Finite Group Databasebeta

← C71×#7735 #31828607

Subgroups — C71×#7735 #31828607

Order 18176.

604 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C71×#2732C211yesmaximal, characteristic, complemented
2C71×#2041C211yesmaximal, characteristic, complemented
3C71×#2014C211yesmaximal, characteristic, complemented
4C71×#2011C211yesmaximal, characteristic
5C71×#1272C211yesmaximal, characteristic, complemented
6C71×#1271C211yesmaximal, characteristic, complemented
7C71×#1270C211yesmaximal, characteristic, complemented
8C22×SD16×C71—21no—
9C22×D8×C71—21no—
10C2×#127×C71C2211yescharacteristic
11C2×C22⋊Q8×C71—21no—
12C2×#102×C71—21no—
13C2×#102×C71—21no—
14C2×#102×C71—21no—
15C2×#102×C71—21no—
16#399×C71—21no—
17#398×C71—21no—
18#396×C71—21no—
19#391×C71C2211yescharacteristic
20#390×C71C2211yescharacteristic
21#389×C71C2211yescharacteristic
22#386×C71—21no—
23#385×C71—21no—
24#336×C71C2211yescharacteristic
25#336×C71C2211yescharacteristic
26#336×C71C2211yescharacteristic
27C22×Q8×C71D411yescharacteristic
28C22×D4×C71—21no—
29C22×D4×C71—21no—
30C22×D4×C71—21no—
31C22×D4×C71D411yescharacteristic
32C23×C4×C71—21no—
33C2×SD16×C71—42no—
34C2×SD16×C71—42no—
35C2×SD16×C71—42no—
36C2×D8×C71—42no—
37C2×D8×C71—42no—
38C2×D8×C71—42no—
39C22×C8×C71—21no—
40C22×C8×C71—21no—
41C22×C8×C71—21no—
42#127×C71—42no—
43C22⋊Q8×C71—42no—
44C22⋊Q8×C71—42no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C4⋊C4×C71D411yescharacteristic
52C2×C4⋊C4×C71D411yescharacteristic
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—21no—
56C2×C22⋊C4×C71—21no—
57C2×C22⋊C4×C71—21no—
58C2×C42×C71C2311yescharacteristic
59#102×C71—42no—
60#102×C71—42no—
61#102×C71—42no—
62#102×C71—42no—
63#95×C71—21no—
64#95×C71—21no—
65#95×C71—21no—
66#95×C71—21no—
67C24×C71—21no—
68C24×C71—41no—
69C2×Q8×C71—21no—
70C2×Q8×C71—41no—
71C2×Q8×C71—41no—
72C2×Q8×C71D812yescomplemented
73C2×D4×C71—22no—
74C2×D4×C71—22no—
75C2×D4×C71—22no—
76C2×D4×C71—21no—
77C2×D4×C71—41no—
78C2×D4×C71—41no—
79C2×D4×C71—41no—
80C2×D4×C71—41no—
81C2×D4×C71—41no—
82C2×D4×C71—41no—
83C2×D4×C71—41no—
84C2×D4×C71—41no—
85C2×D4×C71—41no—
86C2×D4×C71—41no—
87C2×D4×C71—41no—
88C2×D4×C71—41no—
89C2×D4×C71—41no—
90C2×D4×C71—41no—
91C2×D4×C71—41no—
92C2×D4×C71—41no—
93C2×D4×C71—41no—
94C2×D4×C71SD1612yescomplemented
95C22×C4×C71—21no—
96C22×C4×C71—21no—
97C22×C4×C71—21no—
98C22×C4×C71—21no—
99C22×C4×C71—21no—
100C22×C4×C71—42no—
101C22×C4×C71—81no—
102C22×C4×C71—81no—
103C22×C4×C71C2×D411yescharacteristic
104C22×C4×C71C2×D411yescharacteristic
105C22×C4×C71C2×D411yescharacteristic
106SD16×C71—48no—
107D8×C71—48no—
108C2×C8×C71—22no—
109C2×C8×C71—22no—
110C2×C8×C71—41no—
111C2×C8×C71—41no—
112C2×C8×C71—41no—
113C2×C8×C71—41no—
114C2×C8×C71—41no—
115C2×C8×C71—41no—
116C2×C8×C71—41no—
117C4⋊C4×C71—21no—
118C4⋊C4×C71—21no—
119C4⋊C4×C71—41no—
120C4⋊C4×C71—41no—
121C4⋊C4×C71—41no—
122C4⋊C4×C71—42no—
123C4⋊C4×C71—42no—
124C4⋊C4×C71—41no—
125C4⋊C4×C71—41no—
126C4⋊C4×C71—81no—
127C22⋊C4×C71—42no—
128C22⋊C4×C71—41no—
129C22⋊C4×C71—41no—
130C22⋊C4×C71—41no—
131C22⋊C4×C71—41no—
132C22⋊C4×C71—81no—
133C22⋊C4×C71—81no—
