Finite Group Databasebeta

← C4×C31×#557 #21490928

Subgroups — C4×C31×#557 #21490928

Order 7936.

2220 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#557C211yesmaximal, characteristic
2C31×#2370C216yesmaximal
3C31×#2365C213yesmaximal
4C31×#2355C213yesmaximal
5C31×#2351C212yesmaximal
6Q8×C4⋊C4×C31C211yesmaximal, characteristic
7C4×C31×C4⋊Q8C213yesmaximal
8C4×C31×#125C216yesmaximal
9C42×Q8×C31C213yesmaximal
10C42×Q8×C31C212yesmaximal
11C42×Q8×C31C211yesmaximal, characteristic
12C31×#557C4116yesdirect product factor
13C2×C31×C4⋊Q8C2213yes—
14C2×C31×#125C2216yes—
15C2×C4×Q8×C31C2213yes—
16C2×C4×Q8×C31C2212yes—
17C2×C4×Q8×C31C2211yescharacteristic
18C2×C4×Q8×C31C2213yes—
19C2×C4×Q8×C31C2211yescharacteristic
20C2×C4×Q8×C31C2212yes—
21C2×C4×Q8×C31C2213yes—
22C2×C4×Q8×C31C2211yescharacteristic
23C31×#391C2216yes—
24C31×#389C2216yes—
25C31×#389C2216yes—
26C31×#389C2216yes—
27C31×#389C2216yes—
28C31×#387C22112yes—
29C31×#387C22112yes—
30C31×#384C2213yes—
31C31×#382C2216yes—
32C4×C4⋊C4×C31C2216yes—
33C4×C4⋊C4×C31C2216yes—
34C4×C4⋊C4×C31C2213yes—
35C4×C4⋊C4×C31C2216yes—
36C4×C4⋊C4×C31C2213yes—
37C4×C4⋊C4×C31C2213yes—
38C4×C4⋊C4×C31C2216yes—
39C4×C4⋊C4×C31C2216yes—
40C4×C4⋊C4×C31C2216yes—
41C4×C4⋊C4×C31C2213yes—
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C4⋊C4×C31C2216yes—
44C4×C4⋊C4×C31C2213yes—
45C43×C31C2213yes—
46C43×C31C2216yes—
47C22×Q8×C31C2313yes—
48C22×Q8×C31C2311yescharacteristic
49C31×C4⋊Q8C2×C4124yes—
50C31×#125C2×C4148yes—
51C4×Q8×C31—28no—
52C4×Q8×C31—24no—
53C4×Q8×C31—212no—
54C4×Q8×C31—212no—
55C4×Q8×C31C2×C4116yes—
56C4×Q8×C31C2×C4124yes—
57C4×Q8×C31C2×C418yes—
58C4×Q8×C31Q818yescomplemented
59C2×C4⋊C4×C31C2313yes—
60C2×C4⋊C4×C31C2316yes—
61C2×C4⋊C4×C31C2313yes—
62C2×C4⋊C4×C31C2316yes—
63C2×C4⋊C4×C31C2316yes—
64C2×C4⋊C4×C31C2316yes—
65C2×C4⋊C4×C31C2311yescharacteristic
66C2×C4⋊C4×C31C2316yes—
67C2×C4⋊C4×C31C2313yes—
68C2×C4⋊C4×C31C2316yes—
69C2×C4⋊C4×C31C2316yes—
70C2×C4⋊C4×C31C2313yes—
71C2×C4⋊C4×C31C2311yescharacteristic
72C2×C42×C31C2316yes—
73C2×C42×C31C2313yes—
74C2×C42×C31C2316yes—
75C2×C42×C31C2313yes—
76C2×C42×C31C2316yes—
77C2×C42×C31C2312yes—
78C2×C42×C31C2313yes—
79C2×C42×C31C2316yes—
80C2×C42×C31C2313yes—
81C2×C42×C31C2316yes—
82C2×C42×C31C2313yes—
83C2×C42×C31C2313yes—
84C2×C42×C31C2311yescharacteristic
85C31×#95C2312yes—
86C31×#95C2316yes—
87C31×#95C2316yes—
88C31×#95C2316yes—
89C31×#95C23112yes—
90C31×#95C23112yes—
91C2×Q8×C31—26no—
92C2×Q8×C31—26no—
93C2×Q8×C31C22×C4112yes—
94C2×Q8×C31C22×C414yes—
95C2×Q8×C31C2×Q814yes—
96C2×Q8×C31#4114yes—
97C22×C4×C31C2413yes—
