Finite Group Databasebeta

← #1386×D71 #31912365

Subgroups — #1386×D71 #31912365

Order 18176.

630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M32×D71C211yesmaximal, characteristic
2#427×D71C211yesmaximal, characteristic
3C2×#106×D71C211yesmaximal, characteristic
4#22586447C211yesmaximal, characteristic, complemented
5#22586446C211yesmaximal, characteristic, complemented
6#22586445C211yesmaximal, characteristic, complemented
7C71×#1386C211yesmaximal, characteristic, complemented
8C22×C8×D71C2211yescharacteristic
9#117×D71—21no—
10C2×C4⋊C4×D71—21no—
11C2×#14556068C2211yescharacteristic
12C2×#14556067—21no—
13M32×D71—22no—
14M32×D71C414yescomplemented
15C2×C16×D71—21no—
16#17986037C2211yescharacteristic
17#17986036C412yescomplemented
18#17986035—21no—
19#17986034—21no—
20#107×D71C412yescomplemented
21C2×#14556060C2211yescharacteristic
22C2×#14556059—21no—
23#106×D71—22no—
24#106×D71—21no—
25#106×D71C414yescomplemented
26C2×M32×C71C2211yescharacteristic
27#427×C71C2211yescharacteristic
28C2×#106×C71C2211yescharacteristic
29C22×C4×D71—41no—
30C22×C4×D71D411yescharacteristic
31C22×C71⋊C8D411yescharacteristic
32C2×C8×D71C2×C412yes—
33C2×C8×D71C2×C412yes—
34C2×C8×D71C2×C411yescharacteristic
35C2×C8×D71C2×C411yescharacteristic
36C2×C8×D71D411yescharacteristic
37C2×C8×D71D412yes—
38C2×C8×D71D411yescharacteristic
39C2×C8×D71Q812yes—
40#14556087—21no—
41#14556086—22no—
42#14556085—41no—
43#14556084—41no—
44C2×C4⋊Dic71—21no—
45C2×Dic71⋊C4—41no—
46C4⋊C4×D71—24no—
47C4⋊C4×D71—22no—
48C4⋊C4×D71—42no—
49C22⋊C4×D71—41no—
50C42×D71—41no—
51#14556068—21no—
52#14556068C2×C412yescomplemented
53#14556067—24no—
54C2×C71⋊C16—21no—
55C16×D71—24no—
56#14556062C2×C412yescomplemented
57#14556061—22no—
58#14556060—21no—
59#14556060—21no—
60#14556060C2×C412yescomplemented
61#14556059—24no—
62#14556059—22no—
63C22×C8×C71C2311yescharacteristic
64#117×C71—21no—
65C2×C4⋊C4×C71—21no—
66M32×C71—21no—
67M32×C71C2×C412yescomplemented
68C2×C16×C71—21no—
69#107×C71C2×C412yescomplemented
70#106×C71—21no—
71#106×C71—21no—
72#106×C71C2×C412yescomplemented
73C23×D71D811yescharacteristic
74C22×Dic71—41no—
75C22×Dic71D811yescharacteristic
76C2×C4×D71—44no—
77C2×C4×D71—42no—
78C2×C4×D71—84no—
79C2×C4×D71C22⋊C412yes—
80C2×C4×D71C22⋊C411yescharacteristic
81C2×C4×D71C22⋊C411yescharacteristic
82C2×C4×D71C4⋊C412yes—
83C2×C4×D71SD1611yescharacteristic
84C2×C4×D71SD1612yes—
85C2×C4×D71SD1611yescharacteristic
86C2×C4×D71Q1612yes—
87C2×C71⋊C8C22⋊C412yes—
88C2×C71⋊C8C22⋊C411yescharacteristic
89C2×C71⋊C8C22⋊C411yescharacteristic
90C2×C71⋊C8C4⋊C412yes—
91C8×D71C4212yes—
92C8×D71C4212yes—
93C8×D71C22⋊C412yes—
94C8×D71C22⋊C412yes—
95C8×D71C4⋊C412yes—
96C8×D71C4⋊C412yes—
97C8×D71C4⋊C412yes—
98C8×D71C4⋊C412yes—
99C4⋊Dic71—22no—
100C4⋊Dic71—22no—
101C4⋊Dic71—41no—
102Dic71⋊C4—44no—
103Dic71⋊C4—42no—
104C22⋊Dic71—41no—
105(C2×D71)⋊C4—42no—
106C4×Dic71—41no—
107C4×Dic71—42no—
108C71⋊C16—22no—
109C22×C4×C71—41no—
110C22×C4×C71C2×D411yescharacteristic
111C2×C8×C71C22×C411yescharacteristic
112C2×C8×C71C22×C411yescharacteristic
113C2×C8×C71C22×C411yescharacteristic
114C2×C8×C71C2×D411yescharacteristic
115C2×C8×C71C2×D411yescharacteristic
116C2×C8×C71C2×Q811yescharacteristic
117C4⋊C4×C71—22no—
