Finite Group Databasebeta

← C4×#557×C71 #31855041

Subgroups — C4×#557×C71 #31855041

Order 18176.

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×#557×C71C211yesmaximal, characteristic
2C71×#2370C216yesmaximal
3C71×#2365C213yesmaximal
4C71×#2355C213yesmaximal
5C71×#2351C212yesmaximal
6Q8×C4⋊C4×C71C211yesmaximal, characteristic
7C4×C4⋊Q8×C71C213yesmaximal
8C4×#125×C71C216yesmaximal
9C42×Q8×C71C213yesmaximal
10C42×Q8×C71C212yesmaximal
11C42×Q8×C71C211yesmaximal, characteristic
12#557×C71C4116yesdirect product factor
13C2×C4⋊Q8×C71C2213yes—
14C2×#125×C71C2216yes—
15C2×C4×Q8×C71C2213yes—
16C2×C4×Q8×C71C2212yes—
17C2×C4×Q8×C71C2211yescharacteristic
18C2×C4×Q8×C71C2213yes—
19C2×C4×Q8×C71C2211yescharacteristic
20C2×C4×Q8×C71C2212yes—
21C2×C4×Q8×C71C2213yes—
22C2×C4×Q8×C71C2211yescharacteristic
23#391×C71C2216yes—
24#389×C71C2216yes—
25#389×C71C2216yes—
26#389×C71C2216yes—
27#389×C71C2216yes—
28#387×C71C22112yes—
29#387×C71C22112yes—
30#384×C71C2213yes—
31#382×C71C2216yes—
32C4×C4⋊C4×C71C2211yescharacteristic
33C4×C4⋊C4×C71C2216yes—
34C4×C4⋊C4×C71C2213yes—
35C4×C4⋊C4×C71C2216yes—
36C4×C4⋊C4×C71C2213yes—
37C4×C4⋊C4×C71C2216yes—
38C4×C4⋊C4×C71C2213yes—
39C4×C4⋊C4×C71C2213yes—
40C4×C4⋊C4×C71C2216yes—
41C4×C4⋊C4×C71C2216yes—
42C4×C4⋊C4×C71C2216yes—
43C4×C4⋊C4×C71C2213yes—
44C4×C4⋊C4×C71C2216yes—
45C43×C71C2216yes—
46C43×C71C2213yes—
47C22×Q8×C71C2313yes—
48C22×Q8×C71C2311yescharacteristic
49C4⋊Q8×C71C2×C4124yes—
50#125×C71C2×C4148yes—
51C4×Q8×C71—28no—
52C4×Q8×C71—24no—
53C4×Q8×C71—212no—
54C4×Q8×C71—212no—
55C4×Q8×C71C2×C4116yes—
56C4×Q8×C71C2×C4124yes—
57C4×Q8×C71C2×C418yes—
58C4×Q8×C71Q818yescomplemented
59C2×C4⋊C4×C71C2313yes—
60C2×C4⋊C4×C71C2316yes—
61C2×C4⋊C4×C71C2313yes—
62C2×C4⋊C4×C71C2316yes—
63C2×C4⋊C4×C71C2316yes—
64C2×C4⋊C4×C71C2316yes—
65C2×C4⋊C4×C71C2311yescharacteristic
66C2×C4⋊C4×C71C2316yes—
67C2×C4⋊C4×C71C2313yes—
68C2×C4⋊C4×C71C2316yes—
69C2×C4⋊C4×C71C2316yes—
70C2×C4⋊C4×C71C2313yes—
71C2×C4⋊C4×C71C2311yescharacteristic
72C2×C42×C71C2316yes—
73C2×C42×C71C2313yes—
74C2×C42×C71C2316yes—
75C2×C42×C71C2313yes—
76C2×C42×C71C2316yes—
77C2×C42×C71C2312yes—
78C2×C42×C71C2313yes—
79C2×C42×C71C2316yes—
80C2×C42×C71C2313yes—
81C2×C42×C71C2316yes—
82C2×C42×C71C2313yes—
83C2×C42×C71C2313yes—
84C2×C42×C71C2311yescharacteristic
85#95×C71C2312yes—
86#95×C71C2316yes—
87#95×C71C2316yes—
88#95×C71C2316yes—
89#95×C71C23112yes—
90#95×C71C23112yes—
91C2×Q8×C71—26no—
92C2×Q8×C71—26no—
93C2×Q8×C71C22×C4112yes—
94C2×Q8×C71C22×C414yes—
95C2×Q8×C71C2×Q814yes—
