Finite Group Databasebeta

← C71×#34954 #31855826

Subgroups — C71×#34954 #31855826

Order 18176.

1948 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#2593C214yesmaximal, complemented
2C71×#2582C211yesmaximal, characteristic, complemented
3C71×#2571C211yesmaximal, characteristic, complemented
4C71×#2548C212yesmaximal
5C71×#2513C214yesmaximal, complemented
6C71×#2448C214yesmaximal, complemented
7C71×#2395C214yesmaximal, complemented
8C4×#127×C71C211yesmaximal, characteristic
9C4×#124×C71C214yesmaximal, complemented
10C71×#2293C214yesmaximal, complemented
11C42×Q8×C71C212yesmaximal, complemented
12C2×#127×C71C2211yescharacteristic
13C2×#124×C71C2214yes—
14C2×C4×Q8×C71C2212yes—
15C2×C4×Q8×C71C2212yes—
16C2×C4×Q8×C71C2212yes—
17C2×C4×D4×C71C2214yes—
18C2×C4×D4×C71C2212yes—
19C2×#117×C71C2214yes—
20#400×C71C2214yes—
21#399×C71C2214yes—
22#398×C71C2214yes—
23#397×C71C2218yes—
24#397×C71C2214yes—
25#394×C71C2218yes—
26#391×C71C2214yes—
27#390×C71C2214yes—
28#390×C71C2214yes—
29#390×C71C2214yes—
30#390×C71C2212yes—
31#390×C71C2214yes—
32#389×C71C2214yes—
33#389×C71C2214yes—
34#388×C71C2218yes—
35#388×C71C2218yes—
36#388×C71C2218yes—
37#388×C71C2218yes—
38#384×C71C2211yescharacteristic
39#382×C71C2214yes—
40C4×C4⋊C4×C71C2212yes—
41C4×C4⋊C4×C71C2212yes—
42C4×C4⋊C4×C71C2214yes—
43C4×C4⋊C4×C71C2214yes—
44C4×C4⋊C4×C71C2214yes—
45C4×C4⋊C4×C71C2212yes—
46C4×C22⋊C4×C71C2218yes—
47C4×C22⋊C4×C71C2214yes—
48C43×C71C2211yescharacteristic
49C43×C71C2214yes—
50C22×Q8×C71C2312yes—
51C22×D4×C71C2314yes—
52C22×D4×C71C2312yes—
53C23×C4×C71C2314yes—
54#127×C71—24no—
55#124×C71—216no—
56C4×Q8×C71—28no—
57C4×Q8×C71—28no—
58C4×Q8×C71—28no—
59C4×D4×C71—216no—
60C4×D4×C71—44no—
61#117×C71—48no—
62C2×C4⋊C4×C71C2314yes—
63C2×C4⋊C4×C71C2312yes—
64C2×C4⋊C4×C71C2314yes—
65C2×C4⋊C4×C71C2312yes—
66C2×C4⋊C4×C71C2312yes—
67C2×C4⋊C4×C71C2314yes—
68C2×C4⋊C4×C71C2314yes—
69C2×C4⋊C4×C71C2314yes—
70C2×C4⋊C4×C71C2314yes—
71C2×C4⋊C4×C71C2312yes—
72C2×C22⋊C4×C71C2318yes—
73C2×C22⋊C4×C71C2318yes—
74C2×C22⋊C4×C71C2318yes—
75C2×C22⋊C4×C71C2318yes—
76C2×C22⋊C4×C71C2314yes—
77C2×C22⋊C4×C71C2318yes—
78C2×C42×C71C2312yes—
79C2×C42×C71C2314yes—
80C2×C42×C71C2314yes—
81C2×C42×C71C2314yes—
82C2×C42×C71C2314yes—
83C2×C42×C71C2311yescharacteristic
84C2×C42×C71C2314yes—
85C2×C42×C71C2311yescharacteristic
86C2×C42×C71C2314yes—
87C2×C42×C71C2311yescharacteristic
88C2×C42×C71C2312yes—
89#95×C71C2314yes—
90#95×C71C2318yes—
91#95×C71C2318yes—
92#95×C71C2314yes—
93#95×C71C2314yes—
94#95×C71C2314yes—
