Finite Group Databasebeta

← C7×#16522 #13193738

Subgroups — C7×#16522 #13193738

Order 1792.

1414 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#398C212yesmaximal
2C2×C7×#395C211yesmaximal, characteristic
3C2×C7×#395C212yesmaximal
4C2×C7×#384C211yesmaximal, characteristic
5C7×#1444C212yesmaximal
6C7×#1444C212yesmaximal
7C7×#1442C212yesmaximal
8C7×#1441C211yesmaximal, characteristic
9C7×#1434C212yesmaximal
10C23×C7×Q8C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C7×#398—28no—
21C7×#395—24no—
22C7×#395—28no—
23C7×#384—24no—
24C2×C7×#95C2212yes—
25C2×C7×#95C2212yes—
26C2×C7×#95C2212yes—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2212yes—
33C2×C7×#95C2212yes—
34C2×C7×#95C2212yes—
35#16522C711yesmaximal, characteristic, direct product factor, Sylow
36C22×C7×Q8—22no—
37C22×C7×Q8—24no—
38C22×C7×Q8—41no—
39C22×C7×Q8—42no—
40C22×C7×Q8—41no—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C2×C7×C4⋊C4—24no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—24no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—41no—
62C2×C7×C4⋊C4—41no—
63C2×C7×C4⋊C4—42no—
64C2×C7×C4⋊C4—42no—
65C2×C7×C4⋊C4—42no—
66C2×C7×C4⋊C4—41no—
67C2×C7×C4⋊C4—42no—
68C2×C7×C4⋊C4—41no—
69C2×C7×C4⋊C4—41no—
70C2×C7×C4⋊C4—42no—
71C2×C7×C4⋊C4—42no—
72C2×C7×C4⋊C4D414yes—
73C2×C42×C7—22no—
74C2×C42×C7—22no—
75C2×C42×C7Q814yescomplemented
76C7×#95—22no—
77C7×#95—24no—
78C7×#95—24no—
79C7×#95—24no—
80C7×#95—24no—
81C7×#95—24no—
82C7×#95—24no—
83C7×#95—41no—
84C7×#95—42no—
85C7×#95D418yes—
86C7×#95Q818yescomplemented
87C2×#398C2×C712yes—
88C2×#395C2×C711yescharacteristic
89C2×#395C2×C712yes—
90C2×#384C2×C711yescharacteristic
91#1444C2×C712yes—
92#1444C2×C712yes—
93#1442C2×C712yes—
94#1441C2×C711yescharacteristic
95#1434C2×C712yes—
96C24×C7C2411yescharacteristic
97C2×C7×Q8—48no—
98C2×C7×Q8—44no—
99C2×C7×Q8—44no—
100C2×C7×Q8—48no—
101C2×C7×Q8—44no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D414yes—
155C22×C4×C7C2×Q814yes—
156C22×C4×C7C2×Q814yes—
157C22×C4×C7C2×Q812yes—
158C22×C4×C7C2×Q814yes—
159C22×C4×C7C2×Q814yes—
160C22×C4×C7#4112yes—
161C7×C4⋊C4—44no—
162C7×C4⋊C4—48no—
163C7×C4⋊C4—48no—
164C7×C4⋊C4—44no—
165C7×C4⋊C4—48no—
166C7×C4⋊C4—48no—
167C7×C4⋊C4—44no—
168C7×C4⋊C4—48no—
169C42×C7—28no—
170C23×Q8C22×C711yescharacteristic
171C22×C4⋊C4C22×C712yes—
172C22×C4⋊C4C22×C712yes—
173C22×C4⋊C4C22×C712yes—
174C22×C4⋊C4C22×C711yescharacteristic
175C22×C4⋊C4C22×C712yes—
176C22×C4⋊C4C22×C712yes—
177C22×C4⋊C4C22×C711yescharacteristic
178C22×C4⋊C4C22×C711yescharacteristic
179C22×C42C22×C711yescharacteristic
180#398—28no—
181#395—24no—
182#395—28no—
183#384—24no—
184C2×#95C22×C712yes—
185C2×#95C22×C712yes—
186C2×#95C22×C712yes—
187C2×#95C22×C711yescharacteristic
188C2×#95C22×C712yes—
189C2×#95C22×C712yes—
190C2×#95C22×C712yes—
191C2×#95C22×C711yescharacteristic
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C2×#95C22×C712yes—
195C23×C7C22×D411yescharacteristic
196C23×C7C22×D412yes—
197C23×C7C22×Q811yescharacteristic
198C23×C7C22×Q812yes—
199C23×C7C2×#4111yescharacteristic
200C23×C7H(5,𝔽2)11yescharacteristic
201C23×C7H(5,𝔽2)11yescharacteristic
202C23×C7H(5,𝔽2)12yes—
203C23×C72−1+411yescharacteristic
204C23×C72−1+411yescharacteristic
205C23×C72−1+412yes—
206C7×Q8—416no—
207C2×C4×C7—24no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—24no—
226C2×C4×C7—24no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—41no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7C22⋊Q814yes—
265C2×C4×C7C4⋊Q818yes—
266C22×Q8—22no—
267C22×Q8—24no—
268C22×Q8—41no—
269C22×Q8—42no—
270C22×Q8—41no—
271C23×C4C23×C712yes—
272C23×C4C23×C711yescharacteristic
273C23×C4C23×C711yescharacteristic
274C23×C4C23×C712yes—
275C23×C4C23×C712yes—
276C23×C4C23×C712yes—
277C23×C4C23×C712yes—
278C23×C4C23×C712yes—
