Finite Group Databasebeta

← #31895316

Subgroups — #31895316

Order 18176.

454 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 115 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#22593702C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic
3#22586770C212yesmaximal
4#22586199C212yesmaximal
5C71×#2983C211yesmaximal, characteristic
6C2×C4×Dic142C2211yescharacteristic
7#17986023C2212yes—
8C2×#14556047—21no—
9#17985938C412yes—
10#17985919C412yes—
11#17985906—21no—
12#17985895—21no—
13#17985883—22no—
14#17985865C2211yescharacteristic
15#17985768—21no—
16#17985731—24no—
17C2×C4×Q8×C71C2211yescharacteristic
18#446×C71—24no—
19#432×C71—22no—
20#431×C71C2212yes—
21C22×Dic142C2×C411yescharacteristic
22C2×#11765849—22no—
23C4×Dic142—22no—
24C4×Dic142—22no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—21no—
29C2×C4×Dic71—21no—
30C4⋊(C71⋊C8)—22no—
31C4⋊(C71⋊C8)—22no—
32#14556047—42no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C71⋊#95C2×C412yes—
38C22×Q8×C71D411yescharacteristic
39C4×Q8×C71—22no—
40C4×Q8×C71—22no—
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C42×C71D412yes—
44C2×C42×C71C2311yescharacteristic
45C4⋊C8×C71—24no—
46C4⋊C8×C71—22no—
47C22⋊C8×C71—22no—
48#97×C71—24no—
49C4×C8×C71D414yes—
50C2×Dic142—22no—
51C2×Dic142—42no—
52C22×Dic71—21no—
53C22×Dic71—22no—
54C22×Dic71—21no—
55#11765849—42no—
56#11765849—42no—
57C2×C71⋊C8—42no—
58C4⋊Dic71—22no—
59C4⋊Dic71—41no—
60C4⋊Dic71—41no—
61Dic71⋊C4—44no—
62Dic71⋊C4—41no—
63Dic71⋊C4—41no—
64C4×Dic71—42no—
65C2×Q8×C71—22no—
66C2×Q8×C71—21no—
67C2×Q8×C71—21no—
68C2×Q8×C71—22no—
69C22×C4×C71C22⋊C412yes—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—24no—
75C4⋊C4×C71—24no—
76C4⋊C4×C71—22no—
77C42×C71—22no—
78C42×C71—22no—
79C42×C71C2×D412yes—
80C42×C71#4112yes—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—42no—
83Q8×C71—42no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
91C2×C4×C71C4×D412yes—
92C2×C4×C71#12011yescharacteristic
93C2×C4×C71#12112yes—
94C2×C4×C71#12311yescharacteristic
95C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C71#13712yes—
97C2×C4×C71#13711yescharacteristic
98C2×Dic71—22no—
99C2×Dic71—41no—
100C2×Dic71—41no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103C2×Dic71—42no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106Dic142—24no—
107Dic142—42no—
108C8×C71—44no—
109C71⋊C8—82no—
110C22×C71—21no—
111C22×C71—21no—
112C22×C71#38611yescharacteristic
113C22×C71#41711yescharacteristic
114C22×C71#41911yescharacteristic
115C4×C71—22no—
116C4×C71—21no—
117C4×C71—21no—
118C4×C71—42no—
119C4×C71—42no—
120C4×C71#46812yes—
121C4×C71#47012yes—
122Dic71—42no—
123Dic71—82no—
124Dic71—81no—
125#9488—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71#189111yescharacteristic
129C2×C71#189411yescharacteristic
130C2×C71#191211yescharacteristic
131#2983D7111yescharacteristic
132#2581—711no—
133#1801—711no—
134#1487—712no—
135#1286—712no—
136C71#948811yescharacteristic, complemented, Sylow
137C2×C4×Q8—711no—
138C2×C4×Q8C2×D7111yescharacteristic
139#446—24no—
140#432—22no—
141#431C2×D7112yes—
142#423—712no—
143C2×#101—1421no—
144#400—712no—
145#395—712no—
146#391—1421no—
147#389—1421no—
148#387—1422no—
149#382—711no—
150#343—1421no—
151Q8⋊C8—1424no—
152C22×Q8—711no—
153C22×Q8D28411yescharacteristic
154C2×M16—1422no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1422no—
158C4×Q8—1422no—
159C2×C4⋊C4—711no—
160C2×C4⋊C4—1422no—
161C2×C4⋊C4—1421no—
162C2×C4⋊C4—1421no—
163C2×C4⋊C4C4×D7112yes—
164C2×C4⋊C4D28411yescharacteristic
165C2×C42—1421no—
166C2×C42C22⋊D7112yes—
167C2×C42C22×D7111yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—22no—
170C4⋊C8—1422no—
171C4⋊C8—1422no—
172#101—2842no—
173C22⋊C8—22no—
174#97—24no—
175C4×C8C22⋊D7114yes—
176#95—712no—
177#95—1422no—
178#95—1422no—
179#95—1422no—
180#95—1421no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—2842no—
186C2×Q8—1422no—
187C22×C4—1422no—
188C22×C4—1421no—
189C22×C4—1421no—
190C22×C4(C2×D71)⋊C412yes—
191C22×C4(C2×D71)⋊C412yes—
192C22×C4C2×C4×D7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194C22×C4C2×D28411yescharacteristic
195M16—2842no—
196M16—2842no—
197C2×C8—24no—
198C2×C8—2842no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—2844no—
202C4⋊C4—2841no—
203C4⋊C4—2841no—
204C4⋊C4—2841no—
205C4⋊C4—2841no—
206C4⋊C4—1422no—
207C42—22no—
208C42—22no—
209C42—2842no—
210C42C2×C22⋊D7112yes—
211C42#1176586512yes—
212C23C2×(C2×D71)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—2842no—
216Q8—1424no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—2841no—
225C2×C4—2841no—
226C2×C4—2842no—
227C2×C4—2842no—
228C2×C4—2842no—
229C2×C4—2841no—
230C2×C4—2842no—
231C2×C4—1422no—
232C2×C4C4×C22⋊D7112yes—
233C2×C4#1455609711yescharacteristic
234C2×C4#1455610212yes—
235C2×C4#1455611111yescharacteristic
236C2×C4#1455615611yescharacteristic
237C2×C4#1455616112yes—
238C2×C4#1455616111yescharacteristic
239C8—44no—
240C8—5682no—
241C22—21no—
242C22—21no—
243C22#1798587811yescharacteristic
244C22#1798600111yescharacteristic
245C22#1798600911yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—2842no—
252C4—5682no—
253C4—5681no—
254C4#1798624512yes—
255C4#1798625712yes—
256C2—21no—
257C2—21no—
258C2#2258875811yescharacteristic
259C2#2258877311yescharacteristic
260C2#2258887711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2983C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2983D71
hypocenterC71#9488
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
136. C71#9488semidirect

Maximal subgroups

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

subgroupindex
#225937022
#225883362
#225867702
#225861992
C71×#29832
#948871