96C2×Q8×C71#4114yes—
97C22×C4×C71C2413yes—
98C22×C4×C71C2416yes—
99C22×C4×C71C2412yes—
100C22×C4×C71C2413yes—
101C22×C4×C71C2416yes—
102C22×C4×C71C2413yes—
103C22×C4×C71C2412yes—
104C22×C4×C71C2411yescharacteristic
105C22×C4×C71C2413yes—
106C22×C4×C71C2411yescharacteristic
107C22×C4×C71C2411yescharacteristic
108C4⋊C4×C71—212no—
109C4⋊C4×C71—212no—
110C4⋊C4×C71—26no—
111C4⋊C4×C71—212no—
112C4⋊C4×C71—26no—
113C4⋊C4×C71—46no—
114C4⋊C4×C71C22×C414yes—
115C4⋊C4×C71C22×C4124yes—
116C4⋊C4×C71C22×C4112yes—
117C4⋊C4×C71C22×C4124yes—
118C4⋊C4×C71C22×C4124yes—
119C4⋊C4×C71C2×Q8112yes—
120C4⋊C4×C71#4114yes—
121C42×C71—212no—
122C42×C71—26no—
123C42×C71—212no—
124C42×C71—26no—
125C42×C71—212no—
126C42×C71—24no—
127C42×C71—26no—
128C42×C71—43no—
129C42×C71—46no—
130C42×C71C22×C4124yes—
131C42×C71C22×C4112yes—
132C42×C71C2×Q8112yes—
133C42×C71#4114yes—
134C23×C71C2511yescharacteristic
135Q8×C71—224no—
136Q8×C71C4×Q8116yescomplemented
137C2×C4×C71—23no—
138C2×C4×C71—21no—
139C2×C4×C71—22no—
140C2×C4×C71—26no—
141C2×C4×C71—26no—
142C2×C4×C71—22no—
143C2×C4×C71—23no—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—23no—
147C2×C4×C71—21no—
148C2×C4×C71—23no—
149C2×C4×C71—26no—
150C2×C4×C71—26no—
151C2×C4×C71—23no—
152C2×C4×C71—23no—
153C2×C4×C71—23no—
154C2×C4×C71—26no—
155C2×C4×C71C23×C414yes—
156C2×C4×C71C23×C416yes—
157C2×C4×C71C23×C4112yes—
158C2×C4×C71C23×C416yes—
159C2×C4×C71C23×C412yes—
160C2×C4×C71C22×Q816yes—
161C2×C4×C71C22×Q816yes—
162C2×C4×C71C22×Q812yes—
163C2×C4×C71C2×#4114yes—
164C2×C4×C71C2×#4112yes—
165C2×C4×C71C2×#4116yes—
166C2×C4×C71C2×#4112yes—
167C2×C4×C71C2×#4112yes—
168C2×C4×C71C2×#4112yes—
169C2×C4×C712−1+412yes—
170C22×C71C24×C411yescharacteristic
171C22×C71C23×Q811yescharacteristic
172C22×C71C22×#4111yescharacteristic
173C22×C71C22×#4111yescharacteristic
174C22×C71C22×#4111yescharacteristic
175C22×C71C2×2−1+411yescharacteristic
176C22×C71#58511yescharacteristic
177C4×C71—212no—
178C4×C71—24no—
179C4×C71—26no—
180C4×C71—26no—
181C4×C71—22no—
182C4×C71—24no—
183C4×C71—46no—
184C4×C71—43no—
185C4×C71C2×C4×Q8112yes—
186C4×C71C4×#4114yes—
187C4×C71#55714yesdirect product factor
188C4×#557C7111yesmaximal, characteristic, direct product factor, Sylow
189C2×C71C22×C4×Q811yescharacteristic
190C2×C71C2×C4×#4111yescharacteristic
191C2×C71C4×2−1+411yescharacteristic
192C2×C71C2×#55711yescharacteristic
193C2×C71Q8×#4111yescharacteristic
194C2×C71#348011yescharacteristic
195C2×C71#348211yescharacteristic
196C2×#557C2×C7111yescharacteristic
