Finite Group Databasebeta

← C7×#9998 #13187214

Subgroups — C7×#9998 #13187214

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3103C211yesmaximal, characteristic, complemented
2C7×#3036C211yesmaximal, characteristic
3C7×#2580C211yesmaximal, characteristic, complemented
4C7×#1517C211yesmaximal, characteristic, complemented
5C7×#1487C211yesmaximal, characteristic, complemented
6C7×#1478C211yesmaximal, characteristic, complemented
7C7×#1286C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)—21no—
9C7×#532C2211yescharacteristic
10C2×C4×C7×Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#481—21no—
13C7×#475—21no—
14C7×#463—22no—
15C7×#461—21no—
16C7×#460—21no—
17C7×#459—21no—
18C7×#448—21no—
19C4×C7×SD16—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#417—21no—
24C7×#399—21no—
25C7×#393—21no—
26C7×#390—21no—
27C7×#389—21no—
28C7×#385—22no—
29C7×#382C2211yescharacteristic
30C7×#327—22no—
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#9998C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41—21no—
35C22×C7×Q8D411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C23×C4×C7—21no—
38C7×AGL(1,ℤ/8)—42no—
39C7×AGL(1,ℤ/8)—42no—
40C2×C7×SD16—22no—
41C2×C7×SD16—22no—
42C2×C7×D8—22no—
43C2×C7×M16—21no—
44C7×C4⋊Q8D411yescharacteristic
45C7×#127D411yescharacteristic
46C7×#124—21no—
47C7×#121—21no—
48C7×#121—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—22no—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C7×#106—22no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8—21no—
72C7×C4⋊C8—21no—
73C7×C4⋊C8D412yes—
74C7×#103—22no—
75C7×#103—22no—
76C7×#102—22no—
77C7×#102—22no—
78C7×C22⋊C8—21no—
79C7×C22⋊C8—21no—
80C4×C7×C8—21no—
81C4×C7×C8D412yes—
82C7×#95—22no—
83C7×#95—21no—
84C7×#95D412yes—
85#3103C2×C711yescharacteristic, complemented
86#3036C2×C711yescharacteristic
87#2580C2×C711yescharacteristic, complemented
88#1517C2×C711yescharacteristic, complemented
89#1487C2×C711yescharacteristic, complemented
90#1478C2×C711yescharacteristic, complemented
91#1286C2×C711yescharacteristic, complemented
92C24×C7—21no—
93C7×#41—41no—
94C7×#41—41no—
95C7×#41—41no—
96C7×#41—41no—
97C2×C7×Q8—21no—
98C2×C7×Q8—21no—
99C2×C7×Q8—21no—
100C2×C7×Q8—42no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—21no—
104C2×C7×D4—21no—
105C2×C7×D4—21no—
106C2×C7×D4—21no—
107C2×C7×D4—21no—
108C2×C7×D4—21no—
109C2×C7×D4—41no—
110C2×C7×D4SD1612yescomplemented
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—41no—
116C22×C4×C7—41no—
117C22×C4×C7—41no—
118C22×C4×C7—82no—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C7×SD16—44no—
123C7×SD16—44no—
124C7×D8—44no—
125C7×M16—41no—
126C7×M16—41no—
127C2×C7×C8—22no—
128C2×C7×C8—22no—
129C2×C7×C8—41no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—41no—
137C7×C4⋊C4—41no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4—41no—
140C7×C4⋊C4—82no—
141C7×C22⋊C4—44no—
142C7×C22⋊C4—41no—
143C7×C22⋊C4—41no—
144C7×C22⋊C4—41no—
145C7×C22⋊C4—41no—
146C7×C22⋊C4—82no—
147C42×C7—41no—
148C42×C7—41no—
149C42×C7C2×D411yescharacteristic
150C42×C7C2×D411yescharacteristic
151C42×C7C2×D411yescharacteristic
152C42×C7#4111yescharacteristic
153C2×AGL(1,ℤ/8)—21no—
154#532C22×C711yescharacteristic
155C2×C4×Q8C22×C711yescharacteristic
156C2×C4×D4C22×C711yescharacteristic
157#481—21no—
158#475—21no—
159#463—22no—
160#461—21no—
161#460—21no—
162#459—21no—
163#448—21no—
164C4×SD16—22no—
165#431C22×C711yescharacteristic
166#431C22×C711yescharacteristic
167#423C22×C711yescharacteristic
168#417—21no—
169#399—21no—
170#393—21no—
171#390—21no—
172#389—21no—
173#385—22no—
174#382C22×C711yescharacteristic
175#327—22no—
176Q8⋊C8—22no—
177D4⋊C8—22no—
178C23×C7—22no—
179C23×C7—42no—
180C23×C7—42no—
181C23×C7—41no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7#12011yescharacteristic
185C7×Q8—22no—
186C7×Q8—22no—
187C7×Q8—41no—
188C7×Q8—41no—
189C7×Q8—41no—
190C7×Q8—41no—
191C7×D4—22no—
192C7×D4—22no—
193C7×D4—42no—
194C7×D4—41no—
195C7×D4—41no—
196C7×D4—41no—
197C7×D4—41no—
198C7×D4—81no—
199C7×D4—81no—
200C7×D4—81no—
201C2×C4×C7—22no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—42no—
207C2×C4×C7—41no—
208C2×C4×C7—42no—
209C2×C4×C7—41no—
210C2×C4×C7—42no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7—82no—
216C2×C4×C7—82no—
217C2×C4×C7—81no—
218C2×C4×C7—81no—
219C2×C4×C7#12011yescharacteristic
220C2×C4×C7#12011yescharacteristic
