Finite Group Databasebeta

← #13248564

Subgroups — #13248564

Order 1792.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2355C211yesmaximal, characteristic, complemented
2C2×#89799C211yesmaximal, characteristic
3C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4#11727334C211yesmaximal, characteristic
5#11727241C211yesmaximal, characteristic
6#11725440C211yesmaximal, characteristic, complemented
7#11725425C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C22×#1143—21no—
10C2×#6437C2211yescharacteristic
11C7×#391—21no—
12C7×#387—22no—
13C7×#384C2211yescharacteristic
14C7×#382—21no—
15C4×C7×C4⋊C4—21no—
16C4×C7×C4⋊C4—21no—
17C4×C7×C4⋊C4C2211yescharacteristic
18#89799—22no—
19#89799C414yescomplemented
20C2×#6383—21no—
21C2×#6383—21no—
22C2×#6383—21no—
23C2×#6383—22no—
24C2×C4⋊(C7⋊C8)C2211yescharacteristic
25C2×C4⋊(C7⋊C8)C2211yescharacteristic
26C2×C4×C7⋊C8C2211yescharacteristic
27Q8⋊(C7⋊C8)—22no—
28#8978—71nomaximal, Sylow
29C22×C7×Q8D411yescharacteristic
30C22×D28—21no—
31C22×D28—21no—
32C22×D28—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8C2×C412yes—
36C4×C7×Q8D412yes—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C4⋊C4D412yes—
44C2×C42×C7—21no—
45C2×C42×C7—21no—
46C2×C42×C7—22no—
47C2×C42×C7C2311yescharacteristic
48C2×#1143—24no—
49C2×#1143—42no—
50C2×#1143—42no—
51C22×C7⋊C8—21no—
52C22×C7⋊C8—21no—
53C22×C7⋊C8—21no—
54#6437—21no—
55#6437—21no—
56#6437—21no—
57#6437C2×C412yes—
58C7×#95—22no—
59C7×#95—22no—
60C7×#95—22no—
61C7×#95—21no—
62#6383—24no—
63#6383—44no—
64#6383—42no—
65#6383—42no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)—21no—
69C4⋊(C7⋊C8)C2×C412yes—
70C4×C7⋊C8—21no—
71C4×C7⋊C8—21no—
72C2×#460—71no—
73#2355D711yescharacteristic, complemented
74#1982—71no—
75#1957—71no—
76C2×Q8⋊C8—71no—
77#1296—71no—
78#1279—71no—
79C23×D7—41no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×Q8—21no—
85C2×C7×Q8C22⋊C412yes—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C2×D28—22no—
95C2×D28—22no—
96C2×D28—22no—
97C2×D28—42no—
98C2×D28—42no—
99C2×D28—41no—
100C2×D28—41no—
101C2×D28—41no—
102C2×D28—41no—
103C2×D28—41no—
104C2×D28—41no—
105C2×D28—41no—
106C2×D28—41no—
107C2×D28—41no—
108C2×D28—41no—
109C2×D28—41no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—42no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4C22⋊C412yes—
122C7×C4⋊C4SD1612yescomplemented
123C42×C7—22no—
124C42×C7—22no—
125C42×C7—41no—
126C42×C7—41no—
127C42×C7—82no—
128C42×C7C22×C411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C42×C7#4111yescharacteristic
132#1143—48no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8—22no—
135C2×C7⋊C8—41no—
136C2×C7⋊C8—41no—
137C2×C7⋊C8—41no—
138C2×C7⋊C8—41no—
139C2×C7⋊C8—41no—
140C2×C7⋊C8—41no—
141C2×C7⋊C8—41no—
142C22×SD16—141no—
143C2×#127—71no—
144C2×C4×Q8C2×D711yescharacteristic
145#460—74no—
146#460—142no—
147C2×C4⋊C8—71no—
148C2×C4⋊C8—71no—
149C2×#102—141no—
150C2×#102—142no—
151C2×#102—141no—
152C2×#102—141no—
153C2×C4×C8—71no—
154#391—21no—
155#387—22no—
156#384C2×D711yescharacteristic
157#382—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2×D711yescharacteristic
