Finite Group Databasebeta

← C71×#8459 #31829331

Subgroups — C71×#8459 #31829331

Order 18176.

922 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 922 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#3497C211yesmaximal, characteristic
2C71×#1985C211yesmaximal, characteristic, complemented
3C71×#1551C212yesmaximal, complemented
4C71×#1530C212yesmaximal, complemented
5C71×#1467C211yesmaximal, characteristic, complemented
6C2×2−1+4×C71C412yes—
7#550×C71—22no—
8D4×Q8×C71—22no—
9#548×C71—24no—
10#537×C71—22no—
11#537×C71C412yes—
12#532×C71—22no—
13#532×C71C2212yes—
14C2×C4×Q8×C71C2211yescharacteristic
15#432×C71C2212yes—
16C2×C4⋊C8×C71C2211yescharacteristic
17C2×#97×C71C2211yescharacteristic
18#336×C71—22no—
19#333×C71—22no—
20#331×C71—22no—
21#329×C71—24no—
22#324×C71—22no—
232−1+4×C71—48no—
24C2×#41×C71—24no—
25C2×#41×C71—24no—
26C2×#41×C71C2×C414yes—
27C22×Q8×C71—24no—
28C22×Q8×C71C2×C411yescharacteristic
29C22×C8×C71—21no—
30C22×C8×C71—21no—
31C4⋊Q8×C71—22no—
32C4⋊Q8×C71D412yes—
33#127×C71—22no—
34#127×C71D412yes—
35#124×C71—24no—
36#124×C71D414yes—
37C22⋊Q8×C71—28no—
38C22⋊Q8×C71—24no—
39#121×C71—28no—
40#121×C71—24no—
41#121×C71—24no—
42#121×C71—22no—
43#121×C71—22no—
44C4×Q8×C71—22no—
45C4×Q8×C71—22no—
46C4×D4×C71—24no—
47C4×D4×C71—24no—
48C4×D4×C71—22no—
49C4×D4×C71—22no—
50C2×C4⋊C4×C71C2×C411yescharacteristic
51C2×C4⋊C4×C71D412yes—
52C2×C42×C71D412yes—
53C2×C42×C71C2311yescharacteristic
54C4⋊C8×C71—22no—
55C4⋊C8×C71—22no—
56C22⋊C8×C71—22no—
57#97×C71—22no—
58#97×C71—22no—
59#41×C71—48no—
60#41×C71—48no—
61#41×C71—416no—
62#41×C71—48no—
63C2×Q8×C71—24no—
64C2×Q8×C71—24no—
65C2×Q8×C71—22no—
66C2×Q8×C71—416no—
67C2×Q8×C71C22⋊C414yes—
68C2×D4×C71—24no—
69C2×D4×C71—24no—
70C2×D4×C71—24no—
71C2×D4×C71—22no—
72C2×D4×C71—22no—
73C2×D4×C71C22⋊C414yes—
74C22×C4×C71—28no—
75C22×C4×C71—24no—
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C2×C8×C71—22no—
82C2×C8×C71—41no—
83C2×C8×C71—41no—
84C2×C8×C71—42no—
85C2×C8×C71—42no—
86C2×C8×C71—41no—
87C4⋊C4×C71—24no—
88C4⋊C4×C71—22no—
89C4⋊C4×C71—24no—
90C4⋊C4×C71—22no—
91C4⋊C4×C71—22no—
92C22⋊C4×C71—28no—
93C22⋊C4×C71—24no—
94C22⋊C4×C71—24no—
95C42×C71—22no—
96C42×C71—22no—
97C42×C71C2×D412yes—
98C42×C71C2×D412yes—
99C23×C71—24no—
100C23×C71C2×C22⋊C411yescharacteristic
101Q8×C71—28no—
102Q8×C71—44no—
103Q8×C71—48no—
104D4×C71—44no—
105D4×C71—48no—
106D4×C71—48no—
107D4×C71—42no—
108D4×C71—42no—
109D4×C71—82no—
110D4×C71—84no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—22no—
117C2×C4×C71—24no—
118C2×C4×C71—41no—
119C2×C4×C71—416no—
120C2×C4×C71—48no—
121C2×C4×C71—42no—
122C2×C4×C71#10014yes—
123C2×C4×C71C2×C22⋊C412yes—
124C2×C4×C71#12011yescharacteristic
125C2×C4×C71#12012yes—
126C2×C4×C71#12011yescharacteristic
127C8×C71—42no—
128C8×C71—82no—
129C8×C71—81no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—44no—
133C22×C71—44no—
134C22×C71—44no—
135C22×C71#37911yescharacteristic
136C22×C71C2×#10012yes—
137C4×C71—21no—
138C4×C71—22no—
139C4×C71—24no—
140C4×C71—22no—
141C4×C71—42no—
142C4×C71—44no—
143C4×C71#45411yescharacteristic
144C4×C71#45611yescharacteristic
145#8459C7111yesmaximal, characteristic, direct product factor, Sylow
146C2×C71—22no—
147C2×C71—84no—
148C2×C71#178311yescharacteristic
149C2×C71#179311yescharacteristic
