Finite Group Databasebeta

← C71×#9300 #31830172

Subgroups — C71×#9300 #31830172

Order 18176.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3226C211yesmaximal, characteristic
2C71×#2327C211yesmaximal, characteristic, complemented
3C71×#1963C211yesmaximal, characteristic
4C71×#1546C211yesmaximal, characteristic, complemented
5C71×#1517C211yesmaximal, characteristic
6C71×#1286C211yesmaximal, characteristic, complemented
7C71×#1274C211yesmaximal, characteristic, complemented
8#532×C71C2211yescharacteristic
9C2×C4⋊Q8×C71C2211yescharacteristic
10C2×C4×Q8×C71—21no—
11#500×C71C412yescomplemented
12#486×C71C412yescomplemented
13#480×C71—22no—
14#474×C71—22no—
15#431×C71C2211yescharacteristic
16#431×C71C2211yescharacteristic
17#423×C71C2211yescharacteristic
18C2×#104×C71—21no—
19C2×#104×C71—21no—
20#391×C71—21no—
21#387×C71—22no—
22#387×C71—22no—
23#382×C71C2211yescharacteristic
24#376×C71C2211yescharacteristic
25#332×C71—22no—
26#327×C71—22no—
27C2×#41×C71—21no—
28C22×Q8×C71—21no—
29C2×M16×C71—21no—
30C4⋊Q8×C71—21no—
31C4⋊Q8×C71—21no—
32C4⋊Q8×C71C2×C411yescharacteristic
33C4⋊Q8×C71C2×C412yes—
34C4⋊Q8×C71D412yes—
35#127×C71C2×C411yescharacteristic
36#124×C71—21no—
37#121×C71—21no—
38#121×C71—21no—
39C4×Q8×C71—42no—
40C4×Q8×C71—42no—
41C4×D4×C71—21no—
42C4×D4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71D412yes—
47C2×C42×C71—22no—
48C2×C42×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71C2311yescharacteristic
51#107×C71—24no—
52C4⋊C8×C71—21no—
53C4⋊C8×C71—21no—
54C4⋊C8×C71—21no—
55C4⋊C8×C71—21no—
56#104×C71—42no—
57#104×C71—42no—
58#102×C71—24no—
59C22⋊C8×C71—21no—
60C22⋊C8×C71—21no—
61C4×C8×C71—21no—
62C4×C8×C71C2×C412yes—
63#95×C71—24no—
64#95×C71—21no—
65#95×C71—21no—
66#95×C71—21no—
67#95×C71D412yes—
68#41×C71—41no—
69#41×C71—41no—
70#41×C71—41no—
71#41×C71—41no—
72C2×Q8×C71—22no—
73C2×Q8×C71—22no—
74C2×Q8×C71—21no—
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—21no—
84C22×C4×C71—22no—
85C22×C4×C71—21no—
86C22×C4×C71—22no—
87C22×C4×C71—41no—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91M16×C71—41no—
92M16×C71—41no—
93C2×C8×C71—22no—
94C2×C8×C71—41no—
95C2×C8×C71—41no—
96C4⋊C4×C71—22no—
97C4⋊C4×C71—22no—
98C4⋊C4×C71—22no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71—44no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
104C4⋊C4×C71C22⋊C414yes—
105C22⋊C4×C71—41no—
106C22⋊C4×C71—41no—
107C42×C71—21no—
108C42×C71—21no—
109C42×C71—21no—
110C42×C71—44no—
111C42×C71D812yes—
112C42×C71C22×C411yescharacteristic
113C42×C71C2×D411yescharacteristic
114C42×C71#4111yescharacteristic
115C42×C71#4111yescharacteristic
116C23×C71—41no—
117C23×C71#12011yescharacteristic
118Q8×C71—42no—
119Q8×C71—42no—
120Q8×C71—42no—
121Q8×C71—42no—
122Q8×C71—41no—
123Q8×C71—41no—
124D4×C71—42no—
125D4×C71—41no—
126D4×C71—41no—
127D4×C71—81no—
128D4×C71—81no—
129D4×C71—81no—
130C2×C4×C71—24no—
131C2×C4×C71—22no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—41no—
139C2×C4×C71—42no—
140C2×C4×C71—41no—
141C2×C4×C71—42no—
142C2×C4×C71—41no—
143C2×C4×C71—42no—
144C2×C4×C71—41no—
145C2×C4×C71—81no—
146C2×C4×C71—81no—
147C2×C4×C71C2×C22⋊C411yescharacteristic
148C2×C4×C71C4×D411yescharacteristic
149C2×C4×C71C4×D411yescharacteristic
150C2×C4×C71#12111yescharacteristic
151C2×C4×C71#12111yescharacteristic
152C2×C4×C71#12311yescharacteristic
153C2×C4×C71C2×D811yescharacteristic
154C2×C4×C71#13511yescharacteristic
155C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
157C8×C71—42no—
158C8×C71—81no—
159C8×C71—81no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—81no—
163C22×C71—81no—
164C22×C71—81no—
165C22×C71#38611yescharacteristic
166C22×C71#44711yescharacteristic
167C22×C71#44911yescharacteristic
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—21no—
172C4×C71—42no—
173C4×C71—44no—
174C4×C71—82no—
175C4×C71—81no—
176C4×C71—81no—
177C4×C71C4×D811yescharacteristic
178C4×C71#44211yescharacteristic
179C4×C71#45911yescharacteristic
180C4×C71#46311yescharacteristic
