Finite Group Databasebeta

← C7×#8536 #13185752

Subgroups — C7×#8536 #13185752

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3496C211yesmaximal, characteristic
2C7×#1900C212yesmaximal, complemented
3C7×#1870C211yesmaximal, characteristic, complemented
4C7×#1471C212yesmaximal, complemented
5C7×#1465C211yesmaximal, characteristic, complemented
6C2×C7×H(5,𝔽2)C412yes—
7C22×C7×M16C2211yescharacteristic, complemented
8C7×#547—22no—
9C7×#546—24no—
10C7×D42—22no—
11C7×#537—21no—
12C7×#534—21no—
13C7×#534C412yes—
14C7×#532—21no—
15C2×C7×#121—21no—
16C2×C7×#121C2212yes—
17C2×C4×C7×D4C2211yescharacteristic
18C2×C7×#100—22no—
19C7×#407C2212yescomplemented
20C2×C7×C22⋊C8—21no—
21C2×C7×C22⋊C8C2211yescharacteristic, complemented
22C7×#343—22no—
23C7×#336—21no—
24C7×#324—21no—
25C7×#323—21no—
26C7×#323—24no—
27#8536C711yesmaximal, characteristic, direct product factor, Sylow
28C7×H(5,𝔽2)—44no—
29C7×H(5,𝔽2)—44no—
30C2×C7×#41—22no—
31C2×C7×#41—24no—
32C22×C7×D4—22no—
33C22×C7×D4—24no—
34C22×C7×D4—21no—
35C22×C7×D4C2×C411yescharacteristic
36C22×C7×D4C2×C414yes—
37C22×C7×D4D412yes—
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C2×C7×M16—22no—
41C2×C7×M16—22no—
42C2×C7×M16—42no—
43C22×C7×C8—21no—
44C22×C7×C8—21no—
45C7×C4⋊Q8—21no—
46C7×#127—21no—
47C7×#124—22no—
48C7×#123—24no—
49C7×#123—24no—
50C7×C22⋊Q8—24no—
51C7×#121—22no—
52C7×#121—22no—
53C7×#121—24no—
54C7×#121—22no—
55C7×#121—22no—
56C7×#121—22no—
57C7×#121—22no—
58C7×#121—22no—
59C7×#121—22no—
60C7×#120—24no—
61C7×#120—24no—
62C4×C7×D4—22no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C4×C7×D4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D412yes—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4C2×C411yescharacteristic
71C2×C7×C22⋊C4D411yescharacteristic
72C2×C7×C22⋊C4D414yes—
73C2×C42×C7D411yescharacteristic
74C7×#100—44no—
75C7×C22⋊C8—22no—
76C7×C22⋊C8—22no—
77C7×C22⋊C8—22no—
78C7×C22⋊C8—42no—
79#3496C2×C711yescharacteristic
80#1900C2×C712yescomplemented
81#1870C2×C711yescharacteristic, complemented
82#1471C2×C712yescomplemented
83#1465C2×C711yescharacteristic, complemented
84C24×C7—22no—
85C24×C7C22⋊C411yescharacteristic
86C24×C7C22⋊C414yes—
87C24×C7C22×C411yescharacteristic
88C7×#41—44no—
89C7×#41—416no—
90C7×#41—44no—
91C2×C7×Q8—22no—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—24no—
95C2×C7×D4—22no—
96C2×C7×D4—24no—
97C2×C7×D4—24no—
98C2×C7×D4—22no—
99C2×C7×D4—22no—
100C2×C7×D4—22no—
101C2×C7×D4—22no—
102C2×C7×D4—22no—
103C2×C7×D4—42no—
104C2×C7×D4—42no—
105C2×C7×D4—416no—
106C2×C7×D4—44no—
107C2×C7×D4—44no—
108C2×C7×D4—44no—
109C2×C7×D4—44no—
110C2×C7×D4—81no—
111C2×C7×D4—81no—
112C22×C4×C7—22no—
113C22×C4×C7—24no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—41no—
117C22×C4×C7—41no—
118C22×C4×C7—41no—
119C22×C4×C7—41no—
120C22×C4×C7C22⋊C411yescharacteristic
121C22×C4×C7C22⋊C411yescharacteristic
122C22×C4×C7C22⋊C411yescharacteristic
123C22×C4×C7C22⋊C414yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2×D411yescharacteristic
128C7×M16—24no—
129C7×M16—42no—
