Finite Group Databasebeta

← #13256141

Subgroups — #13256141

Order 1792.

628 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#477C211yesmaximal, characteristic, complemented
2C2×#89856C211yesmaximal, characteristic
3#11739945C211yesmaximal, characteristic
4#11729539C211yesmaximal, characteristic, complemented
5C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6C2×#89278C211yesmaximal, characteristic
7#11725508C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C2×#6509—21no—
11C2×C4×D28C2211yescharacteristic
12C7×#477—22no—
13C7×#477—22no—
14C2×C7×#107—21no—
15C2×C7×C4⋊C8C2211yescharacteristic
16C2×C7×#103—21no—
17C2×#6409—22no—
18#89856—22no—
19#89856—22no—
20#89766—22no—
21#89759—21no—
22C2×#6383—21no—
23C2×#6382—21no—
24#89711—21no—
25#89707C2211yescharacteristic
26#89700—21no—
27C2×C4⋊(C7⋊C8)C2211yescharacteristic
28C2×C4×C7⋊C8C2211yescharacteristic
29Q8⋊(C7⋊C8)—22no—
30#89278—22no—
31#10236—71nomaximal, Sylow
32C22×C7×Q8—21no—
33C22×C7×Q8D411yescharacteristic
34C22×D28D411yescharacteristic
35C22×C4×D7—21no—
36C22×C7×C8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8D412yes—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8—21no—
44C4×C7×Q8—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C42×C7—21no—
50C2×C42×C7C2311yescharacteristic
51#6509—42no—
52#6509—42no—
53C2×(C2×D7)⋊C4—22no—
54C2×(C2×D7)⋊C4—21no—
55C2×(C2×D7)⋊C4—22no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7D411yescharacteristic
59C2×C4⋊Dic7D412yes—
60C2×Dic7⋊C4—22no—
61C22×C7⋊C8—21no—
62C22×C7⋊C8—21no—
63C4×D28—21no—
64C4×D28—21no—
65C4×D28—21no—
66C4×D28—21no—
67C7×#107—24no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8D412yes—
70C7×#103—24no—
71#6409—44no—
72C7⋊#95—22no—
73#6383—24no—
74#6382—24no—
75C4⋊(C7⋊C8)—21no—
76C4⋊(C7⋊C8)D412yes—
77C4×C7⋊C8—21no—
78C4×C7⋊C8—21no—
79C2×#477D711yescharacteristic, complemented
80C2×#476—71no—
81#2538—71no—
82#1956—71no—
83C2×Q8⋊C8—71no—
84C2×D4⋊C8—71no—
85#1296—71no—
86C23×D7—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—21no—
90C2×C7×Q8—22no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—42no—
94C2×C7×Q8—42no—
95C2×C7×Q8—41no—
96C2×C7×Q8SD1612yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—41no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×Dic7—21no—
104C22×Dic7—22no—
105C2×D28—21no—
106C2×D28—21no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28Q1612yes—
111C2×C4×D7—41no—
112C2×C4×D7—41no—
113C2×C4×D7—41no—
114C2×C4×D7—41no—
115C2×C4×D7—41no—
116C2×C4×D7—41no—
117C2×C7×C8—22no—
118C2×C7×C8—41no—
119C2×C7×C8—41no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C42×C7—41no—
129C42×C7—41no—
130C42×C7C2×D411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7#4111yescharacteristic
134(C2×D7)⋊C4—44no—
135(C2×D7)⋊C4—41no—
136(C2×D7)⋊C4—41no—
137(C2×D7)⋊C4—82no—
138C4⋊Dic7—21no—
139C4⋊Dic7—21no—
140C4⋊Dic7—41no—
141C4⋊Dic7—41no—
142C4⋊Dic7—41no—
143C4⋊Dic7—41no—
144C4⋊Dic7—42no—
145Dic7⋊C4—44no—
146C2×C7⋊C8—22no—
147C2×C7⋊C8—22no—
148C2×C7⋊C8—41no—
149C2×C7⋊C8—41no—
150C2×C7⋊C8—41no—
151C2×C7⋊C8—41no—
152C2×C4⋊Q8C2×D711yescharacteristic
153C2×C22⋊Q8—141no—
154C2×C4×Q8C2×D711yescharacteristic
155C2×C4×D4—71no—
156#477—22no—
157#477—22no—
158#476—142no—
159#476—142no—
160C2×#107—21no—
161C2×#106—141no—
