Finite Group Databasebeta

← #31997980

Subgroups — #31997980

Order 18176.

982 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Dic71C211yesmaximal, characteristic
2#22590742C212yesmaximal
3#22588495C212yesmaximal, complemented
4#22588466C212yesmaximal, complemented
5#22588365C211yesmaximal, characteristic
6#22588363C212yesmaximal
7#106×Dic71C212yesmaximal, complemented
8#22588228C212yesmaximal, complemented
9C71×#1807C211yesmaximal, characteristic
10C23×C71⋊C8C2211yescharacteristic
11C2×#14556084—22no—
12C22×C4×Dic71C2211yescharacteristic
13C2×#14556060C2212yes—
14C2×#14556059C2214yes—
15C2×C8×Dic71C418yescomplemented
16C2×C8×Dic71C2211yescharacteristic
17C2×C8×Dic71C2211yescharacteristic
18C2×C8×Dic71C2211yescharacteristic
19C2×C8×Dic71C2211yescharacteristic
20#17985868C2214yes—
21#17985863—22no—
22#17985859C2212yes—
23#17985857—22no—
24C4⋊C4×Dic71C2214yes—
25#17985850—22no—
26#17985762C2212yes—
27#17985761C2214yes—
28C23×C8×C71C2211yescharacteristic
29C2×#106×C71C2212yes—
30#380×C71C2212yes—
31#340×C71C2212yes—
32C23×Dic71D411yescharacteristic
33C23×Dic71D411yescharacteristic
34C22×C71⋊C8—24no—
35C22×C71⋊C8C2×C414yes—
36C22×C71⋊C8C2311yescharacteristic
37C22×C71⋊C8C2311yescharacteristic
38#14556084—48no—
39C2×C4⋊Dic71C2314yes—
40C2×Dic71⋊C4—24no—
41C2×Dic71⋊C4—24no—
42C2×C22⋊Dic71—22no—
43C2×C4×Dic71—24no—
44C2×C4×Dic71C2×C414yes—
45C2×C4×Dic71D411yescharacteristic
46C2×C4×Dic71D411yescharacteristic
47C2×C4×Dic71Q814yes—
48C2×C4×Dic71C2311yescharacteristic
49C2×C4×Dic71C2311yescharacteristic
50#14556060—24no—
51#14556059—28no—
52C8×Dic71C2×C414yes—
53C8×Dic71C2×C414yes—
54C8×Dic71C2×C414yes—
55C8×Dic71C2×C414yes—
56C71⋊#95—24no—
57C71⋊#95—22no—
58C23×C4×C71C2311yescharacteristic
59C22×C8×C71—24no—
60C22×C8×C71C2×C414yes—
61C22×C8×C71C2311yescharacteristic
62C22×C8×C71C2311yescharacteristic
63C2×C4⋊C4×C71C2314yes—
64C2×C22⋊C4×C71—22no—
65#106×C71—24no—
66#95×C71—22no—
67C22×Dic71—24no—
68C22×Dic71—24no—
69C22×Dic71C4⋊C414yes—
70C22×Dic71C4⋊C414yes—
71C22×Dic71SD1618yes—
72C22×Dic71C2×D411yescharacteristic
73C22×Dic71C2×D411yescharacteristic
74C22×Dic71C2×D411yescharacteristic
75C22×Dic71C2×D411yescharacteristic
76C2×C71⋊C8—22no—
77C2×C71⋊C8—28no—
78C2×C71⋊C8C22×C412yes—
79C2×C71⋊C8C22×C412yes—
80C2×C71⋊C8#4114yes—
81C4⋊Dic71—24no—
82C4⋊Dic71—24no—
83Dic71⋊C4—48no—
84Dic71⋊C4—48no—
85C22⋊Dic71—44no—
86C4×Dic71—48no—
87C4×Dic71C4⋊C414yes—
88C4×Dic71C4⋊C414yes—
89C4×Dic71C22×C412yes—
90C4×Dic71C22×C412yes—
91C4×Dic71C2×Q812yes—
92C4×Dic71C2×Q812yes—
93C24×C71C2×D411yescharacteristic
94C22×C4×C71—22no—
95C22×C4×C71—22no—
96C22×C4×C71—24no—
97C22×C4×C71C22×C412yes—
98C22×C4×C71C2×D411yescharacteristic
