Finite Group Databasebeta

← #31997197

Subgroups — #31997197

Order 18176.

1546 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1546 conjugacy classes of subgroups — too many to draw. Only its 275 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
1C2×#17986472C211yesmaximal, characteristic
2#22590921C212yesmaximal
3#22588414C211yesmaximal, characteristic, complemented
4#22588413C214yesmaximal, complemented
5#22588412C212yesmaximal, complemented
6#22588071C214yesmaximal, complemented
7C71×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D71C2211yescharacteristic
9#17986472—28no—
10C2×C4×C22⋊D71C414yescomplemented
11C2×(C22⋊D71)⋊C4C414yescomplemented
12C2×#14556084C2214yes—
13C2×#14556084C2212yes—
14C22×C22⋊Dic71C2211yescharacteristic
15C2×C22⋊C4×D71C2212yes—
16#17985899—28no—
17#17985855C2212yes—
18#17985854C2214yes—
19#17985766C2214yes—
20#17985765C2218yes—
21C22×C22⋊C4×C71C2211yescharacteristic
22#379×C71C2212yes—
23#342×C71C2214yes—
24C24×D71C2311yescharacteristic
25C22×C22⋊D71—28no—
26C22×C22⋊D71C2×C412yes—
27C22×C22⋊D71C2×C412yes—
28C22×C22⋊D71D418yes—
29C23×Dic71C2×C412yes—
30C23×Dic71C2311yescharacteristic
31C22×C4×D71C2×C414yes—
32C4×C22⋊D71—28no—
33C4×C22⋊D71—28no—
34(C22⋊D71)⋊C4—216no—
35#14556084—28no—
36#14556084—216no—
37C2×Dic71⋊C4—24no—
38C2×Dic71⋊C4C2×C414yes—
39C2×C22⋊Dic71—24no—
40C2×C22⋊Dic71—24no—
41C2×C22⋊Dic71C2×C412yes—
42C2×C22⋊Dic71C2314yes—
43C2×C22⋊Dic71C2312yes—
44C2×(C2×D71)⋊C4—28no—
45C2×(C2×D71)⋊C4C2×C414yes—
46C22⋊C4×D71—28no—
47C2×C4×Dic71—24no—
48C2×C4×Dic71C2×C414yes—
49C25×C71C2311yescharacteristic
50C23×C4×C71C2×C412yes—
51C2×C22⋊C4×C71—24no—
52C2×C22⋊C4×C71—24no—
53C2×C22⋊C4×C71C2×C412yes—
54C2×C22⋊C4×C71C2314yes—
55C2×C22⋊C4×C71C2312yes—
56C23×D71—24no—
57C23×D71—82no—
58C23×D71C22×C412yes—
59C23×D71C2×D414yes—
60C2×C22⋊D71—28no—
61C2×C22⋊D71—28no—
62C2×C22⋊D71—216no—
63C2×C22⋊D71—216no—
64C2×C22⋊D71C22⋊C418yes—
65C2×C22⋊D71C22⋊C418yes—
66C22×Dic71—24no—
67C22×Dic71—24no—
68C22×Dic71—82no—
69C22×Dic71C22×C414yes—
70C22×Dic71C22×C412yes—
71C22×Dic71C2×D414yes—
72C22×Dic71#4118yes—
73C2×C4×D71—28no—
74C2×C4×D71—84no—
75Dic71⋊C4—28no—
76Dic71⋊C4—48no—
77C22⋊Dic71—24no—
78C22⋊Dic71—24no—
79C22⋊Dic71—28no—
80C22⋊Dic71—84no—
81(C2×D71)⋊C4—28no—
82(C2×D71)⋊C4—88no—
83C4×Dic71—28no—
84C4×Dic71—48no—
85C24×C71—24no—
86C24×C71—82no—
87C24×C71C22×C411yescharacteristic
88C24×C71C22×C411yescharacteristic
89C24×C71C2×D414yes—
90C24×C71C2411yescharacteristic
91C22×C4×C71—22no—
92C22×C4×C71—22no—
93C22×C4×C71—24no—
94C22×C4×C71—82no—
95C22×C4×C71C22×C414yes—
96C22⋊C4×C71—24no—
97C22⋊C4×C71—24no—
98C22⋊C4×C71—28no—
99C22⋊C4×C71—84no—
100C23×C71—22no—
101C23×C71—22no—
102C23×C71—28no—
103C23×C71—42no—
104C23×C71—42no—
105C23×C71—42no—
106C23×C71—44no—
107C23×C71—44no—
108C23×C71—88no—
109C23×C71C2×C22⋊C412yes—
110C23×C71C2×C22⋊C412yes—
111C23×C71C23×C411yescharacteristic
112C23×C71C22×D412yes—
113C23×C71C2×#4114yes—
114C22×D71—24no—
115C22×D71—28no—
116C22×D71—44no—
117C22×D71—48no—
118C22×D71—88no—
119C22×D71C2×C22⋊C414yes—
120C2×C4×C71—24no—
121C2×C4×C71—44no—
122C2×C4×C71—48no—
123C2×C4×C71—44no—
124C2×C4×C71—82no—
125C2×C4×C71—82no—
126C22⋊D71—216no—
127C22⋊D71—216no—
128C2×Dic71—24no—
129C2×Dic71—44no—
130C2×Dic71—48no—
131C2×Dic71—44no—
132C2×Dic71—84no—
133C2×Dic71C2×C22⋊C414yes—
134C2×Dic71#11718yes—
135C2×Dic71#12318yes—
136C2×Dic71#12318yes—
137C4×D71—88no—
138C22×C71—22no—
139C22×C71—24no—
140C22×C71—24no—
141C22×C71—22no—
142C22×C71—22no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—44no—
146C22×C71—44no—
147C22×C71—44no—
148C22×C71—88no—
149C22×C71C22×C22⋊C411yescharacteristic
150C22×C71C2×#11712yes—
151C22×C71C2×#12314yes—
152C2×D71—28no—
153C2×D71—24no—
154C2×D71—48no—
155C2×D71—88no—
156C4×C71—44no—
157C4×C71—84no—
158Dic71—44no—
