Finite Group Databasebeta

← #13269534

Subgroups — #13269534

Order 1792.

928 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#576C211yesmaximal, characteristic
2#11729835C211yesmaximal, characteristic, complemented
3#11729830C211yesmaximal, characteristic, complemented
4C2×#89358C211yesmaximal, characteristic, complemented
5#11729787C211yesmaximal, characteristic, complemented
6#11729767C211yesmaximal, characteristic, complemented
7#11729760C211yesmaximal, characteristic, complemented
8C2×C7×H(5,𝔽2)—21no—
9C7×#576—22no—
10C7×#576—22no—
11C7×#576—22no—
12C7×#576C414yescomplemented
13C2×C7×AGL(1,ℤ/8)—22no—
14C2×C7×AGL(1,ℤ/8)—21no—
15C2×C7×#135—21no—
16C2×C7×#135C2211yescharacteristic
17C22×C7×D8—21no—
18C22×C7×D8C2211yescharacteristic
19C2×C7×#131C2211yescharacteristic
20#90008—21no—
21C2×#6393C2211yescharacteristic, complemented
22#89878C2211yescharacteristic, complemented
23C2×#6386C2211yescharacteristic, complemented
24C2×#6386C2211yescharacteristic, complemented
25#89361—22no—
26#89360—21no—
27#89358—22no—
28#89358—21no—
29#89357—22no—
30#89356—22no—
31#89354—21no—
32#12750—71nomaximal, Sylow
33C7×H(5,𝔽2)—24no—
34C7×H(5,𝔽2)—24no—
35C7×H(5,𝔽2)—42no—
36C7×H(5,𝔽2)—42no—
37C2×C7×#41—22no—
38C2×C7×#41—21no—
39C2×C7×#41—41no—
40C2×C7×#41D411yescharacteristic
41C22×C7×D4—22no—
42C22×C7×D4—21no—
43C22×C7×D4—42no—
44C22×C7×D4—41no—
45C7×AGL(1,ℤ/8)—24no—
46C7×AGL(1,ℤ/8)—28no—
47C7×AGL(1,ℤ/8)—28no—
48C7×AGL(1,ℤ/8)—24no—
49C7×#135—24no—
50C7×#135—24no—
51C7×#135—21no—
52C7×#135—21no—
53C7×#135C2×C412yes—
54C7×#135D412yes—
55C2×C7×Q16D411yescharacteristic
56C2×C7×SD16—22no—
57C2×C7×SD16—21no—
58C2×C7×D8—22no—
59C2×C7×D8—24no—
60C2×C7×D8—24no—
61C2×C7×D8—21no—
62C2×C7×D8—21no—
63C2×C7×D8—21no—
64C2×C7×D8C2×C412yes—
65C2×C7×D8D412yes—
66C2×C7×D8D411yescharacteristic
67C2×C7×D8D412yes—
68C7×#131—21no—
69C7×#131—21no—
70C7×#131C2×C412yes—
71C7×#131D412yes—
72C2×C7×M16—21no—
73C2×C7×M16D411yescharacteristic
74C22×C7×C8—21no—
75C22×C7×C8C2311yescharacteristic
76#6489—21no—
77C2×#1136—21no—
78#6411—42no—
79#6393—21no—
80#6393—21no—
81#6392—21no—
82#6391—21no—
83#6386—21no—
84#6386—21no—
85#6386—21no—
86#6386—21no—
87C2×C7⋊C16—21no—
88C2×C7⋊C16—21no—
89C2×#576D711yescharacteristic
90#2148—71no—
91C2×#361—71no—
92#2145—71no—
93#2140—71no—
94#2117—71no—
95#1814—71no—
96C24×C7—42no—
97C24×C7—41no—
98C7×#41—21no—
99C7×#41—21no—
100C7×#41—28no—
101C7×#41—22no—
102C7×#41—22no—
103C7×#41—44no—
104C7×#41—44no—
105C7×#41—44no—
106C7×#41—41no—
107C7×#41—41no—
108C7×#41C22⋊C412yes—
109C7×#41D812yes—
110C2×C7×Q8—21no—
111C2×C7×Q8SD1611yescharacteristic
112C2×C7×D4—21no—
113C2×C7×D4—28no—
114C2×C7×D4—22no—
115C2×C7×D4—21no—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4—42no—
119C2×C7×D4—44no—
120C2×C7×D4—48no—
121C2×C7×D4—48no—
122C2×C7×D4—44no—
123C2×C7×D4—42no—
124C2×C7×D4—41no—
125C2×C7×D4—44no—
126C2×C7×D4—41no—
127C2×C7×D4—41no—
128C2×C7×D4—41no—
129C2×C7×D4SD1611yescharacteristic
130C22×C4×C7—21no—
131C22×C4×C7—42no—
132C22×C4×C7—41no—
133C22×C4×C7C2×D411yescharacteristic
134C7×Q16—21no—
135C7×Q16C22⋊C412yes—
136C7×SD16—28no—
137C7×SD16—22no—
138C7×SD16—22no—
139C7×D8—28no—
140C7×D8—22no—
141C7×D8—21no—
142C7×D8—22no—
143C7×D8—22no—
144C7×D8C22⋊C412yes—
145C7×D8C22⋊C414yes—
146C7×M16—21no—
147C7×M16—22no—
148C7×M16—22no—
149C7×M16C22⋊C412yes—
150C2×C7×C8—21no—
151C2×C7×C8—22no—
152C2×C7×C8—22no—
153C2×C7×C8C22×C411yescharacteristic