99C22×C4×C71C2×Q812yes—
100C22×C4×C71C2411yescharacteristic
101C2×C8×C71—22no—
102C2×C8×C71—28no—
103C2×C8×C71C22×C412yes—
104C2×C8×C71C22×C412yes—
105C2×C8×C71#4114yes—
106C4⋊C4×C71—24no—
107C4⋊C4×C71—24no—
108C22⋊C4×C71—44no—
109C23×C71—22no—
110C23×C71—22no—
111C23×C71C2×C4⋊C412yes—
112C23×C71C2×SD1614yes—
113C23×C71C22×D411yescharacteristic
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—24no—
118C2×C4×C71—44no—
119C2×C4×C71—44no—
120C2×C4×C71—44no—
121C2×C4×C71C2×C4⋊C412yes—
122C2×C4×C71C23×C411yescharacteristic
123C2×C4×C71C22×Q811yescharacteristic
124C2×C4×C71C2×#4112yes—
125C2×C4×C71C2×#4112yes—
126C2×Dic71—22no—
127C2×Dic71—28no—
128C2×Dic71—44no—
129C2×Dic71—44no—
130C2×Dic71—44no—
131C2×Dic71#106116yescomplemented
132C2×Dic71C2×C4⋊C412yes—
133C2×Dic71C2×C4⋊C412yes—
134C2×Dic71C2×C4⋊C412yes—
135C2×Dic71C2×C4⋊C412yes—
136C2×Dic71C2×SD1614yes—
137C2×Dic71C2×SD1614yes—
138C2×Dic71#13514yes—
139C8×C71—22no—
140C8×C71#11714yescomplemented
141C71⋊C8—22no—
142C71⋊C8#11714yescomplemented
143C22×C71—22no—
144C22×C71—28no—
145C22×C71—22no—
146C22×C71C2×#10614yes—
147C22×C71C22×C4⋊C411yescharacteristic
148C22×C71#53212yes—
149C22×C71C22×SD1612yes—
150C22×C71C2×#13512yes—
151C4×C71—22no—
152C4×C71—84no—
153C4×C71C2×#11711yescharacteristic
154C4×C71C2×#11711yescharacteristic
155C4×C71#53312yes—
156Dic71—22no—
157Dic71—84no—
158Dic71C2×#10614yes—
159Dic71C2×#10614yes—
160Dic71#42714yes—
161#20816—711nomaximal, Sylow
162C2×C71—24no—
163C2×C71#228511yescharacteristic
164C2×C71C22×#10611yescharacteristic
165C2×C71C2×#42711yescharacteristic
166C2×C71#316714yes—
167C22×C4×C8—711no—
168#2285—712no—
169#1844—712no—
170#1834—712no—
171#1807—712no—
172#1807—711no—
173#1807D7111yescharacteristic
174C4×#106—712no—
175C4×#106—712no—
176C71#2081611yescharacteristic, complemented, Sylow
177C23×C8—711no—
178C23×C8C2×D7111yescharacteristic
179C2×#117—1422no—
180C22×C42—711no—
181C2×#106—712no—
182C2×#106—714no—
183C2×#106C2×D7112yes—
184C2×C4×C8—718no—
185C2×C4×C8—711no—
186C2×C4×C8—711no—
187C2×C4×C8—711no—
188C2×C4×C8—711no—
189#384—714no—
190#382—1422no—
191#380—712no—
192#380—1422no—
193#380C2×D7112yes—
194C4×C4⋊C4—714no—
195C4×C4⋊C4—1422no—
196#340—712no—
197#340—714no—
198#340C2×D7112yes—
199C23×C4—711no—
200C23×C4—711no—
201C23×C4C22×D7111yescharacteristic
202C22×C8—24no—
203C22×C8—711no—
204C22×C8—714no—
205C22×C8—711no—
206C22×C8—1424no—
207C22×C8C4×D7114yes—
208C22×C8C22×D7111yescharacteristic
209C22×C8C22×D7111yescharacteristic
210#117—2848no—
211C2×C4⋊C4—714no—
212C2×C4⋊C4—1424no—
213C2×C4⋊C4—1424no—
214C2×C4⋊C4C22×D7114yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1422no—
217C2×C42—714no—
218C2×C42—711no—
