Finite Group Databasebeta

← C7×#12852 #13190068

Subgroups — C7×#12852 #13190068

Order 1792.

992 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#386C211yesmaximal, characteristic
2C7×#1783C211yesmaximal, characteristic
3C7×#1781C211yesmaximal, characteristic
4C7×#1390C214yesmaximal, complemented
5C23×C7×D4C2211yescharacteristic
6C24×C4×C7C411yescharacteristic, complemented
7C22×C7×C22⋊C4—21no—
8C22×C7×C22⋊C4C411yescharacteristic, complemented
9C2×C7×#100—21no—
10C2×C7×#100C2212yes—
11C2×C7×#99—21no—
12C2×C7×#99C2212yes—
13C7×#407—21no—
14C7×#386—21no—
15C7×#386—21no—
16C7×#386C412yescomplemented
17C7×#386C2212yes—
18C7×#379—21no—
19C2×C7×#95—21no—
20#12852C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7C2×C411yescharacteristic
22C25×C7D411yescharacteristic
23C22×C7×D4—24no—
24C22×C7×D4—41no—
25C22×C7×D4—41no—
26C22×C7×D4—42no—
27C22×C7×D4C2×C411yescharacteristic
28C22×C7×D4D411yescharacteristic
29C22×C7×D4D411yescharacteristic
30C22×C7×D4C2311yescharacteristic
31C23×C4×C7—21no—
32C23×C4×C7—21no—
33C23×C4×C7—21no—
34C23×C4×C7—44no—
35C23×C4×C7—41no—
36C23×C4×C7—41no—
37C23×C4×C7C2×C412yes—
38C23×C4×C7D411yescharacteristic
39C2×C7×M16—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—44no—
45C2×C7×C22⋊C4—42no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4C2×C412yes—
49C7×#100—24no—
50C7×#100—42no—
51C7×#99—24no—
52C7×#99—42no—
53C7×C22⋊C8—41no—
54C7×C22⋊C8—41no—
55C7×#95—22no—
56C7×#95—22no—
57C2×#386C2×C711yescharacteristic
58#1783C2×C711yescharacteristic
59#1781C2×C711yescharacteristic
60#1390C2×C714yescomplemented
61C24×C7—22no—
62C24×C7—21no—
63C24×C7—21no—
64C24×C7—22no—
65C24×C7—22no—
66C24×C7—21no—
67C24×C7—21no—
68C24×C7—41no—
69C24×C7—42no—
70C24×C7—42no—
71C24×C7—42no—
72C24×C7—42no—
73C24×C7C22⋊C411yescharacteristic
74C24×C7C22⋊C411yescharacteristic
75C24×C7C22⋊C411yescharacteristic
76C24×C7C22×C411yescharacteristic
77C24×C7C2×D411yescharacteristic
78C2×C7×D4—22no—
79C2×C7×D4—22no—
80C2×C7×D4—22no—
81C2×C7×D4—22no—
82C2×C7×D4—22no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—48no—
90C2×C7×D4—48no—
91C2×C7×D4#4114yes—
92C22×C4×C7—24no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—41no—
99C22×C4×C7—44no—
100C22×C4×C7—44no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7—48no—
104C22×C4×C7—44no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7—42no—
108C22×C4×C7—44no—
109C22×C4×C7—42no—
110C22×C4×C7—42no—
111C22×C4×C7—42no—
112C22×C4×C7—42no—
113C22×C4×C7C22⋊C411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C7×M16—42no—
116C7×M16—42no—
117C2×C7×C8—42no—
118C7×C22⋊C4—44no—
119C7×C22⋊C4—44no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—42no—
122C7×C22⋊C4—42no—
123C7×C22⋊C4—42no—
124C7×C22⋊C4—82no—
125C7×C22⋊C4—82no—
126C23×D4C22×C711yescharacteristic
127C24×C4C4×C711yescharacteristic, complemented
128C22×C22⋊C4—21no—
129C22×C22⋊C4C4×C711yescharacteristic, complemented
130C2×#100—21no—
131C2×#100C22×C712yes—
132C2×#99—21no—
133C2×#99C22×C712yes—
134#407—21no—
135#386—21no—
136#386—21no—
137#386C4×C712yescomplemented
138#386C22×C712yes—
139#379—21no—
140C2×#95—21no—
141C23×C7—22no—
142C23×C7—22no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—24no—
146C23×C7—24no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—42no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—42no—
163C23×C7—48no—
164C23×C7—48no—
165C23×C7—42no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—48no—
171C23×C7—42no—
172C23×C7—42no—
173C23×C7—42no—
174C23×C7—42no—
175C23×C7—81no—
176C23×C7—81no—
177C23×C7—84no—
178C23×C7#9911yescharacteristic
179C23×C7#9911yescharacteristic
180C23×C7#9912yes—
181C23×C7C2×C22⋊C411yescharacteristic
182C23×C7#11712yes—
183C23×C7#12312yes—
184C7×D4—44no—
185C7×D4—44no—
186C7×D4—44no—
187C7×D4—44no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—24no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—44no—
198C2×C4×C7—48no—
199C2×C4×C7—44no—
200C2×C4×C7—44no—
201C2×C4×C7—42no—
202C2×C4×C7—44no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7—84no—
