Finite Group Databasebeta

← #13326780

Subgroups — #13326780

Order 1792.

1262 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3496C211yesmaximal, characteristic
2C2×#89806C212yesmaximal, complemented
3#11739819C211yesmaximal, characteristic, complemented
4#11729629C214yesmaximal, complemented
5#11729598C214yesmaximal, complemented
6#11729429C212yesmaximal, complemented
7#11729354C211yesmaximal, characteristic, complemented
8C2×C7×H(5,𝔽2)—21no—
9C7×#547—22no—
10C7×#546—24no—
11C7×D42—22no—
12C7×#537C2212yes—
13C7×#534—21no—
14C7×#534C2212yes—
15C7×#532—21no—
16C2×C7×#121—21no—
17C2×C7×#121C2212yes—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic, complemented
20#89992—21no—
21C2×#6406—22no—
22#89984—21no—
23#89984C2211yescharacteristic, complemented
24C2×C22⋊(C7⋊C8)C2212yescomplemented
25#89806—24no—
26#89806C2218yescomplemented
27#89758C2212yescomplemented
28C2×#6382—22no—
29#89721C2212yescomplemented
30#89718—21no—
31#89700—21no—
32#89332C2218yescomplemented
33#89323C2212yescomplemented
34#89320C2212yescomplemented
35#18203—71nomaximal, Sylow
36C7×H(5,𝔽2)—44no—
37C7×H(5,𝔽2)—44no—
38C2×C7×#41—24no—
39C2×C7×#41—22no—
40C22×C7×D4—24no—
41C22×C7×D4—22no—
42C22×C7×D4—22no—
43C22×C7×D4—21no—
44C22×C7×D4D411yescharacteristic
45C22×C7×D4D412yes—
46C23×C4×C7D411yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×Dic7—21no—
49C7×C4⋊Q8—21no—
50C7×#127—21no—
51C7×#124—22no—
52C7×#123—24no—
53C7×#123—24no—
54C7×C22⋊Q8—24no—
55C7×#121—22no—
56C7×#121—22no—
57C7×#121—24no—
58C7×#121—22no—
59C7×#121—22no—
60C7×#121—22no—
61C7×#121—22no—
62C7×#121D414yes—
63C7×#121C2314yes—
64C7×#120—24no—
65C7×#120—24no—
66C4×C7×D4—22no—
67C4×C7×D4—22no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4D412yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4D411yescharacteristic
76C2×C7×C22⋊C4C2311yescharacteristic
77C2×C42×C7D411yescharacteristic
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—22no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2312yes—
85C2×Dic7⋊C4—22no—
86C22×C7⋊C8—22no—
87#6406—28no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95C2312yes—
91C22⋊(C7⋊C8)—22no—
92C22⋊(C7⋊C8)C2314yes—
93#6382—28no—
94#3496D711yescharacteristic
95C2×#481—72no—
96#2487—71no—
97#1616—74no—
98#1613—74no—
99#1519—72no—
100#1459—71no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7C2×D411yescharacteristic
104C24×C7C2×D411yescharacteristic
105C7×#41—44no—
106C7×#41—416no—
107C7×#41—44no—
108C2×C7×Q8—22no—
109C2×C7×D4—22no—
110C2×C7×D4—22no—
111C2×C7×D4—24no—
112C2×C7×D4—22no—
113C2×C7×D4—24no—
114C2×C7×D4—24no—
115C2×C7×D4—22no—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4—44no—
119C2×C7×D4—416no—
120C2×C7×D4—44no—
121C2×C7×D4—44no—
122C2×C7×D4—44no—
123C2×C7×D4—44no—
124C2×C7×D4—82no—
125C2×C7×D4SD1614yes—
126C2×C7×D4C2×D414yes—
127C22×C4×C7—22no—
128C22×C4×C7—24no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2411yescharacteristic
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—42no—
144C22×Dic7—82no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4C2×D414yes—
150C7×C22⋊C4—24no—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—22no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—22no—
156C42×C7—21no—
