Finite Group Databasebeta

← C71×#9291 #31830163

Subgroups — C71×#9291 #31830163

Order 18176.

494 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3226C211yesmaximal, characteristic
2C71×#2932C211yesmaximal, characteristic
3C71×#1960C211yesmaximal, characteristic, complemented
4C71×#1546C211yesmaximal, characteristic, complemented
5C71×#1517C211yesmaximal, characteristic
6C71×#1286C211yesmaximal, characteristic, complemented
7C71×#1275C211yesmaximal, characteristic, complemented
8C2×#131×C71—21no—
9#532×C71C2211yescharacteristic
10C2×C4⋊Q8×C71C2211yescharacteristic
11#500×C71C412yescomplemented
12#486×C71C412yescomplemented
13#480×C71—22no—
14#474×C71—22no—
15#435×C71—22no—
16#435×C71—22no—
17#431×C71C2211yescharacteristic
18#431×C71C2211yescharacteristic
19#423×C71C2211yescharacteristic
20C2×#103×C71—21no—
21C2×#103×C71—21no—
22#408×C71—21no—
23C2×#97×C71C2211yescharacteristic
24#382×C71C2211yescharacteristic
25#332×C71—22no—
26#327×C71—22no—
27C2×#41×C71—21no—
28C22×Q8×C71—21no—
29#131×C71—42no—
30#131×C71—42no—
31C2×M16×C71—22no—
32C2×M16×C71—21no—
33C22×C8×C71—22no—
34C22×C8×C71D411yescharacteristic
35C22×C8×C71D411yescharacteristic
36C4⋊Q8×C71—21no—
37C4⋊Q8×C71—21no—
38C4⋊Q8×C71—21no—
39C4⋊Q8×C71C2×C412yes—
40C4⋊Q8×C71C2×C411yescharacteristic
41#127×C71C2×C411yescharacteristic
42#124×C71—21no—
43#121×C71—21no—
44#121×C71D412yes—
45C4×D4×C71—21no—
46C4×D4×C71D412yes—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71—21no—
50C2×C42×C71C2311yescharacteristic
51#107×C71—24no—
52C4⋊C8×C71—21no—
53C4⋊C8×C71—21no—
54C4⋊C8×C71—21no—
55C4⋊C8×C71D412yes—
56#103×C71—42no—
57#103×C71—42no—
58#102×C71—24no—
59C22⋊C8×C71—24no—
60C22⋊C8×C71—21no—
61C22⋊C8×C71—21no—
62#97×C71—21no—
63#97×C71—21no—
64C4×C8×C71—21no—
65C4×C8×C71C2×C412yes—
66#95×C71—21no—
67#95×C71—21no—
68#41×C71—41no—
69#41×C71—41no—
70#41×C71—41no—
71#41×C71—41no—
72C2×Q8×C71—21no—
73C2×Q8×C71—22no—
74C2×Q8×C71—41no—
75C2×Q8×C71—41no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71—41no—
79C2×D4×C71—22no—
80C2×D4×C71—21no—
81C2×D4×C71—21no—
82C2×D4×C71C22⋊C412yes—
83C22×C4×C71—22no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71#4111yescharacteristic
89M16×C71—44no—
90M16×C71—41no—
91M16×C71—41no—
92C2×C8×C71—21no—
93C2×C8×C71—21no—
94C2×C8×C71—22no—
95C2×C8×C71—21no—
96C2×C8×C71—22no—
97C2×C8×C71—21no—
98C2×C8×C71—21no—
99C2×C8×C71—44no—
100C2×C8×C71—41no—
101C2×C8×C71D812yes—
102C4⋊C4×C71—21no—
103C4⋊C4×C71—22no—
104C4⋊C4×C71—22no—
105C4⋊C4×C71—41no—
106C4⋊C4×C71—41no—
107C4⋊C4×C71—41no—
108C4⋊C4×C71—41no—
109C4⋊C4×C71C22⋊C412yes—
110C22⋊C4×C71—22no—
111C22⋊C4×C71—22no—
112C42×C71C22×C411yescharacteristic
113C42×C71C2×D411yescharacteristic
114C42×C71C2×D411yescharacteristic
115C42×C71#4111yescharacteristic
116C23×C71—22no—
117C23×C71#12111yescharacteristic
118Q8×C71—41no—
119Q8×C71—41no—
120Q8×C71—42no—
121Q8×C71—42no—
122Q8×C71—81no—
123Q8×C71—81no—
124D4×C71—42no—
125D4×C71—42no—
126D4×C71—42no—
127D4×C71—41no—
128D4×C71—41no—
129D4×C71—81no—
130C2×C4×C71—22no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—42no—
134C2×C4×C71—42no—
135C2×C4×C71—42no—
136C2×C4×C71—81no—
137C2×C4×C71—81no—
138C2×C4×C71—81no—
139C2×C4×C71—81no—
140C2×C4×C71—81no—
141C2×C4×C71C2×C22⋊C411yescharacteristic
142C2×C4×C71C4×D411yescharacteristic
143C2×C4×C71C4×D411yescharacteristic
144C2×C4×C71#12011yescharacteristic
145C2×C4×C71#12111yescharacteristic
146C2×C4×C71#12311yescharacteristic
147C2×C4×C71C2×D811yescharacteristic
148C2×C4×C71#13511yescharacteristic
149C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
151C8×C71—22no—
152C8×C71—42no—
153C8×C71—42no—
154C8×C71—41no—
155C8×C71—81no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—42no—
159C22×C71—42no—
160C22×C71—42no—
161C22×C71#38611yescharacteristic
162C22×C71#45911yescharacteristic
163C22×C71#46311yescharacteristic
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—42no—
169C4×C71—82no—
170C4×C71—161no—
171C4×C71C4×D811yescharacteristic
172C4×C71#44211yescharacteristic
173C4×C71#44711yescharacteristic
174C4×C71#44911yescharacteristic
175#9291C7111yesmaximal, characteristic, direct product factor, Sylow
176C2×C71—21no—
177C2×C71—21no—
