Finite Group Databasebeta

← #31895052

Subgroups — #31895052

Order 18176.

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

normal subgroups only

This group has 504 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#22597407C211yesmaximal, characteristic
2#22587075C211yesmaximal, characteristic
3#22587025C212yesmaximal, complemented
4#22586230C212yesmaximal, complemented
5C71×#3152C211yesmaximal, characteristic
6#17986077—22no—
7D4×C71⋊C8—22no—
8#17986052C2212yes—
9#17986023C2212yes—
10#17985966—22no—
11#17985748—21no—
12#17985745—22no—
13#17985745—22no—
14#17985743—21no—
15C2×#137×C71—22no—
16#532×C71C2211yescharacteristic
17C2×C4⋊Q8×C71C2211yescharacteristic
18#497×C71C414yes—
19#494×C71—21no—
20#492×C71—21no—
21C4×M16×C71C2211yescharacteristic
22C2×#11765849—22no—
23C2×#11765849—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)—22no—
27C4⋊(C71⋊C8)D412yes—
28C4⋊(C71⋊C8)D412yes—
29C22⋊(C71⋊C8)—24no—
30C22⋊(C71⋊C8)—22no—
31C4×C71⋊C8—22no—
32C2×#41×C71D412yes—
33C22×Q8×C71—21no—
34#137×C71—44no—
35#137×C71—44no—
36C2×Q16×C71—24no—
37C2×SD16×C71—24no—
38C2×M16×C71—21no—
39C4⋊Q8×C71—22no—
40C4⋊Q8×C71—21no—
41C4⋊Q8×C71C2×C412yes—
42C4⋊Q8×C71D411yescharacteristic
43#127×C71D411yescharacteristic
44#124×C71C2×C412yes—
45#121×C71—22no—
46C4×D4×C71—22no—
47C2×C4⋊C4×C71—22no—
48C2×C42×C71C2311yescharacteristic
49#97×C71C2×C412yes—
50C4×C8×C71—21no—
51#11765849—82no—
52#11765849—82no—
53C2×C71⋊C8—22no—
54C2×C71⋊C8—22no—
55C2×C71⋊C8—42no—
56C2×C71⋊C8—82no—
57#41×C71—42no—
58#41×C71—42no—
59C2×Q8×C71—22no—
60C2×Q8×C71—42no—
61C2×Q8×C71—42no—
62C2×Q8×C71—41no—
63C2×Q8×C71C22⋊C412yes—
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71C22⋊C412yes—
67C22×C4×C71—22no—
68C22×C4×C71—41no—
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71#4112yes—
71Q16×C71—48no—
72SD16×C71—48no—
73M16×C71—22no—
74M16×C71—22no—
75C2×C8×C71—22no—
76C4⋊C4×C71—22no—
77C4⋊C4×C71—24no—
78C4⋊C4×C71—42no—
79C22⋊C4×C71—24no—
80C42×C71C22×C411yescharacteristic
81C42×C71C2×D412yes—
82C42×C71C2×D411yescharacteristic
83C23×C71—22no—
84C23×C71#12311yescharacteristic
85Q8×C71—42no—
86Q8×C71—44no—
87Q8×C71—81no—
88Q8×C71—81no—
89D4×C71—42no—
90D4×C71—42no—
91D4×C71—42no—
92D4×C71—82no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—82no—
102C2×C4×C71C2×C22⋊C411yescharacteristic
103C2×C4×C71C4×D412yes—
104C2×C4×C71#12011yescharacteristic
105C2×C4×C71#12112yes—
106C8×C71—24no—
107C71⋊C8—42no—
108C71⋊C8—42no—
109C71⋊C8—82no—
110C22×C71—21no—
111C22×C71—22no—
112C22×C71—42no—
113C22×C71—44no—
114C22×C71#38611yescharacteristic
115C4×C71—21no—
116C4×C71—22no—
117C4×C71—42no—
118C4×C71—82no—
119C4×C71#45311yescharacteristic
120C4×C71#45311yescharacteristic
121C4×C71#47112yes—
122C4×C71#47312yes—
123#9435—711nomaximal, Sylow
124C2×C71—41no—
125C2×C71—82no—
126C2×C71#188711yescharacteristic
127C2×C71#190812yes—
128#3152D7111yescharacteristic
129#2979—711no—
130#1560—711no—
131#1549—712no—
132#1299—712no—
133C71#943511yescharacteristic, complemented, Sylow
134C2×#137—22no—
135#532C2×D7111yescharacteristic
136C2×C4⋊Q8C2×D7111yescharacteristic
137#497Dic7114yes—
138#494—21no—
139#492—21no—
140#436—1422no—
141C8×D4—1422no—
142#431—712no—
143#423—712no—
144#408—1422no—
145C4×M16C2×D7111yescharacteristic
146#333—1421no—
147#332—1422no—
148#332—1422no—
149#331—1421no—
150C2×#41C22⋊D7112yes—
151C22×Q8—21no—
152#137—44no—
153#137—44no—
154C2×Q16—24no—
155C2×SD16—24no—
156C2×M16—21no—
157C2×M16—1422no—
158C2×M16—1422no—
159C22×C8—1422no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8C2×Dic7112yes—
163C4⋊Q8C22⋊D7111yescharacteristic
164#127C22⋊D7111yescharacteristic
165#124C2×Dic7112yes—
166#121—22no—
167C4×D4—22no—
168C2×C4⋊C4—22no—
169C2×C42C22×D7111yescharacteristic
170C4⋊C8—712no—
171C4⋊C8—712no—
172C4⋊C8—1422no—
173C4⋊C8—1422no—
174C22⋊C8—1424no—
175C22⋊C8—1422no—
176#97C2×Dic7112yes—
177C4×C8—21no—
178C4×C8—1422no—
179#41—42no—
180#41—42no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8C22⋊Dic7112yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4C22⋊Dic7112yes—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4C2×C22⋊D7111yescharacteristic
192C22×C4#1176586412yes—
193Q16—48no—
194SD16—48no—
195M16—22no—
196M16—22no—
197M16—5682no—
198M16—5682no—
199C2×C8—22no—
200C2×C8—2842no—
201C2×C8—5682no—
202C2×C8—1422no—
203C2×C8—1422no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—42no—
207C22⋊C4—24no—
208C42C22×Dic7111yescharacteristic
209C42D4×D7112yes—
210C42C2×C22⋊D7111yescharacteristic
211C23—22no—
212C23#1455610911yescharacteristic
213Q8—42no—
214Q8—44no—
215Q8—81no—
216Q8—81no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—82no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4C2×C22⋊Dic7111yescharacteristic
231C2×C4D4×Dic7112yes—
232C2×C4#1455609611yescharacteristic
233C2×C4#1455610012yes—
234C8—24no—
235C8—2842no—
236C8—2842no—
237C8—5682no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—44no—
242C22#1798587911yescharacteristic
243C4—21no—
244C4—22no—
245C4—42no—
246C4—82no—
247C4#1798616611yescharacteristic
248C4#1798616611yescharacteristic
249C4#1798626412yes—
250C4#1798627212yes—
251C2—41no—
252C2—82no—
253C2#2258873911yescharacteristic
254C2#2258885712yes—
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#9435
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
3. #22587025C2semidirect
4. #22586230C2semidirect
133. C71#9435semidirect

Maximal subgroups

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

subgroupindex
#225974072
#225870752
#225870252
#225862302
C71×#31522
#943571