130C7×M16—42no—
131C7×M16—42no—
132C2×C7×C8—22no—
133C2×C7×C8—41no—
134C2×C7×C8—41no—
135C2×C7×C8—41no—
136C2×C7×C8—41no—
137C2×C7×C8—41no—
138C2×C7×C8—42no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—22no—
147C7×C22⋊C4—24no—
148C7×C22⋊C4—22no—
149C7×C22⋊C4—24no—
150C42×C7—21no—
151C42×C7—21no—
152C2×H(5,𝔽2)C4×C712yes—
153C22×M16C22×C711yescharacteristic, complemented
154#547—22no—
155#546—24no—
156D42—22no—
157#537—21no—
158#534—21no—
159#534C4×C712yes—
160#532—21no—
161C2×#121—21no—
162C2×#121C22×C712yes—
163C2×C4×D4C22×C711yescharacteristic
164C2×#100—22no—
165#407C22×C712yescomplemented
166C2×C22⋊C8—21no—
167C2×C22⋊C8C22×C711yescharacteristic, complemented
168#343—22no—
169#336—21no—
170#324—21no—
171#323—21no—
172#323—24no—
173C23×C7—21no—
174C23×C7—21no—
175C23×C7—21no—
176C23×C7—21no—
177C23×C7—24no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7—44no—
183C23×C7—44no—
184C23×C7—44no—
185C23×C7—44no—
186C23×C7—82no—
187C23×C7—82no—
188C23×C7#10012yescomplemented
189C23×C7C2×C22⋊C412yes—
190C23×C7C2×C22⋊C411yescharacteristic
191C7×Q8—44no—
192C7×D4—28no—
193C7×D4—44no—
194C7×D4—48no—
195C7×D4—48no—
196C7×D4—44no—
197C7×D4—44no—
198C7×D4—42no—
199C7×D4—42no—
200C7×D4—82no—
201C7×D4—82no—
202C7×D4—82no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—22no—
207C2×C4×C7—21no—
208C2×C4×C7—22no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—24no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—48no—
215C2×C4×C7—44no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—44no—
221C2×C4×C7—81no—
222C2×C4×C7#10012yescomplemented
223C2×C4×C7#12011yescharacteristic
224C2×C4×C7#12012yes—
225C2×C4×C7#12011yescharacteristic
226C7×C8—42no—
227C7×C8—81no—
228C7×C8—82no—
229H(5,𝔽2)—44no—
230H(5,𝔽2)—44no—
231C2×#41—22no—
232C2×#41—24no—
233C22×D4—22no—
234C22×D4—24no—
235C22×D4—21no—
236C22×D4C2×C4×C711yescharacteristic
237C22×D4C2×C4×C714yes—
238C22×D4C7×D412yes—
239C23×C4C7×D411yescharacteristic
240C23×C4C23×C711yescharacteristic
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—42no—
244C22×C8—21no—
245C22×C8—21no—
246C4⋊Q8—21no—
247#127—21no—
248#124—22no—
249#123—24no—
250#123—24no—
251C22⋊Q8—24no—
252#121—22no—
253#121—22no—
254#121—24no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—22no—
259#121—22no—
260#121—22no—
261#120—24no—
262#120—24no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C7×D411yescharacteristic
269C2×C4⋊C4C7×D412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4C2×C4×C711yescharacteristic
272C2×C22⋊C4C7×D411yescharacteristic
273C2×C22⋊C4C7×D414yes—
274C2×C42C7×D411yescharacteristic
275#100—44no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—42no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—22no—
283C22×C7—22no—
284C22×C7—22no—
285C22×C7—22no—
286C22×C7—42no—
287C22×C7—41no—
288C22×C7—41no—
289C22×C7—44no—
290C22×C7—44no—
291C22×C7—44no—
292C22×C7—81no—
293C22×C7—81no—
294C22×C7—81no—
295C22×C7—164no—
296C22×C7—164no—
297C22×C7#37911yescharacteristic
298C22×C7C2×#10011yescharacteristic
