Finite Group Databasebeta

← #2255×D71 #32152945

Subgroups — #2255×D71 #32152945

Order 18176.

518 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D71C211yesmaximal, characteristic
2#499×D71C211yesmaximal, characteristic
3#22590627C211yesmaximal, characteristic, complemented
4#22590626C211yesmaximal, characteristic, complemented
5#22590625C212yesmaximal, complemented
6#22590624C212yesmaximal, complemented
7#22590623C211yesmaximal, characteristic, complemented
8#367×D71C212yesmaximal
9#366×D71C212yesmaximal
10#346×D71C211yesmaximal, characteristic
11C71×#2255C211yesmaximal, characteristic, complemented
12C4⋊Q8×D71—21no—
13#125×D71—21no—
14C2×C16×D71C2212yes—
15#17986404C2211yescharacteristic
16#17986403—21no—
17#17986403C2212yes—
18#17986402—21no—
19#17986401C2211yescharacteristic
20#17986400C2211yescharacteristic
21#17986400C2211yescharacteristic
22#17986399—21no—
23#17986398—21no—
24#107×D71—21no—
25#107×D71C2211yescharacteristic
26#107×D71C2212yes—
27#107×D71C2211yescharacteristic
28#106×D71—21no—
29C4×C8×D71C2211yescharacteristic
30#17985827C2212yes—
31#17985826C2212yes—
32#17985826C2212yes—
33#17985825C2212yes—
34#17985824C2212yes—
35#17985824C2212yes—
36#17985775C2211yescharacteristic
37#17985774C2212yes—
38#500×C71C2211yescharacteristic
39#499×C71C2211yescharacteristic
40#367×C71C2212yes—
41#366×C71C2212yes—
42#346×C71C2211yescharacteristic
43C2×Q8×D71—41no—
44C2×C8×D71D411yescharacteristic
45C2×C8×D71C2311yescharacteristic
46#14556128—21no—
47#14556127—22no—
48#14556127—41no—
49#14556126—41no—
50#14556120—21no—
51#14556119—21no—
52#14556119—21no—
53#14556118—41no—
54#14556118—41no—
55C4⋊C4×D71—21no—
56C4⋊C4×D71—21no—
57C4⋊C4×D71—22no—
58C4⋊C4×D71—41no—
59C4⋊C4×D71—41no—
60C4⋊C4×D71—41no—
61C42×D71D411yescharacteristic
62C2×C71⋊C16C2312yes—
63C16×D71Q818yescomplemented
64#14556062—21no—
65#14556062C2311yescharacteristic
66#14556062C2312yes—
67#14556062C2311yescharacteristic
68#14556061—21no—
69#14556061—21no—
70#14556061—22no—
71#14556061—22no—
72#14556060—21no—
73#14556059—21no—
74#14556059—21no—
75C4×C71⋊C8D411yescharacteristic
76C8×Dic71D411yescharacteristic
77C8×Dic71C2311yescharacteristic
78C4⋊Q8×C71—21no—
79#125×C71—21no—
80C2×C16×C71C2312yes—
81#107×C71—21no—
82#107×C71C2311yescharacteristic
83#107×C71C2312yes—
84#107×C71C2311yescharacteristic
85#106×C71—21no—
86C4×C8×C71C2311yescharacteristic
87C2×Dic142—41no—
88C2×Dic142—42no—
89Q8×D71—84no—
90C2×C4×D71—42no—
91C2×C4×D71—41no—
92C2×C4×D71—41no—
93C2×C4×D71D811yescharacteristic
94C2×C4×D71D811yescharacteristic
95C2×C4×D71C2×D411yescharacteristic
96C2×C71⋊C8C2×D411yescharacteristic
97C2×C71⋊C8C2×D411yescharacteristic
98C8×D71Q1614yes—
99C8×D71C2×Q814yes—
100C4⋊Dic71—21no—
101C4⋊Dic71—21no—
102C4⋊Dic71—22no—
103C4⋊Dic71—41no—
104C4⋊Dic71—41no—
105C4⋊Dic71—41no—
106Dic71⋊C4—41no—
107Dic71⋊C4—41no—
108Dic71⋊C4—41no—