157C42×C7—21no—
158C22⋊Dic7—42no—
159C22⋊Dic7—82no—
160C4⋊Dic7—24no—
161C4⋊Dic7—42no—
162C4⋊Dic7—41no—
163C4⋊Dic7—41no—
164Dic7⋊C4—82no—
165C2×C7⋊C8—24no—
166C2×C7⋊C8—44no—
167C2×H(5,𝔽2)—21no—
168#547—22no—
169#546—24no—
170D42—22no—
171#537C2×D712yes—
172#534—21no—
173#534C2×D712yes—
174#532—21no—
175C2×#121—21no—
176C2×#121C2×D712yes—
177C2×C4×D4C2×D711yescharacteristic
178C22×C4⋊C4—71no—
179#481—78no—
180#481—144no—
181C2×#106—142no—
182C2×#102—142no—
183C2×C22⋊C8—72no—
184#400—72no—
185#399—72no—
186#389—141no—
187#386—141no—
188#385—141no—
189#380—71no—
190#380—141no—
191#327—78no—
192#324—72no—
193#323—72no—
194C23×C7—21no—
195C23×C7—21no—
196C23×C7—24no—
197C23×C7—41no—
198C23×C7—41no—
199C23×C7—41no—
200C23×C7—41no—
201C23×C7—44no—
202C23×C7—44no—
203C23×C7—82no—
204C23×C7—82no—
205C23×C7—82no—
206C23×C7—82no—
207C23×C7#12012yes—
208C23×C7C2×SD1612yes—
209C23×C7C2×SD1612yes—
210C23×C7AGL(1,ℤ/8)12yes—
211C23×C7C22×D411yescharacteristic
212C7×Q8—44no—
213C7×D4—28no—
214C7×D4—44no—
215C7×D4—44no—
216C7×D4—48no—
217C7×D4—44no—
218C7×D4—44no—
219C7×D4—48no—
220C7×D4—44no—
221C7×D4—82no—
222C7×D4—82no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—24no—
230C2×C4×C7—42no—
231C2×C4×C7—41no—
232C2×C4×C7—48no—
233C2×C4×C7—44no—
234C2×C4×C7—44no—
235C2×C4×C7—42no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—81no—
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7#12011yescharacteristic
241C2×C4×C7C2×SD1612yes—
242C2×C4×C7C22×D412yes—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—84no—
250C7⋊C8—44no—
251H(5,𝔽2)—44no—
252H(5,𝔽2)—44no—
253C2×#41—24no—
254C2×#41—22no—
255C22×D4—24no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4C22⋊D711yescharacteristic
260C22×D4C22⋊D712yes—
261C23×C4—141no—
262C23×C4C22⋊D711yescharacteristic
263C23×C4C22×D711yescharacteristic
264C22×C8—142no—
265C4⋊Q8—21no—
266#127—21no—
267#124—22no—
268#123—24no—
269#123—24no—
270C22⋊Q8—24no—
271#121—22no—
272#121—22no—
273#121—24no—
274#121—22no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121C22⋊D714yes—
279#121C22×D714yes—
280#120—24no—
281#120—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—72no—
288C2×C4⋊C4—142no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4—141no—
291C2×C4⋊C4—141no—
292C2×C4⋊C4—142no—
293C2×C4⋊C4C22⋊D712yes—
294C2×C4⋊C4C22×D711yescharacteristic
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—141no—
298C2×C22⋊C4—142no—
299C2×C22⋊C4C22⋊D711yescharacteristic
300C2×C22⋊C4C22×D711yescharacteristic
301C2×C42C22⋊D711yescharacteristic
302#106—148no—
303#102—148no—
304C22⋊C8—74no—
305C22⋊C8—142no—
306#95—72no—
307#95—142no—
308#95—142no—
309C22×C7—21no—
310C22×C7—22no—
311C22×C7—21no—
312C22×C7—22no—
313C22×C7—22no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7—42no—
317C22×C7—41no—
318C22×C7—41no—
319C22×C7—44no—
320C22×C7—44no—
321C22×C7—44no—
322C22×C7—81no—
323C22×C7—164no—
324C22×C7—164no—
325C22×C7#45014yes—
326C22×C7C2×#12011yescharacteristic
327C22×C7C22×SD1611yescharacteristic
328C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
329C4×C7—22no—
330C4×C7—22no—
331C4×C7—21no—
332C4×C7—44no—
333C4×C7—41no—
334C4×C7—41no—
335C4×C7#45812yes—
336Dic7—82no—
337Dic7—82no—
338C24—22no—