219C2×C42—711no—
220C2×C42—714no—
221C2×C42—711no—
222C2×C42—711no—
223C2×C42—1424no—
224#106—24no—
225#106—1428no—
226#106—1424no—
227C4×C8—714no—
228C4×C8—714no—
229C4×C8—714no—
230C4×C8—714no—
231#95—22no—
232#95—1424no—
233#95—1422no—
234C24D4×D7111yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—711no—
239C22×C4—711no—
240C22×C4—711no—
241C22×C4—714no—
242C22×C4—718no—
243C22×C4—711no—
244C22×C4—714no—
245C22×C4—1424no—
246C22×C4—1424no—
247C22×C4C2×C4×D7112yes—
248C22×C4D4×D7111yescharacteristic
249C22×C4Q8×D7112yes—
250C22×C4C23×D7111yescharacteristic
251C2×C8—22no—
252C2×C8—28no—
253C2×C8—712no—
254C2×C8—714no—
255C2×C8—712no—
256C2×C8—1428no—
257C2×C8—1422no—
258C2×C8C2×C4×D7112yes—
259C2×C8C2×C4×D7112yes—
260C2×C8#1176586414yes—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—2848no—
264C4⋊C4—2848no—
265C4⋊C4—1424no—
266C4⋊C4—1424no—
267C22⋊C4—44no—
268C22⋊C4—2844no—
269C42—2848no—
270C42—712no—
271C42—712no—
272C42—712no—
273C42—712no—
274C42—714no—
275C42—714no—
276C23—22no—
277C23—22no—
278C23C4⋊C4×D7112yes—
279C23SD16×D7114yes—
280C23C2×D4×D7111yescharacteristic
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—2844no—
289C2×C4—2844no—
290C2×C4—2844no—
291C2×C4—712no—
292C2×C4—712no—
293C2×C4—712no—
294C2×C4—714no—
295C2×C4—714no—
296C2×C4—714no—
297C2×C4—7116no—
298C2×C4—712no—
299C2×C4—1428no—
300C2×C4—1422no—
301C2×C4C4⋊C4×D7112yes—
302C2×C4C22×C4×D7111yescharacteristic
303C2×C4C2×Q8×D7111yescharacteristic
304C2×C4#41×D7112yes—
305C2×C4C2×#1176586412yes—
306C8—22no—
307C8—714no—
308C8—1422no—
309C8#1455608414yescomplemented
310C22—22no—
311C22—22no—
312C22—28no—
313C22#106×D7114yes—
314C22C2×C4⋊C4×D7111yescharacteristic
315C22#1798654412yes—
316C22C2×SD16×D7112yes—
317C22#135×D7112yes—
318C4—22no—
319C4—84no—
320C4—5684no—
321C4—714no—
322C4—714no—
323C4—714no—
324C4—1422no—
325C4#117×D7111yescharacteristic
326C4C2×#1455608411yescharacteristic
327C4#1798655112yes—
328C2—24no—
329C2#2259074211yescharacteristic
330C2C2×#106×D7111yescharacteristic
331C2#427×D7111yescharacteristic
332C2#2259889414yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1807C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#1807D71
hypocenterC71#20816
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
3. #22588495C2semidirect
4. #22588466C2semidirect
7. #106×Dic71C2semidirect
8. #22588228C2semidirect
15. C2×C8×Dic71C4semidirect
131. C2×Dic71#106semidirect
140. C8×C71#117semidirect
142. C71⋊C8#117semidirect
176. C71#20816semidirect
309. C8#14556084semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Dic712
#225907422
#225884952
#225884662
#225883652
#225883632
#106×Dic712
#225882282
C71×#18072
#2081671