109Dic71⋊C4—41no—
110Dic71⋊C4—42no—
111Dic71⋊C4—41no—
112Dic71⋊C4—41no—
113Dic71⋊C4—42no—
114C4×Dic71D811yescharacteristic
115C4×Dic71D811yescharacteristic
116C4×Dic71C2×D411yescharacteristic
117C71⋊C16C2×Q814yescomplemented
118C2×Q8×C71—41no—
119C2×C8×C71C2×D411yescharacteristic
120C2×C8×C71C2411yescharacteristic
121C4⋊C4×C71—21no—
122C4⋊C4×C71—21no—
123C4⋊C4×C71—22no—
124C4⋊C4×C71—41no—
125C4⋊C4×C71—41no—
126C4⋊C4×C71—41no—
127C42×C71C2×D411yescharacteristic
128C16×C71C2×Q814yescomplemented
129C22×D71C2×D811yescharacteristic
130Q8×C71—82no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—42no—
134C2×C4×C71C2×D811yescharacteristic
135C2×C4×C71C2×D811yescharacteristic
136C2×C4×C71C22×D411yescharacteristic
137C2×Dic71—41no—
138C2×Dic71—41no—
139C2×Dic71—42no—
140C2×Dic71C2×D811yescharacteristic
141C2×Dic71C2×D811yescharacteristic
142C2×Dic71C2×D811yescharacteristic
143C4×D71—22no—
144C4×D71—22no—
145C4×D71—82no—
146C4×D71—84no—
147C4×D71—82no—
148C4×D71C4⋊Q812yes—
149C4×D71C2×Q1612yes—
150Dic142—84no—
151Dic142—82no—
152C8×C71C2×Q1612yes—
153C8×C71C22×Q812yes—
154C71⋊C8C4⋊Q812yes—
155C71⋊C8C2×Q1612yes—
156C22×C71C22×D811yescharacteristic
157C2×D71#50012yes—
158C2×D71#50912yes—
159C2×D71#51012yes—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—82no—
163C4×C71—81no—
164C4×C71—81no—
165C4×C71C2×C4⋊Q811yescharacteristic
166C4×C71C22×Q1611yescharacteristic
167Dic71—21no—
168Dic71—21no—
169Dic71—82no—
170Dic71—81no—
171Dic71—81no—
172Dic71#50012yes—
173C2×#2255—711nomaximal, Sylow
174C2×C71C2×#50011yescharacteristic
175C2×C71C2×#50911yescharacteristic
176C2×C71C2×#51011yescharacteristic
177D71#225514yesdirect product factor
178C2×#500—711no—
179C2×#499—711no—
180#2255—711no—
181#2255—711no—
182#2255—712no—
183#2255—712no—
184#2255—711no—
185#2255D7111yescharacteristic, direct product factor
186C2×#367—712no—
187C2×#366—712no—
188C2×#346—711no—
189C71C2×#225511yescharacteristic, complemented, Sylow
190C2×C4⋊Q8—1421no—
191C2×#125—1421no—
192C22×C16—712no—
193#500—712no—
194#500—711no—
195#500—1421no—
196#500—1421no—
197#500C2×D7111yescharacteristic
198#499—711no—
199#499—711no—
200#499—711no—
201#499—1421no—
202#499—1421no—
203#499C2×D7111yescharacteristic
204C2×#107—711no—
205C2×#107—711no—
206C2×#107—712no—
207C2×#107—1421no—
208C2×#106—1421no—
209C2×C4×C8—711no—
210#367—712no—
211#367—712no—
212#367—712no—
213#367C2×D7112yes—
214#366—712no—
215#366—712no—
216#366—712no—
217#366C2×D7112yes—
218#346—712no—
219#346—711no—
220#346C2×D7111yescharacteristic
221C22×Q8—2841no—
222C22×C8—711no—
223C22×C8—711no—
224C4⋊Q8—21no—
225C4⋊Q8—2841no—
226C4⋊Q8—2841no—
227C4⋊Q8—1422no—
228C4⋊Q8—1421no—
229#125—21no—
230#125—2841no—
231#125—2841no—
232#125—1421no—
233#125—1421no—
234#125—1421no—
235C2×C4⋊C4—2841no—
236C2×C4⋊C4—2841no—
237C2×C4⋊C4—2841no—
238C2×C4⋊C4—1421no—
239C2×C4⋊C4—1421no—
