Finite Group Databasebeta

← #31894995

Subgroups — #31894995

Order 18176.

422 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 103 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#22591521C211yesmaximal, characteristic
2#22587066C211yesmaximal, characteristic
3#22586891C212yesmaximal
4#22586205C212yesmaximal
5C71×#3139C211yesmaximal, characteristic
6#17986056C2212yes—
7#17985906—22no—
8#17985885—24no—
9#17985867C2212yes—
10#17985748—22no—
11#17985736—24no—
12#533×C71C2211yescharacteristic
13C2×C4⋊Q8×C71C2211yescharacteristic
14#491×C71—22no—
15#487×C71C414yescomplemented
16#419×C71—22no—
17C4×M16×C71C2211yescharacteristic
18C2×C4⋊Dic71—22no—
19C2×C4×Dic71—22no—
20C4⋊(C71⋊C8)—22no—
21C22⋊(C71⋊C8)—22no—
22#14556039—22no—
23C71⋊#95—24no—
24C71⋊#95—22no—
25C71⋊#95—22no—
26C71⋊#95D414yes—
27C22×Q8×C71D412yes—
28C2×M16×C71—21no—
29C4⋊Q8×C71—22no—
30C4⋊Q8×C71C2×C412yes—
31C4⋊Q8×C71C2×C412yes—
32C4⋊Q8×C71D412yes—
33#125×C71—21no—
34C22⋊Q8×C71—22no—
35C4×Q8×C71—22no—
36#117×C71—21no—
37C2×C4⋊C4×C71—22no—
38C2×C42×C71C2311yescharacteristic
39#103×C71—28no—
40#97×C71—21no—
41C4×C8×C71C2×C412yes—
42C22×Dic71—24no—
43C2×C71⋊C8—42no—
44C4⋊Dic71—82no—
45C4×Dic71—82no—
46C2×Q8×C71—22no—
47C2×Q8×C71—22no—
48C2×Q8×C71—42no—
49C2×Q8×C71—42no—
50C2×Q8×C71C22⋊C414yes—
51C22×C4×C71—22no—
52C22×C4×C71C2×D412yes—
53C22×C4×C71C2×D411yescharacteristic
54M16×C71—42no—
55C2×C8×C71—22no—
56C4⋊C4×C71—24no—
57C4⋊C4×C71—24no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—42no—
61C22⋊C4×C71—41no—
62C42×C71—41no—
63C42×C71C22×C411yescharacteristic
64C42×C71C2×D411yescharacteristic
65C42×C71#4112yes—
66C23×C71#12011yescharacteristic
67Q8×C71—44no—
68Q8×C71—82no—
69Q8×C71—82no—
70C2×C4×C71—22no—
71C2×C4×C71—24no—
72C2×C4×C71—42no—
73C2×C4×C71—42no—
74C2×C4×C71—42no—
75C2×C4×C71—42no—
76C2×C4×C71C2×C22⋊C411yescharacteristic
77C2×C4×C71C4×D412yes—
78C2×C4×C71#12112yes—
79C2×C4×C71#12311yescharacteristic
80C2×Dic71—48no—
81C2×Dic71—44no—
82C8×C71—42no—
83C71⋊C8—82no—
84C22×C71—22no—
85C22×C71#38611yescharacteristic
86C22×C71#45512yes—
87C4×C71—21no—
88C4×C71—21no—
89C4×C71—22no—
90C4×C71—22no—
91C4×C71—44no—
92C4×C71—82no—
93Dic71—84no—
94#9426—711nomaximal, Sylow
95C2×C71—22no—
96C2×C71#188511yescharacteristic
97C2×C71#190512yes—
98#3139D7111yescharacteristic
99#2381—711no—
100#1558—711no—
101#1515—712no—
102#1288—712no—
103C71#942611yescharacteristic, complemented, Sylow
104#533C2×D7111yescharacteristic
105C2×C4⋊Q8C2×D7111yescharacteristic
106#491—22no—
107#487Dic7114yescomplemented
108#432—712no—
109#419—22no—
110C4×M16C2×D7111yescharacteristic
111#391—1422no—
112#387—1424no—
113#383—712no—
114#333—1422no—
115#328—1424no—
116C22×Q8C22⋊D7112yes—
117C2×M16—21no—
118C4⋊Q8—22no—
119C4⋊Q8C2×Dic7112yes—
120C4⋊Q8C2×Dic7112yes—
121C4⋊Q8C22⋊D7112yes—
122#125—21no—
123C22⋊Q8—22no—
124C4×Q8—22no—
125#117—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—1422no—
128C2×C42—1422no—
129C2×C42C22×D7111yescharacteristic
130C4⋊C8—1422no—
131#103—28no—
132C22⋊C8—1422no—
133#97—21no—
134#97—1422no—
135C4×C8C2×Dic7112yes—
136#95—714no—
137#95—1424no—
138#95—1422no—
139#95—1422no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C22⋊Dic7114yes—
145C22×C4—22no—
146C22×C4—1424no—
147C22×C4D4×D7112yes—
148C22×C4C2×C22⋊D7111yescharacteristic
149M16—42no—
150C2×C8—22no—
151C2×C8—2842no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—5682no—
158C22⋊C4—41no—
159C42—41no—
160C42—5682no—
161C42C22×Dic7111yescharacteristic
162C42C2×C22⋊D7111yescharacteristic
163C42#1176586412yes—
164C23#1455609611yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—82no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—2848no—
175C2×C4—2844no—
176C2×C4C2×C22⋊Dic7111yescharacteristic
177C2×C4D4×Dic7112yes—
178C2×C4#1455610012yes—
179C2×C4#1455610911yescharacteristic
180C8—42no—
181C8—5682no—
182C22—22no—
183C22#1798587911yescharacteristic
184C22#1798617612yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—82no—
191C4—5684no—
192C2—22no—
193C2#2258872911yescharacteristic
194C2#2258883612yes—
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3139C2
socleC22×C71#386
prime radicalC23×C71#120
cosocleC2×C42×C71C23
hypercenter#3139D71
hypocenterC71#9426
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
15. #487×C71C4semidirect
103. C71#9426semidirect
107. #487Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915212
#225870662
#225868912
#225862052
C71×#31392
#942671