Finite Group Databasebeta

← #31893480

Subgroups — #31893480

Order 18176.

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

normal subgroups only

This group has 478 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#22594106C211yesmaximal, characteristic
2#22587013C212yesmaximal, complemented
3#22586654C211yesmaximal, characteristic
4#22586200C212yesmaximal, complemented
5C71×#2921C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7C2×#14556107—21no—
8#17986052C2212yes—
9#17985929—21no—
10#17985929C412yes—
11#17985924—21no—
12#17985924C412yes—
13#17985905—21no—
14#17985865C2211yescharacteristic
15#17985746—24no—
16#17985725—21no—
17#17985723—21no—
18C2×#117×C71C2211yescharacteristic
19#433×C71—24no—
20#423×C71C2212yes—
21#407×C71—21no—
22C4×M16×C71—21no—
23C22×Dic142—21no—
24#14556128—22no—
25#14556128—22no—
26#14556107—44no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71C2×C412yes—
30C2×Dic71⋊C4—22no—
31C2×C22⋊Dic71—22no—
32C4⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)—22no—
34C4×C71⋊C8—22no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C71⋊#95—21no—
38C71⋊#95C2×C412yes—
39C23×C4×C71D411yescharacteristic
40C2×M16×C71—22no—
41#117×C71—22no—
42#117×C71—22no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C22⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71C2311yescharacteristic
49C4⋊C8×C71—22no—
50C4⋊C8×C71D414yes—
51C22⋊C8×C71—24no—
52#97×C71—22no—
53C4×C8×C71—22no—
54C2×Dic142—41no—
55C2×Dic142—41no—
56C2×Dic142—42no—
57C2×Dic142—42no—
58C22×Dic71—22no—
59C22×Dic71—22no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—24no—
62C4⋊Dic71—42no—
63C4⋊Dic71—42no—
64Dic71⋊C4—44no—
65C22⋊Dic71—44no—
66C24×C71C22⋊C411yescharacteristic
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22×C411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77M16×C71—42no—
78M16×C71—42no—
79C2×C8×C71—24no—
80C4⋊C4×C71—22no—
81C4⋊C4×C71—22no—
82C22⋊C4×C71—22no—
83C22⋊C4×C71—22no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71C2×D412yes—
87C42×C71#4112yes—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—41no—
92C23×C71—41no—
93C23×C71C2×C22⋊C411yescharacteristic
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71C4×D412yes—
106C2×C4×C71#12011yescharacteristic
107C2×C4×C71#12112yes—
108C2×C4×C71#12311yescharacteristic
109C2×C4×C71#13512yes—
110C2×C4×C71AGL(1,ℤ/8)12yes—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71—42no—
116C2×Dic71—42no—
117Dic142—82no—
118Dic142—82no—
119C8×C71—44no—
120C71⋊C8—82no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—42no—
124C22×C71—41no—
125C22×C71—41no—
126C22×C71—41no—
127C22×C71—81no—
128C22×C71#38611yescharacteristic
129C22×C71#41611yescharacteristic
130C22×C71#41811yescharacteristic
131C4×C71—22no—
132C4×C71—22no—
133C4×C71—41no—
134C4×C71—41no—
135C4×C71—41no—
136C4×C71#44912yes—
137C4×C71#48212yes—
138Dic71—82no—
139Dic71—82no—
140#9196—711nomaximal, Sylow
141C2×C71—21no—
142C2×C71—21no—
143C2×C71—81no—
144C2×C71#187511yescharacteristic
145C2×C71#190911yescharacteristic
146C2×C71#191511yescharacteristic
147#2921D7111yescharacteristic
148#2621—711no—
149#1546—712no—
150#1458—711no—
151#1286—712no—
152C71#919611yescharacteristic, complemented, Sylow
153C2×C4⋊Q8—711no—
154C2×C22⋊Q8—1421no—
155C2×#117C2×D7111yescharacteristic
156#433—24no—
157#431—712no—
158#423C2×D7112yes—
159#407—21no—
160C4×M16—21no—
161#398—712no—
162#398—1421no—
163#397—712no—
164#397—1421no—
165#391—1421no—
166#382—711no—
167#332—1424no—
168#324—1421no—
169#323—1421no—
170C22×Q8—1421no—
171C23×C4C22⋊D7111yescharacteristic
172C2×M16—22no—
173C4⋊Q8—1422no—
174C4⋊Q8—1422no—
175C22⋊Q8—2844no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—712no—
179C2×C4⋊C4—1422no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4—1421no—
182C2×C4⋊C4C4×D7111yescharacteristic
183C2×C4⋊C4D28411yescharacteristic
184C2×C22⋊C4—1422no—
185C2×C22⋊C4C4×D7111yescharacteristic
186C2×C22⋊C4D28411yescharacteristic
187C2×C42C22⋊D7111yescharacteristic
188C2×C42C22×D7111yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—1422no—
191C4⋊C8C22⋊D7114yes—
192C22⋊C8—24no—
193C22⋊C8—1422no—
194#97—22no—
195C4×C8—22no—
196C4×C8—1422no—
197#95—712no—
198#95—1422no—
199#95—1422no—
200#95—1421no—
201C24(C2×D71)⋊C411yescharacteristic
202C2×Q8—2841no—
203C2×Q8—2841no—
204C2×Q8—2842no—
205C2×Q8—2842no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—1422no—
211C22×C4—1422no—
212C22×C4(C2×D71)⋊C411yescharacteristic
213C22×C4(C2×D71)⋊C411yescharacteristic
214C22×C4(C2×D71)⋊C411yescharacteristic
215C22×C4C2×C4×D7111yescharacteristic
216C22×C4C2×C22⋊D7111yescharacteristic
217C22×C4C2×D28411yescharacteristic
218M16—42no—
219M16—42no—
220C2×C8—24no—
221C2×C8—2842no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—2844no—
225C4⋊C4—2842no—
226C4⋊C4—2842no—
227C4⋊C4—1424no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—2844no—
231C42—21no—
232C42—21no—
233C42C2×C22⋊D7112yes—
234C42#1176586512yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×(C2×D71)⋊C411yescharacteristic
241Q8—5682no—
242Q8—5682no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—2842no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—2842no—
259C2×C4—2842no—
260C2×C4C4×C22⋊D7112yes—
261C2×C4#1455609711yescharacteristic
262C2×C4#1455610212yes—
263C2×C4#1455611111yescharacteristic
264C2×C4#1455614812yes—
265C2×C4#1455615312yes—
266C8—44no—
267C8—5682no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1798587811yescharacteristic
276C22#1798599311yescharacteristic
277C22#1798600311yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—5682no—
284C4—5682no—
285C4#1798614412yes—
286C4#1798632312yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2258866211yescharacteristic
291C2#2258886211yescharacteristic
292C2#2258889211yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2921C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2921D71
hypocenterC71#9196
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #22587013C2semidirect
4. #22586200C2semidirect
152. C71#9196semidirect

Maximal subgroups

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

subgroupindex
#225941062
#225870132
#225866542
#225862002
C71×#29212
#919671