161Q8⋊C8—142no—
162C23×C7#12011yescharacteristic
163C22×D7—42no—
164C22×D7—81no—
165C22×D7—81no—
166C22×D7—81no—
167C22×D7—81no—
168C22×D7—81no—
169C22×D7—81no—
170C7×Q8—22no—
171C7×Q8—22no—
172C7×Q8—41no—
173C7×Q8—41no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—22no—
177C2×C4×C7—22no—
178C2×C4×C7—22no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—41no—
191C2×C4×C7C2×C22⋊C411yescharacteristic
192C2×C4×C7C4×D411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12311yescharacteristic
197C2×C4×C7C2×SD1611yescharacteristic
198C2×C4×C7#13511yescharacteristic
199C2×C4×C7#13511yescharacteristic
200C2×C4×C7#13511yescharacteristic
201C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C7#13711yescharacteristic
205D28—44no—
206D28—42no—
207D28—42no—
208D28—42no—
209D28—42no—
210D28—82no—
211D28—81no—
212D28—81no—
213D28—81no—
214D28—81no—
215C7⋊C8—42no—
216C7⋊C8—42no—
217C7⋊C8—81no—
218C22×Q8C22⋊D711yescharacteristic
219C22×D4—141no—
220C22×D4—141no—
221C22×D4—141no—
222C2×SD16—144no—
223C2×SD16—282no—
224C2×SD16—282no—
225C22×C8—141no—
226C22×C8—141no—
227C22×C8—141no—
228#127—72no—
229#127—141no—
230#127—141no—
231#127—141no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C4×D712yes—
235C4×Q8D2812yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C4⋊C4C22⋊D711yescharacteristic
241C2×C4⋊C4C22⋊D711yescharacteristic
242C2×C4⋊C4C22⋊D712yes—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42C22×D711yescharacteristic
247C4⋊C8—72no—
248C4⋊C8—141no—
249C4⋊C8—141no—
250C4⋊C8—141no—
251#102—144no—
252#102—282no—
253#102—284no—
254#102—282no—
255C4×C8—141no—
256C4×C8—141no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261C22×C7#38611yescharacteristic
262C22×C7#44811yescharacteristic
263C22×C7#44911yescharacteristic
264C22×C7#44911yescharacteristic
265C22×C7#44911yescharacteristic
266C22×C7#45311yescharacteristic
267C22×C7#45411yescharacteristic
268C2×D7—82no—
269C2×D7—82no—
270C2×D7—82no—
271C2×D7—161no—
272C2×D7—161no—
273C2×D7—161no—
274C2×D7—161no—
275C4×C7—21no—
276C4×C7—21no—
277C4×C7—42no—
278C4×C7—42no—
279C4×C7—42no—
280C4×C7—82no—
281C4×C7—81no—
282C4×C7#41611yescharacteristic
283C4×C7#41811yescharacteristic
284C4×C7C4×SD1611yescharacteristic
285C4×C7#44011yescharacteristic
286C4×C7#46011yescharacteristic
287C4×C7#46411yescharacteristic
288C4×C7#48211yescharacteristic
289C4×C7#48211yescharacteristic
290C24—281no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8(C2×D7)⋊C412yes—
297C2×D4—142no—
298C2×D4—142no—
299C2×D4—142no—
300C2×D4—282no—
301C2×D4—282no—
302C2×D4—281no—
303C2×D4—281no—
304C2×D4—281no—
305C2×D4—281no—
306C2×D4—281no—
307C2×D4—281no—
308C2×D4—281no—
309C2×D4—281no—
310C2×D4—281no—
311C2×D4—281no—
312C2×D4—281no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4C2×C22⋊D711yescharacteristic
319C22×C4C2×C22⋊D711yescharacteristic
320C22×C4C2×C22⋊D711yescharacteristic
321SD16—288no—
322C2×C8—142no—
323C2×C8—142no—
324C2×C8—281no—
325C2×C8—281no—
326C2×C8—281no—
327C2×C8—281no—
328C2×C8—281no—
329C2×C8—281no—
