Finite Group Databasebeta

← C71×#9301 #31830173

Subgroups — C71×#9301 #31830173

Order 18176.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3206C211yesmaximal, characteristic
2C71×#2331C211yesmaximal, characteristic, complemented
3C71×#1963C211yesmaximal, characteristic
4C71×#1548C211yesmaximal, characteristic, complemented
5C71×#1517C211yesmaximal, characteristic
6C71×#1286C211yesmaximal, characteristic, complemented
7C71×#1274C211yesmaximal, characteristic, complemented
8#532×C71C2211yescharacteristic
9C2×C4⋊Q8×C71C2211yescharacteristic
10#492×C71C412yescomplemented
11#487×C71C412yescomplemented
12#461×C71—22no—
13#448×C71—22no—
14#431×C71C2211yescharacteristic
15#431×C71C2211yescharacteristic
16#423×C71C2211yescharacteristic
17C2×#104×C71—21no—
18C2×#104×C71—21no—
19#391×C71—22no—
20#389×C71—21no—
21#387×C71—22no—
22#382×C71C2211yescharacteristic
23C4×C4⋊C4×C71—21no—
24#376×C71C2211yescharacteristic
25#333×C71—22no—
26#327×C71—22no—
27C2×#41×C71—21no—
28C22×Q8×C71D412yes—
29C2×SD16×C71—24no—
30C2×M16×C71—21no—
31C4⋊Q8×C71—21no—
32C4⋊Q8×C71—21no—
33C4⋊Q8×C71C2×C411yescharacteristic
34C4⋊Q8×C71C2×C412yes—
35C4⋊Q8×C71D412yes—
36#127×C71C2×C411yescharacteristic
37#124×C71—21no—
38#121×C71—21no—
39#121×C71—21no—
40C4×D4×C71—21no—
41C4×D4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—22no—
46C2×C42×C71—22no—
47C2×C42×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71C2311yescharacteristic
50C4⋊C8×C71—21no—
51C4⋊C8×C71—21no—
52C4⋊C8×C71—21no—
53C4⋊C8×C71—21no—
54#104×C71—42no—
55#104×C71—42no—
56#103×C71—24no—
57C22⋊C8×C71—21no—
58C22⋊C8×C71—21no—
59C4×C8×C71—21no—
60C4×C8×C71C2×C412yes—
61#95×C71—22no—
62#95×C71—22no—
63#95×C71—21no—
64#95×C71—21no—
65#95×C71—21no—
66#95×C71D412yes—
67#41×C71—41no—
68#41×C71—41no—
69#41×C71—41no—
70#41×C71—41no—
71C2×Q8×C71—22no—
72C2×Q8×C71—21no—
73C2×Q8×C71—82no—
74C2×Q8×C71C22⋊C414yes—
75C2×D4×C71—21no—
76C2×D4×C71—21no—
77C2×D4×C71—21no—
78C2×D4×C71—41no—
79C22×C4×C71—21no—
80C22×C4×C71—22no—
81C22×C4×C71—21no—
82C22×C4×C71—22no—
83C22×C4×C71—41no—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87SD16×C71—84no—
88M16×C71—41no—
89M16×C71—41no—
90C2×C8×C71—22no—
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C4⋊C4×C71—22no—
94C4⋊C4×C71—22no—
95C4⋊C4×C71—22no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—82no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C42×C71—21no—
105C42×C71—21no—
106C42×C71—21no—
107C42×C71—82no—
108C42×C71SD1612yescomplemented
109C42×C71C22×C411yescharacteristic
110C42×C71C2×D411yescharacteristic
111C42×C71#4111yescharacteristic
112C42×C71#4111yescharacteristic
113C23×C71—41no—
114C23×C71#12011yescharacteristic
115Q8×C71—44no—
116Q8×C71—41no—
117Q8×C71—41no—
118Q8×C71—82no—
119D4×C71—42no—
120D4×C71—41no—
121D4×C71—41no—
122D4×C71—81no—
123D4×C71—81no—
124D4×C71—81no—
125C2×C4×C71—24no—
126C2×C4×C71—22no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71—41no—
133C2×C4×C71—42no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—42no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—42no—
140C2×C4×C71—81no—
141C2×C4×C71—81no—
142C2×C4×C71C2×C22⋊C411yescharacteristic
143C2×C4×C71C4×D411yescharacteristic
144C2×C4×C71C4×D411yescharacteristic
145C2×C4×C71#12111yescharacteristic
146C2×C4×C71#12111yescharacteristic
147C2×C4×C71#12311yescharacteristic
148C2×C4×C71C2×SD1611yescharacteristic
149C2×C4×C71#13511yescharacteristic
150C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C71#13711yescharacteristic
152C8×C71—42no—
153C8×C71—81no—
154C8×C71—81no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71—81no—
160C22×C71#38611yescharacteristic
161C22×C71#44811yescharacteristic
162C22×C71#44911yescharacteristic
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—42no—
168C4×C71—44no—
169C4×C71—82no—
170C4×C71—81no—
171C4×C71—81no—
172C4×C71C4×SD1611yescharacteristic
173C4×C71#44011yescharacteristic
174C4×C71#46011yescharacteristic
175C4×C71#46411yescharacteristic
176#9301C7111yesmaximal, characteristic, direct product factor, Sylow
