Finite Group Databasebeta

← #13326773

Subgroups — #13326773

Order 1792.

1282 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3496C211yesmaximal, characteristic
2C2×#89805C212yesmaximal, complemented
3#11739818C211yesmaximal, characteristic, complemented
4#11729628C214yesmaximal, complemented
5#11729597C214yesmaximal, 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
19C2×#6498—21no—
20C22×C4⋊Dic7C2211yescharacteristic, complemented
21C2×#6406—22no—
22#89984—21no—
23#89984C2211yescharacteristic, complemented
24C2×C22⋊(C7⋊C8)C2212yescomplemented
25#89805—24no—
26#89805C2218yescomplemented
27#89755C2212yescomplemented
28C2×#6381—22no—
29#89725C2212yescomplemented
30#89715—21no—
31#89700—21no—
32#89332C2218yescomplemented
33#89323C2212yescomplemented
34#89320C2212yescomplemented
35#18202—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—
80#6498—44no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7C2312yes—
86C2×Dic7⋊C4—22no—
87C22×C7⋊C8—22no—
88#6406—28no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91C7⋊#95C2312yes—
92C22⋊(C7⋊C8)—22no—
93C22⋊(C7⋊C8)C2314yes—
94#6381—28no—
95#3496D711yescharacteristic
96C2×#480—72no—
97#2486—71no—
98#1610—74no—
99#1607—74no—
100#1519—72no—
101#1459—71no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7C2×D411yescharacteristic
105C24×C7C2×D411yescharacteristic
106C7×#41—44no—
107C7×#41—416no—
108C7×#41—44no—
109C2×C7×Q8—22no—
110C2×C7×D4—22no—
111C2×C7×D4—22no—
112C2×C7×D4—24no—
113C2×C7×D4—22no—
114C2×C7×D4—24no—
115C2×C7×D4—24no—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4—44no—
120C2×C7×D4—416no—
121C2×C7×D4—44no—
122C2×C7×D4—44no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
125C2×C7×D4—82no—
126C2×C7×D4D814yes—
127C2×C7×D4C2×D414yes—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—42no—
134C22×C4×C7—42no—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—42no—
145C22×Dic7—42no—
146C22×Dic7—42no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4C2×D414yes—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—22no—
155C7×C22⋊C4—24no—
156C7×C22⋊C4—24no—
157C7×C22⋊C4—22no—
158C42×C7—21no—
159C42×C7—21no—
160C22⋊Dic7—42no—
161C22⋊Dic7—44no—
162C4⋊Dic7—24no—
163C4⋊Dic7—42no—
164C4⋊Dic7—41no—
165C4⋊Dic7—41no—
166Dic7⋊C4—44no—
167C2×C7⋊C8—24no—
168C2×C7⋊C8—44no—
169C2×H(5,𝔽2)—21no—
170#547—22no—
171#546—24no—
172D42—22no—
173#537C2×D712yes—
174#534—21no—
175#534C2×D712yes—
176#532—21no—
177C2×#123—141no—
178C2×#121—21no—
179C2×#121C2×D712yes—
180C2×C4×D4C2×D711yescharacteristic
181C22×C4⋊C4—71no—
182#480—78no—
183#480—144no—
184C2×#107—142no—
185C2×#102—142no—
186C2×C22⋊C8—72no—
187#398—72no—
188#397—72no—
189#389—141no—
190#381—141no—
191#380—71no—
192#380—141no—
193#327—78no—
194#324—72no—
195#323—72no—
196C23×C7—21no—
197C23×C7—21no—
198C23×C7—24no—
199C23×C7—41no—
200C23×C7—41no—
201C23×C7—41no—
202C23×C7—41no—
203C23×C7—44no—
204C23×C7—44no—
205C23×C7—82no—
206C23×C7—82no—
207C23×C7—82no—
208C23×C7—82no—
209C23×C7#12012yes—
210C23×C7C2×D812yes—
211C23×C7C2×D812yes—
212C23×C7AGL(1,ℤ/8)12yes—
213C23×C7C22×D411yescharacteristic
214C7×Q8—44no—
215C7×D4—28no—
216C7×D4—44no—
217C7×D4—44no—
218C7×D4—48no—
219C7×D4—44no—
220C7×D4—44no—
221C7×D4—48no—
222C7×D4—44no—
223C7×D4—82no—
224C7×D4—82no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—24no—
228C2×C4×C7—22no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—24no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—48no—
235C2×C4×C7—44no—
236C2×C4×C7—44no—
237C2×C4×C7—42no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—81no—
241C2×C4×C7#12011yescharacteristic
242C2×C4×C7#12011yescharacteristic
243C2×C4×C7C2×D812yes—
244C2×C4×C7C22×D412yes—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—84no—
252C7⋊C8—44no—
253H(5,𝔽2)—44no—
254H(5,𝔽2)—44no—
255C2×#41—24no—
256C2×#41—22no—
257C22×D4—24no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4C22⋊D711yescharacteristic
262C22×D4C22⋊D712yes—
263C23×C4—141no—
264C23×C4C22⋊D711yescharacteristic
265C23×C4C22×D711yescharacteristic
266C22×C8—142no—
267C4⋊Q8—21no—
268#127—21no—
269#124—22no—
270#123—24no—
271#123—24no—
272#123—284no—
273C22⋊Q8—24no—
274#121—22no—
275#121—22no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281#121C22⋊D714yes—
282#121C22×D714yes—
283#120—24no—
