Finite Group Databasebeta

← C7×#23985 #13201201

Subgroups — C7×#23985 #13201201

Order 1792.

1474 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3490C211yesmaximal, characteristic
2C7×#2849C212yesmaximal, complemented
3C2×C7×#381C211yesmaximal, characteristic, complemented
4C7×#2030C212yesmaximal, complemented
5C7×#2022C212yesmaximal, complemented
6C7×#1898C212yesmaximal, complemented
7C7×#1890C214yesmaximal, complemented
8C7×#1819C211yesmaximal, characteristic, complemented
9C24×C4×C7C2211yescharacteristic
10C7×#551—24no—
11C7×#546—24no—
12C7×#543C2211yescharacteristic
13C7×#538—22no—
14C7×#534C2211yescharacteristic
15C7×#525—22no—
16C2×C7×#123C2214yes—
17C2×C7×#120C2212yes—
18C7×#518C2212yes—
19C22×C7×C22⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2211yescharacteristic
21C2×C7×#99C2214yes—
22C7×#386—24no—
23C7×#385—24no—
24C7×#381—22no—
25C7×#381C2212yes—
26C7×#381C2214yes—
27C7×#381C2212yes—
28C7×#379C2212yes—
29C2×C7×#95—22no—
30C7×#342C2212yes—
31C7×#342C2214yes—
32C7×#342C2212yes—
33#23985C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7C2311yescharacteristic
35C2×C7×#41—22no—
36C22×C7×D4D414yes—
37C22×C7×D4C2312yes—
38C23×C4×C7—21no—
39C23×C4×C7—21no—
40C23×C4×C7—24no—
41C23×C4×C7—21no—
42C23×C4×C7—24no—
43C23×C4×C7—42no—
44C23×C4×C7D412yes—
45C23×C4×C7C2312yes—
46C7×#126—24no—
47C7×#124—24no—
48C7×#123—24no—
49C7×#123—24no—
50C7×#123—22no—
51C7×#123—28no—
52C7×#123—44no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—24no—
56C7×#121—22no—
57C7×#121—24no—
58C7×#121—22no—
59C7×#120—22no—
60C7×#120—22no—
61C7×#120—24no—
62C7×#120—22no—
63C4×C7×D4—24no—
64C4×C7×D4—24no—
65C4×C7×D4—24no—
66C7×#117—22no—
67C7×#117—22no—
68C2×C7×C4⋊C4—24no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—42no—
75C2×C7×C22⋊C4D412yes—
76C2×C7×C22⋊C4D414yes—
77C2×C7×C22⋊C4C2314yes—
78C2×C7×C22⋊C4C2312yes—
79C2×C7×C22⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4C2312yes—
82C7×#99—44no—
83C7×#95—24no—
84C7×#95—24no—
85C7×#95—28no—
86#3490C2×C711yescharacteristic
87#2849C2×C712yescomplemented
88C2×#381C2×C711yescharacteristic, complemented
89#2030C2×C712yescomplemented
90#2022C2×C712yescomplemented
91#1898C2×C712yescomplemented
92#1890C2×C714yescomplemented
93#1819C2×C711yescharacteristic, complemented
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—44no—
97C24×C7C2×D411yescharacteristic
98C24×C7C2×D414yes—
99C24×C7C2×D412yes—
100C24×C7C2×D411yescharacteristic
101C24×C7C2411yescharacteristic
102C7×#41—84no—
103C2×C7×Q8—22no—
104C2×C7×D4—24no—
105C2×C7×D4—22no—
106C2×C7×D4—22no—
107C2×C7×D4—24no—
108C2×C7×D4—24no—
109C2×C7×D4—22no—
110C2×C7×D4—24no—
111C2×C7×D4—48no—
112C22×C4×C7—24no—
113C22×C4×C7—24no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—24no—
117C22×C4×C7—22no—
118C22×C4×C7—24no—
119C22×C4×C7—24no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—24no—
123C22×C4×C7—416no—
124C22×C4×C7—42no—
125C22×C4×C7—42no—
126C22×C4×C7—42no—
127C22×C4×C7—42no—
128C22×C4×C7—44no—
129C22×C4×C7—42no—
