Finite Group Databasebeta

← C71×#13026 #31833898

Subgroups — C71×#13026 #31833898

Order 18176.

486 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#510×C71C211yesmaximal, characteristic
2C71×#2943C211yesmaximal, characteristic
3C2×#360×C71C211yesmaximal, characteristic
4C71×#2141C211yesmaximal, characteristic, complemented
5C71×#1971C211yesmaximal, characteristic, complemented
6C71×#1386C211yesmaximal, characteristic, complemented
7C71×#1377C211yesmaximal, characteristic, complemented
8C2×#135×C71C2211yescharacteristic
9C22×Q16×C71C2211yescharacteristic
10C4×#41×C71—21no—
11#510×C71—22no—
12#510×C71C414yescomplemented
13C2×Q32×C71—21no—
14C2×SD32×C71—21no—
15C2×M32×C71C2211yescharacteristic
16C2×M32×C71C2211yescharacteristic
17#442×C71—21no—
18#441×C71—21no—
19#440×C71—22no—
20#427×C71C2211yescharacteristic
21C2×#106×C71C2211yescharacteristic
22#416×C71—21no—
23C2×#103×C71—21no—
24C2×#97×C71C2211yescharacteristic
25#391×C71—21no—
26#360×C71—22no—
27#358×C71—22no—
28C2×#41×C71—21no—
29C22×Q8×C71—21no—
30#135×C71—21no—
31#135×C71—21no—
32#135×C71C2×C412yes—
33#135×C71D412yes—
34C2×Q16×C71—21no—
35C2×Q16×C71—21no—
36C2×Q16×C71—21no—
37C2×Q16×C71—21no—
38C2×Q16×C71—21no—
39C2×Q16×C71C2×C412yescomplemented
40C2×Q16×C71D411yescharacteristic
41C2×SD16×C71—21no—
42C2×D8×C71D411yescharacteristic
43C22×C8×C71D411yescharacteristic
44C22×C8×C71C2311yescharacteristic
45C4×Q8×C71—22no—
46C4×D4×C71—22no—
47C4×D4×C71—42no—
48#117×C71—21no—
49#117×C71—41no—
50C2×C4⋊C4×C71—21no—
51C2×C42×C71—41no—
52C2×C42×C71D411yescharacteristic
53Q32×C71—24no—
54SD32×C71—24no—
55M32×C71—21no—
56M32×C71—21no—
57M32×C71—21no—
58M32×C71C2×C412yescomplemented
59C2×C16×C71—21no—
60C2×C16×C71—21no—
61#107×C71D412yes—
62#106×C71—21no—
63#106×C71—21no—
64#106×C71—21no—
65#103×C71—22no—
66#103×C71—42no—
67#102×C71—22no—
68#97×C71—21no—
69#97×C71—21no—
70#95×C71—42no—
71#41×C71—22no—
72#41×C71—22no—
73#41×C71—41no—
74#41×C71—41no—
75C2×Q8×C71—22no—
76C2×Q8×C71—21no—
77C2×Q8×C71—41no—
78C2×Q8×C71—41no—
79C2×Q8×C71—41no—
80C2×Q8×C71—41no—
81C2×Q8×C71—41no—
82C2×D4×C71—21no—
83C2×D4×C71—41no—
84C22×C4×C71—42no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71D812yes—
89C22×C4×C71C2×D411yescharacteristic
90Q16×C71—22no—
91Q16×C71—22no—
92Q16×C71—21no—
93Q16×C71—41no—
94Q16×C71—41no—
95Q16×C71C22⋊C412yes—
96SD16×C71—22no—
97SD16×C71—22no—
98D8×C71—21no—
99D8×C71C22⋊C412yes—
100C2×C8×C71—21no—
101C2×C8×C71—21no—
102C2×C8×C71—21no—
103C2×C8×C71C22×C411yescharacteristic
104C2×C8×C71C2×D411yescharacteristic
105C2×C8×C71C2×D411yescharacteristic
106C2×C8×C71C2×D411yescharacteristic
107C2×C8×C71#4111yescharacteristic
108C2×C8×C71#4111yescharacteristic
109C4⋊C4×C71—22no—
110C4⋊C4×C71—42no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C22⋊C4×C71—42no—
114C22⋊C4×C71—41no—
115C42×C71—21no—
116C42×C71—21no—
117C42×C71—42no—
118C42×C71—41no—
119C16×C71—22no—
120C16×C71—41no—
121C23×C71—41no—
122C23×C71C2×D811yescharacteristic
123Q8×C71—22no—
124Q8×C71—41no—
125Q8×C71—42no—
126Q8×C71—42no—
127Q8×C71—81no—
128Q8×C71—81no—
129D4×C71—22no—
130D4×C71—42no—
131D4×C71—42no—
132D4×C71—41no—
133C2×C4×C71—22no—
134C2×C4×C71—41no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—42no—
140C2×C4×C71—84no—
141C2×C4×C71—81no—
142C2×C4×C71—81no—
143C2×C4×C71—82no—
144C2×C4×C71—81no—
145C2×C4×C71C4×D411yescharacteristic
146C2×C4×C71#12011yescharacteristic
147C2×C4×C71#12111yescharacteristic
148C2×C4×C71#13511yescharacteristic
149C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
151C8×C71—41no—
152C8×C71C2×C22⋊C411yescharacteristic
153C8×C71C4×D411yescharacteristic
154C8×C71#12111yescharacteristic
155C8×C71#12311yescharacteristic
156C22×C71—22no—
157C22×C71—42no—
158C22×C71—82no—
159C22×C71C4×D811yescharacteristic
160C22×C71#44711yescharacteristic
161C22×C71#45911yescharacteristic
162C4×C71—42no—
163C4×C71—42no—
164C4×C71—82no—
165C4×C71—82no—
166C4×C71—161no—
167C4×C71#38611yescharacteristic
168C4×C71#44211yescharacteristic
169C4×C71#44911yescharacteristic
170C4×C71#46311yescharacteristic
171#13026C7111yesmaximal, characteristic, direct product factor, Sylow
172C2×C71—21no—
173C2×C71—21no—
174C2×C71—82no—
175C2×C71#187911yescharacteristic
176C2×C71#219112yes—
177C2×#510C2×C7111yescharacteristic
