Finite Group Databasebeta

← C71×#9030 #31829902

Subgroups — C71×#9030 #31829902

Order 18176.

484 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C71×#2983C211yesmaximal, characteristic
2C71×#2399C211yesmaximal, characteristic
3C4×#101×C71C211yesmaximal, characteristic
4C71×#1491C212yesmaximal
5C71×#1300C212yesmaximal
6C2×C4×Q8×C71C2211yescharacteristic
7C2×C4×Q8×C71C2211yescharacteristic
8#446×C71—24no—
9#432×C71—21no—
10#432×C71C2212yes—
11#431×C71—21no—
12C2×#101×C71—21no—
13C4×M16×C71C2212yes—
14#398×C71—21no—
15#398×C71C411yescharacteristic
16#398×C71C411yescharacteristic
17#395×C71—21no—
18#395×C71C411yescharacteristic
19#395×C71C411yescharacteristic
20C4×C4⋊C4×C71—21no—
21C4×C4⋊C4×C71—21no—
22#376×C71C2211yescharacteristic
23#343×C71—21no—
24Q8⋊C8×C71—24no—
25C22×Q8×C71C2×C411yescharacteristic
26C22×Q8×C71D411yescharacteristic
27C2×M16×C71—22no—
28C4×Q8×C71—22no—
29C4×Q8×C71—22no—
30C4×Q8×C71—22no—
31C4×Q8×C71—22no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C42×C71—21no—
42C2×C42×C71—21no—
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C2×C42×C71C2311yescharacteristic
46C4⋊C8×C71—22no—
47C4⋊C8×C71—22no—
48C4⋊C8×C71D414yes—
49#101×C71—24no—
50C22⋊C8×C71—22no—
51#97×C71—22no—
52#97×C71—22no—
53#97×C71—22no—
54C4×C8×C71—22no—
55C4×C8×C71—22no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71C2×C411yescharacteristic
62#95×C71C2×C411yescharacteristic
63C2×Q8×C71—22no—
64C2×Q8×C71—22no—
65C2×Q8×C71—22no—
66C2×Q8×C71—41no—
67C2×Q8×C71—42no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79M16×C71—24no—
80M16×C71—42no—
81C2×C8×C71—24no—
82C2×C8×C71—42no—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—22no—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—22no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C4⋊C4×C71—42no—
90C4⋊C4×C71—42no—
91C4⋊C4×C71—42no—
92C42×C71—24no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—41no—
98C42×C71—41no—
99C42×C71C2×D412yes—
100C42×C71#4112yes—
101C23×C71C2×C22⋊C411yescharacteristic
102Q8×C71—24no—
103Q8×C71—44no—
104Q8×C71—42no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71#10412yes—
131C2×C4×C71C4×D412yes—
132C2×C4×C71#12011yescharacteristic
133C2×C4×C71#12112yes—
134C2×C4×C71#12311yescharacteristic
135C8×C71—44no—
136C8×C71—44no—
137C22×C71—21no—
138C22×C71—21no—
139C22×C71#38611yescharacteristic
140C22×C71C2×#10411yescharacteristic
141C22×C71#42111yescharacteristic
142C4×C71—22no—
143C4×C71—22no—
144C4×C71—21no—
145C4×C71—22no—
146C4×C71—42no—
147C4×C71—42no—
148C4×C71—41no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—42no—
152C4×C71—81no—
153C4×C71#45612yes—
154#9030C7111yesmaximal, characteristic, direct product factor, Sylow
155C2×C71—21no—
156C2×C71—21no—
157C2×C71#187111yescharacteristic
158C2×C71#188611yescharacteristic
159C2×C71#189111yescharacteristic
160#2983C2×C7111yescharacteristic
161#2399C2×C7111yescharacteristic
162C4×#101C2×C7111yescharacteristic
163#1491C2×C7112yes—
164#1300C2×C7112yes—
165C71#903011yescharacteristic, direct product factor, Sylow
166C2×C4×Q8C22×C7111yescharacteristic
167C2×C4×Q8C22×C7111yescharacteristic
168#446—24no—
169#432—21no—
170#432C22×C7112yes—
171#431—21no—
172C2×#101—21no—
173C4×M16C22×C7112yes—
174#398—21no—
175#398C4×C7111yescharacteristic
176#398C4×C7111yescharacteristic
177#395—21no—
178#395C4×C7111yescharacteristic
179#395C4×C7111yescharacteristic
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182#376C22×C7111yescharacteristic
183#343—21no—
184Q8⋊C8—24no—
185C22×Q8C2×C4×C7111yescharacteristic
186C22×Q8D4×C7111yescharacteristic
187C2×M16—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C7111yescharacteristic
198C2×C4⋊C4C2×C4×C7111yescharacteristic
199C2×C4⋊C4C2×C4×C7111yescharacteristic
200C2×C4⋊C4D4×C7111yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42D4×C7111yescharacteristic
204C2×C42D4×C7111yescharacteristic
205C2×C42C23×C7111yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8D4×C7114yes—
209#101—24no—
210C22⋊C8—22no—
211#97—22no—
212#97—22no—
213#97—22no—
214C4×C8—22no—
215C4×C8—22no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C2×C4×C7111yescharacteristic
222#95C2×C4×C7111yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4C22⋊C4×C7111yescharacteristic
233C22×C4C22⋊C4×C7111yescharacteristic
234C22×C4C22⋊C4×C7111yescharacteristic
235C22×C4C22⋊C4×C7111yescharacteristic
236C22×C4C22×C4×C7111yescharacteristic
237C22×C4C2×D4×C7111yescharacteristic
238C22×C4C2×D4×C7111yescharacteristic
239M16—24no—
240M16—42no—
241C2×C8—24no—
242C2×C8—42no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C42—24no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42C2×D4×C7112yes—
260C42#41×C7112yes—
261C23C2×C22⋊C4×C7111yescharacteristic
262Q8—24no—
263Q8—44no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#104×C7112yes—
291C2×C4C4×D4×C7112yes—
292C2×C4#120×C7111yescharacteristic
293C2×C4#121×C7112yes—
294C2×C4#123×C7111yescharacteristic
295C8—44no—
296C8—44no—
297C22—21no—
298C22—21no—
299C22#386×C7111yescharacteristic
300C22C2×#104×C7111yescharacteristic
301C22#421×C7111yescharacteristic
302C4—22no—
303C4—22no—
304C4—21no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4#456×C7112yes—
314C2—21no—
315C2—21no—
316C2C71×#187111yescharacteristic
317C2C71×#188611yescharacteristic
318C2C71×#189111yescharacteristic
subgroupG/H
centerC2×C4×C71#120
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C71 ◁ C2×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
154. #9030C71direct (N × H)
165. C71#9030direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#29832
C71×#23992
C4×#101×C712
C71×#14912
C71×#13002
#903071