Finite Group Databasebeta

← C71×#12854 #31833726

Subgroups — C71×#12854 #31833726

Order 18176.

784 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2308C211yesmaximal, characteristic
2C71×#1783C211yesmaximal, characteristic
3C71×#1781C211yesmaximal, characteristic
4C71×#1390C214yesmaximal, complemented
5C23×D4×C71C2211yescharacteristic
6C22×C22⋊C4×C71C411yescharacteristic, complemented
7C22×C22⋊C4×C71C411yescharacteristic, complemented
8C2×#100×C71—21no—
9C2×#100×C71C2212yes—
10C2×#99×C71—21no—
11C2×#99×C71C2212yes—
12#407×C71—21no—
13#386×C71—21no—
14#386×C71—21no—
15#386×C71C412yescomplemented
16#386×C71C2212yes—
17#381×C71—21no—
18#379×C71—21no—
19#379×C71—21no—
20C25×C71C2×C411yescharacteristic
21C25×C71D411yescharacteristic
22C22×D4×C71—24no—
23C22×D4×C71—42no—
24C22×D4×C71—42no—
25C22×D4×C71C2×C411yescharacteristic
26C22×D4×C71D411yescharacteristic
27C22×D4×C71D411yescharacteristic
28C22×D4×C71C2311yescharacteristic
29C23×C4×C71—21no—
30C23×C4×C71C2×C412yes—
31C23×C4×C71D411yescharacteristic
32C2×M16×C71—22no—
33C2×C22⋊C4×C71—21no—
34C2×C22⋊C4×C71—21no—
35C2×C22⋊C4×C71—21no—
36C2×C22⋊C4×C71—22no—
37C2×C22⋊C4×C71—22no—
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71—42no—
40C2×C22⋊C4×C71—42no—
41C2×C22⋊C4×C71—42no—
42C2×C22⋊C4×C71C2×C412yes—
43#100×C71—24no—
44#100×C71—42no—
45#99×C71—24no—
46#99×C71—42no—
47C22⋊C8×C71—42no—
48#95×C71—22no—
49#95×C71—22no—
50C24×C71—22no—
51C24×C71—22no—
52C24×C71—22no—
53C24×C71—22no—
54C24×C71—22no—
55C24×C71—41no—
56C24×C71—82no—
57C24×C71—82no—
58C24×C71C22⋊C411yescharacteristic
59C24×C71C22⋊C411yescharacteristic
60C24×C71C22⋊C411yescharacteristic
61C24×C71C22×C411yescharacteristic
62C24×C71C2×D411yescharacteristic
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—22no—
67C2×D4×C71—22no—
68C2×D4×C71—41no—
69C2×D4×C71—41no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C2×D4×C71—41no—
73C2×D4×C71—41no—
74C2×D4×C71—416no—
75C2×D4×C71#4114yes—
76C22×C4×C71—22no—
77C22×C4×C71—22no—
78C22×C4×C71—22no—
79C22×C4×C71—22no—
80C22×C4×C71—41no—
81C22×C4×C71—42no—
82C22×C4×C71—42no—
83C22×C4×C71—42no—
84C22×C4×C71—82no—
85C22×C4×C71—82no—
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88M16×C71—42no—
89M16×C71—42no—
90C2×C8×C71—42no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—42no—
94C22⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—82no—
98C22⋊C4×C71—82no—
99C22⋊C4×C71—84no—
100C23×C71—28no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71—42no—
113C23×C71—44no—
114C23×C71—42no—
115C23×C71—42no—
116C23×C71—42no—
117C23×C71—42no—
118C23×C71—44no—
119C23×C71—82no—
120C23×C71—82no—
121C23×C71—82no—
122C23×C71—84no—
123C23×C71—84no—
124C23×C71—84no—
125C23×C71—84no—
126C23×C71#9912yes—
127C23×C71#9912yes—
128C23×C71C2×C22⋊C411yescharacteristic
129C23×C71#11712yes—
130C23×C71#12312yes—
131D4×C71—44no—
132D4×C71—44no—
133D4×C71—44no—
134D4×C71—44no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—44no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
146C2×C4×C71—42no—
147C2×C4×C71—84no—
148C2×C4×C71—84no—
149C2×C4×C71—82no—
150C2×C4×C71—84no—
151C2×C4×C71#12312yes—
152C8×C71—82no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71—22no—
158C22×C71—22no—
159C22×C71—22no—
160C22×C71—42no—
161C22×C71—42no—
162C22×C71—42no—
163C22×C71—42no—
164C22×C71—42no—
165C22×C71—42no—
166C22×C71—42no—
167C22×C71—42no—
168C22×C71—42no—
169C22×C71—42no—
170C22×C71—42no—
171C22×C71—41no—
172C22×C71—42no—
173C22×C71—81no—
174C22×C71—81no—
175C22×C71—88no—
