Finite Group Databasebeta

← C7×#60884 #13238100

Subgroups — C7×#60884 #13238100

Order 1792.

1622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3509C211yesmaximal, characteristic
2C7×#3504C211yesmaximal, characteristic
3C7×#3242C218yesmaximal, complemented
4C7×#3237C214yesmaximal, complemented
5C7×#3236C214yesmaximal, complemented
6C7×#3166C211yesmaximal, characteristic, complemented
7C7×#3162C211yesmaximal, characteristic, complemented
8C7×#3139C211yesmaximal, characteristic, complemented
9C2×C7×#491C211yesmaximal, characteristic, complemented
10C7×#3097C214yesmaximal
11C7×#3082C214yesmaximal, complemented
12C7×#2923C211yesmaximal, characteristic
13C7×#557—22no—
14C7×#557—22no—
15C7×#556—22no—
16C7×#554—24no—
17C7×#554—22no—
18C7×#552—24no—
19C7×#544C2214yes—
20C7×#544C2212yes—
21C7×#541C2214yes—
22C7×#536C2212yes—
23C7×#533C2214yes—
24C7×#533C2211yescharacteristic
25C7×#533C2211yescharacteristic
26C7×#533C2212yes—
27C2×C7×C4⋊Q8C2211yescharacteristic
28C2×C7×#125C2211yescharacteristic
29C2×C7×#125C2212yes—
30C7×#520C2212yes—
31C7×#519C2212yes—
32C4×C7×#41C2211yescharacteristic
33C7×#501C2212yes—
34C7×#501C2212yes—
35C7×#501C2214yes—
36C7×#501C2214yes—
37C7×#499C2214yes—
38C7×#491C2212yes—
39C7×#491C2214yes—
40C7×#491C2212yes—
41C7×#491C2214yes—
42C7×#487C2214yes—
43C7×#485C2218yes—
44C7×#485C2218yes—
45C7×#484C2218yes—
46C7×#483C2218yes—
47C7×#479C2218yes—
48C7×#479C2218yes—
49C7×#477C2218yes—
50C7×#475C2218yes—
51C7×#433C2214yes—
52C7×#432C2214yes—
53C7×#428C2214yes—
54C7×#427C2214yes—
55C7×#424C2212yes—
56C7×#419C2214yes—
57C2×C7×#103C2214yes—
58C7×#408C2212yes—
59C4×C7×M16C2211yescharacteristic
60C2×C7×#97C2211yescharacteristic
61#60884C711yesmaximal, characteristic, direct product factor, Sylow
62C2×C7×#41C2311yescharacteristic
63C22×C7×Q8C2312yes—
64C2×C7×M16C2312yes—
65C22×C7×C8C2312yes—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—24no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8C2312yes—
71C7×C4⋊Q8C2312yes—
72C7×C4⋊Q8C2314yes—
73C7×C4⋊Q8C2312yes—
74C7×C4⋊Q8C2312yes—
75C7×#126—28no—
76C7×#125—28no—
77C7×#125—22no—
78C7×#125—24no—
79C7×#125—24no—
80C7×#125—22no—
81C7×#125—24no—
82C7×#125C2314yes—
83C7×#125C2312yes—
84C7×#125C2311yescharacteristic
85C7×#125C2312yes—
86C7×#125C2311yescharacteristic
87C7×#125C2312yes—
88C7×#124—22no—
89C7×#123—24no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—24no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—24no—
94C7×C22⋊Q8—24no—
95C7×C22⋊Q8—24no—
96C7×C22⋊Q8—24no—
97C7×C22⋊Q8C2318yes—
98C4×C7×Q8—22no—
99C4×C7×Q8—24no—
100C4×C7×Q8—24no—
101C4×C7×Q8—24no—
102C4×C7×Q8—22no—
103C4×C7×Q8—22no—
104C4×C7×Q8D412yes—
105C4×C7×Q8C2318yes—
106C4×C7×D4—24no—
107C4×C7×D4—22no—
108C4×C7×D4D414yes—
109C4×C7×D4D412yes—
110C7×#117—24no—
111C7×#117—24no—
112C7×#117C2314yes—
113C7×#117C2312yes—
114C7×#117C2312yes—
115C7×#117C2311yescharacteristic
116C7×#117C2312yes—
117C2×C7×C4⋊C4—22no—
118C2×C7×C4⋊C4—24no—
119C2×C7×C4⋊C4—22no—
120C2×C7×C4⋊C4C2314yes—
121C2×C7×C4⋊C4C2312yes—
122C2×C42×C7C2311yescharacteristic
123C2×C42×C7C2312yes—
124C7×#107C2314yes—
125C7×#107C2318yes—
126C7×#107C2314yes—
127C7×#106C2314yes—
128C7×#106C2318yes—
129C7×#106C2314yes—
130C7×C4⋊C8C2318yes—
131C7×#103C2318yes—
132C7×#103C2318yes—
133C7×#103C2318yes—
134C7×#103C2318yes—
135C7×C22⋊C8C2318yes—
136C7×#97C2312yes—
137C7×#97C2312yes—
138C7×#97C2312yes—
139C4×C7×C8C2312yes—
140#3509C2×C711yescharacteristic
141#3504C2×C711yescharacteristic
142#3242C2×C718yescomplemented
143#3237C2×C714yescomplemented
144#3236C2×C714yescomplemented