207C2×C4×C7—82no—
208C2×C4×C7—82no—
209C2×C4×C7—82no—
210C2×C4×C7#12312yes—
211C7×C8—82no—
212C25C2×C4×C711yescharacteristic
213C25C7×D411yescharacteristic
214C22×D4—24no—
215C22×D4—41no—
216C22×D4—41no—
217C22×D4—42no—
218C22×D4C2×C4×C711yescharacteristic
219C22×D4C7×D411yescharacteristic
220C22×D4C7×D411yescharacteristic
221C22×D4C23×C711yescharacteristic
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—44no—
226C23×C4—41no—
227C23×C4—41no—
228C23×C4C2×C4×C712yes—
229C23×C4C7×D411yescharacteristic
230C2×M16—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4C2×C4×C712yes—
240#100—24no—
241#100—42no—
242#99—24no—
243#99—42no—
244C22⋊C8—41no—
245C22⋊C8—41no—
246#95—22no—
247#95—22no—
248C22×C7—24no—
249C22×C7—24no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—22no—
257C22×C7—22no—
258C22×C7—22no—
259C22×C7—42no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7—42no—
264C22×C7—48no—
265C22×C7—48no—
266C22×C7—42no—
267C22×C7—41no—
268C22×C7—41no—
269C22×C7—41no—
270C22×C7—41no—
271C22×C7—41no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7—41no—
275C22×C7—41no—
276C22×C7—81no—
277C22×C7—84no—
278C22×C7—84no—
279C22×C7—81no—
280C22×C7AΣL(1,𝔽16)12yes—
281C22×C7#35212yes—
282C22×C7#38111yescharacteristic
283C22×C7C2×#9911yescharacteristic
284C22×C7C2×#9911yescharacteristic
285C4×C7—22no—
286C4×C7—41no—
287C4×C7—44no—
288C4×C7—82no—
289C4×C7—162no—
290C24—22no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C7×C22⋊C411yescharacteristic
303C24C7×C22⋊C411yescharacteristic
304C24C7×C22⋊C411yescharacteristic
305C24C22×C4×C711yescharacteristic
306C24C2×C7×D411yescharacteristic
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4C7×#4114yes—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—44no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4C7×C22⋊C411yescharacteristic
343C22×C4C2×C7×D411yescharacteristic
344M16—42no—
345M16—42no—
346C2×C8—42no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C2×C7—21no—
356C2×C7—21no—
357C2×C7—22no—
358C2×C7—22no—
359C2×C7—42no—
360C2×C7—42no—
361C2×C7—42no—
362C2×C7—81no—
363C2×C7#181911yescharacteristic
364C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
365C2×C7C2×#35211yescharacteristic
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—24no—
371C23—24no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—48no—
389C23—48no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—48no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—81no—
401C23—81no—
402C23—84no—
403C23C7×#9911yescharacteristic
404C23C7×#9911yescharacteristic
405C23C7×#9912yes—
406C23C2×C7×C22⋊C411yescharacteristic
407C23C7×#11712yes—
408C23C7×#12312yes—
409D4—44no—
410D4—44no—
411D4—44no—
412D4—44no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4C7×#12312yes—
436C8—82no—
437C7#1285211yescharacteristic, direct product factor, Sylow
438C22—24no—
439C22—24no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—48no—
455C22—48no—
456C22—42no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—84no—
468C22—84no—
469C22—81no—
470C22C7×AΣL(1,𝔽16)12yes—
471C22C7×#35212yes—
472C22C7×#38111yescharacteristic
473C22C2×C7×#9911yescharacteristic
474C22C2×C7×#9911yescharacteristic
475C4—22no—
476C4—41no—
477C4—44no—
478C4—82no—
479C4—162no—
480C2—21no—
481C2—21no—
482C2—22no—
483C2—22no—
484C2—42no—
485C2—42no—
486C2—42no—
487C2—81no—
488C2C7×#181911yescharacteristic
489C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
490C2C2×C7×#35211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC23×C7×D4C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C23×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1390C2semidirect
6. C24×C4×C7C4semidirect
8. C22×C7×C22⋊C4C4semidirect
16. C7×#386C4semidirect
20. #12852C7direct (N × H)
60. #1390C2×C7semidirect
127. C24×C4C4×C7semidirect
129. C22×C22⋊C4C4×C7semidirect
137. #386C4×C7semidirect
437. C7#12852direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3862
C7×#17832
C7×#17812
C7×#13902
#128527