197#2370C2×C7116yes—
198#2365C2×C7113yes—
199#2355C2×C7113yes—
200#2351C2×C7112yes—
201Q8×C4⋊C4C2×C7111yescharacteristic
202C4×C4⋊Q8C2×C7113yes—
203C4×#125C2×C7116yes—
204C42×Q8C2×C7113yes—
205C42×Q8C2×C7112yes—
206C42×Q8C2×C7111yescharacteristic
207C71C4×#55711yescharacteristic, direct product factor, Sylow
208#557C4×C71116yesdirect product factor
209C2×C4⋊Q8C22×C7113yes—
210C2×#125C22×C7116yes—
211C2×C4×Q8C22×C7113yes—
212C2×C4×Q8C22×C7112yes—
213C2×C4×Q8C22×C7111yescharacteristic
214C2×C4×Q8C22×C7113yes—
215C2×C4×Q8C22×C7111yescharacteristic
216C2×C4×Q8C22×C7112yes—
217C2×C4×Q8C22×C7113yes—
218C2×C4×Q8C22×C7111yescharacteristic
219#391C22×C7116yes—
220#389C22×C7116yes—
221#389C22×C7116yes—
222#389C22×C7116yes—
223#389C22×C7116yes—
224#387C22×C71112yes—
225#387C22×C71112yes—
226#384C22×C7113yes—
227#382C22×C7116yes—
228C4×C4⋊C4C22×C7116yes—
229C4×C4⋊C4C22×C7116yes—
230C4×C4⋊C4C22×C7113yes—
231C4×C4⋊C4C22×C7116yes—
232C4×C4⋊C4C22×C7113yes—
233C4×C4⋊C4C22×C7116yes—
234C4×C4⋊C4C22×C7113yes—
235C4×C4⋊C4C22×C7113yes—
236C4×C4⋊C4C22×C7116yes—
237C4×C4⋊C4C22×C7116yes—
238C4×C4⋊C4C22×C7116yes—
239C4×C4⋊C4C22×C7113yes—
240C4×C4⋊C4C22×C7111yescharacteristic
241C43C22×C7116yes—
242C43C22×C7113yes—
243C22×Q8C23×C7113yes—
244C22×Q8C23×C7111yescharacteristic
245C4⋊Q8C2×C4×C71124yes—
246#125C2×C4×C71148yes—
247C4×Q8—28no—
248C4×Q8—24no—
249C4×Q8—212no—
250C4×Q8—212no—
251C4×Q8C2×C4×C71116yes—
252C4×Q8C2×C4×C71124yes—
253C4×Q8C2×C4×C7118yes—
254C4×Q8Q8×C7118yescomplemented
255C2×C4⋊C4C23×C7113yes—
256C2×C4⋊C4C23×C7116yes—
257C2×C4⋊C4C23×C7113yes—
258C2×C4⋊C4C23×C7116yes—
259C2×C4⋊C4C23×C7116yes—
260C2×C4⋊C4C23×C7116yes—
261C2×C4⋊C4C23×C7111yescharacteristic
262C2×C4⋊C4C23×C7116yes—
263C2×C4⋊C4C23×C7113yes—
264C2×C4⋊C4C23×C7116yes—
265C2×C4⋊C4C23×C7116yes—
266C2×C4⋊C4C23×C7113yes—
267C2×C4⋊C4C23×C7111yescharacteristic
268C2×C42C23×C7116yes—
269C2×C42C23×C7116yes—
270C2×C42C23×C7113yes—
271C2×C42C23×C7116yes—
272C2×C42C23×C7113yes—
273C2×C42C23×C7112yes—
274C2×C42C23×C7113yes—
275C2×C42C23×C7116yes—
276C2×C42C23×C7113yes—
277C2×C42C23×C7116yes—
278C2×C42C23×C7113yes—
279C2×C42C23×C7113yes—
280C2×C42C23×C7111yescharacteristic
281#95C23×C7112yes—
282#95C23×C7116yes—
283#95C23×C7116yes—
284#95C23×C7116yes—
285#95C23×C71112yes—
286#95C23×C71112yes—
287C2×Q8—26no—
288C2×Q8—26no—
289C2×Q8C22×C4×C71112yes—
290C2×Q8C22×C4×C7114yes—
291C2×Q8C2×Q8×C7114yes—
292C2×Q8#41×C7114yes—
293C22×C4C24×C7113yes—
294C22×C4C24×C7116yes—
295C22×C4C24×C7112yes—
296C22×C4C24×C7113yes—
