Finite Group Databasebeta

← C71×#9941 #31830813

Subgroups — C71×#9941 #31830813

Order 18176.

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

normal subgroups only

This group has 600 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×#3094C211yesmaximal, characteristic
2C71×#2591C211yesmaximal, characteristic
3C71×#2484C211yesmaximal, characteristic, complemented
4C71×#1826C211yesmaximal, characteristic, complemented
5C71×#1510C211yesmaximal, characteristic
6C71×#1463C211yesmaximal, characteristic, complemented
7C71×#1284C211yesmaximal, characteristic, complemented
8C2×#121×C71C2211yescharacteristic
9C2×#117×C71C2211yescharacteristic
10C22×C4⋊C4×C71C2211yescharacteristic
11#482×C71—22no—
12#481×C71—21no—
13#480×C71—21no—
14#428×C71—21no—
15#418×C71—21no—
16#407×C71C2211yescharacteristic
17#407×C71C2211yescharacteristic
18#400×C71—21no—
19#400×C71—21no—
20#398×C71—21no—
21#394×C71—21no—
22#388×C71—22no—
23#387×C71—22no—
24#386×C71—21no—
25#386×C71—21no—
26#385×C71—21no—
27#381×C71—21no—
28#381×C71C2211yescharacteristic
29C4×C22⋊C4×C71C2211yescharacteristic
30#339×C71—22no—
31#327×C71—22no—
32#324×C71—22no—
33C22×D4×C71—21no—
34C22×D4×C71D411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C23×C4×C71C2311yescharacteristic
38C2×M16×C71—21no—
39C2×M16×C71—21no—
40#121×C71—21no—
41#121×C71—21no—
42#121×C71—21no—
43#121×C71—21no—
44#117×C71—21no—
45#117×C71—21no—
46#117×C71—21no—
47#117×C71—21no—
48C2×C4⋊C4×C71—22no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71—21no—
51C2×C4⋊C4×C71—21no—
52C2×C4⋊C4×C71—21no—
53C2×C4⋊C4×C71—21no—
54C2×C4⋊C4×C71—21no—
55C2×C4⋊C4×C71D411yescharacteristic
56C2×C4⋊C4×C71D411yescharacteristic
57C2×C4⋊C4×C71D411yescharacteristic
58C2×C22⋊C4×C71—22no—
59C2×C22⋊C4×C71—21no—
60C2×C22⋊C4×C71—21no—
61C2×C22⋊C4×C71—21no—
62C2×C22⋊C4×C71—21no—
63C2×C22⋊C4×C71—22no—
64C2×C42×C71—21no—
65C2×C42×C71D412yes—
66#107×C71—22no—
67#106×C71—22no—
68#102×C71—22no—
69#102×C71—22no—
70C22⋊C8×C71—21no—
71C22⋊C8×C71—21no—
72C22⋊C8×C71—21no—
73C22⋊C8×C71D412yes—
74#95×C71—22no—
75#95×C71—22no—
76#95×C71—22no—
77#95×C71—22no—
78#95×C71—22no—
79#95×C71—21no—
80#95×C71—21no—
81#95×C71D412yes—
82C24×C71—21no—
83C24×C71C2×D411yescharacteristic
84C2×D4×C71—21no—
85C2×D4×C71—21no—
86C2×D4×C71—42no—
87C2×D4×C71—44no—
88C2×D4×C71—41no—
89C2×D4×C71—41no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—22no—
94C22×C4×C71—21no—
95C22×C4×C71—21no—
96C22×C4×C71—21no—
97C22×C4×C71—22no—
98C22×C4×C71—21no—
99C22×C4×C71—21no—
100C22×C4×C71—21no—
101C22×C4×C71—42no—
102C22×C4×C71—42no—
103C22×C4×C71—42no—
104C22×C4×C71C2×D411yescharacteristic
105C22×C4×C71C2×D411yescharacteristic
106C22×C4×C71C2×D411yescharacteristic
107C22×C4×C71C2×D411yescharacteristic
108C22×C4×C71C2×D411yescharacteristic
109C22×C4×C71#4111yescharacteristic
110M16×C71—42no—
111M16×C71—42no—
112C2×C8×C71—22no—
113C2×C8×C71—41no—
114C4⋊C4×C71—21no—
115C4⋊C4×C71—21no—
116C4⋊C4×C71—21no—
117C4⋊C4×C71—21no—
118C4⋊C4×C71—21no—
119C4⋊C4×C71—21no—
120C4⋊C4×C71—44no—
121C4⋊C4×C71—41no—
122C4⋊C4×C71—41no—
123C4⋊C4×C71—41no—
124C4⋊C4×C71—41no—
125C22⋊C4×C71—44no—
126C22⋊C4×C71—42no—
127C22⋊C4×C71—42no—
128C22⋊C4×C71—41no—
129C22⋊C4×C71—41no—
130C22⋊C4×C71—41no—
131C22⋊C4×C71—41no—
132C22⋊C4×C71—82no—
133C42×C71—42no—
134C42×C71—41no—
135C42×C71—41no—
136C23×C71—22no—
137C23×C71—22no—
138C23×C71—41no—
139C23×C71—42no—
140C23×C71—42no—
141C23×C71—41no—
142C23×C71—42no—
143C23×C71#12011yescharacteristic
144C23×C71#12011yescharacteristic
145C23×C71#12111yescharacteristic
146D4×C71—42no—
147D4×C71—81no—
148D4×C71—82no—
149D4×C71—82no—
150C2×C4×C71—22no—
151C2×C4×C71—22no—
152C2×C4×C71—22no—
153C2×C4×C71—22no—
154C2×C4×C71—22no—
155C2×C4×C71—41no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—41no—
160C2×C4×C71—41no—
161C2×C4×C71—41no—
162C2×C4×C71—41no—
163C2×C4×C71—42no—
164C2×C4×C71—42no—
165C2×C4×C71—42no—
166C2×C4×C71—42no—
167C2×C4×C71—42no—
168C2×C4×C71—42no—
169C2×C4×C71—42no—
170C2×C4×C71—41no—
171C2×C4×C71—41no—
172C2×C4×C71—41no—
173C2×C4×C71—41no—
174C2×C4×C71—42no—
175C2×C4×C71—42no—
176C2×C4×C71—42no—
177C2×C4×C71—42no—
178C2×C4×C71—81no—
179C2×C4×C71#12011yescharacteristic
180C2×C4×C71#12111yescharacteristic