134C42×C71—41no—
135C23×C71—22no—
136C23×C71—41no—
137C23×C71—41no—
138C23×C71—41no—
139C23×C71—41no—
140C23×C71—41no—
141C23×C71—41no—
142C23×C71—42no—
143C23×C71—81no—
144C23×C71—81no—
145C23×C71—81no—
146C23×C71—81no—
147C23×C71—81no—
148C23×C71—81no—
149C23×C71#12011yescharacteristic
150Q8×C71—42no—
151Q8×C71—81no—
152D4×C71—42no—
153D4×C71—42no—
154D4×C71—42no—
155D4×C71—42no—
156D4×C71—42no—
157D4×C71—42no—
158D4×C71—42no—
159D4×C71—81no—
160D4×C71—81no—
161D4×C71—81no—
162D4×C71—81no—
163D4×C71—81no—
164D4×C71—81no—
165D4×C71—81no—
166C2×C4×C71—21no—
167C2×C4×C71—21no—
168C2×C4×C71—21no—
169C2×C4×C71—21no—
170C2×C4×C71—21no—
171C2×C4×C71—21no—
172C2×C4×C71—22no—
173C2×C4×C71—41no—
174C2×C4×C71—41no—
175C2×C4×C71—41no—
176C2×C4×C71—41no—
177C2×C4×C71—41no—
178C2×C4×C71—41no—
179C2×C4×C71—41no—
180C2×C4×C71—41no—
181C2×C4×C71—41no—
182C2×C4×C71—41no—
183C2×C4×C71—41no—
184C2×C4×C71—41no—
185C2×C4×C71—41no—
186C2×C4×C71—41no—
187C2×C4×C71—84no—
188C2×C4×C71C2×D811yescharacteristic
189C2×C4×C71C2×SD1611yescharacteristic
190C2×C4×C71#13511yescharacteristic
191C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C71#13711yescharacteristic
194C8×C71—42no—
195C8×C71—42no—
196C8×C71—81no—
197C22×C71—42no—
198C22×C71—42no—
199C22×C71—42no—
200C22×C71—81no—
201C22×C71—81no—
202C22×C71—81no—
203C22×C71—81no—
204C22×C71—82no—
205C22×C71—82no—
206C22×C71—82no—
207C22×C71—161no—
208C22×C71—161no—
209C22×C71—161no—
210C22×C71—161no—
211C22×C71#44711yescharacteristic
212C22×C71#44811yescharacteristic
213C22×C71#44911yescharacteristic
214C22×C71#45311yescharacteristic
215C22×C71#45311yescharacteristic
216C22×C71#45411yescharacteristic
217C22×C71B(4,𝔽2)11yescharacteristic
218C4×C71—21no—
219C4×C71—21no—
220C4×C71—21no—
221C4×C71—21no—
222C4×C71—21no—
223C4×C71—21no—
224C4×C71—42no—
225C4×C71—81no—
226C4×C71—81no—
227C4×C71—81no—
228C4×C71—81no—
229#7735C7111yesmaximal, characteristic, direct product factor, Sylow
230C2×C71—82no—
231C2×C71—162no—
232C2×C71#159511yescharacteristic
233C2×C71#159611yescharacteristic
234C2×C71#159711yescharacteristic
235C2×C71#162111yescharacteristic
236C2×C71#162211yescharacteristic
237C2×C71#162711yescharacteristic
238C2×C71#165611yescharacteristic
239#2732C2×C7111yescharacteristic, complemented
240#2041C2×C7111yescharacteristic, complemented
241#2014C2×C7111yescharacteristic, complemented
242#2011C2×C7111yescharacteristic
243#1272C2×C7111yescharacteristic, complemented
244#1271C2×C7111yescharacteristic, complemented
245#1270C2×C7111yescharacteristic, complemented
246C71#773511yescharacteristic, direct product factor, Sylow
247C22×SD16—21no—
248C22×D8—21no—
249C2×#127C22×C7111yescharacteristic
250C2×C22⋊Q8—21no—
251C2×#102—21no—
252C2×#102—21no—
253C2×#102—21no—
254C2×#102—21no—
255#399—21no—
256#398—21no—
257#396—21no—
258#391C22×C7111yescharacteristic
259#390C22×C7111yescharacteristic
260#389C22×C7111yescharacteristic
261#386—21no—
262#385—21no—
263#336C22×C7111yescharacteristic
264#336C22×C7111yescharacteristic
265#336C22×C7111yescharacteristic