98C22×C4×C31C2416yes—
99C22×C4×C31C2412yes—
100C22×C4×C31C2413yes—
101C22×C4×C31C2416yes—
102C22×C4×C31C2413yes—
103C22×C4×C31C2412yes—
104C22×C4×C31C2411yescharacteristic
105C22×C4×C31C2413yes—
106C22×C4×C31C2411yescharacteristic
107C22×C4×C31C2411yescharacteristic
108C4⋊C4×C31—212no—
109C4⋊C4×C31—212no—
110C4⋊C4×C31—26no—
111C4⋊C4×C31—212no—
112C4⋊C4×C31—26no—
113C4⋊C4×C31—46no—
114C4⋊C4×C31C22×C414yes—
115C4⋊C4×C31C22×C4124yes—
116C4⋊C4×C31C22×C4112yes—
117C4⋊C4×C31C22×C4124yes—
118C4⋊C4×C31C22×C4124yes—
119C4⋊C4×C31C2×Q8112yes—
120C4⋊C4×C31#4114yes—
121C42×C31—212no—
122C42×C31—26no—
123C42×C31—212no—
124C42×C31—26no—
125C42×C31—212no—
126C42×C31—24no—
127C42×C31—26no—
128C42×C31—43no—
129C42×C31—46no—
130C42×C31C22×C4124yes—
131C42×C31C22×C4112yes—
132C42×C31C2×Q8112yes—
133C42×C31#4114yes—
134C4×#557C3111yesmaximal, characteristic, direct product factor, Sylow
135C23×C31C2511yescharacteristic
136Q8×C31—224no—
137Q8×C31C4×Q8116yescomplemented
138C2×C4×C31—23no—
139C2×C4×C31—21no—
140C2×C4×C31—22no—
141C2×C4×C31—26no—
142C2×C4×C31—26no—
143C2×C4×C31—22no—
144C2×C4×C31—23no—
145C2×C4×C31—22no—
146C2×C4×C31—22no—
147C2×C4×C31—23no—
148C2×C4×C31—21no—
149C2×C4×C31—23no—
150C2×C4×C31—26no—
151C2×C4×C31—26no—
152C2×C4×C31—23no—
153C2×C4×C31—23no—
154C2×C4×C31—23no—
155C2×C4×C31—26no—
156C2×C4×C31C23×C414yes—
157C2×C4×C31C23×C416yes—
158C2×C4×C31C23×C4112yes—
159C2×C4×C31C23×C416yes—
160C2×C4×C31C23×C412yes—
161C2×C4×C31C22×Q816yes—
162C2×C4×C31C22×Q816yes—
163C2×C4×C31C22×Q812yes—
164C2×C4×C31C2×#4114yes—
165C2×C4×C31C2×#4112yes—
166C2×C4×C31C2×#4116yes—
167C2×C4×C31C2×#4112yes—
168C2×C4×C31C2×#4112yes—
169C2×C4×C31C2×#4112yes—
170C2×C4×C312−1+412yes—
171C2×#557C2×C3111yescharacteristic
172#2370C2×C3116yes—
173#2365C2×C3113yes—
174#2355C2×C3113yes—
175#2351C2×C3112yes—
176Q8×C4⋊C4C2×C3111yescharacteristic
177C4×C4⋊Q8C2×C3113yes—
178C4×#125C2×C3116yes—
179C42×Q8C2×C3113yes—
180C42×Q8C2×C3112yes—
181C42×Q8C2×C3111yescharacteristic
182C22×C31C24×C411yescharacteristic
183C22×C31C23×Q811yescharacteristic
184C22×C31C22×#4111yescharacteristic
185C22×C31C22×#4111yescharacteristic
186C22×C31C22×#4111yescharacteristic
187C22×C31C2×2−1+411yescharacteristic
188C22×C31#58511yescharacteristic
189C4×C31—212no—
190C4×C31—24no—
191C4×C31—26no—
192C4×C31—26no—
193C4×C31—22no—
194C4×C31—24no—
195C4×C31—46no—
196C4×C31—43no—
197C4×C31C2×C4×Q8112yes—
198C4×C31C4×#4114yes—
199C4×C31#55714yesdirect product factor
200#557C4×C31116yesdirect product factor
201C2×C4⋊Q8C22×C3113yes—
202C2×#125C22×C3116yes—
203C2×C4×Q8C22×C3113yes—
