Finite Group Databasebeta

← C7×#60497 #13237713

Subgroups — C7×#60497 #13237713

Order 1792.

2006 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×#3503C211yesmaximal, characteristic, complemented
2C4×C7×2−1+4C211yesmaximal, characteristic
3C7×#3084C216yesmaximal, complemented
4C7×#3082C216yesmaximal, complemented
5C7×#3079C213yesmaximal, complemented
6C7×#3078C213yesmaximal, complemented
7C7×#3076C213yesmaximal, complemented
8C2×C7×#475C213yesmaximal, complemented
9C7×#2912C212yesmaximal, complemented
10C7×#2906C211yesmaximal, characteristic, complemented
11C7×#2898C212yesmaximal
12C2×C7×2−1+4C2211yescharacteristic
13C2×C7×#131C2212yes—
14C7×Q82—21no—
15C7×#557—21no—
16C7×#554—23no—
17C7×D4×Q8—23no—
18C7×#544C2216yes—
19C7×#533C2213yes—
20C2×C7×C4⋊Q8C2213yes—
21C7×#520—23no—
22C7×#520C2213yes—
23C7×#520C2211yescharacteristic
24C7×#520C2212yes—
25C7×#519—21no—
26C7×#519C2213yes—
27C4×C7×#41—23no—
28C4×C7×#41C2213yes—
29C4×C7×#41C2211yescharacteristic
30C2×C4×C7×Q8—21no—
31C2×C4×C7×Q8C2213yes—
32C7×#479C2216yes—
33C7×#479C22112yes—
34C7×#478C2216yes—
35C7×#477C2213yes—
36C7×#477C2216yes—
37C7×#476C2211yescharacteristic
38C7×#476C2216yes—
39C7×#475C2213yes—
40C7×#475C2216yes—
41C7×#475C22112yes—
42C7×#474C2213yes—
43C7×#428C2216yes—
44C7×#428C2216yes—
45C7×#427C2216yes—
46C2×C7×#106C2216yes—
47C7×#424C2216yes—
48C7×#423C2213yes—
49C2×C7×C4⋊C8C2213yes—
50C7×#419C2216yes—
51C7×#417C2216yes—
52C7×#416C2216yes—
53C2×C7×#103C2216yes—
54#60497C711yesmaximal, characteristic, direct product factor, Sylow
55C7×2−1+4—24no—
56C7×2−1+4Q818yescomplemented
57C2×C7×#41—23no—
58C2×C7×#41C2313yes—
59C2×C7×#41C2311yescharacteristic
60C22×C7×Q8—21no—
61C22×C7×Q8—23no—
62C22×C7×Q8C2313yes—
63C7×#131—28no—
64C2×C7×M16C2316yes—
65C22×C7×C8C2316yes—
66C7×C4⋊Q8—26no—
67C7×C4⋊Q8—23no—
68C7×C4⋊Q8—23no—
69C7×C4⋊Q8C2313yes—
70C7×C4⋊Q8C2316yes—
71C7×C4⋊Q8C2311yescharacteristic
72C7×#125—26no—
73C7×#125C2316yes—
74C7×C22⋊Q8—23no—
75C7×C22⋊Q8—23no—
76C7×C22⋊Q8—212no—
77C4×C7×Q8—21no—
78C4×C7×Q8—22no—
79C4×C7×Q8—23no—
80C4×C7×Q8—23no—
81C4×C7×Q8—23no—
82C4×C7×Q8—23no—
83C4×C7×Q8—23no—
84C4×C7×Q8—23no—
85C4×C7×Q8D411yescharacteristic
86C4×C7×Q8D411yescharacteristic
87C4×C7×Q8C2313yes—
88C4×C7×Q8C2316yes—
89C4×C7×Q8C2313yes—
90C4×C7×D4—23no—
91C4×C7×D4—26no—
92C4×C7×D4—23no—
93C4×C7×D4—23no—
94C4×C7×D4—26no—
95C4×C7×D4D413yes—
96C4×C7×D4D413yes—
97C4×C7×D4C2311yescharacteristic
98C4×C7×D4C2313yes—
99C7×#117—23no—
100C7×#117—26no—
101C7×#117—23no—
102C7×#117C2313yes—
103C7×#117C2316yes—
104C7×#117C2313yes—
105C2×C7×C4⋊C4—23no—
106C2×C7×C4⋊C4—23no—
107C2×C7×C4⋊C4—23no—
108C2×C7×C4⋊C4C2313yes—
109C2×C7×C4⋊C4C2316yes—
110C2×C42×C7—23no—
111C2×C42×C7—23no—
112C2×C42×C7C2313yes—
113C7×#107C2316yes—
114C7×#107C2316yes—
115C7×#106C2316yes—
116C7×#106C23112yes—
117C7×#106C2312yes—
118C7×C4⋊C8C2313yes—
119C7×C4⋊C8C2311yescharacteristic
120C7×C4⋊C8C2316yes—
121C7×C4⋊C8C2316yes—
122C7×#103C2316yes—
123C7×#103C23112yes—
124C7×#103C2316yes—
125C7×#102C2316yes—
126C7×#102C2312yes—
127#3503C2×C711yescharacteristic, complemented
128C4×2−1+4C2×C711yescharacteristic
129#3084C2×C716yescomplemented
130#3082C2×C716yescomplemented
131#3079C2×C713yescomplemented
132#3078C2×C713yescomplemented
133#3076C2×C713yescomplemented
134C2×#475C2×C713yescomplemented
135#2912C2×C712yescomplemented
136#2906C2×C711yescharacteristic, complemented
137#2898C2×C712yes—
