Finite Group Databasebeta

← C31×#8549 #21465308

Subgroups — C31×#8549 #21465308

Order 7936.

1478 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1478 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×C31×#121C211yesmaximal, characteristic
2C31×#1783C212yesmaximal, complemented
3C31×#1463C212yesmaximal, complemented
4C31×#1462C212yesmaximal, complemented
5C23×D4×C31C411yescharacteristic
6C23×D4×C31C411yescharacteristic
7C23×D4×C31C2211yescharacteristic
8C24×C4×C31C411yescharacteristic
9C2×C31×#121—24no—
10C2×C31×#121—24no—
11C2×C31×#121—24no—
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C22⋊C4×C31C411yescharacteristic
14C22×C22⋊C4×C31C2211yescharacteristic
15C2×C31×#100—24no—
16C31×#407C2214yes—
17C31×#324—24no—
18C31×#323—24no—
19C25×C31C2×C411yescharacteristic
20C25×C31C2×C411yescharacteristic
21C25×C31C2×C411yescharacteristic
22C22×D4×C31—22no—
23C22×D4×C31—22no—
24C22×D4×C31—22no—
25C22×D4×C31—22no—
26C22×D4×C31—22no—
27C22×D4×C31—22no—
28C22×D4×C31—22no—
29C22×D4×C31—22no—
30C22×D4×C31—44no—
31C22×D4×C31—44no—
32C22×D4×C31—44no—
33C22×D4×C31D414yes—
34C23×C4×C31—22no—
35C23×C4×C31—22no—
36C23×C4×C31—22no—
37C23×C4×C31—44no—
38C23×C4×C31C2×C411yescharacteristic
39C23×C4×C31C2×C411yescharacteristic
40C23×C4×C31C2×C411yescharacteristic
41C23×C4×C31C2311yescharacteristic
42C2×M16×C31—24no—
43C31×#121—232no—
44C2×C4⋊C4×C31—22no—
45C2×C4⋊C4×C31—22no—
46C2×C4⋊C4×C31D414yes—
47C2×C22⋊C4×C31—22no—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31—22no—
50C2×C22⋊C4×C31—22no—
51C2×C22⋊C4×C31—22no—
52C2×C22⋊C4×C31D414yes—
53C31×#100—48no—
54C31×C22⋊C8—24no—
55C31×C22⋊C8—24no—
56C24×C31—24no—
57C24×C31—21no—
58C24×C31—21no—
59C24×C31—21no—
60C24×C31—21no—
61C24×C31—21no—
62C24×C31—21no—
63C24×C31—42no—
64C24×C31—42no—
65C24×C31—42no—
66C24×C31—42no—
67C24×C31—42no—
68C24×C31—42no—
69C24×C31—42no—
70C24×C31—42no—
71C24×C31C22⋊C412yes—
72C24×C31C22⋊C412yes—
73C24×C31C22⋊C412yes—
74C24×C31C22×C411yescharacteristic
75C2×D4×C31—28no—
76C2×D4×C31—28no—
77C2×D4×C31—28no—
78C2×D4×C31—28no—
79C2×D4×C31—44no—
80C2×D4×C31—48no—
81C2×D4×C31—48no—
82C2×D4×C31—44no—
83C2×D4×C31—48no—
84C2×D4×C31—48no—
85C2×D4×C31—48no—
86C2×D4×C31—44no—
87C2×D4×C31—48no—
88C2×D4×C31—44no—
89C2×D4×C31—44no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—24no—
93C22×C4×C31—28no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—42no—
99C22×C4×C31—48no—
100C22×C4×C31—48no—
101C22×C4×C31—44no—
102C22×C4×C31—44no—
103C22×C4×C31—42no—
104C22×C4×C31—42no—
105C22×C4×C31C22⋊C412yes—
106C22×C4×C31C22⋊C412yes—
107C22×C4×C31C22⋊C412yes—
108C22×C4×C31C2×D412yes—
109C22×C4×C31C2×D412yes—
110C22×C4×C31C2×D412yes—
111M16×C31—48no—
112C2×C8×C31—44no—
113C4⋊C4×C31—28no—
114C22⋊C4×C31—28no—
115C22⋊C4×C31—28no—
116#8549C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—22no—
118C23×C31—22no—
119C23×C31—22no—
120C23×C31—24no—
121C23×C31—24no—
122C23×C31—22no—
123C23×C31—22no—
124C23×C31—22no—
125C23×C31—22no—
126C23×C31—24no—
127C23×C31—24no—
128C23×C31—42no—
129C23×C31—42no—
130C23×C31—42no—
131C23×C31—416no—
132C23×C31—42no—
133C23×C31—42no—
134C23×C31—42no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—42no—
139C23×C31—42no—
140C23×C31—42no—
141C23×C31—42no—
142C23×C31—42no—
143C23×C31—42no—
144C23×C31—42no—
145C23×C31—42no—
146C23×C31—42no—
147C23×C31—42no—
148C23×C31—42no—
149C23×C31—42no—
150C23×C31—42no—
151C23×C31—88no—
152C23×C31—88no—
153C23×C31#9914yes—
154C23×C31C2×C22⋊C411yescharacteristic
155C23×C31C2×C22⋊C411yescharacteristic
156C23×C31C2×C22⋊C411yescharacteristic
157D4×C31—416no—
158D4×C31—416no—
159D4×C31—416no—
160C2×C4×C31—24no—
161C2×C4×C31—24no—
162C2×C4×C31—22no—
163C2×C4×C31—24no—
164C2×C4×C31—22no—
165C2×C4×C31—22no—
166C2×C4×C31—22no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—416no—
170C2×C4×C31—42no—
171C2×C4×C31—42no—
172C2×C4×C31—42no—
173C2×C4×C31—42no—
174C2×C4×C31—42no—
175C2×C4×C31—42no—
176C2×C4×C31#10014yescomplemented
177C2×C4×C31#12014yes—
178C8×C31—84no—
179C22×#121C2×C3111yescharacteristic
