Finite Group Databasebeta

← C7×#8470 #13185686

Subgroups — C7×#8470 #13185686

Order 1792.

754 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3511C211yesmaximal, characteristic
2C7×#1997C211yesmaximal, characteristic, complemented
3C7×#1966C211yesmaximal, characteristic, complemented
4C7×#1560C211yesmaximal, characteristic, complemented
5C7×#1525C211yesmaximal, characteristic, complemented
6C7×#1489C211yesmaximal, characteristic, complemented
7C7×#1489C211yesmaximal, characteristic, complemented
8C7×#556—21no—
9C7×#555—21no—
10C7×#554—21no—
11C7×#553—22no—
12C7×#548—22no—
13C7×#547—21no—
14C7×#542—22no—
15C7×#537C412yes—
16C7×#533—21no—
17C7×#532—21no—
18C7×#532C2211yescharacteristic
19C2×C7×C4⋊Q8C2211yescharacteristic
20C7×#520C412yes—
21C4×C7×#41C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#423C2211yescharacteristic
24C2×C7×#101—22no—
25C4×C7×M16C2211yescharacteristic
26C4×C7×M16C2211yescharacteristic
27C7×#343—22no—
28C7×#333—21no—
29C7×#331—21no—
30C7×#329—22no—
31C7×Q8⋊C8—21no—
32C7×Q8⋊C8—21no—
33C7×D4⋊C8—21no—
34C7×D4⋊C8—21no—
35#8470C711yesmaximal, characteristic, direct product factor, Sylow
36C2×C7×#41—22no—
37C2×C7×#41C2×C411yescharacteristic
38C2×C7×#41C2×C412yes—
39C22×C7×Q8D412yes—
40C2×C7×M16—21no—
41C2×C7×M16—21no—
42C2×C7×M16—21no—
43C2×C7×M16—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8D411yescharacteristic
49C7×#127D411yescharacteristic
50C7×#126—24no—
51C7×#125—22no—
52C7×#124—22no—
53C7×#124—24no—
54C7×#124D412yes—
55C7×#123—24no—
56C7×C22⋊Q8—22no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—22no—
59C7×C22⋊Q8—22no—
60C7×#121—22no—
61C7×#121—22no—
62C7×#121—22no—
63C7×#121—24no—
64C7×#121—21no—
65C7×#121—21no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8D411yescharacteristic
69C4×C7×Q8D411yescharacteristic
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4—24no—
73C4×C7×D4—21no—
74C4×C7×D4—21no—
75C4×C7×D4—21no—
76C4×C7×D4—21no—
77C4×C7×D4D411yescharacteristic
78C4×C7×D4D411yescharacteristic
79C7×#117—22no—
80C7×#117—22no—
81C7×#117—21no—
82C7×#117C2×C411yescharacteristic
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4C2×C412yes—
85C2×C7×C4⋊C4D412yes—
86C2×C42×C7—21no—
87C2×C42×C7C2311yescharacteristic
88C7×C4⋊C8—21no—
89C7×C4⋊C8—21no—
90C7×C4⋊C8—21no—
91C7×C4⋊C8—21no—
92C7×#101—44no—
93C7×#97—21no—
94C7×#97—21no—
95C4×C7×C8—21no—
96C4×C7×C8—21no—
97#3511C2×C711yescharacteristic
98#1997C2×C711yescharacteristic, complemented
99#1966C2×C711yescharacteristic, complemented
100#1560C2×C711yescharacteristic, complemented
101#1525C2×C711yescharacteristic, complemented
102#1489C2×C711yescharacteristic, complemented
103#1489C2×C711yescharacteristic, complemented
104C7×#41—42no—
105C7×#41—44no—
106C7×#41—82no—
107C7×#41—82no—
108C2×C7×Q8—22no—
109C2×C7×Q8—22no—
110C2×C7×Q8—42no—
111C2×C7×Q8—42no—
112C2×C7×Q8C22⋊C411yescharacteristic
113C2×C7×Q8C22⋊C412yes—
114C2×C7×D4—22no—
115C2×C7×D4—21no—
116C2×C7×D4—24no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4C22⋊C411yescharacteristic
120C2×C7×D4C22⋊C412yes—
121C22×C4×C7—24no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7C22⋊C414yes—