266C22×Q8D4×C7111yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4D4×C7111yescharacteristic
271C23×C4—21no—
272C2×SD16—42no—
273C2×SD16—42no—
274C2×SD16—42no—
275C2×D8—42no—
276C2×D8—42no—
277C2×D8—42no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281#127—42no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4D4×C7111yescharacteristic
289C2×C4⋊C4D4×C7111yescharacteristic
290C2×C4⋊C4D4×C7111yescharacteristic
291C2×C4⋊C4D4×C7111yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42C23×C7111yescharacteristic
298#102—42no—
299#102—42no—
300#102—42no—
301#102—42no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306C24—21no—
307C24—41no—
308C2×Q8—21no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8D8×C7112yescomplemented
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4SD16×C7112yescomplemented
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—42no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4C2×D4×C7111yescharacteristic
343C22×C4C2×D4×C7111yescharacteristic
344C22×C4C2×D4×C7111yescharacteristic
345SD16—48no—
346D8—48no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—81no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—81no—
372C22⋊C4—81no—
373C42—41no—
374C23—22no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23#120×C7111yescharacteristic
389Q8—42no—
390Q8—81no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—81no—
404D4—81no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—84no—
427C2×C4C2×D8×C7111yescharacteristic
428C2×C4C2×SD16×C7111yescharacteristic
429C2×C4#135×C7111yescharacteristic
430C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
431C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
432C2×C4#137×C7111yescharacteristic
433C8—42no—
434C8—42no—
435C8—81no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22#447×C7111yescharacteristic
451C22#448×C7111yescharacteristic
452C22#449×C7111yescharacteristic
453C22#453×C7111yescharacteristic
454C22#453×C7111yescharacteristic
455C22#454×C7111yescharacteristic
456C22B(4,𝔽2)×C7111yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C2—82no—
469C2—162no—
470C2C71×#159511yescharacteristic
471C2C71×#159611yescharacteristic
472C2C71×#159711yescharacteristic
473C2C71×#162111yescharacteristic
474C2C71×#162211yescharacteristic
475C2C71×#162711yescharacteristic
476C2C71×#165611yescharacteristic
subgroupG/H
center, socleC23×C71#120
commutator, Frattini, cosocleC2×C42C23×C71
prime radicalC71×#2011C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. C71×#2732C2semidirect
2. C71×#2041C2semidirect
3. C71×#2014C2semidirect
5. C71×#1272C2semidirect
6. C71×#1271C2semidirect
7. C71×#1270C2semidirect
72. C2×Q8×C71D8semidirect
94. C2×D4×C71SD16semidirect
229. #7735C71direct (N × H)
239. #2732C2×C71semidirect
240. #2041C2×C71semidirect
241. #2014C2×C71semidirect
243. #1272C2×C71semidirect
244. #1271C2×C71semidirect
245. #1270C2×C71semidirect
246. C71#7735direct (N × H)
311. C2×Q8D8×C71semidirect
333. C2×D4SD16×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#27322
C71×#20412
C71×#20142
C71×#20112
C71×#12722
C71×#12712
C71×#12702
#773571