Finite Group Databasebeta

← #31895012

Subgroups — #31895012

Order 18176.

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

normal subgroups only

This group has 508 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, complemented
2#22587066C211yesmaximal, characteristic, complemented
3#22586887C212yesmaximal
4#22586205C212yesmaximal, complemented
5C71×#3152C211yesmaximal, characteristic
6#17986056C2212yes—
7#17985906—22no—
8#17985885—24no—
9#17985867C2212yes—
10#17985748—22no—
11#17985733—24no—
12C2×#137×C71—22no—
13#532×C71C2211yescharacteristic
14C2×C4⋊Q8×C71C2211yescharacteristic
15#497×C71—22no—
16#494×C71C412yescomplemented
17#492×C71C412yescomplemented
18C4×M16×C71C2211yescharacteristic
19C2×C4⋊Dic71—22no—
20C2×C4×Dic71—22no—
21C4⋊(C71⋊C8)—22no—
22C22⋊(C71⋊C8)—22no—
23#14556039—22no—
24C71⋊#95—24no—
25C71⋊#95—22no—
26C71⋊#95—22no—
27C71⋊#95D414yes—
28C2×#41×C71—21no—
29C22×Q8×C71D412yes—
30#137×C71—44no—
31#137×C71—44no—
32C2×Q16×C71—24no—
33C2×SD16×C71—24no—
34C2×M16×C71—21no—
35C4⋊Q8×C71—22no—
36C4⋊Q8×C71C2×C411yescharacteristic
37C4⋊Q8×C71C2×C412yes—
38C4⋊Q8×C71D412yes—
39#127×C71C2×C411yescharacteristic
40#124×C71—21no—
41#121×C71—22no—
42C4×D4×C71—22no—
43C2×C4⋊C4×C71—22no—
44C2×C42×C71C2311yescharacteristic
45#97×C71—21no—
46C4×C8×C71C2×C412yes—
47C22×Dic71—24no—
48C2×C71⋊C8—42no—
49C4⋊Dic71—82no—
50C4×Dic71—82no—
51#41×C71—42no—
52#41×C71—42no—
53C2×Q8×C71—22no—
54C2×Q8×C71—21no—
55C2×Q8×C71—42no—
56C2×Q8×C71—42no—
57C2×Q8×C71C22⋊C414yes—
58C2×D4×C71—21no—
59C2×D4×C71—22no—
60C2×D4×C71—41no—
61C22×C4×C71—22no—
62C22×C4×C71—41no—
63C22×C4×C71C2×D412yes—
64C22×C4×C71C2×D411yescharacteristic
65Q16×C71—48no—
66SD16×C71—48no—
67M16×C71—24no—
68C2×C8×C71—22no—
69C4⋊C4×C71—24no—
70C4⋊C4×C71—24no—
71C4⋊C4×C71—22no—
72C22⋊C4×C71—42no—
73C42×C71C22×C411yescharacteristic
74C42×C71C2×D411yescharacteristic
75C42×C71#4112yes—
76C23×C71—41no—
77C23×C71#12011yescharacteristic
78Q8×C71—44no—
79Q8×C71—42no—
80Q8×C71—82no—
81D4×C71—42no—
82D4×C71—82no—
83D4×C71—82no—
84C2×C4×C71—22no—
85C2×C4×C71—24no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71—41no—
90C2×C4×C71—81no—
91C2×C4×C71—81no—
92C2×C4×C71C2×C22⋊C411yescharacteristic
93C2×C4×C71C4×D412yes—
94C2×C4×C71#12112yes—
95C2×C4×C71#12311yescharacteristic
96C2×Dic71—48no—
97C2×Dic71—44no—
98C8×C71—24no—
99C71⋊C8—82no—
100C22×C71—22no—
101C22×C71—81no—
102C22×C71—82no—
103C22×C71#38611yescharacteristic
104C22×C71#45312yes—
105C4×C71—21no—
106C4×C71—21no—
107C4×C71—22no—
108C4×C71—22no—
109C4×C71—44no—
110C4×C71—81no—
111Dic71—84no—
112#9429—711nomaximal, Sylow
113C2×C71—22no—
114C2×C71—161no—
115C2×C71#188711yescharacteristic
116C2×C71#190512yes—
117#3152D7111yescharacteristic
118#2381—711no—
119#1558—711no—
120#1514—712no—
121#1288—712no—
122C71#942911yescharacteristic, complemented, Sylow
123C2×#137—22no—
124#532C2×D7111yescharacteristic
125C2×C4⋊Q8C2×D7111yescharacteristic
126#497—22no—
127#494Dic7112yescomplemented
128#492Dic7112yescomplemented
129#432—712no—
130C4×M16C2×D7111yescharacteristic
131#391—1422no—
132#387—1424no—
133#383—712no—
134#333—1422no—
135#327—1424no—
136C2×#41—21no—
137C22×Q8C22⋊D7112yes—
138#137—44no—
139#137—44no—
140C2×Q16—24no—
141C2×SD16—24no—
142C2×M16—21no—
143C4⋊Q8—22no—
144C4⋊Q8C2×Dic7111yescharacteristic
145C4⋊Q8C2×Dic7112yes—
146C4⋊Q8C22⋊D7112yes—
147#127C2×Dic7111yescharacteristic
148#124—21no—
149#121—22no—
150C4×D4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—1422no—
153C2×C42—1422no—
154C2×C42C22×D7111yescharacteristic
155C4⋊C8—1422no—
156C22⋊C8—1422no—
157#97—21no—
158#97—1422no—
159C4×C8C2×Dic7112yes—
160#95—714no—
161#95—1424no—
162#95—1422no—
163#95—1422no—
164#41—42no—
165#41—42no—
166C2×Q8—22no—
167C2×Q8—21no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8C22⋊Dic7114yes—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—1424no—
177C22×C4D4×D7112yes—
178C22×C4C2×C22⋊D7111yescharacteristic
179Q16—48no—
180SD16—48no—
181M16—24no—
182C2×C8—22no—
183C2×C8—2842no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—5682no—
188C22⋊C4—42no—
189C42—5682no—
190C42C22×Dic7111yescharacteristic
191C42C2×C22⋊D7111yescharacteristic
192C42#1176586412yes—
193C23—41no—
194C23#1455609611yescharacteristic
195Q8—44no—
196Q8—42no—
197Q8—82no—
198D4—42no—
199D4—82no—
200D4—82no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—2848no—
210C2×C4—2844no—
211C2×C4C2×C22⋊Dic7111yescharacteristic
212C2×C4D4×Dic7112yes—
213C2×C4#1455610012yes—
214C2×C4#1455610911yescharacteristic
215C8—24no—
216C8—5682no—
217C22—22no—
218C22—81no—
219C22—82no—
220C22#1798587911yescharacteristic
221C22#1798616612yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—81no—
228C4—5684no—
229C2—22no—
230C2—161no—
231C2#2258873911yescharacteristic
232C2#2258883612yes—
subgroupG/H
centerC22#17985879
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#3152C2
socleC22×C71#386
prime radical#532×C71C22
cosocleC2×C42×C71C23
hypercenter#3152D71
hypocenterC71#9429
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. #22591521C2semidirect
2. #22587066C2semidirect
4. #22586205C2semidirect
16. #494×C71C4semidirect
17. #492×C71C4semidirect
122. C71#9429semidirect
127. #494Dic71semidirect
128. #492Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915212
#225870662
#225868872
#225862052
C71×#31522
#942971