154C2×C7×C8C2×D411yescharacteristic
155C2×C7×C8C2×D411yescharacteristic
156C2×C7×C8C2×D411yescharacteristic
157C2×C7×C8C2×D411yescharacteristic
158C2×C7×C8C2×D411yescharacteristic
159C22⋊Dic7—41no—
160C4⋊Dic7—41no—
161C4×Dic7—41no—
162#1136—41no—
163#1136—41no—
164C2×C7⋊C8—41no—
165C7⋊C16—41no—
166C7⋊C16—41no—
167C2×H(5,𝔽2)—21no—
168#576—22no—
169#576—22no—
170#576—22no—
171#576Dic714yescomplemented
172C2×AGL(1,ℤ/8)—22no—
173C2×AGL(1,ℤ/8)—21no—
174C2×#135—21no—
175C2×#135C2×D711yescharacteristic
176C22×D8—21no—
177C22×D8C2×D711yescharacteristic
178C2×#131C2×D711yescharacteristic
179C2×M32—71no—
180C2×M32—71no—
181C2×#108—71no—
182#427—71no—
183#421—141no—
184#362—141no—
185#361—142no—
186#361—141no—
187#360—142no—
188#357—142no—
189#350—142no—
190#344—141no—
191C23×C7—21no—
192C23×C7—48no—
193C23×C7—42no—
194C23×C7—41no—
195C23×C7—42no—
196C23×C7—42no—
197C23×C7—162no—
198C23×C7—164no—
199C23×C7AGL(1,ℤ/8)11yescharacteristic
200C7×Q8—21no—
201C7×Q8—22no—
202C7×Q8—41no—
203C7×Q8#10212yes—
204C7×D4—22no—
205C7×D4—22no—
206C7×D4—22no—
207C7×D4—24no—
208C7×D4—21no—
209C7×D4—48no—
210C7×D4—48no—
211C7×D4—42no—
212C7×D4—42no—
213C7×D4—42no—
214C7×D4—41no—
215C7×D4—42no—
216C7×D4—42no—
217C7×D4#10212yes—
218C2×C4×C7—21no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—48no—
222C2×C4×C7—41no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7C2×C22⋊C411yescharacteristic
226C2×C4×C7#12011yescharacteristic
227C2×C4×C7#12011yescharacteristic
228C2×C4×C7C2×D811yescharacteristic
229C2×C4×C7C2×SD1611yescharacteristic
230C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
231C2×Dic7—81no—
232C7×C8—22no—
233C7×C8C2×C22⋊C411yescharacteristic
234C7×C8C2×C22⋊C411yescharacteristic
235C7×C8#12011yescharacteristic
236C7×C8#12011yescharacteristic
237C7⋊C8—81no—
238H(5,𝔽2)—24no—
239H(5,𝔽2)—24no—
240H(5,𝔽2)—42no—
241H(5,𝔽2)—42no—
242C2×#41—22no—
243C2×#41—21no—
244C2×#41—41no—
245C2×#41C22⋊D711yescharacteristic
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—42no—
249C22×D4—41no—
250AGL(1,ℤ/8)—24no—
251AGL(1,ℤ/8)—28no—
252AGL(1,ℤ/8)—28no—
253AGL(1,ℤ/8)—24no—
254#135—24no—
255#135—24no—
256#135—21no—
257#135—21no—
258#135C2×Dic712yes—
259#135C22⋊D712yes—
260C2×Q16C22⋊D711yescharacteristic
261C2×SD16—22no—
262C2×SD16—21no—
263C2×D8—22no—
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—21no—
267C2×D8—21no—
268C2×D8—21no—
269C2×D8C2×Dic712yes—
270C2×D8C22⋊D711yescharacteristic
271C2×D8C22⋊D712yes—
272C2×D8C22⋊D712yes—
273#131—21no—
274#131—21no—
275#131C2×Dic712yes—
276#131C22⋊D712yes—
277C2×M16—21no—
278C2×M16—141no—
279C2×M16C22⋊D711yescharacteristic
280C22×C8—21no—
281C22×C8C22×D711yescharacteristic
282#117—141no—
283M32—141no—
284M32—141no—
285M32—141no—
286M32—141no—
287C2×C16—141no—
288C2×C16—141no—
289#108—141no—
290#108—141no—
291#107—141no—
292#106—141no—
293#104—282no—
294C22×C7—22no—
295C22×C7—41no—
296C22×C7—42no—
297C22×C7—44no—
298C22×C7—42no—
299C22×C7—84no—
300C22×C7—82no—
301C22×C7—162no—
302C22×C7—162no—
303C22×C7—168no—
304C22×C7#41811yescharacteristic
305C22×C7#44711yescharacteristic
306C22×C7#45011yescharacteristic
307C4×C7—22no—
308C4×C7—42no—
309C4×C7#37911yescharacteristic
310C4×C7C2×#10211yescharacteristic
311C4×C7#44711yescharacteristic
312C4×C7#45011yescharacteristic
313Dic7—161no—
314C24—42no—
315C24—41no—
316#41—21no—
317#41—21no—
318#41—28no—
319#41—22no—
320#41—22no—
321#41—44no—
322#41—44no—