297C22×C4C24×C7116yes—
298C22×C4C24×C7113yes—
299C22×C4C24×C7112yes—
300C22×C4C24×C7111yescharacteristic
301C22×C4C24×C7113yes—
302C22×C4C24×C7111yescharacteristic
303C22×C4C24×C7111yescharacteristic
304C4⋊C4—212no—
305C4⋊C4—212no—
306C4⋊C4—26no—
307C4⋊C4—212no—
308C4⋊C4—26no—
309C4⋊C4—46no—
310C4⋊C4C22×C4×C7114yes—
311C4⋊C4C22×C4×C71124yes—
312C4⋊C4C22×C4×C71112yes—
313C4⋊C4C22×C4×C71124yes—
314C4⋊C4C22×C4×C71124yes—
315C4⋊C4C2×Q8×C71112yes—
316C4⋊C4#41×C7114yes—
317C42—212no—
318C42—26no—
319C42—212no—
320C42—26no—
321C42—212no—
322C42—24no—
323C42—26no—
324C42—43no—
325C42—46no—
326C42C22×C4×C71124yes—
327C42C22×C4×C71112yes—
328C42C2×Q8×C71112yes—
329C42#41×C7114yes—
330C23C25×C7111yescharacteristic
331Q8—224no—
332Q8C4×Q8×C71116yescomplemented
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×C7114yes—
352C2×C4C23×C4×C7116yes—
353C2×C4C23×C4×C71112yes—
354C2×C4C23×C4×C7116yes—
355C2×C4C23×C4×C7112yes—
356C2×C4C22×Q8×C7116yes—
357C2×C4C22×Q8×C7116yes—
358C2×C4C22×Q8×C7112yes—
359C2×C4C2×#41×C7114yes—
360C2×C4C2×#41×C7112yes—
361C2×C4C2×#41×C7116yes—
362C2×C4C2×#41×C7112yes—
363C2×C4C2×#41×C7112yes—
364C2×C4C2×#41×C7112yes—
365C2×C42−1+4×C7112yes—
366C22C24×C4×C7111yescharacteristic
367C22C23×Q8×C7111yescharacteristic
368C22C22×#41×C7111yescharacteristic
369C22C22×#41×C7111yescharacteristic
370C22C22×#41×C7111yescharacteristic
371C22C2×2−1+4×C7111yescharacteristic
372C22#585×C7111yescharacteristic
373C4—212no—
374C4—24no—
375C4—26no—
376C4—26no—
377C4—22no—
378C4—24no—
379C4—46no—
380C4—43no—
381C4C2×C4×Q8×C71112yes—
382C4C4×#41×C7114yes—
383C4#557×C7114yesdirect product factor
384C2C22×C4×Q8×C7111yescharacteristic
385C2C2×C4×#41×C7111yescharacteristic
386C2C4×2−1+4×C7111yescharacteristic
387C2C2×#557×C7111yescharacteristic
388C2Q8×#41×C7111yescharacteristic
389C2C71×#348011yescharacteristic
390C2C71×#348211yescharacteristic
subgroupG/H
centerC22×C4×C71C24
commutatorC22C24×C4×C71
Frattini, cosocleC23C25×C71
socle, prime radicalC23×C71C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
12. #557×C71C4direct (N × H)
58. C4×Q8×C71Q8semidirect
136. Q8×C71C4×Q8semidirect
187. C4×C71#557direct (N × H)
188. C4×#557C71direct (N × H)
207. C71C4×#557direct (N × H)
208. #557C4×C71direct (N × H)
254. C4×Q8Q8×C71semidirect
332. Q8C4×Q8×C71semidirect
383. C4#557×C71direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#557×C712
C71×#23702
C71×#23652
C71×#23552
C71×#23512
Q8×C4⋊C4×C712
C4×C4⋊Q8×C712
C4×#125×C712
C42×Q8×C712
C4×#55771