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—161no—
180C2×C71#188011yescharacteristic
181C2×C71#190511yescharacteristic
182C2×C71#191111yescharacteristic
183#3206C2×C7111yescharacteristic
184#2331C2×C7111yescharacteristic, complemented
185#1963C2×C7111yescharacteristic
186#1548C2×C7111yescharacteristic, complemented
187#1517C2×C7111yescharacteristic
188#1286C2×C7111yescharacteristic, complemented
189#1274C2×C7111yescharacteristic, complemented
190C71#930111yescharacteristic, direct product factor, Sylow
191#532C22×C7111yescharacteristic
192C2×C4⋊Q8C22×C7111yescharacteristic
193#492C4×C7112yescomplemented
194#487C4×C7112yescomplemented
195#461—22no—
196#448—22no—
197#431C22×C7111yescharacteristic
198#431C22×C7111yescharacteristic
199#423C22×C7111yescharacteristic
200C2×#104—21no—
201C2×#104—21no—
202#391—22no—
203#389—21no—
204#387—22no—
205#382C22×C7111yescharacteristic
206C4×C4⋊C4—21no—
207#376C22×C7111yescharacteristic
208#333—22no—
209#327—22no—
210C2×#41—21no—
211C22×Q8D4×C7112yes—
212C2×SD16—24no—
213C2×M16—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8C2×C4×C7111yescharacteristic
217C4⋊Q8C2×C4×C7112yes—
218C4⋊Q8D4×C7112yes—
219#127C2×C4×C7111yescharacteristic
220#124—21no—
221#121—21no—
222#121—21no—
223C4×D4—21no—
224C4×D4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C42—22no—
230C2×C42D4×C7111yescharacteristic
231C2×C42D4×C7111yescharacteristic
232C2×C42C23×C7111yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237#104—42no—
238#104—42no—
239#103—24no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C4×C8—21no—
243C4×C8C2×C4×C7112yes—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95D4×C7112yes—
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—82no—
257C2×Q8C22⋊C4×C7114yes—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4C2×D4×C7111yescharacteristic
268C22×C4C2×D4×C7111yescharacteristic
269C22×C4C2×D4×C7111yescharacteristic
270SD16—84no—
271M16—41no—
272M16—41no—
273C2×C8—22no—
274C2×C8—41no—
275C2×C8—41no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—82no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—82no—
291C42SD16×C7112yescomplemented
292C42C22×C4×C7111yescharacteristic
293C42C2×D4×C7111yescharacteristic
294C42#41×C7111yescharacteristic
295C42#41×C7111yescharacteristic
296C23—41no—
297C23#120×C7111yescharacteristic
298Q8—44no—
299Q8—41no—
300Q8—41no—
301Q8—82no—
302D4—42no—
303D4—41no—
304D4—41no—
305D4—81no—
306D4—81no—
307D4—81no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4C2×C22⋊C4×C7111yescharacteristic
326C2×C4C4×D4×C7111yescharacteristic
327C2×C4C4×D4×C7111yescharacteristic
328C2×C4#121×C7111yescharacteristic
329C2×C4#121×C7111yescharacteristic
330C2×C4#123×C7111yescharacteristic
331C2×C4C2×SD16×C7111yescharacteristic
332C2×C4#135×C7111yescharacteristic
333C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
334C2×C4#137×C7111yescharacteristic
335C8—42no—
336C8—81no—
337C8—81no—
338C22—21no—
339C22—21no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22#386×C7111yescharacteristic
344C22#448×C7111yescharacteristic
345C22#449×C7111yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—44no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4C4×SD16×C7111yescharacteristic
356C4#440×C7111yescharacteristic
357C4#460×C7111yescharacteristic
358C4#464×C7111yescharacteristic
359C2—21no—
360C2—21no—
361C2—161no—
362C2C71×#188011yescharacteristic
363C2C71×#190511yescharacteristic
364C2C71×#191111yescharacteristic
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
2. C71×#2331C2semidirect
4. C71×#1548C2semidirect
6. C71×#1286C2semidirect
7. C71×#1274C2semidirect
10. #492×C71C4semidirect
11. #487×C71C4semidirect
108. C42×C71SD16semidirect
176. #9301C71direct (N × H)
184. #2331C2×C71semidirect
186. #1548C2×C71semidirect
188. #1286C2×C71semidirect
189. #1274C2×C71semidirect
190. C71#9301direct (N × H)
193. #492C4×C71semidirect
194. #487C4×C71semidirect
291. C42SD16×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#32062
C71×#23312
C71×#19632
C71×#15482
C71×#15172
C71×#12862
C71×#12742
#930171