130C22×C4×C7—44no—
131C22×C4×C7C2×D412yes—
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×D414yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7#4114yescomplemented
136C7×C4⋊C4—24no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—24no—
140C7×C4⋊C4—48no—
141C7×C22⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—22no—
147C7×C22⋊C4—22no—
148C7×C22⋊C4—22no—
149C7×C22⋊C4—22no—
150C7×C22⋊C4—24no—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—48no—
154C7×C22⋊C4—48no—
155C7×C22⋊C4—42no—
156C42×C7—24no—
157C24×C4C22×C711yescharacteristic
158#551—24no—
159#546—24no—
160#543C22×C711yescharacteristic
161#538—22no—
162#534C22×C711yescharacteristic
163#525—22no—
164C2×#123C22×C714yes—
165C2×#120C22×C712yes—
166#518C22×C712yes—
167C22×C22⋊C4C22×C711yescharacteristic
168C22×C22⋊C4C22×C711yescharacteristic
169C2×#99C22×C714yes—
170#386—24no—
171#385—24no—
172#381—22no—
173#381C22×C714yes—
174#381C22×C712yes—
175#381C22×C712yes—
176#379C22×C712yes—
177C2×#95—22no—
178#342C22×C712yes—
179#342C22×C712yes—
180#342C22×C714yes—
181C23×C7—21no—
182C23×C7—21no—
183C23×C7—24no—
184C23×C7—21no—
185C23×C7—21no—
186C23×C7—28no—
187C23×C7—28no—
188C23×C7—22no—
189C23×C7—24no—
190C23×C7—42no—
191C23×C7—44no—
192C23×C7—42no—
193C23×C7—48no—
194C23×C7—48no—
195C23×C7—44no—
196C23×C7#12012yes—
197C23×C7#12012yes—
198C23×C7C22×D411yescharacteristic
199C23×C7C22×D412yes—
200C23×C7C2×#4112yes—
201C23×C7H(5,𝔽2)12yes—
202C7×Q8—82no—
203C7×D4—42no—
204C7×D4—42no—
205C7×D4—44no—
206C7×D4—48no—
207C7×D4—84no—
208C7×D4—82no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—24no—
212C2×C4×C7—22no—
213C2×C4×C7—24no—
214C2×C4×C7—28no—
215C2×C4×C7—24no—
216C2×C4×C7—48no—
217C2×C4×C7—48no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—44no—
221C2×C4×C7—44no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—44no—
225C2×C4×C7—48no—
226C2×C4×C7—44no—
227C2×C4×C7—42no—
228C2×C4×C7—44no—
229C2×C4×C7—82no—
230C2×C4×C7—84no—
231C2×C4×C7—84no—
232C25C23×C711yescharacteristic
233C2×#41—22no—
234C22×D4C7×D414yes—
235C22×D4C23×C712yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—42no—
242C23×C4C7×D412yes—
243C23×C4C23×C712yes—
244#126—24no—
245#124—24no—
246#123—24no—
247#123—24no—
248#123—22no—
249#123—28no—
250#123—44no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—24no—
254#121—22no—
255#121—24no—
256#121—22no—
257#120—22no—
258#120—22no—
259#120—24no—
260#120—22no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264#117—22no—
265#117—22no—
266C2×C4⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4C7×D412yes—
274C2×C22⋊C4C7×D414yes—
275C2×C22⋊C4C23×C712yes—
276C2×C22⋊C4C23×C714yes—
277C2×C22⋊C4C23×C711yescharacteristic
278C2×C22⋊C4C23×C711yescharacteristic
279C2×C22⋊C4C23×C712yes—
280#99—44no—
281#95—24no—
282#95—24no—
283#95—28no—
284C22×C7—24no—
285C22×C7—24no—
286C22×C7—22no—
287C22×C7—22no—
288C22×C7—24no—
289C22×C7—22no—
290C22×C7—22no—
291C22×C7—24no—
292C22×C7—42no—
293C22×C7—42no—
294C22×C7—416no—
295C22×C7—42no—