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C7×M16—22no—
131C7×M16—22no—
132C7×M16—41no—
133C7×M16—41no—
134C7×M16—42no—
135C7×M16—42no—
136C2×C7×C8—41no—
137C2×C7×C8—41no—
138C2×C7×C8—41no—
139C2×C7×C8—41no—
140C7×C4⋊C4—24no—
141C7×C4⋊C4—24no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4C22⋊C411yescharacteristic
151C7×C4⋊C4C22⋊C411yescharacteristic
152C7×C22⋊C4—24no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—21no—
156C7×C22⋊C4—21no—
157C7×C22⋊C4—21no—
158C7×C22⋊C4—22no—
159C7×C22⋊C4—22no—
160C42×C7—24no—
161C42×C7—21no—
162C42×C7—21no—
163C42×C7—21no—
164C42×C7C2×D411yescharacteristic
165C42×C7C2×D411yescharacteristic
166C42×C7C2×D411yescharacteristic
167C42×C7C2×D411yescharacteristic
168#556—21no—
169#555—21no—
170#554—21no—
171#553—22no—
172#548—22no—
173#547—21no—
174#542—22no—
175#537C4×C712yes—
176#533—21no—
177#532—21no—
178#532C22×C711yescharacteristic
179C2×C4⋊Q8C22×C711yescharacteristic
180#520C4×C712yes—
181C4×#41C22×C711yescharacteristic
182#423C22×C711yescharacteristic
183#423C22×C711yescharacteristic
184C2×#101—22no—
185C4×M16C22×C711yescharacteristic
186C4×M16C22×C711yescharacteristic
187#343—22no—
188#333—21no—
189#331—21no—
190#329—22no—
191Q8⋊C8—21no—
192Q8⋊C8—21no—
193D4⋊C8—21no—
194D4⋊C8—21no—
195C23×C7—21no—
196C23×C7—22no—
197C23×C7C2×C22⋊C411yescharacteristic
198C7×Q8—22no—
199C7×Q8—42no—
200C7×Q8—42no—
201C7×Q8—42no—
202C7×Q8—82no—
203C7×D4—22no—
204C7×D4—42no—
205C7×D4—42no—
206C7×D4—44no—
207C7×D4—41no—
208C7×D4—41no—
209C7×D4—41no—
210C7×D4—41no—
211C7×D4—84no—
212C2×C4×C7—22no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—22no—
220C2×C4×C7—21no—
221C2×C4×C7—24no—
222C2×C4×C7—42no—
223C2×C4×C7—44no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—44no—
229C2×C4×C7—84no—
230C2×C4×C7C2×C22⋊C411yescharacteristic
231C2×C4×C7C2×C22⋊C411yescharacteristic
232C2×C4×C7#12011yescharacteristic
233C2×C4×C7#12011yescharacteristic
234C2×C4×C7#12011yescharacteristic
235C2×C4×C7#12011yescharacteristic
236C7×C8—42no—
237C7×C8—42no—
238C7×C8—81no—
239C7×C8—81no—
240C2×#41—22no—
241C2×#41C2×C4×C711yescharacteristic
242C2×#41C2×C4×C712yes—
243C22×Q8C7×D412yes—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C7×D411yescharacteristic
253#127C7×D411yescharacteristic
254#126—24no—
255#125—22no—
256#124—22no—
257#124—24no—
258#124C7×D412yes—
259#123—24no—
260C22⋊Q8—22no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264#121—22no—
265#121—22no—
266#121—22no—
267#121—24no—
268#121—21no—
269#121—21no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8C7×D411yescharacteristic
273C4×Q8C7×D411yescharacteristic
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C7×D411yescharacteristic
282C4×D4C7×D411yescharacteristic
283#117—22no—
284#117—22no—
285#117—21no—
286#117C2×C4×C711yescharacteristic
287C2×C4⋊C4—22no—
288C2×C4⋊C4C2×C4×C712yes—
289C2×C4⋊C4C7×D412yes—
290C2×C42—21no—
291C2×C42C23×C711yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296#101—44no—
297#97—21no—
298#97—21no—
299C4×C8—21no—
300C4×C8—21no—
301C22×C7—21no—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7—41no—
305C22×C7—42no—
306C22×C7—41no—
307C22×C7—42no—