118C4⋊C4×C71—22no—
119C4⋊C4×C71—41no—
120C22⋊C4×C71—41no—
121C42×C71—41no—
122C16×C71—22no—
123C23×C71C2×D811yescharacteristic
124C22×D71—24no—
125C22×D71#10211yescharacteristic
126C22×D71#10212yes—
127C22×D71#10211yescharacteristic
128C22×D71#10712yes—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—82no—
132C2×C4×C71C2×C22⋊C411yescharacteristic
133C2×C4×C71C2×C22⋊C411yescharacteristic
134C2×C4×C71C2×C4⋊C411yescharacteristic
135C2×C4×C71C2×SD1611yescharacteristic
136C2×C4×C71C2×SD1611yescharacteristic
137C2×C4×C71C2×Q1611yescharacteristic
138C2×Dic71—42no—
139C2×Dic71—42no—
140C2×Dic71—82no—
141C2×Dic71#10211yescharacteristic
142C2×Dic71#10212yes—
143C2×Dic71#10211yescharacteristic
144C2×Dic71#10712yes—
145C4×D71—88no—
146C4×D71—84no—
147C4×D71#9512yes—
148C4×D71#9512yes—
149C4×D71#10312yes—
150C4×D71#10312yes—
151C4×D71#10312yes—
152C4×D71#10312yes—
153C4×D71#10612yes—
154C4×D71#10612yes—
155C8×C71C2×C4211yescharacteristic
156C8×C71C2×C22⋊C411yescharacteristic
157C8×C71C2×C4⋊C411yescharacteristic
158C8×C71C2×C4⋊C411yescharacteristic
159C71⋊C8#9512yes—
160C71⋊C8#9512yes—
161C22×C71—22no—
162C22×C71C2×#10211yescharacteristic
163C22×C71C2×#10211yescharacteristic
164C22×C71C2×#10711yescharacteristic
165C2×D71—22no—
166C2×D71—24no—
167C2×D71—24no—
168C2×D71#34012yes—
169C2×D71#34012yes—
170C2×D71#36014yes—
171C4×C71—82no—
172C4×C71—82no—
173C4×C71C2×#9511yescharacteristic
174C4×C71C2×#10311yescharacteristic
175C4×C71C2×#10311yescharacteristic
176C4×C71C2×#10611yescharacteristic
177Dic71—82no—
178Dic71—82no—
179Dic71#34012yes—
180Dic71#34012yes—
181C2×#1386—711nomaximal, Sylow
182C2×C71—21no—
183C2×C71—21no—
184C2×C71C2×#34011yescharacteristic
185C2×C71C2×#36012yes—
186D71—22no—
187D71#138614yesdirect product factor
188C22×M32—711no—
189C2×#427—711no—
190C22×#106—711no—
191#1386—711no—
192#1386—711no—
193#1386—711no—
194#1386D7111yescharacteristic, direct product factor
195C71C2×#138611yescharacteristic, complemented, Sylow
196C23×C8—711no—
197C2×#117—1421no—
198C22×C4⋊C4—1421no—
199C2×M32—711no—
200C2×M32—714no—
201C2×M32—1421no—
202C2×M32—1422no—
203C2×M32C2×D7111yescharacteristic
204C22×C16—1421no—
205#427—711no—
206#427—712no—
207#427—1421no—
208#427—1421no—
209#427C2×D7111yescharacteristic
210C2×#107—712no—
211C2×#106—711no—
212C2×#106—714no—
213C2×#106—1422no—
214C2×#106—1421no—
215C2×#106—1421no—
216C2×#106C2×D7111yescharacteristic
217C23×C4—2841no—
218C23×C4—711no—
219C22×C8—711no—
220C22×C8—712no—
221C22×C8—711no—
222C22×C8—711no—
223C22×C8—712no—
224C22×C8—711no—
225C22×C8—712no—
226C22×C8—711no—
227C22×C8—712no—
228C22×C8C22×D7111yescharacteristic
229#117—21no—
230#117—2841no—
231#117—2841no—
232#117—1422no—
233#117—1421no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—2842no—
236C2×C4⋊C4—2841no—
237C2×C4⋊C4—1424no—
238C2×C4⋊C4—1422no—
239C2×C4⋊C4—1421no—
240C2×C22⋊C4—2841no—
241C2×C42—2841no—
242M32—21no—
243M32—712no—
244M32—1424no—
245M32—1421no—
246M32C4×D7112yescomplemented
247C2×C16—21no—
248C2×C16—1424no—
249C2×C16—1421no—
250#107—712no—
251#107—1422no—
252#107C4×D7112yescomplemented
253#106—21no—
254#106—21no—
255#106—712no—
256#106—1424no—
257#106—1422no—
258#106—1421no—
259#106—1421no—