145#3166C2×C711yescharacteristic, complemented
146#3162C2×C711yescharacteristic, complemented
147#3139C2×C711yescharacteristic, complemented
148C2×#491C2×C711yescharacteristic, complemented
149#3097C2×C714yes—
150#3082C2×C714yescomplemented
151#2923C2×C711yescharacteristic
152C7×#41—24no—
153C2×C7×Q8—22no—
154C2×C7×Q8—22no—
155C2×C7×Q8—24no—
156C2×C7×Q8—22no—
157C2×C7×Q8—28no—
158C2×C7×Q8C2×D411yescharacteristic
159C2×C7×Q8C2414yes—
160C2×C7×D4C2×D412yes—
161C2×C7×D4C2×D411yescharacteristic
162C22×C4×C7—22no—
163C22×C4×C7—24no—
164C22×C4×C7—22no—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D414yes—
167C22×C4×C7C2412yes—
168C22×C4×C7C2411yescharacteristic
169C7×M16—24no—
170C2×C7×C8—24no—
171C2×C7×C8C2414yes—
172C2×C7×C8C2414yes—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—28no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—28no—
189C7×C4⋊C4C2×D412yes—
190C7×C4⋊C4C2×D414yes—
191C7×C4⋊C4C2414yes—
192C7×C4⋊C4C2412yes—
193C7×C4⋊C4C2412yes—
194C7×C4⋊C4C2414yes—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—28no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4C2×D412yes—
200C7×C22⋊C4C2×D414yes—
201C42×C7—28no—
202C42×C7—24no—
203C42×C7—22no—
204C42×C7—22no—
205C42×C7C2×D414yes—
206C42×C7C2×D412yes—
207C42×C7C2411yescharacteristic
208C42×C7C2411yescharacteristic
209C42×C7C2412yes—
210#557—22no—
211#557—22no—
212#556—22no—
213#554—24no—
214#554—22no—
215#552—24no—
216#544C22×C712yes—
217#544C22×C714yes—
218#541C22×C714yes—
219#536C22×C712yes—
220#533C22×C714yes—
221#533C22×C711yescharacteristic
222#533C22×C711yescharacteristic
223#533C22×C712yes—
224C2×C4⋊Q8C22×C711yescharacteristic
225C2×#125C22×C711yescharacteristic
226C2×#125C22×C712yes—
227#520C22×C712yes—
228#519C22×C712yes—
229C4×#41C22×C711yescharacteristic
230#501C22×C712yes—
231#501C22×C712yes—
232#501C22×C714yes—
233#501C22×C714yes—
234#499C22×C714yes—
235#491C22×C714yes—
236#491C22×C714yes—
237#491C22×C712yes—
238#491C22×C712yes—
239#487C22×C714yes—
240#485C22×C718yes—
241#485C22×C718yes—
242#484C22×C718yes—
243#483C22×C718yes—
244#479C22×C718yes—
245#479C22×C718yes—
246#477C22×C718yes—
247#475C22×C718yes—
248#433C22×C714yes—
249#432C22×C714yes—
250#428C22×C714yes—
251#427C22×C714yes—
252#424C22×C712yes—
253#419C22×C714yes—
254C2×#103C22×C714yes—
255#408C22×C712yes—
256C4×M16C22×C711yescharacteristic
257C2×#97C22×C711yescharacteristic
258C23×C7C22×D412yes—
259C23×C7C22×D411yescharacteristic
260C7×Q8—22no—
261C7×Q8—28no—
262C7×Q8—44no—
263C7×Q8—44no—
264C7×Q8—84no—
265C7×Q8—82no—
266C7×D4—24no—
267C7×D4—22no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—22no—
274C2×C4×C7—42no—
275C2×C4×C7—44no—
276C2×C4×C7—44no—
277C2×C4×C7—82no—
278C2×C4×C7—84no—
279C2×C4×C7C22×D412yes—
280C2×C4×C7C22×D414yes—
281C2×C4×C7C22×D412yes—
282C2×C4×C7C22×D412yes—
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C72−1+414yes—
285C2×C4×C7C2511yescharacteristic
286C7×C8—24no—
287C7×C8—24no—
288C2×#41C23×C711yescharacteristic
289C22×Q8C23×C712yes—
290C2×M16C23×C712yes—
291C22×C8C23×C712yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—24no—
295C4⋊Q8—22no—
296C4⋊Q8C23×C712yes—
297C4⋊Q8C23×C712yes—
298C4⋊Q8C23×C712yes—
299C4⋊Q8C23×C714yes—
300C4⋊Q8C23×C712yes—
301#126—28no—
302#125—28no—
303#125—22no—
304#125—24no—
305#125—24no—
306#125—22no—
307#125—24no—
308#125C23×C712yes—
309#125C23×C711yescharacteristic
310#125C23×C714yes—
311#125C23×C712yes—
312#125C23×C711yescharacteristic
313#125C23×C712yes—
314#124—22no—
315#123—24no—
316C22⋊Q8—22no—
317C22⋊Q8—24no—
318C22⋊Q8—22no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8C23×C718yes—