330C2×C8—281no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4(C2×D7)⋊C412yes—
343C4⋊C4#114312yescomplemented
344C42—22no—
345C42—22no—
346C42—41no—
347C42—41no—
348C42—82no—
349C42C2×C4×D711yescharacteristic
350C42C2×D2811yescharacteristic
351C42#115711yescharacteristic
352C42#115711yescharacteristic
353C2×C7#162011yescharacteristic
354C2×C7#162211yescharacteristic
355C2×C7#162611yescharacteristic
356C2×C7#162811yescharacteristic
357C2×C7#187511yescharacteristic
358C2×C7#188011yescharacteristic
359C2×C7#188311yescharacteristic
360D7—162no—
361C23—282no—
362C23—561no—
363C23—561no—
364C23—561no—
365C23—561no—
366C23—561no—
367C23—561no—
368C23#651611yescharacteristic
369Q8—22no—
370Q8—22no—
371Q8—41no—
372Q8—41no—
373D4—284no—
374D4—282no—
375D4—282no—
376D4—282no—
377D4—282no—
378D4—562no—
379D4—561no—
380D4—561no—
381D4—561no—
382D4—561no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4C2×(C2×D7)⋊C411yescharacteristic
401C2×C4C4×C22⋊D711yescharacteristic
402C2×C4C4×C22⋊D711yescharacteristic
403C2×C4#649211yescharacteristic
404C2×C4#649311yescharacteristic
405C2×C4#649311yescharacteristic
406C2×C4C2×#114311yescharacteristic
407C2×C4#650511yescharacteristic
408C2×C4#651211yescharacteristic
409C2×C4#651211yescharacteristic
410C2×C4#651211yescharacteristic
411C2×C4#651311yescharacteristic
412C2×C4#651311yescharacteristic
413C2×C4#651311yescharacteristic
414C8—282no—
415C8—282no—
416C8—561no—
417C7#897811yescharacteristic, complemented, Sylow
418C22—562no—
419C22—562no—
420C22—562no—
421C22—1121no—
422C22—1121no—
423C22—1121no—
424C22—1121no—
425C22#8998511yescharacteristic
426C22#8999911yescharacteristic
427C22#9001011yescharacteristic
428C22#9001011yescharacteristic
429C22#9001011yescharacteristic
430C22#9001311yescharacteristic
431C22#9001711yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—82no—
439C4#8977311yescharacteristic
440C4#8977411yescharacteristic
441C4#8977511yescharacteristic
442C4#8977511yescharacteristic
443C4C4×#114311yescharacteristic
444C4#8979811yescharacteristic
445C4#8979911yescharacteristic
446C4#8980011yescharacteristic
447C2—1122no—
448C2#1172934011yescharacteristic
449C2#1172938711yescharacteristic
450C2#1172942111yescharacteristic
451C2#1172967711yescharacteristic
452C2#1172968511yescharacteristic
453C2#1172968911yescharacteristic
454C2#1172969711yescharacteristic
subgroupG/H
centerC23#6516
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2355C2
socleC23×C7#120
prime radicalC2×#6437C22
cosocleC2×C42×C7C23
hypercenter#2355D7
hypocenterC7#8978
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2355

Complements

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

NH ≅ G/Ntype
1. C7×#2355C2semidirect
3. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11725440C2semidirect
7. #11725425C2semidirect
19. #89799C4semidirect
73. #2355D7semidirect
122. C7×C4⋊C4SD16semidirect
343. C4⋊C4#1143semidirect
417. C7#8978semidirect

Maximal subgroups

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

subgroupindex
C7×#23552
C2×#897992
C2×Q8⋊(C7⋊C8)2
#117273342
#117272412
#117254402
#117254252
#89787