284#120—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—72no—
291C2×C4⋊C4—142no—
292C2×C4⋊C4—142no—
293C2×C4⋊C4—141no—
294C2×C4⋊C4—142no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4C22⋊D712yes—
297C2×C4⋊C4C22×D711yescharacteristic
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—141no—
301C2×C22⋊C4—142no—
302C2×C22⋊C4C22⋊D711yescharacteristic
303C2×C22⋊C4C22×D711yescharacteristic
304C2×C42C22⋊D711yescharacteristic
305#107—148no—
306#102—148no—
307C22⋊C8—74no—
308C22⋊C8—142no—
309#95—72no—
310#95—142no—
311#95—142no—
312C22×C7—21no—
313C22×C7—22no—
314C22×C7—21no—
315C22×C7—22no—
316C22×C7—22no—
317C22×C7—42no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—41no—
321C22×C7—41no—
322C22×C7—44no—
323C22×C7—44no—
324C22×C7—44no—
325C22×C7—81no—
326C22×C7—164no—
327C22×C7—164no—
328C22×C7#44714yes—
329C22×C7C2×#12011yescharacteristic
330C22×C7C22×D811yescharacteristic
331C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
332C4×C7—22no—
333C4×C7—22no—
334C4×C7—21no—
335C4×C7—44no—
336C4×C7—41no—
337C4×C7—41no—
338C4×C7#45812yes—
339Dic7—82no—
340Dic7—82no—
341C24—22no—
342C24—22no—
343C24C2×C22⋊D711yescharacteristic
344C24C2×C22⋊D711yescharacteristic
345#41—44no—
346#41—416no—
347#41—44no—
348C2×Q8—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—44no—
359C2×D4—416no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4#114114yes—
366C2×D4C2×C22⋊D714yes—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—142no—
375C22×C4—142no—
376C22×C4—282no—
377C22×C4—282no—
378C22×C4—282no—
379C22×C4C2×C22⋊D711yescharacteristic
380C22×C4C2×C22⋊D711yescharacteristic
381C22×C4C2×C22⋊D712yes—
382C22×C4C2×C22⋊D711yescharacteristic
383C22×C4C2×C22⋊D711yescharacteristic
384C22×C4C2×C22⋊D712yes—
385C22×C4C23×D711yescharacteristic
386C2×C8—144no—
387C2×C8—284no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—144no—
393C4⋊C4—282no—
394C4⋊C4—284no—
395C4⋊C4—281no—
396C4⋊C4—281no—
397C4⋊C4C2×C22⋊D714yes—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—22no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—282no—
405C22⋊C4—284no—
406C42—21no—
407C42—21no—
408C2×C7—21no—
409C2×C7—21no—
410C2×C7—24no—
411C2×C7—42no—
412C2×C7—84no—
413C2×C7C2×#44711yescharacteristic
414C2×C7#302211yescharacteristic
415C2×C7C2×#45811yescharacteristic
416C23—21no—
417C23—21no—
418C23—24no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—44no—
424C23—44no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23C2×#114112yes—
430C23C2×#114112yes—
431C23#649512yes—
432C23#651612yes—
433C23C22×C22⋊D711yescharacteristic
434Q8—44no—
435D4—28no—
436D4—44no—
437D4—44no—
438D4—48no—
439D4—44no—
440D4—44no—
441D4—48no—
442D4—44no—
443D4—82no—
444D4—82no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—81no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4—282no—
464C2×C4—282no—
465C2×C4—282no—
466C2×C4—282no—
467C2×C4—564no—
468C2×C4C2×#114112yes—
469C2×C4#651611yescharacteristic
470C2×C4#651611yescharacteristic
471C2×C4C22×C22⋊D712yes—
472C8—284no—
473C7#1820211yescharacteristic, complemented, Sylow
474C22—21no—
475C22—22no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—81no—
488C22—164no—
489C22—164no—
490C22#8998914yes—
491C22C22×#114111yescharacteristic
492C22C2×#649511yescharacteristic
493C22C2×#651611yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—44no—
498C4—41no—
499C4—41no—
500C4—562no—
501C4—562no—
502C4#9001512yes—
503C2—21no—
504C2—21no—
505C2—24no—
506C2—42no—
507C2—84no—
508C2C2×#8998911yescharacteristic
509C2C2×#9001511yescharacteristic
510C2#1174156811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#3496C2
socleC22×C7C2×#120
hypercenter#3496D7
hypocenterC7#18202
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×#89805C2semidirect
3. #11739818C2semidirect
4. #11729628C2semidirect
5. #11729597C2semidirect
6. #11729429C2semidirect
7. #11729354C2semidirect
20. C22×C4⋊Dic7C22semidirect
23. #89984C22semidirect
24. C2×C22⋊(C7⋊C8)C22semidirect
26. #89805C22semidirect
27. #89755C22semidirect
29. #89725C22semidirect
32. #89332C22semidirect
33. #89323C22semidirect
34. #89320C22semidirect
473. C7#18202semidirect

Maximal subgroups

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

subgroupindex
C7×#34962
C2×#898052
#117398182
#117296282
#117295972
#117294292
#117293542
#182027