162C2×C4⋊C8—71no—
163C2×C4⋊C8C2×D711yescharacteristic
164C2×#103—21no—
165C2×#103—142no—
166C2×#102—141no—
167C2×C4×C8—71no—
168#396—142no—
169#390—141no—
170#389—141no—
171#384—71no—
172#380—141no—
173Q8⋊C8—142no—
174D4⋊C8—142no—
175C23×C7#12011yescharacteristic
176C22×D7—22no—
177C22×D7—41no—
178C22×D7—41no—
179C22×D7—41no—
180C22×D7—41no—
181C22×D7—41no—
182C22×D7—41no—
183C7×Q8—22no—
184C7×Q8—22no—
185C7×Q8—44no—
186C7×Q8—41no—
187C7×Q8—41no—
188C7×Q8—82no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—22no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—81no—
199C2×C4×C7—81no—
200C2×C4×C7#12011yescharacteristic
201C2×C4×C7#12011yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×C4×C7#12111yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C2×C4×C7#12711yescharacteristic
206C2×C4×C7C2×SD1611yescharacteristic
207C2×C4×C7C2×Q1611yescharacteristic
208C2×C4×C7#13511yescharacteristic
209C2×C4×C7#13511yescharacteristic
210C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
211C2×C4×C7#13711yescharacteristic
212C2×C4×C7#13711yescharacteristic
213C2×C4×C7#13711yescharacteristic
214C2×Dic7—41no—
215C2×Dic7—41no—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—41no—
219C2×Dic7—42no—
220D28—22no—
221D28—22no—
222D28—41no—
223D28—41no—
224C4×D7—81no—
225C4×D7—81no—
226C4×D7—81no—
227C4×D7—81no—
228C7×C8—42no—
229C7⋊C8—42no—
230C7⋊C8—42no—
231C22×Q8—21no—
232C22×Q8C22⋊D711yescharacteristic
233C22×D4—71no—
234C23×C4—141no—
235C22×C8—21no—
236C22×C8—141no—
237C22×C8—141no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C22⋊D712yes—
242C22⋊Q8—282no—
243C22⋊Q8—282no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×D4—141no—
249C4×D4—141no—
250C4×D4—141no—
251C4×D4—141no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—71no—
256C2×C4⋊C4—72no—
257C2×C4⋊C4—142no—
258C2×C4⋊C4—141no—
259C2×C4⋊C4—141no—
260C2×C4⋊C4C22⋊D711yescharacteristic
261C2×C22⋊C4—142no—
262C2×C22⋊C4—141no—
263C2×C22⋊C4—142no—
264C2×C42—21no—
265C2×C42C22×D711yescharacteristic
266#107—24no—
267#106—144no—
268C4⋊C8—21no—
269C4⋊C8—72no—
270C4⋊C8—141no—
271C4⋊C8C22⋊D712yes—
272#103—24no—
273#103—284no—
274#102—144no—
275C4×C8—141no—
276C4×C8—141no—
277#95—142no—
278C22×C7#39211yescharacteristic
279C22×C7#45011yescharacteristic
280C22×C7#45111yescharacteristic
281C22×C7#45211yescharacteristic
282C22×C7#45211yescharacteristic
283C22×C7#45311yescharacteristic
284C22×C7#45411yescharacteristic
285C2×D7—42no—
286C2×D7—42no—
287C2×D7—42no—
288C2×D7—81no—
289C2×D7—81no—
290C2×D7—81no—
291C2×D7—81no—
292C4×C7—21no—
293C4×C7—21no—
294C4×C7—42no—
295C4×C7—81no—
296C4×C7—82no—
297C4×C7#44811yescharacteristic
298C4×C7#44911yescharacteristic
299C4×C7#45111yescharacteristic
300C4×C7#45211yescharacteristic
301C4×C7#46511yescharacteristic
302C4×C7#46711yescharacteristic
303C4×C7#46811yescharacteristic
304C4×C7#47011yescharacteristic
305Dic7—82no—
306Dic7—81no—
307C24—141no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8#114312yes—
318C2×D4—72no—
319C2×D4—141no—
320C2×D4—141no—
321C2×D4—141no—
322C2×D4—141no—
323C2×D4—141no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—141no—
328C22×C4—142no—
329C22×C4—281no—
330C22×C4—281no—
331C22×C4—281no—
332C22×C4—281no—
333C22×C4—281no—
334C22×C4—281no—