159Dic71—84no—
160Dic71#38118yes—
161#20690—711nomaximal, Sylow
162C2×C71—24no—
163C2×C71—22no—
164C2×C71—42no—
165C2×C71—42no—
166C2×C71C2×#38111yescharacteristic
167C2×C71#288312yes—
168D71—44no—
169C2×#518—711no—
170#2309—712no—
171#1819—712no—
172#1819—714no—
173#1819—711no—
174#1819D7111yescharacteristic, complemented
175#1740—714no—
176C71#2069011yescharacteristic, complemented, Sylow
177C23×D4—711no—
178#518—1428no—
179C2×C4×D4—714no—
180C2×C4×D4—714no—
181C2×#117—712no—
182C2×#117—714no—
183C22×C22⋊C4—711no—
184C22×C22⋊C4—712no—
185C22×C22⋊C4C2×D7111yescharacteristic
186#390—1428no—
187#379—712no—
188#379—714no—
189#379C2×D7112yes—
190#342—718no—
191#342—714no—
192#342C2×D7114yes—
193C25—711no—
194C25C22×D7111yescharacteristic
195C22×D4—712no—
196C22×D4—712no—
197C22×D4—718no—
198C22×D4—1428no—
199C23×C4—712no—
200C23×C4—714no—
201C23×C4—711no—
202C23×C4C4×D7112yes—
203C4×D4—14216no—
204C4×D4—1428no—
205C4×D4—1428no—
206#117—14216no—
207#117—1428no—
208C2×C4⋊C4—714no—
209C2×C4⋊C4—1424no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—712no—
213C2×C22⋊C4—712no—
214C2×C22⋊C4—714no—
215C2×C22⋊C4—714no—
216C2×C22⋊C4—1428no—
217C2×C22⋊C4—1428no—
218C2×C22⋊C4—1424no—
219C2×C22⋊C4—1424no—
220C2×C22⋊C4C4×D7112yes—
221C2×C22⋊C4C22×D7114yes—
222C2×C22⋊C4C22×D7112yes—
223C2×C42—714no—
224C2×C42—1424no—
225C24—24no—
226C24—82no—
227C24—5682no—
228C24—712no—
229C24—714no—
230C24—1424no—
231C24C2×C4×D7111yescharacteristic
232C24C2×C4×D7111yescharacteristic
233C24D4×D7114yes—
234C24C23×D7111yescharacteristic
235C2×D4—718no—
236C2×D4—718no—
237C2×D4—1428no—
238C2×D4—1428no—
239C2×D4—14216no—
240C2×D4—14216no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—82no—
245C22×C4—5684no—
246C22×C4—5682no—
247C22×C4—714no—
248C22×C4—712no—
249C22×C4—714no—
250C22×C4—718no—
251C22×C4—1428no—
252C22×C4—1424no—
253C22×C4—1424no—
254C22×C4C2×C4×D7114yes—
255C4⋊C4—2848no—
256C4⋊C4—1428no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—28no—
260C22⋊C4—84no—
261C22⋊C4—5688no—
262C22⋊C4—5684no—
263C22⋊C4—1428no—
264C22⋊C4—1428no—
265C22⋊C4—1424no—
266C22⋊C4—1424no—
267C42—2848no—
268C42—1428no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23—88no—
278C23—2844no—
279C23—2848no—
280C23—5688no—
281C23—714no—
282C23—1424no—
283C23—1428no—
284C23C22⋊C4×D7112yes—
285C23C22⋊C4×D7112yes—
286C23C22×C4×D7111yescharacteristic
287C23C2×D4×D7112yes—
288C23#41×D7114yes—
289D4—14216no—
290D4—14216no—
291C2×C4—24no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—2844no—
298C2×C4—2848no—
299C2×C4—2844no—
300C2×C4—5688no—
301C2×C4—5684no—
302C2×C4—714no—
303C2×C4—718no—
304C2×C4—718no—
305C2×C4—718no—
306C2×C4—1424no—
307C22—22no—
308C22—24no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—44no—
315C22—44no—
316C22—44no—
317C22—88no—
318C22—2848no—
319C22—5688no—
320C22—1428no—
321C22—1424no—
322C22C2×C22⋊C4×D7111yescharacteristic
323C22#117×D7112yes—
324C22#123×D7114yes—
325C4—44no—
326C4—84no—
327C4—2844no—
328C4—5684no—
329C4—718no—
330C2—24no—
331C2—22no—
332C2—42no—
333C2—42no—
334C2—2844no—
335C2#381×D7111yescharacteristic
336C2#2259679812yes—
subgroupG/H
centerC22C2×C22⋊C4×D71
commutatorC23×C71C23×C4
FrattiniC24C23×D71
FittingC71×#1819C2
socleC22×C71C22×C22⋊C4
prime radicalC23×C22⋊D71C22
cosocleC24×C71C24
hypercenter#1819D71
hypocenterC71#20690
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #22588414C2semidirect
4. #22588413C2semidirect
5. #22588412C2semidirect
6. #22588071C2semidirect
7. C71×#1819C2semidirect
10. C2×C4×C22⋊D71C4semidirect
11. C2×(C22⋊D71)⋊C4C4semidirect
174. #1819D71semidirect
176. C71#20690semidirect

Maximal subgroups

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

subgroupindex
C2×#179864722
#225909212
#225884142
#225884132
#225884122
#225880712
C71×#18192
#2069071