Finite Group Databasebeta

← C71×#10550 #31831422

Subgroups — C71×#10550 #31831422

Order 18176.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3152C211yesmaximal, characteristic
2C71×#1950C211yesmaximal, characteristic, complemented
3C71×#1547C211yesmaximal, characteristic
4C71×#1546C211yesmaximal, characteristic, complemented
5C2×#332×C71C211yesmaximal, characteristic, complemented
6C71×#1480C211yesmaximal, characteristic
7C71×#1283C211yesmaximal, characteristic, complemented
8C2×#137×C71—21no—
9C2×#137×C71—21no—
10#532×C71C2211yescharacteristic
11C2×C4⋊Q8×C71C2211yescharacteristic
12#497×C71—22no—
13#494×C71—21no—
14#492×C71—21no—
15#431×C71C2211yescharacteristic
16C2×#108×C71—21no—
17#423×C71C2211yescharacteristic
18C2×C4⋊C8×C71C2211yescharacteristic
19C2×C4⋊C8×C71C2211yescharacteristic
20C4×M16×C71C2211yescharacteristic
21#341×C71—21no—
22#332×C71—22no—
23#332×C71—21no—
24#332×C71—22no—
25#331×C71—21no—
26D4⋊C8×C71—22no—
27C2×#41×C71—21no—
28C22×Q8×C71—21no—
29#137×C71—42no—
30#137×C71—42no—
31#137×C71—42no—
32#137×C71—42no—
33C2×Q16×C71—22no—
34C2×Q16×C71—22no—
35C2×SD16×C71—22no—
36C2×SD16×C71—22no—
37C2×M16×C71—21no—
38C2×M16×C71D411yescharacteristic
39C2×M16×C71D411yescharacteristic
40C22×C8×C71—21no—
41C22×C8×C71—21no—
42C4⋊Q8×C71—21no—
43C4⋊Q8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71D411yescharacteristic
46C4⋊Q8×C71D412yes—
47#127×C71D411yescharacteristic
48#124×C71—21no—
49#121×C71—21no—
50#121×C71—21no—
51C4×D4×C71—21no—
52C4×D4×C71—21no—
53C2×C4⋊C4×C71—21no—
54C2×C4⋊C4×C71—21no—
55C2×C42×C71C2311yescharacteristic
56#108×C71—42no—
57C4⋊C8×C71—21no—
58C4⋊C8×C71—21no—
59C4⋊C8×C71—21no—
60C4⋊C8×C71—21no—
61C4⋊C8×C71—21no—
62C4⋊C8×C71—21no—
63C4⋊C8×C71D412yes—
64C22⋊C8×C71—21no—
65#97×C71—21no—
66C4×C8×C71—21no—
67C4×C8×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—21no—
80C2×D4×C71—21no—
81C2×D4×C71—21no—
82C2×D4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71—41no—
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4111yescharacteristic
87C22×C4×C71#4111yescharacteristic
88Q16×C71—44no—
89Q16×C71—44no—
90SD16×C71—44no—
91SD16×C71—44no—
92M16×C71—21no—
93M16×C71—21no—
94M16×C71—41no—
95M16×C71—41no—
96M16×C71D812yes—
97M16×C71SD1612yes—
98C2×C8×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—22no—
101C2×C8×C71—41no—
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—22no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C22⋊C4×C71—41no—
114C22⋊C4×C71—41no—
115C42×C71C2×D411yescharacteristic
116C42×C71C2×D411yescharacteristic
117C42×C71C2×D411yescharacteristic
118C42×C71#4111yescharacteristic
119C23×C71—41no—
120C23×C71#12411yescharacteristic
121Q8×C71—42no—
122Q8×C71—41no—
123Q8×C71—42no—
124Q8×C71—41no—
125Q8×C71—81no—
126Q8×C71—81no—
127D4×C71—41no—
128D4×C71—41no—
129D4×C71—41no—
130D4×C71—41no—
131D4×C71—81no—
132D4×C71—81no—
133D4×C71—81no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—42no—
139C2×C4×C71—81no—
140C2×C4×C71—81no—
141C2×C4×C71—81no—
142C2×C4×C71—81no—
143C2×C4×C71#12011yescharacteristic
144C2×C4×C71#12111yescharacteristic
145C2×C4×C71#12111yescharacteristic
146C2×C4×C71#12111yescharacteristic
147C2×C4×C71#12311yescharacteristic
148C2×C4×C71#12311yescharacteristic
149C2×C4×C71C2×D811yescharacteristic
150C2×C4×C71C2×SD1611yescharacteristic
151C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
153C8×C71—22no—
154C8×C71—22no—
155C8×C71—42no—
156C8×C71—81no—
157C8×C71—81no—
158C8×C71—81no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—81no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71#39411yescharacteristic
165C22×C71#48611yescharacteristic
166C22×C71#49011yescharacteristic
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—81no—
171C4×C71—82no—
172C4×C71#44711yescharacteristic
173C4×C71#45011yescharacteristic
174C4×C71#45911yescharacteristic
175C4×C71#46011yescharacteristic
176C4×C71#47111yescharacteristic
177C4×C71#47311yescharacteristic
