Finite Group Databasebeta

← #31895057

Subgroups — #31895057

Order 18176.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#22597442C211yesmaximal, characteristic
2#22587078C211yesmaximal, characteristic
3#22587025C212yesmaximal
4#22586230C212yesmaximal
5C71×#3139C211yesmaximal, characteristic
6#17986130—22no—
7Q8×C71⋊C8—22no—
8#17986060—22no—
9#17986052C2212yes—
10#17986023C2212yes—
11#17985745—22no—
12#17985745—22no—
13#17985745—22no—
14#533×C71C2211yescharacteristic
15C2×C4⋊Q8×C71C2211yescharacteristic
16#491×C71C414yes—
17#487×C71—22no—
18#419×C71—22no—
19C4×M16×C71C2211yescharacteristic
20C2×#11765849—22no—
21C4⋊(C71⋊C8)—24no—
22C4⋊(C71⋊C8)—22no—
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)D412yes—
25C4⋊(C71⋊C8)D412yes—
26C22⋊(C71⋊C8)—22no—
27#14556039—22no—
28C4×C71⋊C8—22no—
29C4×C71⋊C8—22no—
30C22×Q8×C71—21no—
31C2×M16×C71—21no—
32C4⋊Q8×C71—22no—
33C4⋊Q8×C71—21no—
34C4⋊Q8×C71C2×C412yes—
35C4⋊Q8×C71D412yes—
36#125×C71C2×C412yes—
37C22⋊Q8×C71—22no—
38C4×Q8×C71—22no—
39#117×C71D412yes—
40C2×C4⋊C4×C71—22no—
41C2×C42×C71C2311yescharacteristic
42#103×C71—28no—
43#97×C71C2×C412yes—
44C4×C8×C71—21no—
45#11765849—82no—
46C2×C71⋊C8—22no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—42no—
49C2×Q8×C71—22no—
50C2×Q8×C71—22no—
51C2×Q8×C71—44no—
52C2×Q8×C71—41no—
53C22×C4×C71—41no—
54C22×C4×C71C2×D411yescharacteristic
55C22×C4×C71#4112yes—
56M16×C71—42no—
57C2×C8×C71—22no—
58C4⋊C4×C71—24no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—24no—
61C4⋊C4×C71—42no—
62C4⋊C4×C71C22⋊C414yes—
63C22⋊C4×C71—22no—
64C42×C71—22no—
65C42×C71C22×C411yescharacteristic
66C42×C71C2×D412yes—
67C42×C71C2×D411yescharacteristic
68C23×C71#12311yescharacteristic
69Q8×C71—44no—
70Q8×C71—42no—
71Q8×C71—42no—
72Q8×C71—82no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—24no—
77C2×C4×C71—42no—
78C2×C4×C71—82no—
79C2×C4×C71C2×C22⋊C411yescharacteristic
80C2×C4×C71C4×D412yes—
81C2×C4×C71#12011yescharacteristic
82C2×C4×C71#12112yes—
83C8×C71—42no—
84C71⋊C8—42no—
85C71⋊C8—42no—
86C71⋊C8—82no—
87C22×C71—21no—
88C22×C71—22no—
89C22×C71#38611yescharacteristic
90C4×C71—21no—
91C4×C71—22no—
92C4×C71—44no—
93C4×C71—82no—
94C4×C71#45412yes—
95C4×C71#47112yes—
96C4×C71#47312yes—
97#9436—711nomaximal, Sylow
98C2×C71—41no—
99C2×C71#188811yescharacteristic
100C2×C71#190812yes—
101#3139D7111yescharacteristic
102#2986—711no—
103#1561—711no—
104#1549—712no—
105#1299—712no—
106C71#943611yescharacteristic, complemented, Sylow
107#533C2×D7111yescharacteristic
108C2×C4⋊Q8C2×D7111yescharacteristic
109#491Dic7114yes—
110#487—22no—
111#446—1422no—
112C8×Q8—1422no—
113#433—1422no—
114#431—712no—
115#423—712no—
116#419—22no—
117C4×M16C2×D7111yescharacteristic
118#332—1422no—
119#332—1422no—
120#332—1422no—
121C22×Q8—21no—
122C2×M16—21no—
123C2×M16—1422no—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8C2×Dic7112yes—
127C4⋊Q8C22⋊D7112yes—
128#125C2×Dic7112yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D7112yes—
132C2×C4⋊C4—22no—
133C2×C42C22×D7111yescharacteristic
134C4⋊C8—712no—
135C4⋊C8—712no—
136C4⋊C8—1424no—
137C4⋊C8—1422no—
138C4⋊C8—1422no—
139#103—28no—
140C22⋊C8—1422no—
141#97—1422no—
142#97C2×Dic7112yes—
143C4×C8—21no—
144C4×C8—1422no—
145C4×C8—1422no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—44no—
149C2×Q8—41no—
150C22×C4—41no—
151C22×C4C2×C22⋊D7111yescharacteristic
152C22×C4#1176586412yes—
153M16—42no—
154M16—5682no—
155C2×C8—22no—
156C2×C8—2842no—
157C2×C8—1422no—
158C2×C8—1422no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—24no—
162C4⋊C4—42no—
163C4⋊C4C22⋊Dic7114yes—
164C22⋊C4—22no—
165C42—22no—
166C42C22×Dic7111yescharacteristic
167C42D4×D7112yes—
168C42C2×C22⋊D7111yescharacteristic
169C23#1455610911yescharacteristic
170Q8—44no—
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—82no—
180C2×C4C2×C22⋊Dic7111yescharacteristic
181C2×C4D4×Dic7112yes—
182C2×C4#1455609611yescharacteristic
183C2×C4#1455610012yes—
184C8—42no—
185C8—2842no—
186C8—2842no—
187C8—5682no—
188C22—21no—
189C22—22no—
190C22#1798587911yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—82no—
195C4#1798617112yes—
196C4#1798626412yes—
197C4#1798627212yes—
198C2—41no—
199C2#2258874411yescharacteristic
200C2#2258885712yes—
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3139C2
socleC22×C71#386
prime radicalC23×C71#123
cosocleC2×C42×C71C23
hypercenter#3139D71
hypocenterC71#9436
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
106. C71#9436semidirect

Maximal subgroups

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

subgroupindex
#225974422
#225870782
#225870252
#225862302
C71×#31392
#943671