335C22×C4D4×D711yescharacteristic
336C22×C4D4×D711yescharacteristic
337C22×C4C2×C22⋊D711yescharacteristic
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—142no—
342C2×C8—142no—
343C2×C8—281no—
344C2×C8—281no—
345C2×C8—281no—
346C2×C8—281no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—141no—
356C4⋊C4—141no—
357C4⋊C4—282no—
358C4⋊C4—284no—
359C4⋊C4—281no—
360C4⋊C4—281no—
361C4⋊C4—281no—
362C4⋊C4—281no—
363C22⋊C4—284no—
364C22⋊C4—281no—
365C22⋊C4—281no—
366C22⋊C4—562no—
367C42—41no—
368C42—41no—
369C42D4×D711yescharacteristic
370C42#115911yescharacteristic
371C42C2×C22⋊D711yescharacteristic
372C42C2×C22⋊D711yescharacteristic
373C2×C7#163111yescharacteristic
374C2×C7#163211yescharacteristic
375C2×C7#163711yescharacteristic
376C2×C7#163811yescharacteristic
377C2×C7#200011yescharacteristic
378C2×C7#200111yescharacteristic
379C2×C7#200611yescharacteristic
380D7—82no—
381C23—142no—
382C23—281no—
383C23—281no—
384C23—281no—
385C23—281no—
386C23—281no—
387C23—281no—
388C23#650011yescharacteristic
389Q8—22no—
390Q8—22no—
391Q8—44no—
392Q8—41no—
393Q8—41no—
394Q8—82no—
395D4—142no—
396D4—142no—
397D4—281no—
398D4—281no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—282no—
411C2×C4—282no—
412C2×C4—281no—
413C2×C4—282no—
414C2×C4—281no—
415C2×C4—281no—
416C2×C4—561no—
417C2×C4—561no—
418C2×C4—561no—
419C2×C4—561no—
420C2×C4D7×Q1611yescharacteristic
421C2×C4#648111yescharacteristic
422C2×C4#648111yescharacteristic
423C2×C4#648211yescharacteristic
424C2×C4#650011yescharacteristic
425C2×C4#650111yescharacteristic
426C2×C4#650211yescharacteristic
427C2×C4#650211yescharacteristic
428C2×C4#650311yescharacteristic
429C2×C4C2×#114311yescharacteristic
430C2×C4#650511yescharacteristic
431C2×C4#651211yescharacteristic
432C2×C4#651311yescharacteristic
433C2×C4#651611yescharacteristic
434C8—42no—
435C8—282no—
436C8—282no—
437C7#1023611yescharacteristic, complemented, Sylow
438C22—282no—
439C22—282no—
440C22—282no—
441C22—561no—
442C22—561no—
443C22—561no—
444C22—561no—
445C22#8981811yescharacteristic
446C22#8981911yescharacteristic
447C22#8995811yescharacteristic
448C22#8995911yescharacteristic
449C22#8997011yescharacteristic
450C22#8997311yescharacteristic
451C22#8999511yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—82no—
457C4—562no—
458C4—561no—
459C4#8982011yescharacteristic
460C4#8982111yescharacteristic
461C4#8995811yescharacteristic
462C4#8995911yescharacteristic
463C4#8996011yescharacteristic
464C4#8996111yescharacteristic
465C4#8999911yescharacteristic
466C4#9001011yescharacteristic
467C2—562no—
468C2#1172891011yescharacteristic
469C2#1172891111yescharacteristic
470C2#1172891411yescharacteristic
471C2#1172891711yescharacteristic
472C2#1172945311yescharacteristic
473C2#1172982511yescharacteristic
474C2#1172985211yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC2×C7×#477C2
socleC23×C7#120
prime radicalC22×D28D4
hypercenterC2×#477D7
hypocenterC7#10236
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×C7×#477C2semidirect
4. #11729539C2semidirect
5. C2×Q8⋊(C7⋊C8)C2semidirect
7. #11725508C2semidirect
79. C2×#477D7semidirect
437. C7#10236semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4772
C2×#898562
#117399452
#117295392
C2×Q8⋊(C7⋊C8)2
C2×#892782
#117255082
#102367