Finite Group Databasebeta

← C71×#9274 #31830146

Subgroups — C71×#9274 #31830146

Order 18176.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3208C211yesmaximal, characteristic
2C71×#2962C211yesmaximal, characteristic, complemented
3C71×#1965C211yesmaximal, characteristic
4C71×#1550C211yesmaximal, characteristic, complemented
5C2×#331×C71C211yesmaximal, characteristic
6C71×#1301C211yesmaximal, characteristic, complemented
7C71×#1283C211yesmaximal, characteristic, complemented
8#533×C71C2211yescharacteristic
9C2×#127×C71C2211yescharacteristic
10#492×C71—21no—
11#486×C71—21no—
12#460×C71—22no—
13#450×C71C414yes—
14#446×C71—22no—
15#446×C71—21no—
16C8×Q8×C71—21no—
17#431×C71C2211yescharacteristic
18#424×C71—21no—
19#423×C71C2211yescharacteristic
20C2×C4⋊C8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22C2×#100×C71—21no—
23C2×#100×C71—21no—
24#405×C71—21no—
25C4×M16×C71C2211yescharacteristic
26#333×C71—21no—
27#332×C71—21no—
28#331×C71—21no—
29#331×C71—21no—
30C22×D4×C71—21no—
31C22×D4×C71—21no—
32C22×D4×C71C2×C412yes—
33C2×SD16×C71—24no—
34C2×M16×C71—21no—
35C2×M16×C71D411yescharacteristic
36C2×M16×C71D411yescharacteristic
37C22×C8×C71—21no—
38C22×C8×C71—21no—
39C4⋊Q8×C71D411yescharacteristic
40C4⋊Q8×C71D411yescharacteristic
41#127×C71—21no—
42#127×C71—21no—
43#127×C71—21no—
44#127×C71—21no—
45#125×C71—21no—
46C22⋊Q8×C71—21no—
47C22⋊Q8×C71C2×C412yes—
48C4×Q8×C71—21no—
49C4×Q8×C71D412yes—
50#117×C71—21no—
51C2×C42×C71C2311yescharacteristic
52C4⋊C8×C71—22no—
53C4⋊C8×C71—22no—
54C4⋊C8×C71—21no—
55C4⋊C8×C71—21no—
56C4⋊C8×C71—21no—
57C4⋊C8×C71—21no—
58C4⋊C8×C71—21no—
59C4⋊C8×C71—21no—
60C4⋊C8×C71D411yescharacteristic
61C4⋊C8×C71D411yescharacteristic
62#102×C71—24no—
63#100×C71—42no—
64#100×C71—42no—
65C22⋊C8×C71C2×C412yes—
66#97×C71—22no—
67#97×C71—21no—
68C4×C8×C71—22no—
69C4×C8×C71—21no—
70C4×C8×C71—21no—
71C24×C71—22no—
72C2×Q8×C71—21no—
73C2×Q8×C71C22⋊C412yes—
74C2×D4×C71—22no—
75C2×D4×C71—22no—
76C2×D4×C71—41no—
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—81no—
85C2×D4×C71—81no—
86C2×D4×C71—82no—
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71#4111yescharacteristic
90SD16×C71—84no—
91M16×C71—21no—
92M16×C71—21no—
93M16×C71—21no—
94M16×C71—41no—
95M16×C71—41no—
96M16×C71SD1612yescomplemented
97C2×C8×C71—21no—
98C2×C8×C71—22no—
99C2×C8×C71—21no—
100C2×C8×C71—21no—
101C2×C8×C71—21no—
102C2×C8×C71—41no—
103C2×C8×C71—41no—
104C2×C8×C71—41no—
105C2×C8×C71—41no—
106C2×C8×C71—81no—
107C4⋊C4×C71—22no—
108C4⋊C4×C71—21no—
109C4⋊C4×C71—21no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—22no—
112C4⋊C4×C71C22⋊C412yes—
113C22⋊C4×C71—22no—
114C42×C71—22no—
115C42×C71C2×D411yescharacteristic
116C42×C71C2×D411yescharacteristic
117C42×C71C2×D411yescharacteristic
118C42×C71#4111yescharacteristic
119C23×C71—42no—
120C23×C71—42no—
121C23×C71—42no—
122C23×C71—82no—
123C23×C71—82no—
124C23×C71C4×D411yescharacteristic
125Q8×C71—42no—
126Q8×C71—41no—
127Q8×C71—41no—
128D4×C71—44no—
129D4×C71—82no—
130D4×C71—82no—
131D4×C71—81no—
132D4×C71—81no—
133D4×C71—81no—
134D4×C71—81no—
135D4×C71—81no—
136D4×C71—81no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—41no—
142C2×C4×C71C2×C22⋊C411yescharacteristic
143C2×C4×C71C4×D411yescharacteristic
144C2×C4×C71#12011yescharacteristic
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—22no—
153C8×C71—42no—
154C8×C71—41no—
155C8×C71—41no—
156C8×C71—41no—
157C8×C71—81no—
158C8×C71—81no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—82no—
162C22×C71—82no—
163C22×C71—82no—
164C22×C71—162no—
165C22×C71—162no—
166C22×C71#38611yescharacteristic
167C22×C71C4×SD1611yescharacteristic
168C22×C71#44011yescharacteristic
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—42no—
173C4×C71—42no—
174C4×C71#44811yescharacteristic
175C4×C71#44911yescharacteristic
176C4×C71#46011yescharacteristic
177C4×C71#46411yescharacteristic
178C4×C71#47111yescharacteristic
179C4×C71#47211yescharacteristic
180#9274C7111yesmaximal, characteristic, direct product factor, Sylow
181C2×C71—21no—
182C2×C71—21no—
183C2×C71—162no—
184C2×C71#188011yescharacteristic
