Finite Group Databasebeta

← #31890905

Subgroups — #31890905

Order 18176.

404 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 107 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#22590838C211yesmaximal, characteristic
2#22586654C211yesmaximal, characteristic
3#22586172C212yesmaximal, complemented
4#22586168C212yesmaximal, complemented
5C71×#1458C211yesmaximal, characteristic
6#17986052C2212yes—
7#17985892—21no—
8#17985892—21no—
9#17985885—21no—
10#17985885—21no—
11C4×C4⋊Dic71C411yescharacteristic
12C4×C4⋊Dic71C411yescharacteristic
13C4×C22⋊Dic71C411yescharacteristic
14C4×C22⋊Dic71C411yescharacteristic
15#17985845C2211yescharacteristic
16C42×Dic71C2211yescharacteristic
17#17985725—21no—
18#17985723—21no—
19C2×#117×C71C2211yescharacteristic
20#431×C71C2212yes—
21#324×C71—21no—
22#323×C71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C22⋊Dic71—21no—
26C2×C22⋊Dic71—21no—
27C2×C4×Dic71—21no—
28C2×C4×Dic71—21no—
29C2×C4×Dic71C2×C411yescharacteristic
30C2×C4×Dic71C2×C411yescharacteristic
31C2×C4×Dic71C2×C411yescharacteristic
32C2×C4×Dic71C2×C411yescharacteristic
33C4⋊(C71⋊C8)—22no—
34C22⋊(C71⋊C8)—22no—
35C4×C71⋊C8—22no—
36C71⋊#95—21no—
37C71⋊#95—21no—
38C71⋊#95—21no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C23×C4×C71C2×C411yescharacteristic
43#117×C71—22no—
44#117×C71—22no—
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71C2311yescharacteristic
51C4⋊C8×C71—22no—
52C22⋊C8×C71—22no—
53C4×C8×C71—22no—
54C22×Dic71—21no—
55C22×Dic71—21no—
56C22×Dic71—21no—
57C22×Dic71—21no—
58C2×C71⋊C8—42no—
59C4⋊Dic71—42no—
60C4⋊Dic71—42no—
61C22⋊Dic71—42no—
62C22⋊Dic71—42no—
63C4×Dic71—22no—
64C4×Dic71—22no—
65C4×Dic71—22no—
66C4×Dic71—22no—
67C4×Dic71—42no—
68C4×Dic71—42no—
69C24×C71C22⋊C411yescharacteristic
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C2×C8×C71—42no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—22no—
83C22⋊C4×C71—22no—
84C22⋊C4×C71—22no—
85C42×C71—21no—
86C42×C71—21no—
87C42×C71#4112yes—
88C42×C71#4112yes—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71—41no—
93C23×C71—41no—
94C23×C71C2×C22⋊C411yescharacteristic
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71#11711yescharacteristic
107C2×C4×C71#11711yescharacteristic
108C2×C4×C71#12312yes—
109C2×C4×C71#12312yes—
110C2×C4×C71#13112yes—
111C2×C4×C71#13112yes—
112C2×Dic71—22no—
113C2×Dic71—22no—
114C2×Dic71—41no—
115C2×Dic71—41no—
116C2×Dic71—41no—
117C2×Dic71—41no—
118C2×Dic71—41no—
119C2×Dic71—41no—
120C2×Dic71—41no—
121C2×Dic71—41no—
122C2×Dic71—41no—
123C2×Dic71—41no—
124C8×C71—82no—
125C71⋊C8—82no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—41no—
129C22×C71—41no—
130C22×C71—41no—
131C22×C71—81no—
132C22×C71—81no—
133C22×C71#38111yescharacteristic
134C22×C71#40811yescharacteristic
135C22×C71#40811yescharacteristic
136C4×C71—22no—
137C4×C71—22no—
138C4×C71—41no—
139C4×C71—41no—
140C4×C71—41no—
141C4×C71#43312yes—
142C4×C71#43312yes—
143Dic71—42no—
144Dic71—42no—
145Dic71—81no—
146Dic71—81no—
147#8733—711nomaximal, Sylow
148C2×C71—21no—
149C2×C71—21no—
150C2×C71—81no—
151C2×C71#181811yescharacteristic
152C2×C71#182711yescharacteristic
153C2×C71#182711yescharacteristic
154#2298—711no—
155#1458—711no—
156#1458D7111yescharacteristic