308C22×C7—41no—
309C22×C7—42no—
310C22×C7#37911yescharacteristic
311C4×C7—21no—
312C4×C7—21no—
313C4×C7—21no—
314C4×C7—41no—
315C4×C7—44no—
316C4×C7—41no—
317C4×C7—41no—
318C4×C7—42no—
319C4×C7#41312yes—
320C4×C7#45311yescharacteristic
321C4×C7#45411yescharacteristic
322C4×C7#45511yescharacteristic
323C4×C7#45611yescharacteristic
324#41—42no—
325#41—44no—
326#41—82no—
327#41—82no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8C7×C22⋊C411yescharacteristic
333C2×Q8C7×C22⋊C412yes—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—24no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4C7×C22⋊C411yescharacteristic
340C2×D4C7×C22⋊C412yes—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4C7×C22⋊C414yes—
347C22×C4C22×C4×C711yescharacteristic
348C22×C4C2×C7×D411yescharacteristic
349C22×C4C2×C7×D411yescharacteristic
350M16—22no—
351M16—22no—
352M16—41no—
353M16—41no—
354M16—42no—
355M16—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4C7×C22⋊C411yescharacteristic
371C4⋊C4C7×C22⋊C411yescharacteristic
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C42—24no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42C2×C7×D411yescharacteristic
385C42C2×C7×D411yescharacteristic
386C42C2×C7×D411yescharacteristic
387C42C2×C7×D411yescharacteristic
388C2×C7—41no—
389C2×C7—81no—
390C2×C7—82no—
391C2×C7#178511yescharacteristic
392C2×C7#179211yescharacteristic
393C2×C7#179311yescharacteristic
394C23—21no—
395C23—22no—
396C23C2×C7×C22⋊C411yescharacteristic
397Q8—22no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401Q8—82no—
402D4—22no—
403D4—42no—
404D4—42no—
405D4—44no—
406D4—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—84no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—84no—
429C2×C4C2×C7×C22⋊C411yescharacteristic
430C2×C4C2×C7×C22⋊C411yescharacteristic
431C2×C4C7×#12011yescharacteristic
432C2×C4C7×#12011yescharacteristic
433C2×C4C7×#12011yescharacteristic
434C2×C4C7×#12011yescharacteristic
435C8—42no—
436C8—42no—
437C8—81no—
438C8—81no—
439C7#847011yescharacteristic, direct product factor, Sylow
440C22—21no—
441C22—21no—
442C22—21no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—41no—
448C22—42no—
449C22C7×#37911yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4C7×#41312yes—
459C4C7×#45311yescharacteristic
460C4C7×#45411yescharacteristic
461C4C7×#45511yescharacteristic
462C4C7×#45611yescharacteristic
463C2—41no—
464C2—81no—
465C2—82no—
466C2C7×#178511yescharacteristic
467C2C7×#179211yescharacteristic
468C2C7×#179311yescharacteristic
subgroupG/H
center, socleC22×C7#379
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3511C2
LCSG ▷ C22×C4 ▷ 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×#1997C2semidirect
3. C7×#1966C2semidirect
4. C7×#1560C2semidirect
5. C7×#1525C2semidirect
6. C7×#1489C2semidirect
7. C7×#1489C2semidirect
35. #8470C7direct (N × H)
98. #1997C2×C7semidirect
99. #1966C2×C7semidirect
100. #1560C2×C7semidirect
101. #1525C2×C7semidirect
102. #1489C2×C7semidirect
103. #1489C2×C7semidirect
439. C7#8470direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35112
C7×#19972
C7×#19662
C7×#15602
C7×#15252
C7×#14892
#84707