339C24—22no—
340C24C2×C22⋊D711yescharacteristic
341C24C2×C22⋊D711yescharacteristic
342#41—44no—
343#41—416no—
344#41—44no—
345C2×Q8—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—44no—
356C2×D4—416no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—82no—
362C2×D4#114214yes—
363C2×D4C2×C22⋊D714yes—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—282no—
374C22×C4—562no—
375C22×C4C2×C22⋊D711yescharacteristic
376C22×C4C2×C22⋊D711yescharacteristic
377C22×C4C2×C22⋊D712yes—
378C22×C4C2×C22⋊D711yescharacteristic
379C22×C4C2×C22⋊D711yescharacteristic
380C22×C4C2×C22⋊D712yes—
381C22×C4C23×D711yescharacteristic
382C2×C8—144no—
383C2×C8—284no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—144no—
389C4⋊C4—282no—
390C4⋊C4—281no—
391C4⋊C4—281no—
392C4⋊C4—562no—
393C4⋊C4C2×C22⋊D714yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—22no—
400C22⋊C4—282no—
401C22⋊C4—562no—
402C42—21no—
403C42—21no—
404C2×C7—21no—
405C2×C7—21no—
406C2×C7—24no—
407C2×C7—42no—
408C2×C7—84no—
409C2×C7C2×#45011yescharacteristic
410C2×C7#302111yescharacteristic
411C2×C7C2×#45811yescharacteristic
412C23—21no—
413C23—21no—
414C23—24no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—44no—
420C23—44no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23#649512yes—
426C23C2×#114212yes—
427C23C2×#114212yes—
428C23#651612yes—
429C23C22×C22⋊D711yescharacteristic
430Q8—44no—
431D4—28no—
432D4—44no—
433D4—44no—
434D4—48no—
435D4—44no—
436D4—44no—
437D4—48no—
438D4—44no—
439D4—82no—
440D4—82no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—48no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—81no—
457C2×C4—282no—
458C2×C4—282no—
459C2×C4—282no—
460C2×C4—282no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4—564no—
464C2×C4C2×#114212yes—
465C2×C4#651611yescharacteristic
466C2×C4#651611yescharacteristic
467C2×C4C22×C22⋊D712yes—
468C8—284no—
469C7#1820311yescharacteristic, complemented, Sylow
470C22—21no—
471C22—22no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—44no—
481C22—44no—
482C22—44no—
483C22—81no—
484C22—164no—
485C22—164no—
486C22#8999014yes—
487C22C2×#649511yescharacteristic
488C22C22×#114211yescharacteristic
489C22C2×#651611yescharacteristic
490C4—22no—
491C4—22no—
492C4—21no—
493C4—44no—
494C4—41no—
495C4—41no—
496C4—562no—
497C4—562no—
498C4#9001512yes—
499C2—21no—
500C2—21no—
501C2—24no—
502C2—42no—
503C2—84no—
504C2C2×#8999011yescharacteristic
505C2C2×#9001511yescharacteristic
506C2#1174156711yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#3496C2
socleC22×C7C2×#120
hypercenter#3496D7
hypocenterC7#18203
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
2. C2×#89806C2semidirect
3. #11739819C2semidirect
4. #11729629C2semidirect
5. #11729598C2semidirect
6. #11729429C2semidirect
7. #11729354C2semidirect
19. C22×C4⋊Dic7C22semidirect
23. #89984C22semidirect
24. C2×C22⋊(C7⋊C8)C22semidirect
26. #89806C22semidirect
27. #89758C22semidirect
29. #89721C22semidirect
32. #89332C22semidirect
33. #89323C22semidirect
34. #89320C22semidirect
469. C7#18203semidirect

Maximal subgroups

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

subgroupindex
C7×#34962
C2×#898062
#117398192
#117296292
#117295982
#117294292
#117293542
#182037