299C22×C7C2×#10011yescharacteristic
300C4×C7—22no—
301C4×C7—21no—
302C4×C7—22no—
303C4×C7—44no—
304C4×C7—41no—
305C4×C7—41no—
306C4×C7B(4,𝔽2)11yescharacteristic
307C4×C7#45811yescharacteristic
308C24—22no—
309C24C7×C22⋊C411yescharacteristic
310C24C7×C22⋊C414yes—
311C24C22×C4×C711yescharacteristic
312#41—44no—
313#41—416no—
314#41—44no—
315C2×Q8—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—416no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—81no—
335C2×D4—81no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C7×C22⋊C411yescharacteristic
345C22×C4C7×C22⋊C411yescharacteristic
346C22×C4C7×C22⋊C411yescharacteristic
347C22×C4C7×C22⋊C414yes—
348C22×C4C2×C7×D411yescharacteristic
349C22×C4C2×C7×D412yes—
350C22×C4C2×C7×D412yes—
351C22×C4C2×C7×D411yescharacteristic
352M16—24no—
353M16—42no—
354M16—42no—
355M16—42no—
356C2×C8—22no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C42—21no—
375C42—21no—
376C2×C7—21no—
377C2×C7—21no—
378C2×C7—42no—
379C2×C7—42no—
380C2×C7—84no—
381C2×C7#178311yescharacteristic
382C2×C7#179011yescharacteristic
383C2×C7#179111yescharacteristic
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—24no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—82no—
398C23—82no—
399C23C7×#10012yescomplemented
400C23C2×C7×C22⋊C412yes—
401C23C2×C7×C22⋊C411yescharacteristic
402Q8—44no—
403D4—28no—
404D4—44no—
405D4—48no—
406D4—48no—
407D4—44no—
408D4—44no—
409D4—42no—
410D4—42no—
411D4—82no—
412D4—82no—
413D4—82no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—81no—
433C2×C4C7×#10012yescomplemented
434C2×C4C7×#12011yescharacteristic
435C2×C4C7×#12012yes—
436C2×C4C7×#12011yescharacteristic
437C8—42no—
438C8—81no—
439C8—82no—
440C7#853611yescharacteristic, direct product factor, Sylow
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—164no—
457C22—164no—
458C22C7×#37911yescharacteristic
459C22C2×C7×#10011yescharacteristic
460C22C2×C7×#10011yescharacteristic
461C4—22no—
462C4—21no—
463C4—22no—
464C4—44no—
465C4—41no—
466C4—41no—
467C4C7×B(4,𝔽2)11yescharacteristic
468C4C7×#45811yescharacteristic
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—84no—
474C2C7×#178311yescharacteristic
475C2C7×#179011yescharacteristic
476C2C7×#179111yescharacteristic
subgroupG/H
center, socleC22×C7#379
commutatorC24C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#3496C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1900C2semidirect
3. C7×#1870C2semidirect
4. C7×#1471C2semidirect
5. C7×#1465C2semidirect
7. C22×C7×M16C22semidirect
19. C7×#407C22semidirect
21. C2×C7×C22⋊C8C22semidirect
27. #8536C7direct (N × H)
80. #1900C2×C7semidirect
81. #1870C2×C7semidirect
82. #1471C2×C7semidirect
83. #1465C2×C7semidirect
153. C22×M16C22×C7semidirect
165. #407C22×C7semidirect
167. C2×C22⋊C8C22×C7semidirect
188. C23×C7#100semidirect
222. C2×C4×C7#100semidirect
399. C23C7×#100semidirect
433. C2×C4C7×#100semidirect
440. C7#8536direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34962
C7×#19002
C7×#18702
C7×#14712
C7×#14652
#85367