185C2×C71#190711yescharacteristic
186C2×C71#191411yescharacteristic
187#3208C2×C7111yescharacteristic
188#2962C2×C7111yescharacteristic, complemented
189#1965C2×C7111yescharacteristic
190#1550C2×C7111yescharacteristic, complemented
191C2×#331C2×C7111yescharacteristic
192#1301C2×C7111yescharacteristic, complemented
193#1283C2×C7111yescharacteristic, complemented
194C71#927411yescharacteristic, direct product factor, Sylow
195#533C22×C7111yescharacteristic
196C2×#127C22×C7111yescharacteristic
197#492—21no—
198#486—21no—
199#460—22no—
200#450C4×C7114yes—
201#446—22no—
202#446—21no—
203C8×Q8—21no—
204#431C22×C7111yescharacteristic
205#424—21no—
206#423C22×C7111yescharacteristic
207C2×C4⋊C8C22×C7111yescharacteristic
208C2×C4⋊C8C22×C7111yescharacteristic
209C2×#100—21no—
210C2×#100—21no—
211#405—21no—
212C4×M16C22×C7111yescharacteristic
213#333—21no—
214#332—21no—
215#331—21no—
216#331—21no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4C2×C4×C7112yes—
220C2×SD16—24no—
221C2×M16—21no—
222C2×M16D4×C7111yescharacteristic
223C2×M16D4×C7111yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C4⋊Q8D4×C7111yescharacteristic
227C4⋊Q8D4×C7111yescharacteristic
228#127—21no—
229#127—21no—
230#127—21no—
231#127—21no—
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8C2×C4×C7112yes—
235C4×Q8—21no—
236C4×Q8D4×C7112yes—
237#117—21no—
238C2×C42C23×C7111yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C7111yescharacteristic
248C4⋊C8D4×C7111yescharacteristic
249#102—24no—
250#100—42no—
251#100—42no—
252C22⋊C8C2×C4×C7112yes—
253#97—22no—
254#97—21no—
255C4×C8—22no—
256C4×C8—21no—
257C4×C8—21no—
258C24—22no—
259C2×Q8—21no—
260C2×Q8C22⋊C4×C7112yes—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4—82no—
274C22×C4C22×C4×C7111yescharacteristic
275C22×C4C2×D4×C7111yescharacteristic
276C22×C4#41×C7111yescharacteristic
277SD16—84no—
278M16—21no—
279M16—21no—
280M16—21no—
281M16—41no—
282M16—41no—
283M16SD16×C7112yescomplemented
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—81no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4C22⋊C4×C7112yes—
300C22⋊C4—22no—
301C42—22no—
302C42C2×D4×C7111yescharacteristic
303C42C2×D4×C7111yescharacteristic
304C42C2×D4×C7111yescharacteristic
305C42#41×C7111yescharacteristic
306C23—42no—
307C23—42no—
308C23—42no—
309C23—82no—
310C23—82no—
311C23C4×D4×C7111yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315D4—44no—
316D4—82no—
317D4—82no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4C2×C22⋊C4×C7111yescharacteristic
330C2×C4C4×D4×C7111yescharacteristic
331C2×C4#120×C7111yescharacteristic
332C2×C4#121×C7111yescharacteristic
333C2×C4#121×C7111yescharacteristic
334C2×C4#123×C7111yescharacteristic
335C2×C4C2×SD16×C7111yescharacteristic
336C2×C4#135×C7111yescharacteristic
337C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
338C2×C4#137×C7111yescharacteristic
339C8—22no—
340C8—42no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—81no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—162no—
352C22—162no—
353C22#386×C7111yescharacteristic
354C22C4×SD16×C7111yescharacteristic
355C22#440×C7111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4#448×C7111yescharacteristic
362C4#449×C7111yescharacteristic
363C4#460×C7111yescharacteristic
364C4#464×C7111yescharacteristic
365C4#471×C7111yescharacteristic
366C4#472×C7111yescharacteristic
367C2—21no—
368C2—21no—
369C2—162no—
370C2C71×#188011yescharacteristic
371C2C71×#190711yescharacteristic
372C2C71×#191411yescharacteristic
subgroupG/H
center, socleC22×C71#386
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC2×#127×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
2. C71×#2962C2semidirect
4. C71×#1550C2semidirect
6. C71×#1301C2semidirect
7. C71×#1283C2semidirect
96. M16×C71SD16semidirect
180. #9274C71direct (N × H)
188. #2962C2×C71semidirect
190. #1550C2×C71semidirect
192. #1301C2×C71semidirect
193. #1283C2×C71semidirect
194. C71#9274direct (N × H)
283. M16SD16×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#32082
C71×#29622
C71×#19652
C71×#15502
C2×#331×C712
C71×#13012
C71×#12832
#927471