260#106C4×D7112yescomplemented
261C24—711no—
262C22×C4—41no—
263C22×C4—2842no—
264C22×C4—2844no—
265C22×C4—2841no—
266C22×C4—5684no—
267C22×C4—711no—
268C22×C4—712no—
269C22×C4—711no—
270C22×C4—711no—
271C22×C4—712no—
272C22×C4—711no—
273C22×C4—712no—
274C22×C4—711no—
275C22×C4—712no—
276C22×C4D4×D7111yescharacteristic
277C2×C8—712no—
278C2×C8—711no—
279C2×C8—711no—
280C2×C8—712no—
281C2×C8—712no—
282C2×C8—712no—
283C2×C8—712no—
284C2×C8—712no—
285C2×C8—712no—
286C2×C8—712no—
287C2×C8—712no—
288C2×C8—712no—
289C2×C8C2×C4×D7111yescharacteristic
290C2×C8C2×C4×D7111yescharacteristic
291C2×C8C2×C4×D7111yescharacteristic
292C2×C8D4×D7111yescharacteristic
293C2×C8D4×D7111yescharacteristic
294C2×C8Q8×D7111yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—2844no—
299C4⋊C4—2842no—
300C4⋊C4—2841no—
301C4⋊C4—1422no—
302C4⋊C4—1422no—
303C22⋊C4—41no—
304C22⋊C4—2842no—
305C22⋊C4—2841no—
306C42—41no—
307C42—2842no—
308C42—2841no—
309C16—22no—
310C16—1422no—
311C23—711no—
312C23—712no—
313C23—711no—
314C23—712no—
315C23—1424no—
316C23D8×D7111yescharacteristic
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—2842no—
321C2×C4—2842no—
322C2×C4—5684no—
323C2×C4—5688no—
324C2×C4—5682no—
325C2×C4—712no—
326C2×C4—711no—
327C2×C4—711no—
328C2×C4—712no—
329C2×C4—712no—
330C2×C4—712no—
331C2×C4—712no—
332C2×C4—712no—
333C2×C4—712no—
334C2×C4—712no—
335C2×C4—712no—
336C2×C4—712no—
337C2×C4C22⋊C4×D7111yescharacteristic
338C2×C4C22⋊C4×D7111yescharacteristic
339C2×C4C4⋊C4×D7111yescharacteristic
340C2×C4SD16×D7111yescharacteristic
341C2×C4SD16×D7111yescharacteristic
342C2×C4Q16×D7111yescharacteristic
343C8—712no—
344C8—712no—
345C8C42×D7111yescharacteristic
346C8C22⋊C4×D7111yescharacteristic
347C8C4⋊C4×D7111yescharacteristic
348C8C4⋊C4×D7111yescharacteristic
349C22—22no—
350C22—714no—
351C22—712no—
352C22—712no—
353C22—1424no—
354C22—1424no—
355C22—1422no—
356C22#102×D7111yescharacteristic
357C22#102×D7111yescharacteristic
358C22#107×D7111yescharacteristic
359C4—82no—
360C4—82no—
361C4—5682no—
362C4—5682no—
363C4—712no—
364C4—712no—
365C4#95×D7111yescharacteristic
366C4#103×D7111yescharacteristic
367C4#103×D7111yescharacteristic
368C4#106×D7111yescharacteristic
369C2—21no—
370C2—21no—
371C2—714no—
372C2—1422no—
373C2#340×D7111yescharacteristic
374C2#360×D7112yes—
subgroupG/H
centerC22#102×D71
commutatorC8×C71C2×C42
FrattiniC22×C8C22×D71
FittingC71×#1386C2
socleC22×C71C2×#102
prime radicalC23×D71D8
cosocleC22×C8×C71C23
hypercenter#1386D71
hypocenterC71C2×#1386
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
4. #22586447C2semidirect
5. #22586446C2semidirect
6. #22586445C2semidirect
7. C71×#1386C2semidirect
14. M32×D71C4semidirect
17. #17986036C4semidirect
20. #107×D71C4semidirect
25. #106×D71C4semidirect
52. #14556068C2×C4semidirect
56. #14556062C2×C4semidirect
60. #14556060C2×C4semidirect
67. M32×C71C2×C4semidirect
69. #107×C71C2×C4semidirect
72. #106×C71C2×C4semidirect
187. D71#1386direct (N × H)
194. #1386D71direct (N × H)
195. C71C2×#1386semidirect
246. M32C4×D71semidirect
252. #107C4×D71semidirect
260. #106C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×M32×D712
#427×D712
C2×#106×D712
#225864472
#225864462
#225864452
C71×#13862
C2×#138671