176C22×C71—88no—
177C22×C71#38111yescharacteristic
178C22×C71C2×#9911yescharacteristic
179C22×C71C2×#9911yescharacteristic
180C4×C71—22no—
181C4×C71—41no—
182C4×C71—82no—
183C4×C71—82no—
184C4×C71—162no—
185#12854C7111yesmaximal, characteristic, direct product factor, Sylow
186C2×C71—21no—
187C2×C71—21no—
188C2×C71—42no—
189C2×C71—42no—
190C2×C71—42no—
191C2×C71—42no—
192C2×C71—81no—
193C2×C71#181911yescharacteristic
194C2×C71#212112yes—
195#2308C2×C7111yescharacteristic
196#1783C2×C7111yescharacteristic
197#1781C2×C7111yescharacteristic
198#1390C2×C7114yescomplemented
199C71#1285411yescharacteristic, direct product factor, Sylow
200C23×D4C22×C7111yescharacteristic
201C22×C22⋊C4C4×C7111yescharacteristic, complemented
202C22×C22⋊C4C4×C7111yescharacteristic, complemented
203C2×#100—21no—
204C2×#100C22×C7112yes—
205C2×#99—21no—
206C2×#99C22×C7112yes—
207#407—21no—
208#386—21no—
209#386—21no—
210#386C4×C7112yescomplemented
211#386C22×C7112yes—
212#381—21no—
213#379—21no—
214#379—21no—
215C25C2×C4×C7111yescharacteristic
216C25D4×C7111yescharacteristic
217C22×D4—24no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4C2×C4×C7111yescharacteristic
221C22×D4D4×C7111yescharacteristic
222C22×D4D4×C7111yescharacteristic
223C22×D4C23×C7111yescharacteristic
224C23×C4—21no—
225C23×C4C2×C4×C7112yes—
226C23×C4D4×C7111yescharacteristic
227C2×M16—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4C2×C4×C7112yes—
238#100—24no—
239#100—42no—
240#99—24no—
241#99—42no—
242C22⋊C8—42no—
243#95—22no—
244#95—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—41no—
251C24—82no—
252C24—82no—
253C24C22⋊C4×C7111yescharacteristic
254C24C22⋊C4×C7111yescharacteristic
255C24C22⋊C4×C7111yescharacteristic
256C24C22×C4×C7111yescharacteristic
257C24C2×D4×C7111yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—416no—
270C2×D4#41×C7114yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C22⋊C4×C7111yescharacteristic
282C22×C4C2×D4×C7111yescharacteristic
283M16—42no—
284M16—42no—
285C2×C8—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—84no—
295C23—28no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—42no—
308C23—44no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—44no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—84no—
318C23—84no—
319C23—84no—
320C23—84no—
321C23#99×C7112yes—
322C23#99×C7112yes—
323C23C2×C22⋊C4×C7111yescharacteristic
324C23#117×C7112yes—
325C23#123×C7112yes—
326D4—44no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4#123×C7112yes—
347C8—82no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—88no—
371C22—88no—
372C22#381×C7111yescharacteristic
373C22C2×#99×C7111yescharacteristic
374C22C2×#99×C7111yescharacteristic
375C4—22no—
376C4—41no—
377C4—82no—
378C4—82no—
379C4—162no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—42no—
384C2—42no—
385C2—42no—
386C2—81no—
387C2C71×#181911yescharacteristic
388C2C71×#212112yes—
subgroupG/H
center, socleC22×C71C2×#99
commutatorC24C22×C4×C71
Frattini, cosocleC22×D4C23×C71
prime radicalC23×D4×C71C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C24×C71 ◁ C23×D4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1390C2semidirect
6. C22×C22⋊C4×C71C4semidirect
7. C22×C22⋊C4×C71C4semidirect
15. #386×C71C4semidirect
185. #12854C71direct (N × H)
198. #1390C2×C71semidirect
199. C71#12854direct (N × H)
201. C22×C22⋊C4C4×C71semidirect
202. C22×C22⋊C4C4×C71semidirect
210. #386C4×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#23082
C71×#17832
C71×#17812
C71×#13902
#1285471