178C2×C71—82no—
179C2×C71#187911yescharacteristic
180C2×C71#190211yescharacteristic
181C2×C71#190911yescharacteristic
182#3226C2×C7111yescharacteristic
183#2932C2×C7111yescharacteristic
184#1960C2×C7111yescharacteristic, complemented
185#1546C2×C7111yescharacteristic, complemented
186#1517C2×C7111yescharacteristic
187#1286C2×C7111yescharacteristic, complemented
188#1275C2×C7111yescharacteristic, complemented
189C71#929111yescharacteristic, direct product factor, Sylow
190C2×#131—21no—
191#532C22×C7111yescharacteristic
192C2×C4⋊Q8C22×C7111yescharacteristic
193#500C4×C7112yescomplemented
194#486C4×C7112yescomplemented
195#480—22no—
196#474—22no—
197#435—22no—
198#435—22no—
199#431C22×C7111yescharacteristic
200#431C22×C7111yescharacteristic
201#423C22×C7111yescharacteristic
202C2×#103—21no—
203C2×#103—21no—
204#408—21no—
205C2×#97C22×C7111yescharacteristic
206#382C22×C7111yescharacteristic
207#332—22no—
208#327—22no—
209C2×#41—21no—
210C22×Q8—21no—
211#131—42no—
212#131—42no—
213C2×M16—22no—
214C2×M16—21no—
215C22×C8—22no—
216C22×C8D4×C7111yescharacteristic
217C22×C8D4×C7111yescharacteristic
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8C2×C4×C7112yes—
222C4⋊Q8C2×C4×C7111yescharacteristic
223#127C2×C4×C7111yescharacteristic
224#124—21no—
225#121—21no—
226#121D4×C7112yes—
227C4×D4—21no—
228C4×D4D4×C7112yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C42C23×C7111yescharacteristic
233#107—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8D4×C7112yes—
238#103—42no—
239#103—42no—
240#102—24no—
241C22⋊C8—24no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244#97—21no—
245#97—21no—
246C4×C8—21no—
247C4×C8C2×C4×C7112yes—
248#95—21no—
249#95—21no—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4C22⋊C4×C7112yes—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C2×D4×C7111yescharacteristic
269C22×C4C2×D4×C7111yescharacteristic
270C22×C4#41×C7111yescharacteristic
271M16—44no—
272M16—41no—
273M16—41no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—44no—
282C2×C8—41no—
283C2×C8D8×C7112yes—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4C22⋊C4×C7112yes—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C42C22×C4×C7111yescharacteristic
295C42C2×D4×C7111yescharacteristic
296C42C2×D4×C7111yescharacteristic
297C42#41×C7111yescharacteristic
298C23—22no—
299C23#121×C7111yescharacteristic
300Q8—41no—
301Q8—41no—
302Q8—42no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306D4—42no—
307D4—42no—
308D4—42no—
309D4—41no—
310D4—41no—
311D4—81no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×C22⋊C4×C7111yescharacteristic
324C2×C4C4×D4×C7111yescharacteristic
325C2×C4C4×D4×C7111yescharacteristic
326C2×C4#120×C7111yescharacteristic
327C2×C4#121×C7111yescharacteristic
328C2×C4#123×C7111yescharacteristic
329C2×C4C2×D8×C7111yescharacteristic
330C2×C4#135×C7111yescharacteristic
331C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
332C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
333C8—22no—
334C8—42no—
335C8—42no—
336C8—41no—
337C8—81no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22#386×C7111yescharacteristic
344C22#459×C7111yescharacteristic
345C22#463×C7111yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—82no—
352C4—161no—
353C4C4×D8×C7111yescharacteristic
354C4#442×C7111yescharacteristic
355C4#447×C7111yescharacteristic
356C4#449×C7111yescharacteristic
357C2—21no—
358C2—21no—
359C2—82no—
360C2C71×#187911yescharacteristic
361C2C71×#190211yescharacteristic
362C2C71×#190911yescharacteristic
subgroupG/H
center, socleC22×C71#386
commutatorC42C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radical#532×C71C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
3. C71×#1960C2semidirect
4. C71×#1546C2semidirect
6. C71×#1286C2semidirect
7. C71×#1275C2semidirect
11. #500×C71C4semidirect
12. #486×C71C4semidirect
175. #9291C71direct (N × H)
184. #1960C2×C71semidirect
185. #1546C2×C71semidirect
187. #1286C2×C71semidirect
188. #1275C2×C71semidirect
189. C71#9291direct (N × H)
193. #500C4×C71semidirect
194. #486C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#32262
C71×#29322
C71×#19602
C71×#15462
C71×#15172
C71×#12862
C71×#12752
#929171