178#2943C2×C7111yescharacteristic
179C2×#360C2×C7111yescharacteristic
180#2141C2×C7111yescharacteristic, complemented
181#1971C2×C7111yescharacteristic, complemented
182#1386C2×C7111yescharacteristic, complemented
183#1377C2×C7111yescharacteristic, complemented
184C71#1302611yescharacteristic, direct product factor, Sylow
185C2×#135C22×C7111yescharacteristic
186C22×Q16C22×C7111yescharacteristic
187C4×#41—21no—
188#510—22no—
189#510C4×C7114yescomplemented
190C2×Q32—21no—
191C2×SD32—21no—
192C2×M32C22×C7111yescharacteristic
193C2×M32C22×C7111yescharacteristic
194#442—21no—
195#441—21no—
196#440—22no—
197#427C22×C7111yescharacteristic
198C2×#106C22×C7111yescharacteristic
199#416—21no—
200C2×#103—21no—
201C2×#97C22×C7111yescharacteristic
202#391—21no—
203#360—22no—
204#358—22no—
205C2×#41—21no—
206C22×Q8—21no—
207#135—21no—
208#135—21no—
209#135C2×C4×C7112yes—
210#135D4×C7112yes—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16C2×C4×C7112yescomplemented
217C2×Q16D4×C7111yescharacteristic
218C2×SD16—21no—
219C2×D8D4×C7111yescharacteristic
220C22×C8D4×C7111yescharacteristic
221C22×C8C23×C7111yescharacteristic
222C4×Q8—22no—
223C4×D4—22no—
224C4×D4—42no—
225#117—21no—
226#117—41no—
227C2×C4⋊C4—21no—
228C2×C42—41no—
229C2×C42D4×C7111yescharacteristic
230Q32—24no—
231SD32—24no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32C2×C4×C7112yescomplemented
236C2×C16—21no—
237C2×C16—21no—
238#107D4×C7112yes—
239#106—21no—
240#106—21no—
241#106—21no—
242#103—22no—
243#103—42no—
244#102—22no—
245#97—21no—
246#97—21no—
247#95—42no—
248#41—22no—
249#41—22no—
250#41—41no—
251#41—41no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4D8×C7112yes—
266C22×C4C2×D4×C7111yescharacteristic
267Q16—22no—
268Q16—22no—
269Q16—21no—
270Q16—41no—
271Q16—41no—
272Q16C22⋊C4×C7112yes—
273SD16—22no—
274SD16—22no—
275D8—21no—
276D8C22⋊C4×C7112yes—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8C22×C4×C7111yescharacteristic
281C2×C8C2×D4×C7111yescharacteristic
282C2×C8C2×D4×C7111yescharacteristic
283C2×C8C2×D4×C7111yescharacteristic
284C2×C8#41×C7111yescharacteristic
285C2×C8#41×C7111yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C42—21no—
293C42—21no—
294C42—42no—
295C42—41no—
296C16—22no—
297C16—41no—
298C23—41no—
299C23C2×D8×C7111yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—42no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306D4—22no—
307D4—42no—
308D4—42no—
309D4—41no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—84no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4C4×D4×C7111yescharacteristic
323C2×C4#120×C7111yescharacteristic
324C2×C4#121×C7111yescharacteristic
325C2×C4#135×C7111yescharacteristic
326C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
327C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
328C8—41no—
329C8C2×C22⋊C4×C7111yescharacteristic
330C8C4×D4×C7111yescharacteristic
331C8#121×C7111yescharacteristic
332C8#123×C7111yescharacteristic
333C22—22no—
334C22—42no—
335C22—82no—
336C22C4×D8×C7111yescharacteristic
337C22#447×C7111yescharacteristic
338C22#459×C7111yescharacteristic
339C4—42no—
340C4—42no—
341C4—82no—
342C4—82no—
343C4—161no—
344C4#386×C7111yescharacteristic
345C4#442×C7111yescharacteristic
346C4#449×C7111yescharacteristic
347C4#463×C7111yescharacteristic
348C2—21no—
349C2—21no—
350C2—82no—
351C2C71×#187911yescharacteristic
352C2C71×#219112yes—
subgroupG/H
center, socleC22×C71#447
commutatorC2×C8C22×C4×C71
Frattini, cosocleC22×C8C23×C71
prime radicalC2×#135×C71C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×#97×C71 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C71×#2141C2semidirect
5. C71×#1971C2semidirect
6. C71×#1386C2semidirect
7. C71×#1377C2semidirect
12. #510×C71C4semidirect
39. C2×Q16×C71C2×C4semidirect
58. M32×C71C2×C4semidirect
171. #13026C71direct (N × H)
180. #2141C2×C71semidirect
181. #1971C2×C71semidirect
182. #1386C2×C71semidirect
183. #1377C2×C71semidirect
184. C71#13026direct (N × H)
189. #510C4×C71semidirect
216. C2×Q16C2×C4×C71semidirect
235. M32C2×C4×C71semidirect

Maximal subgroups

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

subgroupindex
C2×#510×C712
C71×#29432
C2×#360×C712
C71×#21412
C71×#19712
C71×#13862
C71×#13772
#1302671