Finite Group Databasebeta

← C7×#9078 #13186294

Subgroups — C7×#9078 #13186294

Order 1792.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×M16C211yesmaximal, characteristic, complemented
2C4×C7×#121C211yesmaximal, characteristic
3C7×#1797C211yesmaximal, characteristic, complemented
4C2×C7×D4⋊C8C212yesmaximal, complemented
5C7×#1279C212yesmaximal, complemented
6C22×C7×M16—21no—
7C2×C7×#121C411yescharacteristic
8C2×C4×C7×D4—21no—
9C2×C4×C7×D4C2211yescharacteristic
10C22×C42×C7C2211yescharacteristic
11C2×C7×C4⋊C8C2212yes—
12C2×C7×C22⋊C8—21no—
13C4×C7×M16—24no—
14C2×C7×#97—21no—
15C2×C4×C7×C8C2212yes—
16C7×#390C411yescharacteristic
17C7×#389C411yescharacteristic
18C7×#388—21no—
19C7×#386—21no—
20C7×#380C411yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22C4×C7×C22⋊C4—21no—
23C7×#336—21no—
24C7×D4⋊C8—24no—
25#9078C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C2×C7×M16—24no—
33C2×C7×M16—24no—
34C2×C7×M16—24no—
35C22×C7×C8—22no—
36C22×C7×C8—22no—
37C7×#121—22no—
38C7×#121—22no—
39C4×C7×D4—22no—
40C4×C7×D4—22no—
41C4×C7×D4—44no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4C2×C411yescharacteristic
44C2×C7×C4⋊C4C2×C411yescharacteristic
45C2×C7×C4⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C42×C7—22no—
52C2×C42×C7—22no—
53C2×C42×C7—21no—
54C2×C42×C7C2×C411yescharacteristic
55C2×C42×C7D411yescharacteristic
56C2×C42×C7D411yescharacteristic
57C2×C42×C7C2311yescharacteristic
58C7×C4⋊C8—22no—
59C7×C4⋊C8—22no—
60C7×C22⋊C8—42no—
61C7×#97—24no—
62C4×C7×C8—22no—
63C4×C7×C8D414yescomplemented
64C7×#95—21no—
65C7×#95—21no—
66C7×#95—21no—
67C2×C4×M16C2×C711yescharacteristic, complemented
68C4×#121C2×C711yescharacteristic
69#1797C2×C711yescharacteristic, complemented
70C2×D4⋊C8C2×C712yescomplemented
71#1279C2×C712yescomplemented
72C24×C7—21no—
73C24×C7C22⋊C411yescharacteristic
74C2×C7×D4—24no—
75C2×C7×D4—22no—
76C2×C7×D4—21no—
77C2×C7×D4—22no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4M1612yescomplemented
81C22×C4×C7—22no—
82C22×C4×C7—22no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—42no—
97C22×C4×C7—42no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C22×C411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C7×M16—216no—
107C2×C7×C8—24no—
108C2×C7×C8—24no—
109C2×C7×C8—24no—
110C2×C7×C8—42no—
111C2×C7×C8—42no—
112C2×C7×C8—42no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—42no—
117C7×C4⋊C4M1612yescomplemented
118C7×C22⋊C4—24no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—42no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C42×C7—22no—
125C42×C7—21no—
126C42×C7—22no—
127C42×C7—21no—
128C42×C7—42no—
129C42×C7C2×D412yes—
130C42×C7#4112yes—
131C22×M16—21no—
132C2×#121C4×C711yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4C22×C711yescharacteristic
135C22×C42C22×C711yescharacteristic
136C2×C4⋊C8C22×C712yes—
137C2×C22⋊C8—21no—
138C4×M16—24no—
139C2×#97—21no—
140C2×C4×C8C22×C712yes—
141#390C4×C711yescharacteristic
142#389C4×C711yescharacteristic
143#388—21no—
144#386—21no—
145#380C4×C711yescharacteristic
146C4×C4⋊C4C22×C711yescharacteristic
147C4×C22⋊C4—21no—
148#336—21no—
149D4⋊C8—24no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—22no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7#10412yes—
164C23×C7C2×C22⋊C411yescharacteristic
165C7×D4—24no—
166C7×D4—48no—
167C7×D4—42no—
168C2×C4×C7—24no—
169C2×C4×C7—24no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—22no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—42no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—82no—
194C2×C4×C7—82no—
195C2×C4×C7#10412yes—
196C2×C4×C7C4×D412yes—
197C2×C4×C7#12011yescharacteristic
198C2×C4×C7#12112yes—
199C2×C4×C7#12311yescharacteristic
200C2×C4×C7C2×M1611yescharacteristic
201C2×C4×C7C2×M1611yescharacteristic
202C2×C4×C7#13112yes—
203C2×C4×C7AGL(1,ℤ/8)12yes—
204C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C7#13711yescharacteristic
206C7×C8—28no—
207C7×C8—82no—
208C22×D4—21no—
209C22×D4C2×C4×C711yescharacteristic
210C23×C4—21no—
211C23×C4C2×C4×C711yescharacteristic
212C23×C4C7×D411yescharacteristic
213C23×C4C7×D411yescharacteristic