221C2×C4×C7#12111yescharacteristic
222C2×C4×C7#12111yescharacteristic
223C2×C4×C7#12111yescharacteristic
224C2×C4×C7#12711yescharacteristic
225C2×C4×C7C2×SD1611yescharacteristic
226C2×C4×C7#13511yescharacteristic
227C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
228C2×C4×C7#13711yescharacteristic
229C7×C8—42no—
230C7×C8—42no—
231C7×C8—81no—
232C2×#41—21no—
233C22×Q8C7×D411yescharacteristic
234C22×D4C7×D411yescharacteristic
235C23×C4—21no—
236AGL(1,ℤ/8)—42no—
237AGL(1,ℤ/8)—42no—
238C2×SD16—22no—
239C2×SD16—22no—
240C2×D8—22no—
241C2×M16—21no—
242C4⋊Q8C7×D411yescharacteristic
243#127C7×D411yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C7×D411yescharacteristic
261C2×C4⋊C4C7×D411yescharacteristic
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C42—21no—
266C2×C42C23×C711yescharacteristic
267#106—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8C7×D412yes—
272#103—22no—
273#103—22no—
274#102—22no—
275#102—22no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C4×C8—21no—
279C4×C8C7×D412yes—
280#95—22no—
281#95—21no—
282#95C7×D412yes—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—44no—
286C22×C7—42no—
287C22×C7—82no—
288C22×C7—81no—
289C22×C7—81no—
290C22×C7—81no—
291C22×C7—81no—
292C22×C7—81no—
293C22×C7#39211yescharacteristic
294C22×C7#44811yescharacteristic
295C22×C7#44911yescharacteristic
296C4×C7—21no—
297C4×C7—21no—
298C4×C7—21no—
299C4×C7—21no—
300C4×C7—42no—
301C4×C7—82no—
302C4×C7—81no—
303C4×C7—81no—
304C4×C7—81no—
305C4×C7#45011yescharacteristic
306C4×C7#45211yescharacteristic
307C4×C7#46511yescharacteristic
308C4×C7#46811yescharacteristic
309C24—21no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—42no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4C7×SD1612yescomplemented
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—82no—
336C22×C4C2×C7×D411yescharacteristic
337C22×C4C2×C7×D411yescharacteristic
338C22×C4C2×C7×D411yescharacteristic
339SD16—44no—
340SD16—44no—
341D8—44no—
342M16—41no—
343M16—41no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—41no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—82no—
358C22⋊C4—44no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—82no—
364C42—41no—
365C42—41no—
366C42C2×C7×D411yescharacteristic
367C42C2×C7×D411yescharacteristic
368C42C2×C7×D411yescharacteristic
369C42C7×#4111yescharacteristic
370C2×C7—21no—
371C2×C7—21no—
372C2×C7—82no—
373C2×C7—161no—
374C2×C7#200011yescharacteristic
375C2×C7#200811yescharacteristic
376C2×C7#201011yescharacteristic
377C23—22no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23C7×#12011yescharacteristic
384Q8—22no—
385Q8—22no—
386Q8—41no—
387Q8—41no—
388Q8—41no—
389Q8—41no—
390D4—22no—
391D4—22no—
392D4—42no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—81no—
398D4—81no—
399D4—81no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4C7×#12011yescharacteristic
419C2×C4C7×#12011yescharacteristic
420C2×C4C7×#12111yescharacteristic
421C2×C4C7×#12111yescharacteristic
422C2×C4C7×#12111yescharacteristic
423C2×C4C7×#12711yescharacteristic
424C2×C4C2×C7×SD1611yescharacteristic
425C2×C4C7×#13511yescharacteristic
426C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
427C2×C4C7×#13711yescharacteristic
428C8—42no—
429C8—42no—
430C8—81no—
431C7#999811yescharacteristic, direct product factor, Sylow
432C22—21no—
433C22—21no—
434C22—44no—
435C22—42no—
436C22—82no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22C7×#39211yescharacteristic
443C22C7×#44811yescharacteristic
444C22C7×#44911yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4C7×#45011yescharacteristic
455C4C7×#45211yescharacteristic
456C4C7×#46511yescharacteristic
457C4C7×#46811yescharacteristic
458C2—21no—
459C2—21no—
460C2—82no—
461C2—161no—
462C2C7×#200011yescharacteristic
463C2C7×#200811yescharacteristic
464C2C7×#201011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3103C2semidirect
3. C7×#2580C2semidirect
4. C7×#1517C2semidirect
5. C7×#1487C2semidirect
6. C7×#1478C2semidirect
7. C7×#1286C2semidirect
33. #9998C7direct (N × H)
85. #3103C2×C7semidirect
87. #2580C2×C7semidirect
88. #1517C2×C7semidirect
89. #1487C2×C7semidirect
90. #1478C2×C7semidirect
91. #1286C2×C7semidirect
110. C2×C7×D4SD16semidirect
327. C2×D4C7×SD16semidirect
431. C7#9998direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31032
C7×#30362
C7×#25802
C7×#15172
C7×#14872
C7×#14782
C7×#12862
#99987