95#95×C71C2314yes—
96C24×C71C2414yes—
97C2×Q8×C71—24no—
98C2×Q8×C71—24no—
99C2×Q8×C71#4118yes—
100C2×D4×C71—28no—
101C2×D4×C71—28no—
102C2×D4×C71—24no—
103C2×D4×C71—28no—
104C2×D4×C71—42no—
105C2×D4×C71—42no—
106C22×C4×C71—216no—
107C22×C4×C71—28no—
108C22×C4×C71C2414yes—
109C22×C4×C71C2414yes—
110C22×C4×C71C2414yes—
111C22×C4×C71C2411yescharacteristic
112C22×C4×C71C2414yes—
113C22×C4×C71C2414yes—
114C22×C4×C71C2411yescharacteristic
115C22×C4×C71C2412yes—
116C22×C4×C71C2412yes—
117C22×C4×C71C2411yescharacteristic
118C4⋊C4×C71—28no—
119C4⋊C4×C71—28no—
120C4⋊C4×C71—28no—
121C4⋊C4×C71—28no—
122C4⋊C4×C71—24no—
123C4⋊C4×C71—24no—
124C4⋊C4×C71—24no—
125C4⋊C4×C71—44no—
126C4⋊C4×C71—44no—
127C4⋊C4×C71—42no—
128C22⋊C4×C71—216no—
129C22⋊C4×C71—216no—
130C22⋊C4×C71—216no—
131C22⋊C4×C71—48no—
132C22⋊C4×C71—48no—
133C22⋊C4×C71—44no—
134C42×C71—28no—
135C42×C71—24no—
136C42×C71—28no—
137C42×C71—28no—
138C42×C71—28no—
139C42×C71—28no—
140C42×C71—22no—
141C42×C71—41no—
142C42×C71—44no—
143C42×C71#4118yes—
144C42×C71#4114yes—
145C23×C71—28no—
146C23×C71—28no—
147C23×C71—24no—
148C23×C71—24no—
149C23×C71—24no—
150C23×C71C2511yescharacteristic
151Q8×C71—216no—
152D4×C71—48no—
153D4×C71—416no—
154C2×C4×C71—22no—
155C2×C4×C71—24no—
156C2×C4×C71—24no—
157C2×C4×C71—21no—
158C2×C4×C71—22no—
159C2×C4×C71—21no—
160C2×C4×C71—24no—
161C2×C4×C71—24no—
162C2×C4×C71—24no—
163C2×C4×C71—24no—
164C2×C4×C71—24no—
165C2×C4×C71—22no—
166C2×C4×C71—28no—
167C2×C4×C71—21no—
168C2×C4×C71—24no—
169C2×C4×C71—24no—
170C2×C4×C71—24no—
171C2×C4×C71—24no—
172C2×C4×C71—24no—
173C2×C4×C71—416no—
174C2×C4×C71C2×#4118yes—
175C2×C4×C71C2×#4114yes—
176C2×C4×C71C2×#4112yes—
177C2×C4×C71C2×#4114yes—
178C2×C4×C71C2×#4114yes—
179C2×C4×C71C2×#4114yes—
180C2×C4×C71C2×#4114yes—
181C2×C4×C71H(5,𝔽2)12yes—
182C22×C71—48no—
183C22×C71—44no—
184C22×C71—44no—
185C22×C71—48no—
186C22×C71—44no—
187C22×C71C22×#4111yescharacteristic
188C22×C71C22×#4112yes—
189C22×C71C22×#4112yes—
190C22×C71C2×H(5,𝔽2)11yescharacteristic
191C22×C71#58511yescharacteristic
192C4×C71—28no—
193C4×C71—22no—
194C4×C71—24no—
195C4×C71—44no—
196C4×C71—44no—
197C4×C71—44no—
198C4×C71—44no—
199C4×C71—41no—
200C4×C71#52918yes—
201C4×C71#55914yes—
202#34954C7111yesmaximal, characteristic, direct product factor, Sylow
203C2×C71—84no—
204C2×C71C2×#52912yes—
205C2×C71#346011yescharacteristic
206C2×C71C2×#55911yescharacteristic
207C2×C71#348012yes—
208C2×C71#348311yescharacteristic
209#2593C2×C7114yescomplemented
210#2582C2×C7111yescharacteristic, complemented