181#9300C7111yesmaximal, characteristic, direct product factor, Sylow
182C2×C71—21no—
183C2×C71—21no—
184C2×C71—161no—
185C2×C71#187911yescharacteristic
186C2×C71#190511yescharacteristic
187C2×C71#191211yescharacteristic
188#3226C2×C7111yescharacteristic
189#2327C2×C7111yescharacteristic, complemented
190#1963C2×C7111yescharacteristic
191#1546C2×C7111yescharacteristic, complemented
192#1517C2×C7111yescharacteristic
193#1286C2×C7111yescharacteristic, complemented
194#1274C2×C7111yescharacteristic, complemented
195C71#930011yescharacteristic, direct product factor, Sylow
196#532C22×C7111yescharacteristic
197C2×C4⋊Q8C22×C7111yescharacteristic
198C2×C4×Q8—21no—
199#500C4×C7112yescomplemented
200#486C4×C7112yescomplemented
201#480—22no—
202#474—22no—
203#431C22×C7111yescharacteristic
204#431C22×C7111yescharacteristic
205#423C22×C7111yescharacteristic
206C2×#104—21no—
207C2×#104—21no—
208#391—21no—
209#387—22no—
210#387—22no—
211#382C22×C7111yescharacteristic
212#376C22×C7111yescharacteristic
213#332—22no—
214#327—22no—
215C2×#41—21no—
216C22×Q8—21no—
217C2×M16—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8C2×C4×C7111yescharacteristic
221C4⋊Q8C2×C4×C7112yes—
222C4⋊Q8D4×C7112yes—
223#127C2×C4×C7111yescharacteristic
224#124—21no—
225#121—21no—
226#121—21no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×D4—21no—
230C4×D4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4D4×C7112yes—
235C2×C42—22no—
236C2×C42D4×C7111yescharacteristic
237C2×C42D4×C7111yescharacteristic
238C2×C42C23×C7111yescharacteristic
239#107—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244#104—42no—
245#104—42no—
246#102—24no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C4×C8—21no—
250C4×C8C2×C4×C7112yes—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95D4×C7112yes—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4C2×D4×C7111yescharacteristic
277C22×C4C2×D4×C7111yescharacteristic
278C22×C4C2×D4×C7111yescharacteristic
279M16—41no—
280M16—41no—
281C2×C8—22no—
282C2×C8—41no—
283C2×C8—41no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4C22⋊C4×C7114yes—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—44no—
299C42D8×C7112yes—
300C42C22×C4×C7111yescharacteristic
301C42C2×D4×C7111yescharacteristic
302C42#41×C7111yescharacteristic
303C42#41×C7111yescharacteristic
304C23—41no—
305C23#120×C7111yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—42no—
310Q8—41no—
311Q8—41no—
312D4—42no—
313D4—41no—
314D4—41no—
315D4—81no—
316D4—81no—
317D4—81no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4C2×C22⋊C4×C7111yescharacteristic
336C2×C4C4×D4×C7111yescharacteristic
337C2×C4C4×D4×C7111yescharacteristic
338C2×C4#121×C7111yescharacteristic
339C2×C4#121×C7111yescharacteristic
340C2×C4#123×C7111yescharacteristic
341C2×C4C2×D8×C7111yescharacteristic
342C2×C4#135×C7111yescharacteristic
343C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
344C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
345C8—42no—
346C8—81no—
347C8—81no—
348C22—21no—
349C22—21no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22#386×C7111yescharacteristic
354C22#447×C7111yescharacteristic
355C22#449×C7111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—81no—
364C4—81no—
365C4C4×D8×C7111yescharacteristic
366C4#442×C7111yescharacteristic
367C4#459×C7111yescharacteristic
368C4#463×C7111yescharacteristic
369C2—21no—
370C2—21no—
371C2—161no—
372C2C71×#187911yescharacteristic
373C2C71×#190511yescharacteristic
374C2C71×#191211yescharacteristic
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×#2327C2semidirect
4. C71×#1546C2semidirect
6. C71×#1286C2semidirect
7. C71×#1274C2semidirect
11. #500×C71C4semidirect
12. #486×C71C4semidirect
181. #9300C71direct (N × H)
189. #2327C2×C71semidirect
191. #1546C2×C71semidirect
193. #1286C2×C71semidirect
194. #1274C2×C71semidirect
195. C71#9300direct (N × H)
199. #500C4×C71semidirect
200. #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×#23272
C71×#19632
C71×#15462
C71×#15172
C71×#12862
C71×#12742
#930071