324C4×Q8—22no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8C7×D412yes—
331C4×Q8C23×C718yes—
332C4×D4—24no—
333C4×D4—22no—
334C4×D4C7×D414yes—
335C4×D4C7×D412yes—
336#117—24no—
337#117—24no—
338#117C23×C712yes—
339#117C23×C712yes—
340#117C23×C714yes—
341#117C23×C711yescharacteristic
342#117C23×C712yes—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4C23×C712yes—
347C2×C4⋊C4C23×C714yes—
348C2×C42C23×C711yescharacteristic
349C2×C42C23×C712yes—
350#107C23×C714yes—
351#107C23×C714yes—
352#107C23×C718yes—
353#106C23×C714yes—
354#106C23×C714yes—
355#106C23×C718yes—
356C4⋊C8C23×C718yes—
357#103C23×C718yes—
358#103C23×C718yes—
359#103C23×C718yes—
360#103C23×C718yes—
361C22⋊C8C23×C718yes—
362#97C23×C712yes—
363#97C23×C712yes—
364#97C23×C712yes—
365C4×C8C23×C712yes—
366C22×C7—22no—
367C22×C7—24no—
368C22×C7—22no—
369C22×C7#53612yes—
370C22×C7C23×D411yescharacteristic
371C4×C7—22no—
372C4×C7—24no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—44no—
376C4×C7—42no—
377C4×C7—44no—
378C4×C7—42no—
379C4×C7—44no—
380C4×C7#53612yes—
381C4×C7C2×2−1+412yes—
382#41—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8C2×C7×D411yescharacteristic
389C2×Q8C24×C714yes—
390C2×D4C2×C7×D412yes—
391C2×D4C2×C7×D411yescharacteristic
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4C2×C7×D412yes—
396C22×C4C2×C7×D414yes—
397C22×C4C24×C712yes—
398C22×C4C24×C711yescharacteristic
399M16—24no—
400C2×C8—24no—
401C2×C8C24×C714yes—
402C2×C8C24×C714yes—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—28no—
419C4⋊C4C2×C7×D412yes—
420C4⋊C4C2×C7×D414yes—
421C4⋊C4C24×C714yes—
422C4⋊C4C24×C712yes—
423C4⋊C4C24×C712yes—
424C4⋊C4C24×C714yes—
425C22⋊C4—22no—
426C22⋊C4—28no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4C2×C7×D412yes—
430C22⋊C4C2×C7×D414yes—
431C42—28no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42C2×C7×D414yes—
436C42C2×C7×D412yes—
437C42C24×C711yescharacteristic
438C42C24×C711yescharacteristic
439C42C24×C712yes—
440C2×C7—22no—
441C2×C7—42no—
442C2×C7C2×#53611yescharacteristic
443C2×C7#358612yes—
444C23C22×C7×D412yes—
445C23C22×C7×D411yescharacteristic
446Q8—22no—
447Q8—28no—
448Q8—44no—
449Q8—44no—
450Q8—84no—
451Q8—82no—
452D4—24no—
453D4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—82no—
464C2×C4—84no—
465C2×C4C22×C7×D412yes—
466C2×C4C22×C7×D414yes—
467C2×C4C22×C7×D412yes—
468C2×C4C22×C7×D412yes—
469C2×C4C22×C7×D411yescharacteristic
470C2×C4C7×2−1+414yes—
471C2×C4C25×C711yescharacteristic
472C8—24no—
473C8—24no—
474C7#6088411yescharacteristic, direct product factor, Sylow
475C22—22no—
476C22—24no—
477C22—22no—
478C22C7×#53612yes—
479C22C23×C7×D411yescharacteristic
480C4—22no—
481C4—24no—
482C4—22no—
483C4—22no—
484C4—44no—
485C4—42no—
486C4—44no—
487C4—42no—
488C4—44no—
489C4C7×#53612yes—
490C4C2×C7×2−1+412yes—
491C2—22no—
492C2—42no—
493C2C2×C7×#53611yescharacteristic
494C2C7×#358612yes—
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#3242C2semidirect
4. C7×#3237C2semidirect
5. C7×#3236C2semidirect
6. C7×#3166C2semidirect
7. C7×#3162C2semidirect
8. C7×#3139C2semidirect
9. C2×C7×#491C2semidirect
11. C7×#3082C2semidirect
61. #60884C7direct (N × H)
142. #3242C2×C7semidirect
143. #3237C2×C7semidirect
144. #3236C2×C7semidirect
145. #3166C2×C7semidirect
146. #3162C2×C7semidirect
147. #3139C2×C7semidirect
148. C2×#491C2×C7semidirect
150. #3082C2×C7semidirect
474. C7#60884direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35092
C7×#35042
C7×#32422
C7×#32372
C7×#32362
C7×#31662
C7×#31622
C7×#31392
C2×C7×#4912
C7×#30972
C7×#30822
C7×#29232
#608847