279C23×C4C23×C711yescharacteristic
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4C7×D414yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C7×Q814yescomplemented
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—41no—
314#95—42no—
315#95C7×D418yes—
316#95C7×Q818yescomplemented
317C22×C7C2×C22⋊Q811yescharacteristic
318C22×C7C2×C4⋊Q812yes—
319C22×C7#53411yescharacteristic
320C22×C7#53512yes—
321C22×C7#53611yescharacteristic
322C22×C7#53612yes—
323C22×C7#53911yescharacteristic
324C22×C7#53911yescharacteristic
325C22×C7#54411yescharacteristic
326C22×C7#54412yes—
327C22×C7#54411yescharacteristic
328C22×C7#54412yes—
329C22×C7#54412yes—
330C22×C7D4212yes—
331C22×C7D4211yescharacteristic
332C22×C7#54712yes—
333C22×C7D4×Q812yes—
334C22×C7D4×Q812yes—
335C22×C7#55412yes—
336C22×C7Q8211yescharacteristic
337C22×C7#56011yescharacteristic
338C22×C7#56312yes—
339C22×C7#56311yescharacteristic
340C4×C7—24no—
341C4×C7—28no—
342C4×C7—42no—
343C4×C7—42no—
344C4×C7—44no—
345C4×C7—44no—
346C4×C7—44no—
347C4×C7—44no—
348C4×C7—82no—
349C24C24×C711yescharacteristic
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×Q8—44no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C2×C7×D414yes—
403C22×C4C2×C7×D412yes—
404C22×C4C2×C7×D414yes—
405C22×C4C2×C7×D412yes—
406C22×C4C2×C7×D412yes—
407C22×C4C2×C7×D414yes—
408C22×C4C2×C7×Q814yes—
409C22×C4C2×C7×Q814yes—
410C22×C4C2×C7×Q812yes—
411C22×C4C2×C7×Q814yes—
412C22×C4C2×C7×Q814yes—
413C22×C4C7×#4112yes—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C42—28no—
423C2×C7#244012yes—
424C2×C7#255111yescharacteristic
425C2×C7#262711yescharacteristic
426C2×C7#266012yes—
427C2×C7#266911yescharacteristic
428C2×C7#267712yes—
429C2×C7#268011yescharacteristic
430C2×C7#272012yes—
431C2×C7#273411yescharacteristic
432C2×C7#279211yescharacteristic
433C2×C7#284311yescharacteristic
434C23C22×C7×D411yescharacteristic
435C23C22×C7×D412yes—
436C23C22×C7×Q811yescharacteristic
437C23C22×C7×Q812yes—
438C23C2×C7×#4111yescharacteristic
439C23C7×H(5,𝔽2)11yescharacteristic
440C23C7×H(5,𝔽2)11yescharacteristic
441C23C7×H(5,𝔽2)12yes—
442C23C7×2−1+411yescharacteristic
443C23C7×2−1+411yescharacteristic
444C23C7×2−1+412yes—
445Q8—416no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4C7×C22⋊Q814yes—
504C2×C4C7×C4⋊Q818yes—
505C7#1652211yescharacteristic, direct product factor, Sylow
506C22C2×C7×C22⋊Q811yescharacteristic
507C22C2×C7×C4⋊Q812yes—
508C22C7×#53411yescharacteristic
509C22C7×#53512yes—
510C22C7×#53612yes—
511C22C7×#53611yescharacteristic
512C22C7×#53911yescharacteristic
513C22C7×#53911yescharacteristic
514C22C7×#54411yescharacteristic
515C22C7×#54412yes—
516C22C7×#54412yes—
517C22C7×#54412yes—
518C22C7×#54411yescharacteristic
519C22C7×D4212yes—
520C22C7×D4211yescharacteristic
521C22C7×#54712yes—
522C22C7×D4×Q812yes—
523C22C7×D4×Q812yes—
524C22C7×#55412yes—
525C22C7×Q8211yescharacteristic
526C22C7×#56011yescharacteristic
527C22C7×#56312yes—
528C22C7×#56311yescharacteristic
529C4—28no—
530C4—24no—
531C4—42no—
532C4—44no—
533C4—42no—
534C4—44no—
535C4—44no—
536C4—44no—
537C4—82no—
538C2C7×#244012yes—
539C2C7×#255111yescharacteristic
540C2C7×#262711yescharacteristic
541C2C7×#266012yes—
542C2C7×#266911yescharacteristic
543C2C7×#267712yes—
544C2C7×#268011yescharacteristic
545C2C7×#272012yes—
546C2C7×#273411yescharacteristic
547C2C7×#279211yescharacteristic
548C2C7×#284311yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C7C24
commutator, Frattini, cosocleC24C24×C7
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
35. #16522C7direct (N × H)
75. C2×C42×C7Q8semidirect
86. C7×#95Q8semidirect
305. C2×C42C7×Q8semidirect
316. #95C7×Q8semidirect
505. C7#16522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3982
C2×C7×#3952
C2×C7×#3842
C7×#14442
C7×#14422
C7×#14412
C7×#14342
#165227