180#1783C2×C3112yescomplemented
181#1463C2×C3112yescomplemented
182#1462C2×C3112yescomplemented
183C22×C31—28no—
184C22×C31—21no—
185C22×C31—21no—
186C22×C31—21no—
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—21no—
190C22×C31—24no—
191C22×C31—44no—
192C22×C31—44no—
193C22×C31—48no—
194C22×C31—48no—
195C22×C31—42no—
196C22×C31—42no—
197C22×C31—42no—
198C22×C31—42no—
199C22×C31—42no—
200C22×C31—42no—
201C22×C31—42no—
202C22×C31—42no—
203C22×C31—84no—
204C22×C31—84no—
205C22×C31—84no—
206C22×C31—84no—
207C22×C31—84no—
208C22×C31—84no—
209C22×C31#37911yescharacteristic
210C22×C31C2×#9912yes—
211C22×C31C2×#10012yes—
212C22×C31#41312yes—
213C4×C31—24no—
214C4×C31—24no—
215C4×C31—42no—
216C4×C31—42no—
217C23×D4C4×C3111yescharacteristic
218C23×D4C4×C3111yescharacteristic
219C23×D4C22×C3111yescharacteristic
220C24×C4C4×C3111yescharacteristic
221C2×#121—24no—
222C2×#121—24no—
223C2×#121—24no—
224C22×C4⋊C4C22×C3111yescharacteristic
225C22×C22⋊C4C4×C3111yescharacteristic
226C22×C22⋊C4C22×C3111yescharacteristic
227C2×#100—24no—
228#407C22×C3114yes—
229#324—24no—
230#323—24no—
231C2×C31—22no—
232C2×C31—22no—
233C2×C31—22no—
234C2×C31—44no—
235C2×C31—82no—
236C2×C31—82no—
237C2×C31#178111yescharacteristic
238C2×C31#178311yescharacteristic
239C2×C31#178511yescharacteristic
240C25C2×C4×C3111yescharacteristic
241C25C2×C4×C3111yescharacteristic
242C25C2×C4×C3111yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4D4×C3114yes—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4C2×C4×C3111yescharacteristic
260C23×C4C2×C4×C3111yescharacteristic
261C23×C4C2×C4×C3111yescharacteristic
262C23×C4C23×C3111yescharacteristic
263C2×M16—24no—
264#121—232no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4D4×C3114yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4D4×C3114yes—
274#100—48no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C31#854911yescharacteristic, direct product factor, Sylow
278C24—24no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24C22⋊C4×C3112yes—
294C24C22⋊C4×C3112yes—
295C24C22⋊C4×C3112yes—
296C24C22×C4×C3111yescharacteristic
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—44no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—44no—
311C2×D4—44no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C22⋊C4×C3112yes—
328C22×C4C22⋊C4×C3112yes—
329C22×C4C22⋊C4×C3112yes—
330C22×C4C2×D4×C3112yes—
331C22×C4C2×D4×C3112yes—
332C22×C4C2×D4×C3112yes—
333M16—48no—
334C2×C8—44no—
335C4⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—28no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—416no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—88no—
373C23—88no—
374C23C31×#9914yes—
375C23C2×C22⋊C4×C3111yescharacteristic
376C23C2×C22⋊C4×C3111yescharacteristic
377C23C2×C22⋊C4×C3111yescharacteristic
378D4—416no—
379D4—416no—
380D4—416no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—416no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4C31×#10014yescomplemented
398C2×C4C31×#12014yes—
399C8—84no—
400C22—28no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—24no—
408C22—44no—
409C22—44no—
410C22—48no—
411C22—48no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22—84no—
424C22—84no—
425C22—84no—
426C22C31×#37911yescharacteristic
427C22C2×C31×#9912yes—
428C22C2×C31×#10012yes—
429C22C31×#41312yes—
430C4—24no—
431C4—24no—
432C4—42no—
433C4—42no—
434C2—22no—
435C2—22no—
436C2—22no—
437C2—44no—
438C2—82no—
439C2—82no—
440C2C31×#178111yescharacteristic
441C2C31×#178311yescharacteristic
442C2C31×#178511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC22×C31×#121C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1783C2semidirect
3. C31×#1463C2semidirect
4. C31×#1462C2semidirect
116. #8549C31direct (N × H)
176. C2×C4×C31#100semidirect
180. #1783C2×C31semidirect
181. #1463C2×C31semidirect
182. #1462C2×C31semidirect
277. C31#8549direct (N × H)
397. C2×C4C31×#100semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1212
C31×#17832
C31×#14632
C31×#14622
#854931