204C2×C4×Q8C22×C3112yes—
205C2×C4×Q8C22×C3111yescharacteristic
206C2×C4×Q8C22×C3113yes—
207C2×C4×Q8C22×C3111yescharacteristic
208C2×C4×Q8C22×C3112yes—
209C2×C4×Q8C22×C3113yes—
210C2×C4×Q8C22×C3111yescharacteristic
211#391C22×C3116yes—
212#389C22×C3116yes—
213#389C22×C3116yes—
214#389C22×C3116yes—
215#389C22×C3116yes—
216#387C22×C31112yes—
217#387C22×C31112yes—
218#384C22×C3113yes—
219#382C22×C3116yes—
220C4×C4⋊C4C22×C3116yes—
221C4×C4⋊C4C22×C3116yes—
222C4×C4⋊C4C22×C3113yes—
223C4×C4⋊C4C22×C3116yes—
224C4×C4⋊C4C22×C3113yes—
225C4×C4⋊C4C22×C3116yes—
226C4×C4⋊C4C22×C3113yes—
227C4×C4⋊C4C22×C3113yes—
228C4×C4⋊C4C22×C3116yes—
229C4×C4⋊C4C22×C3116yes—
230C4×C4⋊C4C22×C3116yes—
231C4×C4⋊C4C22×C3113yes—
232C4×C4⋊C4C22×C3111yescharacteristic
233C43C22×C3116yes—
234C43C22×C3113yes—
235C2×C31C22×C4×Q811yescharacteristic
236C2×C31C2×C4×#4111yescharacteristic
237C2×C31C4×2−1+411yescharacteristic
238C2×C31C2×#55711yescharacteristic
239C2×C31Q8×#4111yescharacteristic
240C2×C31#348011yescharacteristic
241C2×C31#348211yescharacteristic
242C22×Q8C23×C3113yes—
243C22×Q8C23×C3111yescharacteristic
244C4⋊Q8C2×C4×C31124yes—
245#125C2×C4×C31148yes—
246C4×Q8—28no—
247C4×Q8—24no—
248C4×Q8—212no—
249C4×Q8—212no—
250C4×Q8C2×C4×C31116yes—
251C4×Q8C2×C4×C31124yes—
252C4×Q8C2×C4×C3118yes—
253C4×Q8Q8×C3118yescomplemented
254C2×C4⋊C4C23×C3113yes—
255C2×C4⋊C4C23×C3116yes—
256C2×C4⋊C4C23×C3113yes—
257C2×C4⋊C4C23×C3116yes—
258C2×C4⋊C4C23×C3116yes—
259C2×C4⋊C4C23×C3116yes—
260C2×C4⋊C4C23×C3111yescharacteristic
261C2×C4⋊C4C23×C3116yes—
262C2×C4⋊C4C23×C3113yes—
263C2×C4⋊C4C23×C3116yes—
264C2×C4⋊C4C23×C3116yes—
265C2×C4⋊C4C23×C3113yes—
266C2×C4⋊C4C23×C3111yescharacteristic
267C2×C42C23×C3116yes—
268C2×C42C23×C3116yes—
269C2×C42C23×C3113yes—
270C2×C42C23×C3116yes—
271C2×C42C23×C3113yes—
272C2×C42C23×C3112yes—
273C2×C42C23×C3113yes—
274C2×C42C23×C3116yes—
275C2×C42C23×C3113yes—
276C2×C42C23×C3116yes—
277C2×C42C23×C3113yes—
278C2×C42C23×C3113yes—
279C2×C42C23×C3111yescharacteristic
280#95C23×C3112yes—
281#95C23×C3116yes—
282#95C23×C3116yes—
283#95C23×C3116yes—
284#95C23×C31112yes—
285#95C23×C31112yes—
286C31C4×#55711yescharacteristic, direct product factor, Sylow
287C2×Q8—26no—
288C2×Q8—26no—
289C2×Q8C22×C4×C31112yes—
290C2×Q8C22×C4×C3114yes—
291C2×Q8C2×Q8×C3114yes—
292C2×Q8#41×C3114yes—
293C22×C4C24×C3113yes—
294C22×C4C24×C3116yes—
295C22×C4C24×C3112yes—
296C22×C4C24×C3113yes—
297C22×C4C24×C3116yes—
298C22×C4C24×C3113yes—
299C22×C4C24×C3112yes—
300C22×C4C24×C3111yescharacteristic
301C22×C4C24×C3113yes—
302C22×C4C24×C3111yescharacteristic