150C2×C71#179411yescharacteristic
151#3497C2×C7111yescharacteristic
152#1985C2×C7111yescharacteristic, complemented
153#1551C2×C7112yescomplemented
154#1530C2×C7112yescomplemented
155#1467C2×C7111yescharacteristic, complemented
156C71#845911yescharacteristic, direct product factor, Sylow
157C2×2−1+4C4×C7112yes—
158#550—22no—
159D4×Q8—22no—
160#548—24no—
161#537—22no—
162#537C4×C7112yes—
163#532—22no—
164#532C22×C7112yes—
165C2×C4×Q8C22×C7111yescharacteristic
166#432C22×C7112yes—
167C2×C4⋊C8C22×C7111yescharacteristic
168C2×#97C22×C7111yescharacteristic
169#336—22no—
170#333—22no—
171#331—22no—
172#329—24no—
173#324—22no—
1742−1+4—48no—
175C2×#41—24no—
176C2×#41—24no—
177C2×#41C2×C4×C7114yes—
178C22×Q8—24no—
179C22×Q8C2×C4×C7111yescharacteristic
180C22×C8—21no—
181C22×C8—21no—
182C4⋊Q8—22no—
183C4⋊Q8D4×C7112yes—
184#127—22no—
185#127D4×C7112yes—
186#124—24no—
187#124D4×C7114yes—
188C22⋊Q8—28no—
189C22⋊Q8—24no—
190#121—28no—
191#121—24no—
192#121—24no—
193#121—22no—
194#121—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—22no—
200C4×D4—22no—
201C2×C4⋊C4C2×C4×C7111yescharacteristic
202C2×C4⋊C4D4×C7112yes—
203C2×C42D4×C7112yes—
204C2×C42C23×C7111yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C22⋊C8—22no—
208#97—22no—
209#97—22no—
210#41—48no—
211#41—48no—
212#41—416no—
213#41—48no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—416no—
218C2×Q8C22⋊C4×C7114yes—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4C22⋊C4×C7114yes—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4C22⋊C4×C7112yes—
228C22×C4C22⋊C4×C7112yes—
229C22×C4C22×C4×C7111yescharacteristic
230C22×C4C2×D4×C7111yescharacteristic
231C22×C4C2×D4×C7111yescharacteristic
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—41no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C22⋊C4—28no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C42—22no—
247C42—22no—
248C42C2×D4×C7112yes—
249C42C2×D4×C7112yes—
250C23—24no—
251C23C2×C22⋊C4×C7111yescharacteristic
252Q8—28no—
253Q8—44no—
254Q8—48no—
255D4—44no—
256D4—48no—
257D4—48no—
258D4—42no—
259D4—42no—
260D4—82no—
261D4—84no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—41no—
270C2×C4—416no—
271C2×C4—48no—
272C2×C4—42no—
273C2×C4#100×C7114yes—
274C2×C4C2×C22⋊C4×C7112yes—
275C2×C4#120×C7111yescharacteristic
276C2×C4#120×C7112yes—
277C2×C4#120×C7111yescharacteristic
278C8—42no—
279C8—82no—
280C8—81no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22#379×C7111yescharacteristic
287C22C2×#100×C7112yes—
288C4—24no—
289C4—22no—
290C4—21no—
291C4—22no—
292C4—44no—
293C4—42no—
294C4#454×C7111yescharacteristic
295C4#456×C7111yescharacteristic
296C2—22no—
297C2—84no—
298C2C71×#178311yescharacteristic
299C2C71×#179311yescharacteristic
300C2C71×#179411yescharacteristic
subgroupG/H
center, socleC22×C71#379
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC71×#3497C2
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×#1985C2semidirect
3. C71×#1551C2semidirect
4. C71×#1530C2semidirect
5. C71×#1467C2semidirect
145. #8459C71direct (N × H)
152. #1985C2×C71semidirect
153. #1551C2×C71semidirect
154. #1530C2×C71semidirect
155. #1467C2×C71semidirect
156. C71#8459direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#34972
C71×#19852
C71×#15512
C71×#15302
C71×#14672
#845971