Finite Group Databasebeta

← C7×#9997 #13187213

Subgroups — C7×#9997 #13187213

Order 1792.

622 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3100C211yesmaximal, characteristic
2C7×#3037C211yesmaximal, characteristic, complemented
3C7×#2536C211yesmaximal, characteristic, complemented
4C7×#1517C211yesmaximal, characteristic, complemented
5C7×#1486C211yesmaximal, characteristic, complemented
6C7×#1479C211yesmaximal, characteristic, complemented
7C7×#1286C211yesmaximal, characteristic, complemented
8C2×C7×#137—21no—
9C7×#532C2211yescharacteristic
10C2×C7×C22⋊Q8—21no—
11C2×C4×C7×Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C7×#480—21no—
14C7×#474—21no—
15C7×#464—22no—
16C7×#462—21no—
17C7×#460—21no—
18C7×#459—21no—
19C7×#447—21no—
20C4×C7×D8—22no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#423C2211yescharacteristic
24C7×#417—21no—
25C7×#398—21no—
26C7×#394—21no—
27C7×#388—22no—
28C7×#382C2211yescharacteristic
29C7×#380—21no—
30C7×#327—22no—
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#9997C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41—21no—
35C22×C7×Q8D411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C23×C4×C7—21no—
38C7×#137—42no—
39C7×#137—42no—
40C2×C7×Q16—22no—
41C2×C7×SD16—22no—
42C2×C7×D8—22no—
43C2×C7×M16—21no—
44C7×C4⋊Q8D411yescharacteristic
45C7×#127D411yescharacteristic
46C7×#124—21no—
47C7×C22⋊Q8—42no—
48C7×C22⋊Q8—42no—
49C7×#121—21no—
50C7×#121—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C4×C7×D4—21no—
60C4×C7×D4—21no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—21no—
69C2×C42×C7—21no—
70C2×C42×C7C2311yescharacteristic
71C7×#107—22no—
72C7×C4⋊C8—21no—
73C7×C4⋊C8—21no—
74C7×C4⋊C8—21no—
75C7×C4⋊C8D412yes—
76C7×#103—22no—
77C7×#102—22no—
78C7×#102—22no—
79C7×#102—22no—
80C7×C22⋊C8—21no—
81C7×C22⋊C8—21no—
82C4×C7×C8—21no—
83C4×C7×C8D412yes—
84C7×#95—22no—
85C7×#95—21no—
86C7×#95D412yes—
87#3100C2×C711yescharacteristic
88#3037C2×C711yescharacteristic, complemented
89#2536C2×C711yescharacteristic, complemented
90#1517C2×C711yescharacteristic, complemented
91#1486C2×C711yescharacteristic, complemented
92#1479C2×C711yescharacteristic, complemented
93#1286C2×C711yescharacteristic, complemented
94C24×C7—21no—
95C7×#41—41no—
96C7×#41—41no—
97C7×#41—41no—
98C7×#41—41no—
99C2×C7×Q8—21no—
100C2×C7×Q8—21no—
101C2×C7×Q8—21no—
102C2×C7×Q8—21no—
103C2×C7×Q8—21no—
104C2×C7×Q8—21no—
105C2×C7×Q8D812yescomplemented
106C2×C7×D4—21no—
107C2×C7×D4—21no—
108C2×C7×D4—21no—
109C2×C7×D4—21no—
110C2×C7×D4—21no—
111C2×C7×D4—42no—
112C2×C7×D4—41no—
113C22×C4×C7—21no—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—41no—
118C22×C4×C7—41no—
119C22×C4×C7—41no—
120C22×C4×C7—41no—
121C22×C4×C7—41no—
122C22×C4×C7—41no—
123C22×C4×C7—41no—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D411yescharacteristic
127C7×Q16—44no—
128C7×SD16—44no—
129C7×D8—44no—
130C7×M16—41no—
131C7×M16—41no—
132C2×C7×C8—22no—
133C2×C7×C8—22no—
134C2×C7×C8—41no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—21no—
141C7×C4⋊C4—44no—
142C7×C4⋊C4—41no—
143C7×C4⋊C4—41no—
144C7×C4⋊C4—41no—
145C7×C4⋊C4—41no—
146C7×C22⋊C4—44no—
147C7×C22⋊C4—44no—
148C7×C22⋊C4—41no—
149C7×C22⋊C4—41no—
150C7×C22⋊C4—41no—
151C7×C22⋊C4—41no—
152C42×C7—41no—
153C42×C7—41no—
154C42×C7C2×D411yescharacteristic
155C42×C7C2×D411yescharacteristic
156C42×C7C2×D411yescharacteristic
157C42×C7#4111yescharacteristic
158C2×#137—21no—
159#532C22×C711yescharacteristic
160C2×C22⋊Q8—21no—
161C2×C4×Q8C22×C711yescharacteristic
162C2×C4×D4C22×C711yescharacteristic
163#480—21no—
164#474—21no—
165#464—22no—
166#462—21no—
167#460—21no—
168#459—21no—
169#447—21no—
170C4×D8—22no—
171#431C22×C711yescharacteristic
172#431C22×C711yescharacteristic
173#423C22×C711yescharacteristic
174#417—21no—
175#398—21no—
176#394—21no—
177#388—22no—
178#382C22×C711yescharacteristic
179#380—21no—
180#327—22no—
181Q8⋊C8—22no—
182D4⋊C8—22no—
183C23×C7—22no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7—41no—
191C23×C7#12011yescharacteristic
192C7×Q8—22no—
193C7×Q8—22no—
194C7×Q8—41no—
195C7×Q8—41no—
196C7×Q8—41no—
197C7×Q8—41no—
198C7×D4—22no—
199C7×D4—22no—
200C7×D4—41no—
201C7×D4—41no—
202C7×D4—42no—
203C7×D4—41no—
204C7×D4—41no—
205C7×D4—81no—
206C7×D4—81no—
207C7×D4—81no—
208C2×C4×C7—22no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—42no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—81no—
223C2×C4×C7—81no—
224C2×C4×C7—81no—
225C2×C4×C7—81no—
226C2×C4×C7—81no—
227C2×C4×C7—81no—
228C2×C4×C7#12011yescharacteristic