211#2571C2×C7111yescharacteristic, complemented
212#2548C2×C7112yes—
213#2513C2×C7114yescomplemented
214#2448C2×C7114yescomplemented
215#2395C2×C7114yescomplemented
216C4×#127C2×C7111yescharacteristic
217C4×#124C2×C7114yescomplemented
218#2293C2×C7114yescomplemented
219C42×Q8C2×C7112yescomplemented
220C71#3495411yescharacteristic, direct product factor, Sylow
221C2×#127C22×C7111yescharacteristic
222C2×#124C22×C7114yes—
223C2×C4×Q8C22×C7112yes—
224C2×C4×Q8C22×C7112yes—
225C2×C4×Q8C22×C7112yes—
226C2×C4×D4C22×C7114yes—
227C2×C4×D4C22×C7112yes—
228C2×#117C22×C7114yes—
229#400C22×C7114yes—
230#399C22×C7114yes—
231#398C22×C7114yes—
232#397C22×C7118yes—
233#397C22×C7114yes—
234#394C22×C7118yes—
235#391C22×C7114yes—
236#390C22×C7112yes—
237#390C22×C7114yes—
238#390C22×C7114yes—
239#390C22×C7114yes—
240#390C22×C7114yes—
241#389C22×C7114yes—
242#389C22×C7114yes—
243#388C22×C7118yes—
244#388C22×C7118yes—
245#388C22×C7118yes—
246#388C22×C7118yes—
247#384C22×C7111yescharacteristic
248#382C22×C7114yes—
249C4×C4⋊C4C22×C7112yes—
250C4×C4⋊C4C22×C7114yes—
251C4×C4⋊C4C22×C7114yes—
252C4×C4⋊C4C22×C7114yes—
253C4×C4⋊C4C22×C7112yes—
254C4×C4⋊C4C22×C7112yes—
255C4×C22⋊C4C22×C7118yes—
256C4×C22⋊C4C22×C7114yes—
257C43C22×C7111yescharacteristic
258C43C22×C7114yes—
259C22×Q8C23×C7112yes—
260C22×D4C23×C7114yes—
261C22×D4C23×C7112yes—
262C23×C4C23×C7114yes—
263#127—24no—
264#124—216no—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—28no—
268C4×D4—216no—
269C4×D4—44no—
270#117—48no—
271C2×C4⋊C4C23×C7114yes—
272C2×C4⋊C4C23×C7112yes—
273C2×C4⋊C4C23×C7114yes—
274C2×C4⋊C4C23×C7112yes—
275C2×C4⋊C4C23×C7112yes—
276C2×C4⋊C4C23×C7114yes—
277C2×C4⋊C4C23×C7114yes—
278C2×C4⋊C4C23×C7114yes—
279C2×C4⋊C4C23×C7114yes—
280C2×C4⋊C4C23×C7112yes—
281C2×C22⋊C4C23×C7118yes—
282C2×C22⋊C4C23×C7118yes—
283C2×C22⋊C4C23×C7118yes—
284C2×C22⋊C4C23×C7118yes—
285C2×C22⋊C4C23×C7114yes—
286C2×C22⋊C4C23×C7118yes—
287C2×C42C23×C7114yes—
288C2×C42C23×C7111yescharacteristic
289C2×C42C23×C7114yes—
290C2×C42C23×C7112yes—
291C2×C42C23×C7114yes—
292C2×C42C23×C7114yes—
293C2×C42C23×C7114yes—
294C2×C42C23×C7111yescharacteristic
295C2×C42C23×C7114yes—
296C2×C42C23×C7112yes—
297C2×C42C23×C7111yescharacteristic
298#95C23×C7114yes—
299#95C23×C7118yes—
300#95C23×C7118yes—
301#95C23×C7114yes—
302#95C23×C7114yes—
303#95C23×C7114yes—
304#95C23×C7114yes—
305C24C24×C7114yes—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8#41×C7118yes—
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—42no—
