Finite Group Databasebeta

← C71×#9269 #31830141

Subgroups — C71×#9269 #31830141

Order 18176.

480 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3227C211yesmaximal, characteristic
2C71×#2935C211yesmaximal, characteristic
3C71×#1997C211yesmaximal, characteristic, complemented
4C71×#1547C211yesmaximal, characteristic
5C2×#332×C71C211yesmaximal, characteristic, complemented
6C71×#1301C211yesmaximal, characteristic, complemented
7C71×#1283C211yesmaximal, characteristic, complemented
8C2×#131×C71—21no—
9#532×C71C2211yescharacteristic
10C2×C4⋊Q8×C71C2211yescharacteristic
11#498×C71—21no—
12#486×C71—21no—
13#481×C71C414yes—
14#476×C71—22no—
15#436×C71—21no—
16#435×C71—22no—
17C8×D4×C71—21no—
18#431×C71C2211yescharacteristic
19#423×C71C2211yescharacteristic
20C2×C4⋊C8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22#408×C71—21no—
23C4×M16×C71C2211yescharacteristic
24#343×C71—21no—
25#343×C71—21no—
26#332×C71—21no—
27#332×C71—21no—
28#332×C71—21no—
29#331×C71—21no—
30C2×#41×C71—21no—
31C22×Q8×C71—21no—
32#131×C71—42no—
33#131×C71—42no—
34C2×M16×C71—22no—
35C2×M16×C71—21no—
36C2×M16×C71D411yescharacteristic
37C2×M16×C71D411yescharacteristic
38C22×C8×C71—22no—
39C22×C8×C71—21no—
40C22×C8×C71—21no—
41C4⋊Q8×C71—21no—
42C4⋊Q8×C71—21no—
43C4⋊Q8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71D411yescharacteristic
46#127×C71D411yescharacteristic
47#124×C71—21no—
48#121×C71—21no—
49#121×C71C2×C412yes—
50C4×D4×C71—21no—
51C4×D4×C71D412yes—
52C2×C4⋊C4×C71—21no—
53C2×C4⋊C4×C71C2×C412yes—
54C2×C42×C71C2311yescharacteristic
55#106×C71—24no—
56C4⋊C8×C71—21no—
57C4⋊C8×C71—21no—
58C4⋊C8×C71—21no—
59C4⋊C8×C71—21no—
60C4⋊C8×C71—21no—
61C4⋊C8×C71—21no—
62C4⋊C8×C71D411yescharacteristic
63C4⋊C8×C71D411yescharacteristic
64#102×C71—24no—
65C22⋊C8×C71—22no—
66C22⋊C8×C71—22no—
67C22⋊C8×C71C2×C412yes—
68#97×C71—21no—
69C4×C8×C71—21no—
70C4×C8×C71—21no—
71#41×C71—41no—
72#41×C71—41no—
73#41×C71—41no—
74#41×C71—41no—
75C2×Q8×C71—21no—
76C2×Q8×C71—41no—
77C2×Q8×C71—41no—
78C2×Q8×C71—81no—
79C2×Q8×C71—81no—
80C2×D4×C71—22no—
81C2×D4×C71—21no—
82C2×D4×C71—21no—
83C2×D4×C71C22⋊C412yes—
84C22×C4×C71—22no—
85C22×C4×C71—22no—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71#4111yescharacteristic
89M16×C71—21no—
90M16×C71—21no—
91M16×C71—21no—
92M16×C71—44no—
93M16×C71—41no—
94M16×C71—41no—
95M16×C71SD1612yes—
96C2×C8×C71—21no—
97C2×C8×C71—22no—
98C2×C8×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—21no—
101C2×C8×C71—44no—
102C2×C8×C71—41no—
103C2×C8×C71—41no—
104C2×C8×C71—41no—
105C2×C8×C71—41no—
106C2×C8×C71—81no—
107C4⋊C4×C71—21no—
108C4⋊C4×C71—22no—
109C4⋊C4×C71—22no—
110C4⋊C4×C71—41no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71C22⋊C412yes—
113C22⋊C4×C71—22no—
114C22⋊C4×C71—22no—
115C42×C71C2×D411yescharacteristic
116C42×C71C2×D411yescharacteristic
117C42×C71C2×D411yescharacteristic
118C42×C71#4111yescharacteristic
119C23×C71—22no—
120C23×C71C4×D411yescharacteristic
121Q8×C71—41no—
122Q8×C71—41no—
123Q8×C71—82no—
124Q8×C71—81no—
125Q8×C71—81no—
126D4×C71—42no—
127D4×C71—42no—
128D4×C71—42no—
129D4×C71—41no—
130D4×C71—41no—
131D4×C71—41no—
132D4×C71—41no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—22no—
136C2×C4×C71—41no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—42no—
140C2×C4×C71—82no—
141C2×C4×C71C2×C22⋊C411yescharacteristic
142C2×C4×C71C4×D411yescharacteristic
143C2×C4×C71#12011yescharacteristic
144C2×C4×C71#12111yescharacteristic
145C2×C4×C71#12111yescharacteristic
146C2×C4×C71#12311yescharacteristic
147C2×C4×C71C2×SD1611yescharacteristic
148C2×C4×C71#13511yescharacteristic
149C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C71#13711yescharacteristic
151C8×C71—22no—
152C8×C71—42no—
153C8×C71—41no—
154C8×C71—41no—
155C8×C71—41no—
156C8×C71—81no—
157C8×C71—81no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—42no—
163C22×C71#38611yescharacteristic
164C22×C71C4×SD1611yescharacteristic
165C22×C71#44011yescharacteristic
166C4×C71—21no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—42no—
170C4×C71—82no—
171C4×C71#45011yescharacteristic
172C4×C71#45211yescharacteristic
173C4×C71#46111yescharacteristic