296C22×C7—42no—
297C22×C7—42no—
298C22×C7—44no—
299C22×C7—84no—
300C22×C7#45814yes—
301C22×C7C2×#12011yescharacteristic
302C22×C7#52511yescharacteristic
303C22×C7#53511yescharacteristic
304C22×C7#54614yes—
305C4×C7—44no—
306C4×C7—42no—
307C4×C7—42no—
308C4×C7—42no—
309C4×C7—44no—
310C4×C7—84no—
311C4×C7—82no—
312C24—22no—
313C24—22no—
314C24—44no—
315C24C2×C7×D411yescharacteristic
316C24C2×C7×D414yes—
317C24C2×C7×D412yes—
318C24C2×C7×D411yescharacteristic
319C24C24×C711yescharacteristic
320#41—84no—
321C2×Q8—22no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—48no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—416no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4C2×C7×D412yes—
350C22×C4C2×C7×D412yes—
351C22×C4C2×C7×D414yes—
352C22×C4C2×C7×D412yes—
353C22×C4C7×#4114yescomplemented
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—48no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—48no—
372C22⋊C4—42no—
373C22⋊C4—48no—
374C42—24no—
375C2×C7—21no—
376C2×C7—24no—
377C2×C7—21no—
378C2×C7—24no—
379C2×C7—42no—
380C2×C7—82no—
381C2×C7#240411yescharacteristic
382C2×C7C2×#45812yes—
383C23—21no—
384C23—21no—
385C23—24no—
386C23—21no—
387C23—21no—
388C23—28no—
389C23—28no—
390C23—22no—
391C23—24no—
392C23—42no—
393C23—44no—
394C23—42no—
395C23—48no—
396C23—48no—
397C23—44no—
398C23C7×#12012yes—
399C23C7×#12012yes—
400C23C22×C7×D411yescharacteristic
401C23C22×C7×D412yes—
402C23C2×C7×#4112yes—
403C23C7×H(5,𝔽2)12yes—
404Q8—82no—
405D4—42no—
406D4—42no—
407D4—44no—
408D4—48no—
409D4—84no—
410D4—82no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—28no—
417C2×C4—24no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—84no—
434C7#2398511yescharacteristic, direct product factor, Sylow
435C22—24no—
436C22—24no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—22no—
441C22—22no—
442C22—24no—
443C22—42no—
444C22—42no—
445C22—416no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—84no—
451C22C7×#45814yes—
452C22C2×C7×#12011yescharacteristic
453C22C7×#52511yescharacteristic
454C22C7×#53511yescharacteristic
455C22C7×#54614yes—
456C4—44no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—44no—
461C4—82no—
462C4—84no—
463C2—21no—
464C2—24no—
465C2—21no—
466C2—24no—
467C2—42no—
468C2—82no—
469C2C7×#240411yescharacteristic
470C2C2×C7×#45812yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#3490C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2849C2semidirect
3. C2×C7×#381C2semidirect
4. C7×#2030C2semidirect
5. C7×#2022C2semidirect
6. C7×#1898C2semidirect
7. C7×#1890C2semidirect
8. C7×#1819C2semidirect
33. #23985C7direct (N × H)
87. #2849C2×C7semidirect
88. C2×#381C2×C7semidirect
89. #2030C2×C7semidirect
90. #2022C2×C7semidirect
91. #1898C2×C7semidirect
92. #1890C2×C7semidirect
93. #1819C2×C7semidirect
135. C22×C4×C7#41semidirect
353. C22×C4C7×#41semidirect
434. C7#23985direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34902
C7×#28492
C2×C7×#3812
C7×#20302
C7×#20222
C7×#18982
C7×#18902
C7×#18192
#239857