303C22×C4C24×C3111yescharacteristic
304C4⋊C4—212no—
305C4⋊C4—212no—
306C4⋊C4—26no—
307C4⋊C4—212no—
308C4⋊C4—26no—
309C4⋊C4—46no—
310C4⋊C4C22×C4×C3114yes—
311C4⋊C4C22×C4×C31124yes—
312C4⋊C4C22×C4×C31112yes—
313C4⋊C4C22×C4×C31124yes—
314C4⋊C4C22×C4×C31124yes—
315C4⋊C4C2×Q8×C31112yes—
316C4⋊C4#41×C3114yes—
317C42—212no—
318C42—26no—
319C42—212no—
320C42—26no—
321C42—212no—
322C42—24no—
323C42—26no—
324C42—43no—
325C42—46no—
326C42C22×C4×C31124yes—
327C42C22×C4×C31112yes—
328C42C2×Q8×C31112yes—
329C42#41×C3114yes—
330C23C25×C3111yescharacteristic
331Q8—224no—
332Q8C4×Q8×C31116yescomplemented
333C2×C4—23no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—26no—
337C2×C4—26no—
338C2×C4—22no—
339C2×C4—23no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—23no—
343C2×C4—21no—
344C2×C4—23no—
345C2×C4—26no—
346C2×C4—26no—
347C2×C4—23no—
348C2×C4—23no—
349C2×C4—23no—
350C2×C4—26no—
351C2×C4C23×C4×C3114yes—
352C2×C4C23×C4×C3116yes—
353C2×C4C23×C4×C31112yes—
354C2×C4C23×C4×C3116yes—
355C2×C4C23×C4×C3112yes—
356C2×C4C22×Q8×C3116yes—
357C2×C4C22×Q8×C3116yes—
358C2×C4C22×Q8×C3112yes—
359C2×C4C2×#41×C3114yes—
360C2×C4C2×#41×C3112yes—
361C2×C4C2×#41×C3116yes—
362C2×C4C2×#41×C3112yes—
363C2×C4C2×#41×C3112yes—
364C2×C4C2×#41×C3112yes—
365C2×C4C31×2−1+412yes—
366C22C24×C4×C3111yescharacteristic
367C22C23×Q8×C3111yescharacteristic
368C22C22×#41×C3111yescharacteristic
369C22C22×#41×C3111yescharacteristic
370C22C22×#41×C3111yescharacteristic
371C22C2×C31×2−1+411yescharacteristic
372C22C31×#58511yescharacteristic
373C4—212no—
374C4—24no—
375C4—26no—
376C4—26no—
377C4—22no—
378C4—24no—
379C4—46no—
380C4—43no—
381C4C2×C4×Q8×C31112yes—
382C4C4×#41×C3114yes—
383C4C31×#55714yesdirect product factor
384C2C22×C4×Q8×C3111yescharacteristic
385C2C2×C4×#41×C3111yescharacteristic
386C2C4×C31×2−1+411yescharacteristic
387C2C2×C31×#55711yescharacteristic
388C2Q8×#41×C3111yescharacteristic
389C2C31×#348011yescharacteristic
390C2C31×#348211yescharacteristic
subgroupG/H
centerC22×C4×C31C24
commutatorC22C24×C4×C31
Frattini, cosocleC23C25×C31
socle, prime radicalC23×C31C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
12. C31×#557C4direct (N × H)
58. C4×Q8×C31Q8semidirect
134. C4×#557C31direct (N × H)
137. Q8×C31C4×Q8semidirect
199. C4×C31#557direct (N × H)
200. #557C4×C31direct (N × H)
253. C4×Q8Q8×C31semidirect
286. C31C4×#557direct (N × H)
332. Q8C4×Q8×C31semidirect
383. C4C31×#557direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5572
C31×#23702
C31×#23652
C31×#23552
C31×#23512
Q8×C4⋊C4×C312
C4×C31×C4⋊Q82
C4×C31×#1252
C42×Q8×C312
C4×#55731