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C22×C8—22no—
218C22×C8—22no—
219#121—22no—
220#121—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C2×C4×C711yescharacteristic
226C2×C4⋊C4C2×C4×C711yescharacteristic
227C2×C4⋊C4C2×C4×C711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42C2×C4×C711yescharacteristic
237C2×C42C7×D411yescharacteristic
238C2×C42C7×D411yescharacteristic
239C2×C42C23×C711yescharacteristic
240C4⋊C8—22no—
241C4⋊C8—22no—
242C22⋊C8—42no—
243#97—24no—
244C4×C8—22no—
245C4×C8C7×D414yescomplemented
246#95—21no—
247#95—21no—
248#95—21no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—41no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—42no—
257C22×C7—42no—
258C22×C7—42no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7#38611yescharacteristic
264C22×C7#40711yescharacteristic
265C22×C7#40811yescharacteristic
266C22×C7#41711yescharacteristic
267C22×C7#41811yescharacteristic
268C22×C7C2×#10411yescharacteristic
269C22×C7C2×#10411yescharacteristic
270C4×C7—21no—
271C4×C7—22no—
272C4×C7—21no—
273C4×C7—41no—
274C4×C7—41no—
275C4×C7—42no—
276C4×C7—81no—
277C4×C7—81no—
278C4×C7#43512yes—
279C4×C7#43612yes—
280C4×C7#46812yes—
281C4×C7#46912yes—
282C24—21no—
283C24C7×C22⋊C411yescharacteristic
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4C7×M1612yescomplemented
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C7×C22⋊C411yescharacteristic
311C22×C4C7×C22⋊C411yescharacteristic
312C22×C4C7×C22⋊C411yescharacteristic
313C22×C4C22×C4×C711yescharacteristic
314C22×C4C2×C7×D411yescharacteristic
315C22×C4C2×C7×D411yescharacteristic
316M16—216no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—42no—
327C4⋊C4C7×M1612yescomplemented
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C42—22no—
335C42—21no—
336C42—22no—
337C42—21no—
338C42—42no—
339C42C2×C7×D412yes—
340C42C7×#4112yes—
341C2×C7—42no—
342C2×C7—82no—
343C2×C7#148311yescharacteristic
344C2×C7#148411yescharacteristic
345C2×C7#148911yescharacteristic
346C2×C7#149011yescharacteristic
347C2×C7#187011yescharacteristic
348C2×C7#189511yescharacteristic
349C2×C7#189611yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C7×#10412yes—
364C23C2×C7×C22⋊C411yescharacteristic
365D4—24no—
366D4—48no—
367D4—42no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4C7×#10412yes—
396C2×C4C4×C7×D412yes—
397C2×C4C7×#12011yescharacteristic
398C2×C4C7×#12112yes—
399C2×C4C7×#12311yescharacteristic
400C2×C4C2×C7×M1611yescharacteristic
401C2×C4C2×C7×M1611yescharacteristic
402C2×C4C7×#13112yes—
403C2×C4C7×AGL(1,ℤ/8)12yes—
404C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
405C2×C4C7×#13711yescharacteristic
406C8—28no—
407C8—82no—
408C7#907811yescharacteristic, direct product factor, Sylow
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22C7×#38611yescharacteristic
424C22C7×#40711yescharacteristic
425C22C7×#40811yescharacteristic
426C22C7×#41711yescharacteristic
427C22C7×#41811yescharacteristic
428C22C2×C7×#10411yescharacteristic
429C22C2×C7×#10411yescharacteristic
430C4—21no—
431C4—22no—
432C4—21no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4—81no—
438C4C7×#43512yes—
439C4C7×#43612yes—
440C4C7×#46812yes—
441C4C7×#46912yes—
442C2—42no—
443C2—82no—
444C2C7×#148311yescharacteristic
445C2C7×#148411yescharacteristic
446C2C7×#148911yescharacteristic
447C2C7×#149011yescharacteristic
448C2C7×#187011yescharacteristic
449C2C7×#189511yescharacteristic
450C2C7×#189611yescharacteristic
subgroupG/H
centerC22×C4×C7C2×D4
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
socleC23×C7C2×C22⋊C4
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×M16C2semidirect
3. C7×#1797C2semidirect
4. C2×C7×D4⋊C8C2semidirect
5. C7×#1279C2semidirect
25. #9078C7direct (N × H)
63. C4×C7×C8D4semidirect
67. C2×C4×M16C2×C7semidirect
69. #1797C2×C7semidirect
70. C2×D4⋊C8C2×C7semidirect
71. #1279C2×C7semidirect
80. C2×C7×D4M16semidirect
117. C7×C4⋊C4M16semidirect
245. C4×C8C7×D4semidirect
290. C2×D4C7×M16semidirect
327. C4⋊C4C7×M16semidirect
408. C7#9078direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×M162
C4×C7×#1212
C7×#17972
C2×C7×D4⋊C82
C7×#12792
#90787