Finite Group Databasebeta

← C71×#7641 #31828513

Subgroups — C71×#7641 #31828513

Order 18176.

582 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389×C71C211yesmaximal, characteristic
2C71×#1797C211yesmaximal, characteristic
3C71×#1797C211yesmaximal, characteristic
4C71×#1271C214yesmaximal, complemented
5C22×C4⋊C4×C71—21no—
6C22×C4⋊C4×C71C411yescharacteristic
7C22×C4⋊C4×C71C411yescharacteristic
8C22×C42×C71C2211yescharacteristic
9C2×C4⋊C8×C71—21no—
10C2×C4⋊C8×C71—21no—
11C2×C22⋊C8×C71—21no—
12C2×C22⋊C8×C71—21no—
13#389×C71—21no—
14#389×C71—21no—
15#389×C71C412yes—
16#389×C71C2212yes—
17C2×#95×C71—21no—
18#336×C71C2212yes—
19#336×C71C2212yes—
20C23×C4×C71—21no—
21C23×C4×C71—21no—
22C23×C4×C71C2×C411yescharacteristic
23C23×C4×C71D411yescharacteristic
24C23×C4×C71D411yescharacteristic
25C22×C8×C71—22no—
26C22×C8×C71—22no—
27C2×C4⋊C4×C71—21no—
28C2×C4⋊C4×C71—24no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—42no—
33C2×C4⋊C4×C71—42no—
34C2×C4⋊C4×C71—42no—
35C2×C4⋊C4×C71C2×C412yes—
36C2×C4⋊C4×C71C2×C412yes—
37C2×C42×C71—42no—
38C2×C42×C71C2×C411yescharacteristic
39C2×C42×C71D411yescharacteristic
40C2×C42×C71D411yescharacteristic
41C2×C42×C71C2311yescharacteristic
42C4⋊C8×C71—42no—
43C4⋊C8×C71—42no—
44C22⋊C8×C71—42no—
45C22⋊C8×C71—42no—
46#95×C71—22no—
47#95×C71—22no—
48C24×C71C22⋊C411yescharacteristic
49C22×C4×C71—22no—
50C22×C4×C71—22no—
51C22×C4×C71—22no—
52C22×C4×C71—24no—
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71—41no—
56C22×C4×C71—41no—
57C22×C4×C71—41no—
58C22×C4×C71—41no—
59C22×C4×C71—42no—
60C22×C4×C71—42no—
61C22×C4×C71—42no—
62C22×C4×C71—42no—
63C22×C4×C71—42no—
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71D812yes—
68C22×C4×C71SD1612yes—
69C22×C4×C71SD1612yes—
70C22×C4×C71Q1612yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C22×C4×C71C2×D411yescharacteristic
74C2×C8×C71—42no—
75C2×C8×C71—42no—
76C2×C8×C71—42no—
77C2×C8×C71—42no—
78C2×C8×C71—42no—
79C2×C8×C71—42no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—44no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71M1614yes—
88C42×C71—22no—
89C42×C71—22no—
90C42×C71—41no—
91C42×C71—41no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71#10112yes—
97C23×C71#10212yes—
98C23×C71#10312yes—
99C23×C71C2×C22⋊C411yescharacteristic
100C2×C4×C71—28no—
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—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—24no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—48no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—84no—
125C2×C4×C71#9912yes—
126C2×C4×C71#10212yes—
127C2×C4×C71#10312yes—
128C2×C4×C71C2×M1612yes—
129C2×C4×C71C2×D811yescharacteristic
130C2×C4×C71C2×SD1611yescharacteristic
131C2×C4×C71C2×SD1611yescharacteristic
132C2×C4×C71C2×Q1611yescharacteristic
133C8×C71—82no—
134C8×C71—82no—
135C22×C71—22no—
136C22×C71—22no—
137C22×C71—22no—
138C22×C71—22no—
139C22×C71—22no—
140C22×C71—22no—
141C22×C71#33214yes—
142C22×C71#40711yescharacteristic
143C22×C71C2×#9911yescharacteristic
144C22×C71C2×#10111yescharacteristic
145C22×C71C2×#10211yescharacteristic
146C22×C71C2×#10311yescharacteristic
147C22×C71#42111yescharacteristic
148C22×C71#42111yescharacteristic
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—21no—
152C4×C71—22no—
153C4×C71—21no—
154C4×C71—21no—
155C4×C71—22no—
156C4×C71—21no—
157C4×C71—44no—
158C4×C71—82no—