157#1276—712no—
158#1274—712no—
159C71#873311yescharacteristic, complemented, Sylow
160C2×#117C2×D7111yescharacteristic
161#431—712no—
162#431C2×D7112yes—
163#388—1421no—
164#388—1421no—
165#387—1421no—
166#387—1421no—
167C4×C4⋊C4—711no—
168C4×C4⋊C4—711no—
169C4×C22⋊C4—711no—
170C4×C22⋊C4—711no—
171#376—711no—
172C43—711no—
173#324—21no—
174#324—1421no—
175#323—21no—
176#323—1421no—
177C23×C4C4×D7111yescharacteristic
178#117—22no—
179#117—22no—
180C2×C4⋊C4—1421no—
181C2×C4⋊C4—1421no—
182C2×C4⋊C4D28411yescharacteristic
183C2×C4⋊C4C22⋊D7111yescharacteristic
184C2×C22⋊C4—1421no—
185C2×C22⋊C4—1421no—
186C2×C22⋊C4D28411yescharacteristic
187C2×C22⋊C4C22⋊D7111yescharacteristic
188C2×C42—711no—
189C2×C42—711no—
190C2×C42—711no—
191C2×C42—711no—
192C2×C42—1421no—
193C2×C42—1421no—
194C2×C42C4×D7111yescharacteristic
195C2×C42C22×D7111yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—1422no—
198C22⋊C8—22no—
199C22⋊C8—1422no—
200C4×C8—22no—
201C4×C8—1422no—
202#95—1421no—
203#95—1421no—
204#95—1421no—
205#95—1421no—
206#95—1421no—
207#95—1421no—
208C24(C2×D71)⋊C411yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—1421no—
214C22×C4—1421no—
215C22×C4—1421no—
216C22×C4—1421no—
217C22×C4(C2×D71)⋊C411yescharacteristic
218C22×C4(C2×D71)⋊C411yescharacteristic
219C22×C4(C2×D71)⋊C411yescharacteristic
220C22×C4C2×C4×D7111yescharacteristic
221C22×C4C2×C22⋊D7111yescharacteristic
222C22×C4C2×D28411yescharacteristic
223C2×C8—42no—
224C2×C8—2842no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—2842no—
228C4⋊C4—2842no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2842no—
232C22⋊C4—2842no—
233C42—21no—
234C42—21no—
235C42—2842no—
236C42—2842no—
237C42—1422no—
238C42—1422no—
239C42—1422no—
240C42—1422no—
241C42#1176586412yes—
242C42#1176586412yes—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23C2×(C2×D71)⋊C411yescharacteristic
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—2841no—
261C2×C4—2841no—
262C2×C4—2841no—
263C2×C4—2841no—
264C2×C4—2841no—
265C2×C4—2841no—
266C2×C4—2841no—
267C2×C4—2841no—
268C2×C4—2841no—
269C2×C4—2841no—
270C2×C4—1422no—
271C2×C4—1422no—
272C2×C4#1455608411yescharacteristic
273C2×C4#1455608411yescharacteristic
274C2×C4#1455610912yes—
275C2×C4#1455611212yes—
276C2×C4#1455613412yes—
277C2×C4#1455613612yes—
278C8—82no—
279C8—5682no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1798586011yescharacteristic
288C22#1798596511yescharacteristic
289C22#1798596711yescharacteristic
290C4—22no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—2842no—
296C4—2842no—
297C4—5681no—
298C4—5681no—
299C4#1798605712yes—
300C4#1798605712yes—
301C2—21no—
302C2—21no—
303C2—81no—
304C2#2258840911yescharacteristic
305C2#2258844111yescharacteristic
306C2#2258844511yescharacteristic
subgroupG/H
centerC2×C4#14556084
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1458C2
socleC22×C71#381
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1458D71
hypocenterC71#8733
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #22586172C2semidirect
4. #22586168C2semidirect
159. C71#8733semidirect

Maximal subgroups

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

subgroupindex
#225908382
#225866542
#225861722
#225861682
C71×#14582
#873371