Finite Group Databasebeta

← C2×C71×#1353 #31832591

Subgroups — C2×C71×#1353 #31832591

Order 18176.

500 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C71C211yesmaximal, characteristic
2C2×#343×C71C211yesmaximal, characteristic
3C2×#324×C71C211yesmaximal, characteristic
4C71×#1353C214yesmaximal, direct product factor
5C22×M16×C71—21no—
6C22×C4⋊C4×C71C411yescharacteristic
7C22×C4⋊C4×C71C2211yescharacteristic
8C2×C22⋊C8×C71—21no—
9#400×C71—21no—
10#400×C71—21no—
11#400×C71C412yes—
12#400×C71C2212yes—
13C2×#95×C71—21no—
14C2×#95×C71—21no—
15C2×#95×C71C411yescharacteristic
16#343×C71—21no—
17#343×C71C2212yes—
18#324×C71—21no—
19#324×C71C2212yes—
20C23×C4×C71—21no—
21C23×C4×C71—21no—
22C23×C4×C71C2×C411yescharacteristic
23C23×C4×C71D411yescharacteristic
24C23×C4×C71D411yescharacteristic
25C2×M16×C71—24no—
26C2×M16×C71—22no—
27C2×M16×C71—41no—
28C2×M16×C71—42no—
29C22×C8×C71—41no—
30C22×C8×C71—41no—
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71—41no—
33C2×C4⋊C4×C71—41no—
34C2×C4⋊C4×C71—41no—
35C2×C4⋊C4×C71—41no—
36C2×C4⋊C4×C71C2×C412yes—
37C2×C4⋊C4×C71C2×C411yescharacteristic
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71C2311yescharacteristic
41C22⋊C8×C71—22no—
42C22⋊C8×C71—41no—
43#95×C71—22no—
44#95×C71—22no—
45#95×C71—21no—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71C2×C412yes—
49C24×C71C22⋊C411yescharacteristic
50C22×C4×C71—21no—
51C22×C4×C71—21no—
52C22×C4×C71—22no—
53C22×C4×C71—21no—
54C22×C4×C71—21no—
55C22×C4×C71—21no—
56C22×C4×C71—21no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—42no—
60C22×C4×C71—42no—
61C22×C4×C71—42no—
62C22×C4×C71—42no—
63C22×C4×C71—44no—
64C22×C4×C71—41no—
65C22×C4×C71—41no—
66C22×C4×C71—42no—
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71SD1612yes—
71C22×C4×C71Q1612yes—
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75M16×C71—24no—
76M16×C71—44no—
77M16×C71—42no—
78C2×C8×C71—44no—
79C2×C8×C71—42no—
80C2×C8×C71—42no—
81C2×C8×C71—82no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C23×C71—22no—
89C23×C71—22no—
90C23×C71#9911yescharacteristic
91C23×C71#9911yescharacteristic
92C23×C71#10312yes—
93C23×C71#10412yes—
94C23×C71C2×C22⋊C411yescharacteristic
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—41no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—44no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—81no—
120C2×C4×C71—84no—
121C2×C4×C71—82no—
122C2×C4×C71—82no—
123C2×C4×C71#10312yes—
124C2×C4×C71C2×SD1611yescharacteristic
125C2×C4×C71C2×Q1611yescharacteristic
126C8×C71—44no—
127C8×C71—82no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—24no—
131C22×C71—24no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71#32812yes—
135C22×C71#32812yes—
136C22×C71#35212yes—
137C22×C71#35512yes—
138C22×C71C2×#9911yescharacteristic
139C22×C71C2×#10311yescharacteristic
140C22×C71C2×#10411yescharacteristic
141C4×C71—21no—
142C4×C71—22no—
143C4×C71—21no—
144C4×C71—41no—
145C4×C71—41no—
146C4×C71—41no—
147C4×C71—41no—
148C4×C71—82no—
149C4×C71—82no—
150C2×#1353C7111yesmaximal, characteristic, direct product factor, Sylow
151C2×C71—21no—
152C2×C71—22no—
153C2×C71—21no—