229C2×C4×C7#12011yescharacteristic
230C2×C4×C7#12111yescharacteristic
231C2×C4×C7#12111yescharacteristic
232C2×C4×C7#12111yescharacteristic
233C2×C4×C7#12711yescharacteristic
234C2×C4×C7C2×D811yescharacteristic
235C2×C4×C7#13511yescharacteristic
236C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
237C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
238C7×C8—42no—
239C7×C8—42no—
240C7×C8—81no—
241C2×#41—21no—
242C22×Q8C7×D411yescharacteristic
243C22×D4C7×D411yescharacteristic
244C23×C4—21no—
245#137—42no—
246#137—42no—
247C2×Q16—22no—
248C2×SD16—22no—
249C2×D8—22no—
250C2×M16—21no—
251C4⋊Q8C7×D411yescharacteristic
252#127C7×D411yescharacteristic
253#124—21no—
254C22⋊Q8—42no—
255C22⋊Q8—42no—
256#121—21no—
257#121—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C7×D411yescharacteristic
272C2×C4⋊C4C7×D411yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C42—21no—
277C2×C42C23×C711yescharacteristic
278#107—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8C7×D412yes—
283#103—22no—
284#102—22no—
285#102—22no—
286#102—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C4×C8—21no—
290C4×C8C7×D412yes—
291#95—22no—
292#95—21no—
293#95C7×D412yes—
294C22×C7—21no—
295C22×C7—21no—
296C22×C7—42no—
297C22×C7—42no—
298C22×C7—42no—
299C22×C7—81no—
300C22×C7—81no—
301C22×C7—81no—
302C22×C7—81no—
303C22×C7—81no—
304C22×C7—81no—
305C22×C7—81no—
306C22×C7#39211yescharacteristic
307C22×C7#44711yescharacteristic
308C22×C7#44911yescharacteristic
309C4×C7—21no—
310C4×C7—21no—
311C4×C7—21no—
312C4×C7—21no—
313C4×C7—42no—
314C4×C7—82no—
315C4×C7—81no—
316C4×C7—81no—
317C4×C7—81no—
318C4×C7#44711yescharacteristic
319C4×C7#44911yescharacteristic
320C4×C7#46611yescharacteristic
321C4×C7#46911yescharacteristic
322C24—21no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8C7×D812yescomplemented
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—42no—
340C2×D4—41no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×C7×D411yescharacteristic
353C22×C4C2×C7×D411yescharacteristic
354C22×C4C2×C7×D411yescharacteristic
355Q16—44no—
356SD16—44no—
357D8—44no—
358M16—41no—
359M16—41no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—41no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—44no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42—41no—
381C42—41no—
382C42C2×C7×D411yescharacteristic
383C42C2×C7×D411yescharacteristic
384C42C2×C7×D411yescharacteristic
385C42C7×#4111yescharacteristic
386C2×C7—21no—
387C2×C7—21no—
388C2×C7—82no—
389C2×C7—161no—
390C2×C7#199911yescharacteristic
391C2×C7#200911yescharacteristic
392C2×C7#201011yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C7×#12011yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—42no—
413D4—41no—
414D4—41no—
415D4—81no—
416D4—81no—
417D4—81no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4C7×#12011yescharacteristic
439C2×C4C7×#12011yescharacteristic
440C2×C4C7×#12111yescharacteristic
441C2×C4C7×#12111yescharacteristic
442C2×C4C7×#12111yescharacteristic
443C2×C4C7×#12711yescharacteristic
444C2×C4C2×C7×D811yescharacteristic
445C2×C4C7×#13511yescharacteristic
446C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
447C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
448C8—42no—
449C8—42no—
450C8—81no—
451C7#999711yescharacteristic, direct product factor, Sylow
452C22—21no—
453C22—21no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C7×#39211yescharacteristic
465C22C7×#44711yescharacteristic
466C22C7×#44911yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—82no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4C7×#44711yescharacteristic
477C4C7×#44911yescharacteristic
478C4C7×#46611yescharacteristic
479C4C7×#46911yescharacteristic
480C2—21no—
481C2—21no—
482C2—82no—
483C2—161no—
484C2C7×#199911yescharacteristic
485C2C7×#200911yescharacteristic
486C2C7×#201011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3100C2
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
2. C7×#3037C2semidirect
3. C7×#2536C2semidirect
4. C7×#1517C2semidirect
5. C7×#1486C2semidirect
6. C7×#1479C2semidirect
7. C7×#1286C2semidirect
33. #9997C7direct (N × H)
88. #3037C2×C7semidirect
89. #2536C2×C7semidirect
90. #1517C2×C7semidirect
91. #1486C2×C7semidirect
92. #1479C2×C7semidirect
93. #1286C2×C7semidirect
105. C2×C7×Q8D8semidirect
333. C2×Q8C7×D8semidirect
451. C7#9997direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31002
C7×#30372
C7×#25362
C7×#15172
C7×#14862
C7×#14792
C7×#12862
#99977