Finite Group Databasebeta

← #31997192

Subgroups — #31997192

Order 18176.

1602 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1602 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#22590916C212yesmaximal
3#22588416C214yesmaximal, complemented
4#22588414C211yesmaximal, characteristic, complemented
5#22588412C212yesmaximal, complemented
6#342×D71C214yesmaximal, complemented
7C71×#1819C211yesmaximal, characteristic, complemented
8C23×C22⋊D71C2211yescharacteristic
9#17986472—28no—
10C2×C4×C22⋊D71C414yescomplemented
11C2×(C22⋊D71)⋊C4C414yescomplemented
12C2×#14556084C2212yes—
13C22×C22⋊Dic71C2211yescharacteristic
14C2×C22⋊C4×D71C2214yes—
15C2×C22⋊C4×D71C2212yes—
16#17985875—28no—
17#17985860C2214yes—
18#17985855C2212yes—
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×C412yescomplemented
30C23×Dic71C2311yescharacteristic
31C22×C4×D71—24no—
32C22×C4×D71C2×C414yes—
33C4×C22⋊D71—28no—
34C4×C22⋊D71—28no—
35(C22⋊D71)⋊C4—216no—
36#14556084—28no—
37C2×Dic71⋊C4C2×C414yescomplemented
38C2×C22⋊Dic71—24no—
39C2×C22⋊Dic71—24no—
40C2×C22⋊Dic71C2×C412yescomplemented
41C2×C22⋊Dic71C2314yes—
42C2×C22⋊Dic71C2312yes—
43C2×(C2×D71)⋊C4—24no—
44C2×(C2×D71)⋊C4C2×C414yes—
45C22⋊C4×D71—28no—
46C22⋊C4×D71—216no—
47C2×C4×Dic71C2×C414yescomplemented
48C71⋊#95—28no—
49C25×C71C2311yescharacteristic
50C23×C4×C71C2×C412yescomplemented
51C2×C22⋊C4×C71—24no—
52C2×C22⋊C4×C71—24no—
53C2×C22⋊C4×C71C2×C412yescomplemented
54C2×C22⋊C4×C71C2314yes—
55C2×C22⋊C4×C71C2312yes—
56C23×D71—82no—
57C23×D71C22×C412yes—
58C23×D71C2×D414yes—
59C23×D71#4118yes—
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—24no—
69C22×Dic71—82no—
70C22×Dic71C22×C412yes—
71C22×Dic71C22×C414yes—
72C22×Dic71C2×D414yes—
73C2×C4×D71—28no—
74C2×C4×D71—48no—
75C2×C4×D71—88no—
76C2×C4×D71—84no—
77Dic71⋊C4—28no—
78C22⋊Dic71—24no—
79C22⋊Dic71—24no—
80C22⋊Dic71—28no—
81C22⋊Dic71—84no—
82(C2×D71)⋊C4—28no—
83(C2×D71)⋊C4—48no—
84C4×Dic71—28no—
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—44no—
115C22×D71—48no—
116C22×D71—88no—
117C22×D71C2×C22⋊C414yes—
118C22×D71#11718yes—
119C22×D71#12318yes—
120C22×D71#12318yes—
121C2×C4×C71—24no—
122C2×C4×C71—44no—
123C2×C4×C71—48no—
124C2×C4×C71—44no—
125C2×C4×C71—82no—
126C2×C4×C71—82no—
127C22⋊D71—216no—
128C22⋊D71—216no—
129C2×Dic71—24no—
130C2×Dic71—28no—
131C2×Dic71—24no—
132C2×Dic71—44no—
133C2×Dic71—48no—
134C2×Dic71—44no—
135C2×Dic71—84no—
136C2×Dic71C2×C22⋊C414yes—
137C4×D71—88no—
138C4×D71—816no—
139C22×C71—22no—
140C22×C71—24no—
141C22×C71—24no—
142C22×C71—22no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—44no—
147C22×C71—44no—
148C22×C71—44no—
149C22×C71—88no—
150C22×C71C22×C22⋊C411yescharacteristic
151C22×C71C2×#11712yes—
152C22×C71C2×#12314yes—
153C2×D71—28no—
154C2×D71—48no—
155C2×D71—48no—
156C2×D71—48no—
157C2×D71#38118yes—
158C4×C71—44no—
159C4×C71—84no—
160Dic71—24no—
161Dic71—44no—
162Dic71—84no—
163#20689—711nomaximal, Sylow
164C2×C71—24no—
165C2×C71—22no—
166C2×C71—42no—
167C2×C71—42no—
168C2×C71C2×#38111yescharacteristic
169C2×C71#288312yes—
170D71—28no—
171C2×#518—711no—
172#2308—712no—
173#1820—714no—
174#1819—712no—
175#1819—711no—
176#1819D7111yescharacteristic, complemented
177C2×#342—714no—
178C71#2068911yescharacteristic, complemented, Sylow
179C23×D4—711no—
180#518—1428no—
181C2×C4×D4—714no—
182C2×C4×D4—714no—
183C2×#117—712no—
184C22×C22⋊C4—711no—