314C2×D4—42no—
315C22×C4—216no—
316C22×C4—28no—
317C22×C4C24×C7114yes—
318C22×C4C24×C7114yes—
319C22×C4C24×C7114yes—
320C22×C4C24×C7111yescharacteristic
321C22×C4C24×C7114yes—
322C22×C4C24×C7114yes—
323C22×C4C24×C7111yescharacteristic
324C22×C4C24×C7112yes—
325C22×C4C24×C7112yes—
326C22×C4C24×C7111yescharacteristic
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C22⋊C4—216no—
338C22⋊C4—216no—
339C22⋊C4—216no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—44no—
343C42—28no—
344C42—24no—
345C42—28no—
346C42—28no—
347C42—28no—
348C42—28no—
349C42—22no—
350C42—41no—
351C42—44no—
352C42#41×C7118yes—
353C42#41×C7114yes—
354C23—28no—
355C23—28no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23C25×C7111yescharacteristic
360Q8—216no—
361D4—48no—
362D4—416no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—28no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—416no—
383C2×C4C2×#41×C7118yes—
384C2×C4C2×#41×C7114yes—
385C2×C4C2×#41×C7112yes—
386C2×C4C2×#41×C7114yes—
387C2×C4C2×#41×C7114yes—
388C2×C4C2×#41×C7114yes—
389C2×C4C2×#41×C7114yes—
390C2×C4H(5,𝔽2)×C7112yes—
391C22—48no—
392C22—44no—
393C22—44no—
394C22—48no—
395C22—44no—
396C22C22×#41×C7111yescharacteristic
397C22C22×#41×C7112yes—
398C22C22×#41×C7112yes—
399C22C2×H(5,𝔽2)×C7111yescharacteristic
400C22#585×C7111yescharacteristic
401C4—28no—
402C4—22no—
403C4—24no—
404C4—44no—
405C4—44no—
406C4—44no—
407C4—44no—
408C4—41no—
409C4#529×C7118yes—
410C4#559×C7114yes—
411C2—84no—
412C2C2×#529×C7112yes—
413C2C71×#346011yescharacteristic
414C2C2×#559×C7111yescharacteristic
415C2C71×#348012yes—
416C2C71×#348311yescharacteristic
subgroupG/H
centerC22×C4×C71C24
commutator, Frattini, cosocleC23C25×C71
socleC23×C71C25
prime radicalC2×#127×C71C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#2593C2semidirect
2. C71×#2582C2semidirect
3. C71×#2571C2semidirect
5. C71×#2513C2semidirect
6. C71×#2448C2semidirect
7. C71×#2395C2semidirect
9. C4×#124×C71C2semidirect
10. C71×#2293C2semidirect
11. C42×Q8×C71C2semidirect
202. #34954C71direct (N × H)
209. #2593C2×C71semidirect
210. #2582C2×C71semidirect
211. #2571C2×C71semidirect
213. #2513C2×C71semidirect
214. #2448C2×C71semidirect
215. #2395C2×C71semidirect
217. C4×#124C2×C71semidirect
218. #2293C2×C71semidirect
219. C42×Q8C2×C71semidirect
220. C71#34954direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#25932
C71×#25822
C71×#25712
C71×#25482
C71×#25132
C71×#24482
C71×#23952
C4×#127×C712
C4×#124×C712
C71×#22932
C42×Q8×C712
#3495471