181C2×C4×C71#12111yescharacteristic
182C2×C4×C71#12711yescharacteristic
183C8×C71—42no—
184C8×C71—81no—
185C22×C71—21no—
186C22×C71—21no—
187C22×C71—21no—
188C22×C71—21no—
189C22×C71—21no—
190C22×C71—21no—
191C22×C71—21no—
192C22×C71—44no—
193C22×C71—84no—
194C22×C71—81no—
195C22×C71—81no—
196C22×C71—81no—
197C22×C71#39211yescharacteristic
198C22×C71#45412yes—
199C22×C71#45812yes—
200C4×C71—21no—
201C4×C71—21no—
202C4×C71—21no—
203C4×C71—21no—
204C4×C71—42no—
205C4×C71—82no—
206C4×C71—82no—
207C4×C71—81no—
208C4×C71—81no—
209C4×C71—81no—
210C4×C71—81no—
211#9941C7111yesmaximal, characteristic, direct product factor, Sylow
212C2×C71—22no—
213C2×C71—22no—
214C2×C71—41no—
215C2×C71—161no—
216C2×C71#200311yescharacteristic
217C2×C71#201011yescharacteristic
218C2×C71#202211yescharacteristic
219#3094C2×C7111yescharacteristic
220#2591C2×C7111yescharacteristic
221#2484C2×C7111yescharacteristic, complemented
222#1826C2×C7111yescharacteristic, complemented
223#1510C2×C7111yescharacteristic
224#1463C2×C7111yescharacteristic, complemented
225#1284C2×C7111yescharacteristic, complemented
226C71#994111yescharacteristic, direct product factor, Sylow
227C2×#121C22×C7111yescharacteristic
228C2×#117C22×C7111yescharacteristic
229C22×C4⋊C4C22×C7111yescharacteristic
230#482—22no—
231#481—21no—
232#480—21no—
233#428—21no—
234#418—21no—
235#407C22×C7111yescharacteristic
236#407C22×C7111yescharacteristic
237#400—21no—
238#400—21no—
239#398—21no—
240#394—21no—
241#388—22no—
242#387—22no—
243#386—21no—
244#386—21no—
245#385—21no—
246#381—21no—
247#381C22×C7111yescharacteristic
248C4×C22⋊C4C22×C7111yescharacteristic
249#339—22no—
250#327—22no—
251#324—22no—
252C22×D4—21no—
253C22×D4D4×C7111yescharacteristic
254C23×C4D4×C7111yescharacteristic
255C23×C4D4×C7111yescharacteristic
256C23×C4C23×C7111yescharacteristic
257C2×M16—21no—
258C2×M16—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4D4×C7111yescharacteristic
275C2×C4⋊C4D4×C7111yescharacteristic
276C2×C4⋊C4D4×C7111yescharacteristic
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C42—21no—
284C2×C42D4×C7112yes—
285#107—22no—
286#106—22no—
287#102—22no—
288#102—22no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8D4×C7112yes—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95D4×C7112yes—
301C24—21no—
302C24C2×D4×C7111yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C2×D4×C7111yescharacteristic
324C22×C4C2×D4×C7111yescharacteristic
325C22×C4C2×D4×C7111yescharacteristic
326C22×C4C2×D4×C7111yescharacteristic
327C22×C4C2×D4×C7111yescharacteristic
328C22×C4#41×C7111yescharacteristic
329M16—42no—
330M16—42no—
331C2×C8—22no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—44no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—82no—
352C42—42no—
353C42—41no—
354C42—41no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23#120×C7111yescharacteristic
363C23#120×C7111yescharacteristic
364C23#121×C7111yescharacteristic
365D4—42no—
366D4—81no—
367D4—82no—
368D4—82no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4#120×C7111yescharacteristic
399C2×C4#121×C7111yescharacteristic
400C2×C4#121×C7111yescharacteristic
401C2×C4#127×C7111yescharacteristic
402C8—42no—
403C8—81no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—44no—
412C22—84no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#392×C7111yescharacteristic
417C22#454×C7112yes—
418C22#458×C7112yes—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—82no—
425C4—82no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C2—22no—
431C2—22no—
432C2—41no—
433C2—161no—
434C2C71×#200311yescharacteristic
435C2C71×#201011yescharacteristic
436C2C71×#202211yescharacteristic
subgroupG/H
center, socleC22×C71#392
commutator, Frattini, cosocleC23×C4C23×C71
prime radicalC2×#121×C71C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
3. C71×#2484C2semidirect
4. C71×#1826C2semidirect
6. C71×#1463C2semidirect
7. C71×#1284C2semidirect
211. #9941C71direct (N × H)
221. #2484C2×C71semidirect
222. #1826C2×C71semidirect
224. #1463C2×C71semidirect
225. #1284C2×C71semidirect
226. C71#9941direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#30942
C71×#25912
C71×#24842
C71×#18262
C71×#15102
C71×#14632
C71×#12842
#994171