185C22×C22⋊C4—712no—
186C22×C22⋊C4—714no—
187C22×C22⋊C4C2×D7111yescharacteristic
188#386—1428no—
189#381—714no—
190#379—712no—
191#379C2×D7112yes—
192#342—718no—
193#342—714no—
194#342C2×D7114yes—
195C25—711no—
196C25C22×D7111yescharacteristic
197C22×D4—712no—
198C22×D4—712no—
199C22×D4—718no—
200C22×D4—1428no—
201C23×C4—712no—
202C23×C4—714no—
203C23×C4—711no—
204C23×C4—1424no—
205C23×C4C4×D7112yescomplemented
206C4×D4—14216no—
207C4×D4—1428no—
208C4×D4—1428no—
209#117—1428no—
210C2×C4⋊C4—714no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—712no—
214C2×C22⋊C4—712no—
215C2×C22⋊C4—714no—
216C2×C22⋊C4—714no—
217C2×C22⋊C4—14216no—
218C2×C22⋊C4—1428no—
219C2×C22⋊C4—1424no—
220C2×C22⋊C4—1424no—
221C2×C22⋊C4—1424no—
222C2×C22⋊C4C4×D7112yescomplemented
223C2×C22⋊C4C22×D7114yes—
224C2×C22⋊C4C22×D7112yes—
225C2×C42—714no—
226#95—1428no—
227C24—24no—
228C24—82no—
229C24—5682no—
230C24—712no—
231C24—714no—
232C24—718no—
233C24C2×C4×D7111yescharacteristic
234C24C2×C4×D7111yescharacteristic
235C24D4×D7114yes—
236C24C23×D7111yescharacteristic
237C2×D4—718no—
238C2×D4—718no—
239C2×D4—1428no—
240C2×D4—1428no—
241C2×D4—14216no—
242C2×D4—14216no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—82no—
247C22×C4—2848no—
248C22×C4—5688no—
249C22×C4—5684no—
250C22×C4—5682no—
251C22×C4—714no—
252C22×C4—712no—
253C22×C4—714no—
254C22×C4—1428no—
255C22×C4—1424no—
256C22×C4—1424no—
257C22×C4—1424no—
258C22×C4C2×C4×D7114yes—
259C4⋊C4—1428no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—28no—
263C22⋊C4—84no—
264C22⋊C4—2848no—
265C22⋊C4—5684no—
266C22⋊C4—1428no—
267C22⋊C4—1428no—
268C22⋊C4—1424no—
269C22⋊C4—1424no—
270C42—1428no—
271C23—22no—
272C23—22no—
273C23—28no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—44no—
278C23—44no—
279C23—88no—
280C23—2844no—
281C23—2848no—
282C23—5688no—
283C23—718no—
284C23—718no—
285C23—718no—
286C23—714no—
287C23C22⋊C4×D7112yes—
288C23C22⋊C4×D7112yes—
289C23C22×C4×D7111yescharacteristic
290C23C2×D4×D7112yes—
291C23#41×D7114yes—
292D4—14216no—
293D4—14216no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—2844no—
301C2×C4—2848no—
302C2×C4—2844no—
303C2×C4—5688no—
304C2×C4—56816no—
305C2×C4—5684no—
306C2×C4—714no—
307C2×C4—1424no—
308C2×C4—1424no—
309C2×C4—1428no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—88no—
321C22—2848no—
322C22—2848no—
323C22—2848no—
324C22—718no—
325C22—1428no—
326C22C2×C22⋊C4×D7111yescharacteristic
327C22#117×D7112yes—
328C22#123×D7114yes—
329C4—44no—
330C4—84no—
331C4—2844no—
332C4—5684no—
333C4—1424no—
334C2—24no—
335C2—22no—
336C2—42no—
337C2—42no—
338C2—1428no—
339C2#381×D7111yescharacteristic
340C2#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#20689
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. #22588416C2semidirect
4. #22588414C2semidirect
5. #22588412C2semidirect
6. #342×D71C2semidirect
7. C71×#1819C2semidirect
10. C2×C4×C22⋊D71C4semidirect
11. C2×(C22⋊D71)⋊C4C4semidirect
29. C23×Dic71C2×C4semidirect
37. C2×Dic71⋊C4C2×C4semidirect
40. C2×C22⋊Dic71C2×C4semidirect
47. C2×C4×Dic71C2×C4semidirect
50. C23×C4×C71C2×C4semidirect
53. C2×C22⋊C4×C71C2×C4semidirect
176. #1819D71semidirect
178. C71#20689semidirect
205. C23×C4C4×D71semidirect
222. C2×C22⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864722
#225909162
#225884162
#225884142
#225884122
#342×D712
C71×#18192
#2068971