240C2×C4⋊C4—1422no—
241C2×C42—711no—
242C2×C16—712no—
243C2×C16—718no—
244C2×C16C22×D7112yes—
245#107—21no—
246#107—711no—
247#107—712no—
248#107—711no—
249#107—1421no—
250#107—1421no—
251#107—1422no—
252#107—1422no—
253#107—1421no—
254#107C22×D7111yescharacteristic
255#107C22×D7112yes—
256#107C22×D7111yescharacteristic
257#106—21no—
258#106—1421no—
259#106—1421no—
260#106—1421no—
261C4×C8—711no—
262C4×C8—711no—
263C4×C8—711no—
264C4×C8C22×D7111yescharacteristic
265C2×Q8—41no—
266C2×Q8—2842no—
267C2×Q8—2841no—
268C2×Q8—5684no—
269C22×C4—2841no—
270C22×C4—2841no—
271C22×C4—2842no—
272C22×C4—711no—
273C22×C4—711no—
274C22×C4—711no—
275C2×C8—711no—
276C2×C8—711no—
277C2×C8—714no—
278C2×C8—714no—
279C2×C8D4×D7111yescharacteristic
280C2×C8C23×D7111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—2841no—
288C4⋊C4—2841no—
289C4⋊C4—2842no—
290C4⋊C4—2841no—
291C4⋊C4—2841no—
292C4⋊C4—2842no—
293C4⋊C4—2841no—
294C4⋊C4—2841no—
295C4⋊C4—2841no—
296C4⋊C4—2841no—
297C4⋊C4—2841no—
298C4⋊C4—1421no—
299C4⋊C4—1421no—
300C4⋊C4—1422no—
301C42—711no—
302C42—711no—
303C42—711no—
304C42D4×D7111yescharacteristic
305C16—714no—
306C16Q8×D7114yescomplemented
307C23—711no—
308Q8—82no—
309Q8—5682no—
310Q8—5684no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—2841no—
315C2×C4—2841no—
316C2×C4—2842no—
317C2×C4—5684no—
318C2×C4—5682no—
319C2×C4—5682no—
320C2×C4—711no—
321C2×C4—712no—
322C2×C4—712no—
323C2×C4—711no—
324C2×C4—711no—
325C2×C4—1422no—
326C2×C4—1422no—
327C2×C4D8×D7111yescharacteristic
328C2×C4D8×D7111yescharacteristic
329C2×C4C2×D4×D7111yescharacteristic
330C8—712no—
331C8—712no—
332C8Q16×D7112yes—
333C8C2×Q8×D7112yes—
334C22—712no—
335C22—712no—
336C22—712no—
337C22C2×D8×D7111yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—81no—
342C4—81no—
343C4—5682no—
344C4—5681no—
345C4—5681no—
346C4—712no—
347C4—1421no—
348C4—1421no—
349C4C4⋊Q8×D7111yescharacteristic
350C4C2×Q16×D7111yescharacteristic
351C2—714no—
352C2#500×D7111yescharacteristic
353C2#509×D7111yescharacteristic
354C2#510×D7111yescharacteristic
subgroupG/H
centerC22C2×D8×D71
commutator, cosocleC2×C8×C71C24
FrattiniC2×C8C23×D71
FittingC71×#2255C2
socleC22×C71C22×D8
prime radicalC22×D71C2×D8
hypercenter#2255D71
hypocenterC71C2×#2255
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #22590627C2semidirect
4. #22590626C2semidirect
5. #22590625C2semidirect
6. #22590624C2semidirect
7. #22590623C2semidirect
11. C71×#2255C2semidirect
63. C16×D71Q8semidirect
117. C71⋊C16C2×Q8semidirect
128. C16×C71C2×Q8semidirect
177. D71#2255direct (N × H)
185. #2255D71direct (N × H)
189. C71C2×#2255semidirect
306. C16Q8×D71semidirect

Maximal subgroups

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

subgroupindex
#500×D712
#499×D712
#225906272
#225906262
#225906252
#225906242
#225906232
#367×D712
#366×D712
#346×D712
C71×#22552
C2×#225571