323#41—44no—
324#41—41no—
325#41—41no—
326#41#114112yes—
327#41C22⋊Dic712yes—
328C2×Q8—21no—
329C2×Q8#114211yescharacteristic
330C2×D4—21no—
331C2×D4—28no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4—44no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—44no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4#114211yescharacteristic
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4C2×C22⋊D711yescharacteristic
352Q16—21no—
353Q16C22⋊Dic712yes—
354SD16—28no—
355SD16—22no—
356SD16—22no—
357D8—28no—
358D8—22no—
359D8—21no—
360D8—22no—
361D8—22no—
362D8C22⋊Dic712yes—
363D8C22⋊Dic714yes—
364M16—21no—
365M16—22no—
366M16—22no—
367M16—281no—
368M16—281no—
369M16C22⋊Dic712yes—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—281no—
374C2×C8C22×Dic711yescharacteristic
375C2×C8C2×C22⋊D711yescharacteristic
376C2×C8C2×C22⋊D711yescharacteristic
377C2×C8C2×C22⋊D711yescharacteristic
378C2×C8C2×C22⋊D711yescharacteristic
379C2×C8C2×C22⋊D711yescharacteristic
380C4⋊C4—281no—
381C22⋊C4—281no—
382C42—281no—
383C16—281no—
384C16—281no—
385C2×C7—21no—
386C2×C7—21no—
387C2×C7—21no—
388C2×C7—42no—
389C2×C7—84no—
390C2×C7—82no—
391C2×C7#178611yescharacteristic
392C23—21no—
393C23—48no—
394C23—42no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—162no—
399C23—164no—
400C23#649511yescharacteristic
401Q8—21no—
402Q8—22no—
403Q8—41no—
404Q8#640612yes—
405D4—22no—
406D4—22no—
407D4—21no—
408D4—22no—
409D4—24no—
410D4—48no—
411D4—48no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—41no—
416D4—42no—
417D4—42no—
418D4#640612yes—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—48no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—561no—
427C2×C4C2×#114111yescharacteristic
428C2×C4#649511yescharacteristic
429C2×C4C2×#114211yescharacteristic
430C2×C4C2×C22⋊Dic711yescharacteristic
431C2×C4#651611yescharacteristic
432C2×C4#651611yescharacteristic
433C8—22no—
434C8—561no—
435C8C2×C22⋊Dic711yescharacteristic
436C8C2×C22⋊Dic711yescharacteristic
437C8#651611yescharacteristic
438C8#651611yescharacteristic
439C7#1275011yescharacteristic, complemented, Sylow
440C22—22no—
441C22—41no—
442C22—42no—
443C22—44no—
444C22—42no—
445C22—84no—
446C22—82no—
447C22—162no—
448C22—162no—
449C22—168no—
450C22#8998711yescharacteristic
451C22#8998911yescharacteristic
452C22#8999011yescharacteristic
453C4—22no—
454C4—42no—
455C4—1121no—
456C4C2×#640611yescharacteristic
457C4#8998911yescharacteristic
458C4#8999011yescharacteristic
459C4#9001911yescharacteristic
460C2—21no—
461C2—21no—
462C2—21no—
463C2—42no—
464C2—84no—
465C2—82no—
466C2#1172988211yescharacteristic
subgroupG/H
centerC2#11729882
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
Fitting, prime radicalC2×C7×#576C2
socleC2×C7#1786
cosocleC22×C7×C8C23
hypercenterC2×#576D7
hypocenterC7#12750
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#576

Complements

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

NH ≅ G/Ntype
2. #11729835C2semidirect
3. #11729830C2semidirect
4. C2×#89358C2semidirect
5. #11729787C2semidirect
6. #11729767C2semidirect
7. #11729760C2semidirect
12. C7×#576C4semidirect
21. C2×#6393C22semidirect
22. #89878C22semidirect
23. C2×#6386C22semidirect
24. C2×#6386C22semidirect
171. #576Dic7semidirect
439. C7#12750semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5762
#117298352
#117298302
C2×#893582
#117297872
#117297672
#117297602
#127507