159#7641C7111yesmaximal, characteristic, direct product factor, Sylow
160C2×C71—24no—
161C2×C71#146311yescharacteristic
162C2×C71#148611yescharacteristic
163C2×C71#148711yescharacteristic
164C2×C71#151911yescharacteristic
165C2×C71#152011yescharacteristic
166C2×C71#153411yescharacteristic
167C2×C71C2×#33211yescharacteristic
168C2×#389C2×C7111yescharacteristic
169#1797C2×C7111yescharacteristic
170#1797C2×C7111yescharacteristic
171#1271C2×C7114yescomplemented
172C71#764111yescharacteristic, direct product factor, Sylow
173C22×C4⋊C4—21no—
174C22×C4⋊C4C4×C7111yescharacteristic
175C22×C4⋊C4C4×C7111yescharacteristic
176C22×C42C22×C7111yescharacteristic
177C2×C4⋊C8—21no—
178C2×C4⋊C8—21no—
179C2×C22⋊C8—21no—
180C2×C22⋊C8—21no—
181#389—21no—
182#389—21no—
183#389C4×C7112yes—
184#389C22×C7112yes—
185C2×#95—21no—
186#336C22×C7112yes—
187#336C22×C7112yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C2×C4×C7111yescharacteristic
191C23×C4D4×C7111yescharacteristic
192C23×C4D4×C7111yescharacteristic
193C22×C8—22no—
194C22×C8—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×C4×C7112yes—
204C2×C4⋊C4C2×C4×C7112yes—
205C2×C42—42no—
206C2×C42C2×C4×C7111yescharacteristic
207C2×C42D4×C7111yescharacteristic
208C2×C42D4×C7111yescharacteristic
209C2×C42C23×C7111yescharacteristic
210C4⋊C8—42no—
211C4⋊C8—42no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214#95—22no—
215#95—22no—
216C24C22⋊C4×C7111yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C22⋊C4×C7111yescharacteristic
233C22×C4C22⋊C4×C7111yescharacteristic
234C22×C4C22⋊C4×C7111yescharacteristic
235C22×C4D8×C7112yes—
236C22×C4SD16×C7112yes—
237C22×C4SD16×C7112yes—
238C22×C4Q16×C7112yes—
239C22×C4C22×C4×C7111yescharacteristic
240C22×C4C2×D4×C7111yescharacteristic
241C22×C4C2×D4×C7111yescharacteristic
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4M16×C7114yes—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#101×C7112yes—
265C23#102×C7112yes—
266C23#103×C7112yes—
267C23C2×C22⋊C4×C7111yescharacteristic
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—84no—
293C2×C4#99×C7112yes—
294C2×C4#102×C7112yes—
295C2×C4#103×C7112yes—
296C2×C4C2×M16×C7112yes—
297C2×C4C2×D8×C7111yescharacteristic
298C2×C4C2×SD16×C7111yescharacteristic
299C2×C4C2×SD16×C7111yescharacteristic
300C2×C4C2×Q16×C7111yescharacteristic
301C8—82no—
302C8—82no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22#332×C7114yes—
310C22#407×C7111yescharacteristic
311C22C2×#99×C7111yescharacteristic
312C22C2×#101×C7111yescharacteristic
313C22C2×#102×C7111yescharacteristic
314C22C2×#103×C7111yescharacteristic
315C22#421×C7111yescharacteristic
316C22#421×C7111yescharacteristic
317C4—21no—
318C4—22no—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—82no—
327C2—24no—
328C2C71×#146311yescharacteristic
329C2C71×#148611yescharacteristic
330C2C71×#148711yescharacteristic
331C2C71×#151911yescharacteristic
332C2C71×#152011yescharacteristic
333C2C71×#153411yescharacteristic
334C2C2×#332×C7111yescharacteristic
subgroupG/H
center, socleC23×C71C2×C22⋊C4
commutatorC22×C4C22×C4×C71
Frattini, cosocleC2×C42C23×C71
prime radicalC24×C71C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C71 ◁ C22×C42×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1271C2semidirect
159. #7641C71direct (N × H)
171. #1271C2×C71semidirect
172. C71#7641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#389×C712
C71×#17972
C71×#12712
#764171