174C4×C71#46311yescharacteristic
175C4×C71#47111yescharacteristic
176C4×C71#47211yescharacteristic
177#9269C7111yesmaximal, characteristic, direct product factor, Sylow
178C2×C71—21no—
179C2×C71—21no—
180C2×C71—82no—
181C2×C71#187711yescharacteristic
182C2×C71#190711yescharacteristic
183C2×C71#191511yescharacteristic
184#3227C2×C7111yescharacteristic
185#2935C2×C7111yescharacteristic
186#1997C2×C7111yescharacteristic, complemented
187#1547C2×C7111yescharacteristic
188C2×#332C2×C7111yescharacteristic, complemented
189#1301C2×C7111yescharacteristic, complemented
190#1283C2×C7111yescharacteristic, complemented
191C71#926911yescharacteristic, direct product factor, Sylow
192C2×#131—21no—
193#532C22×C7111yescharacteristic
194C2×C4⋊Q8C22×C7111yescharacteristic
195#498—21no—
196#486—21no—
197#481C4×C7114yes—
198#476—22no—
199#436—21no—
200#435—22no—
201C8×D4—21no—
202#431C22×C7111yescharacteristic
203#423C22×C7111yescharacteristic
204C2×C4⋊C8C22×C7111yescharacteristic
205C2×C4⋊C8C22×C7111yescharacteristic
206#408—21no—
207C4×M16C22×C7111yescharacteristic
208#343—21no—
209#343—21no—
210#332—21no—
211#332—21no—
212#332—21no—
213#331—21no—
214C2×#41—21no—
215C22×Q8—21no—
216#131—42no—
217#131—42no—
218C2×M16—22no—
219C2×M16—21no—
220C2×M16D4×C7111yescharacteristic
221C2×M16D4×C7111yescharacteristic
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8D4×C7111yescharacteristic
230#127D4×C7111yescharacteristic
231#124—21no—
232#121—21no—
233#121C2×C4×C7112yes—
234C4×D4—21no—
235C4×D4D4×C7112yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C2×C4×C7112yes—
238C2×C42C23×C7111yescharacteristic
239#106—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8D4×C7111yescharacteristic
247C4⋊C8D4×C7111yescharacteristic
248#102—24no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8C2×C4×C7112yes—
252#97—21no—
253C4×C8—21no—
254C4×C8—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—81no—
263C2×Q8—81no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4C22⋊C4×C7112yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4C22×C4×C7111yescharacteristic
271C22×C4C2×D4×C7111yescharacteristic
272C22×C4#41×C7111yescharacteristic
273M16—21no—
274M16—21no—
275M16—21no—
276M16—44no—
277M16—41no—
278M16—41no—
279M16SD16×C7112yes—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—44no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—81no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4C22⋊C4×C7112yes—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C42C2×D4×C7111yescharacteristic
300C42C2×D4×C7111yescharacteristic
301C42C2×D4×C7111yescharacteristic
302C42#41×C7111yescharacteristic
303C23—22no—
304C23C4×D4×C7111yescharacteristic
305Q8—41no—
306Q8—41no—
307Q8—82no—
308Q8—81no—
309Q8—81no—
310D4—42no—
311D4—42no—
312D4—42no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4C2×C22⋊C4×C7111yescharacteristic
326C2×C4C4×D4×C7111yescharacteristic
327C2×C4#120×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—22no—
336C8—42no—
337C8—41no—
338C8—41no—
339C8—41no—
340C8—81no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22#386×C7111yescharacteristic
348C22C4×SD16×C7111yescharacteristic
349C22#440×C7111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—82no—
355C4#450×C7111yescharacteristic
356C4#452×C7111yescharacteristic
357C4#461×C7111yescharacteristic
358C4#463×C7111yescharacteristic
359C4#471×C7111yescharacteristic
360C4#472×C7111yescharacteristic
361C2—21no—
362C2—21no—
363C2—82no—
364C2C71×#187711yescharacteristic
365C2C71×#190711yescharacteristic
366C2C71×#191511yescharacteristic
subgroupG/H
center, socleC22×C71#386
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radical#532×C71C22
LCSG ▷ C22×C4 ▷ 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×#1997C2semidirect
5. C2×#332×C71C2semidirect
6. C71×#1301C2semidirect
7. C71×#1283C2semidirect
177. #9269C71direct (N × H)
186. #1997C2×C71semidirect
188. C2×#332C2×C71semidirect
189. #1301C2×C71semidirect
190. #1283C2×C71semidirect
191. C71#9269direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32272
C71×#29352
C71×#19972
C71×#15472
C2×#332×C712
C71×#13012
C71×#12832
#926971