178#10550C7111yesmaximal, characteristic, direct product factor, Sylow
179C2×C71—21no—
180C2×C71—21no—
181C2×C71—161no—
182C2×C71#204111yescharacteristic
183C2×C71#204911yescharacteristic
184C2×C71#205211yescharacteristic
185#3152C2×C7111yescharacteristic
186#1950C2×C7111yescharacteristic, complemented
187#1547C2×C7111yescharacteristic
188#1546C2×C7111yescharacteristic, complemented
189C2×#332C2×C7111yescharacteristic, complemented
190#1480C2×C7111yescharacteristic
191#1283C2×C7111yescharacteristic, complemented
192C71#1055011yescharacteristic, direct product factor, Sylow
193C2×#137—21no—
194C2×#137—21no—
195#532C22×C7111yescharacteristic
196C2×C4⋊Q8C22×C7111yescharacteristic
197#497—22no—
198#494—21no—
199#492—21no—
200#431C22×C7111yescharacteristic
201C2×#108—21no—
202#423C22×C7111yescharacteristic
203C2×C4⋊C8C22×C7111yescharacteristic
204C2×C4⋊C8C22×C7111yescharacteristic
205C4×M16C22×C7111yescharacteristic
206#341—21no—
207#332—22no—
208#332—21no—
209#332—22no—
210#331—21no—
211D4⋊C8—22no—
212C2×#41—21no—
213C22×Q8—21no—
214#137—42no—
215#137—42no—
216#137—42no—
217#137—42no—
218C2×Q16—22no—
219C2×Q16—22no—
220C2×SD16—22no—
221C2×SD16—22no—
222C2×M16—21no—
223C2×M16D4×C7111yescharacteristic
224C2×M16D4×C7111yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8D4×C7112yes—
231C4⋊Q8D4×C7111yescharacteristic
232#127D4×C7111yescharacteristic
233#124—21no—
234#121—21no—
235#121—21no—
236C4×D4—21no—
237C4×D4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C42C23×C7111yescharacteristic
241#108—42no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8D4×C7112yes—
249C22⋊C8—21no—
250#97—21no—
251C4×C8—21no—
252C4×C8—21no—
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×D4×C7111yescharacteristic
271C22×C4#41×C7111yescharacteristic
272C22×C4#41×C7111yescharacteristic
273Q16—44no—
274Q16—44no—
275SD16—44no—
276SD16—44no—
277M16—21no—
278M16—21no—
279M16—41no—
280M16—41no—
281M16D8×C7112yes—
282M16SD16×C7112yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—81no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42C2×D4×C7111yescharacteristic
301C42C2×D4×C7111yescharacteristic
302C42C2×D4×C7111yescharacteristic
303C42#41×C7111yescharacteristic
304C23—41no—
305C23#124×C7111yescharacteristic
306Q8—42no—
307Q8—41no—
308Q8—42no—
309Q8—41no—
310Q8—81no—
311Q8—81no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—81no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4#120×C7111yescharacteristic
329C2×C4#121×C7111yescharacteristic
330C2×C4#121×C7111yescharacteristic
331C2×C4#121×C7111yescharacteristic
332C2×C4#123×C7111yescharacteristic
333C2×C4#123×C7111yescharacteristic
334C2×C4C2×D8×C7111yescharacteristic
335C2×C4C2×SD16×C7111yescharacteristic
336C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
337C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
338C8—22no—
339C8—22no—
340C8—42no—
341C8—81no—
342C8—81no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#394×C7111yescharacteristic
350C22#486×C7111yescharacteristic
351C22#490×C7111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—82no—
357C4#447×C7111yescharacteristic
358C4#450×C7111yescharacteristic
359C4#459×C7111yescharacteristic
360C4#460×C7111yescharacteristic
361C4#471×C7111yescharacteristic
362C4#473×C7111yescharacteristic
363C2—21no—
364C2—21no—
365C2—161no—
366C2C71×#204111yescharacteristic
367C2C71×#204911yescharacteristic
368C2C71×#205211yescharacteristic
subgroupG/H
center, socleC22×C71#394
commutator, Frattini, cosocleC2×C42C23×C71
prime radical#532×C71C22
LCSG ▷ C2×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×#1950C2semidirect
4. C71×#1546C2semidirect
5. C2×#332×C71C2semidirect
7. C71×#1283C2semidirect
178. #10550C71direct (N × H)
186. #1950C2×C71semidirect
188. #1546C2×C71semidirect
189. C2×#332C2×C71semidirect
191. #1283C2×C71semidirect
192. C71#10550direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#31522
C71×#19502
C71×#15472
C71×#15462
C2×#332×C712
C71×#14802
C71×#12832
#1055071