154C2×C71#135314yesdirect product factor
155C2×C71C2×#32811yescharacteristic
156C2×C71C2×#35211yescharacteristic
157C2×C71C2×#35511yescharacteristic
158C2×#400C2×C7111yescharacteristic
159C2×#343C2×C7111yescharacteristic
160C2×#324C2×C7111yescharacteristic
161#1353C2×C7114yesdirect product factor
162C71C2×#135311yescharacteristic, direct product factor, Sylow
163C22×M16—21no—
164C22×C4⋊C4C4×C7111yescharacteristic
165C22×C4⋊C4C22×C7111yescharacteristic
166C2×C22⋊C8—21no—
167#400—21no—
168#400—21no—
169#400C4×C7112yes—
170#400C22×C7112yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95C4×C7111yescharacteristic
174#343—21no—
175#343C22×C7112yes—
176#324—21no—
177#324C22×C7112yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×C4×C7111yescharacteristic
181C23×C4D4×C7111yescharacteristic
182C23×C4D4×C7111yescharacteristic
183C2×M16—24no—
184C2×M16—22no—
185C2×M16—41no—
186C2×M16—42no—
187C22×C8—41no—
188C22×C8—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C2×C4×C7112yes—
195C2×C4⋊C4C2×C4×C7111yescharacteristic
196C2×C4⋊C4D4×C7111yescharacteristic
197C2×C4⋊C4D4×C7111yescharacteristic
198C2×C4⋊C4C23×C7111yescharacteristic
199C22⋊C8—22no—
200C22⋊C8—41no—
201#95—22no—
202#95—22no—
203#95—21no—
204#95—22no—
205#95—22no—
206#95C2×C4×C7112yes—
207C24C22⋊C4×C7111yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4C22⋊C4×C7111yescharacteristic
226C22×C4C22⋊C4×C7111yescharacteristic
227C22×C4C22⋊C4×C7111yescharacteristic
228C22×C4SD16×C7112yes—
229C22×C4Q16×C7112yes—
230C22×C4C22×C4×C7111yescharacteristic
231C22×C4C2×D4×C7111yescharacteristic
232C22×C4C2×D4×C7111yescharacteristic
233M16—24no—
234M16—44no—
235M16—42no—
236C2×C8—42no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—82no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C23—22no—
247C23—22no—
248C23#99×C7111yescharacteristic
249C23#99×C7111yescharacteristic
250C23#103×C7112yes—
251C23#104×C7112yes—
252C23C2×C22⋊C4×C7111yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#103×C7112yes—
282C2×C4C2×SD16×C7111yescharacteristic
283C2×C4C2×Q16×C7111yescharacteristic
284C8—44no—
285C8—82no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22—21no—
291C22—21no—
292C22#328×C7112yes—
293C22#328×C7112yes—
294C22#352×C7112yes—
295C22#355×C7112yes—
296C22C2×#99×C7111yescharacteristic
297C22C2×#103×C7111yescharacteristic
298C22C2×#104×C7111yescharacteristic
299C4—21no—
300C4—22no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4—82no—
308C2—21no—
309C2—21no—
310C2—22no—
311C2C71×#135314yesdirect product factor
312C2C2×#328×C7111yescharacteristic
313C2C2×#352×C7111yescharacteristic
314C2C2×#355×C7111yescharacteristic
subgroupG/H
center, socleC23×C71#99
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C4⋊C4C23×C71
prime radicalC24×C71C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C71 ◁ C24×C71 ◁ C22×C4⋊C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1353C2direct (N × H)
150. C2×#1353C71direct (N × H)
154. C2×C71#1353direct (N × H)
161. #1353C2×C71direct (N × H)
162. C71C2×#1353direct (N × H)
311. C2C71×#1353direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C712
C2×#343×C712
C2×#324×C712
C71×#13532
C2×#135371