC71#10516
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
107. C71#10516semidirect

Maximal subgroups

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

subgroupindex
#225884932
#225870402
#225868522
#225862102
C71×#32492
#1051671