138C7×#41—224no—
139C7×#41—26no—
140C7×#41C2×Q814yes—
141C7×#41C2×Q8112yescomplemented
142C7×#41#4114yes—
143C2×C7×Q8—28no—
144C2×C7×Q8—23no—
145C2×C7×Q8—21no—
146C2×C7×Q8—23no—
147C2×C7×Q8—26no—
148C2×C7×Q8—26no—
149C2×C7×Q8—86no—
150C2×C7×Q8C2×D411yescharacteristic
151C2×C7×Q8C2×Q8112yescomplemented
152C2×C7×Q8C2413yes—
153C2×C7×D4—23no—
154C2×C7×D4C2×D413yes—
155C2×C7×D4C2411yescharacteristic
156C22×C4×C7—23no—
157C22×C4×C7—23no—
158C22×C4×C7—23no—
159C22×C4×C7—26no—
160C22×C4×C7—21no—
161C22×C4×C7—23no—
162C22×C4×C7C2×D413yes—
163C22×C4×C7C2×D413yes—
164C22×C4×C7C2413yes—
165C7×M16—212no—
166C2×C7×C8—212no—
167C2×C7×C8C2412yes—
168C2×C7×C8C2416yes—
169C7×C4⋊C4—26no—
170C7×C4⋊C4—23no—
171C7×C4⋊C4—23no—
172C7×C4⋊C4—23no—
173C7×C4⋊C4—26no—
174C7×C4⋊C4—26no—
175C7×C4⋊C4—23no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—26no—
178C7×C4⋊C4—26no—
179C7×C4⋊C4C2×D413yes—
180C7×C4⋊C4C2×D413yes—
181C7×C4⋊C4C2416yes—
182C7×C4⋊C4C2412yes—
183C7×C4⋊C4C2413yes—
184C7×C4⋊C4C2411yescharacteristic
185C7×C22⋊C4—23no—
186C7×C22⋊C4—26no—
187C7×C22⋊C4—21no—
188C7×C22⋊C4—23no—
189C7×C22⋊C4C2×D416yes—
190C42×C7—23no—
191C42×C7—23no—
192C42×C7—26no—
193C42×C7—23no—
194C42×C7—26no—
195C42×C7C2×D413yes—
196C42×C7C2×D413yes—
197C42×C7C2411yescharacteristic
198C42×C7C2413yes—
199C2×2−1+4C22×C711yescharacteristic
200C2×#131C22×C712yes—
201Q82—21no—
202#557—21no—
203#554—23no—
204D4×Q8—23no—
205#544C22×C716yes—
206#533C22×C713yes—
207C2×C4⋊Q8C22×C713yes—
208#520—23no—
209#520C22×C713yes—
210#520C22×C712yes—
211#520C22×C711yescharacteristic
212#519—21no—
213#519C22×C713yes—
214C4×#41—23no—
215C4×#41C22×C713yes—
216C4×#41C22×C711yescharacteristic
217C2×C4×Q8—21no—
218C2×C4×Q8C22×C713yes—
219#479C22×C716yes—
220#479C22×C7112yes—
221#478C22×C716yes—
222#477C22×C713yes—
223#477C22×C716yes—
224#476C22×C711yescharacteristic
225#476C22×C716yes—
226#475C22×C713yes—
227#475C22×C716yes—
228#475C22×C7112yes—
229#474C22×C713yes—
230#428C22×C716yes—
231#428C22×C716yes—
232#427C22×C716yes—
233C2×#106C22×C716yes—
234#424C22×C716yes—
235#423C22×C713yes—
236C2×C4⋊C8C22×C713yes—
237#419C22×C716yes—
238#417C22×C716yes—
239#416C22×C716yes—
240C2×#103C22×C716yes—
241C23×C7—21no—
242C23×C7C22×D413yes—
243C7×Q8—212no—
244C7×Q8—23no—
245C7×Q8—42no—
246C7×Q8—46no—
247C7×Q8—86no—
248C7×Q8C22⋊Q814yes—
249C7×Q8C22×Q816yes—
250C7×D4—212no—
251C7×D4—21no—
252C7×D4C22⋊Q8112yes—
253C7×D4C22×Q812yes—
254C2×C4×C7—212no—
255C2×C4×C7—23no—
256C2×C4×C7—212no—
257C2×C4×C7—26no—
258C2×C4×C7—21no—
259C2×C4×C7—23no—
260C2×C4×C7—22no—
261C2×C4×C7—26no—
262C2×C4×C7—43no—
263C2×C4×C7—42no—
264C2×C4×C7—83no—
265C2×C4×C7—83no—
266C2×C4×C7—86no—
267C2×C4×C7C22×D413yes—
268C2×C4×C7C22×D413yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D413yes—
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×Q816yes—
273C2×C4×C7C2×#4116yes—
274C2×C4×C7C2511yescharacteristic
275C7×C8—22no—
276C7×C8—26no—
2772−1+4—24no—
2782−1+4C7×Q818yescomplemented
279C2×#41—23no—
280C2×#41C23×C713yes—
281C2×#41C23×C711yescharacteristic
282C22×Q8—21no—
283C22×Q8—23no—
284C22×Q8C23×C713yes—
285#131—28no—
286C2×M16C23×C716yes—
287C22×C8C23×C716yes—
288C4⋊Q8—26no—
289C4⋊Q8—23no—
290C4⋊Q8—23no—
291C4⋊Q8C23×C713yes—
292C4⋊Q8C23×C716yes—
293C4⋊Q8C23×C711yescharacteristic
294#125—26no—
295#125C23×C716yes—
296C22⋊Q8—23no—
297C22⋊Q8—23no—
298C22⋊Q8—212no—
299C4×Q8—21no—
300C4×Q8—22no—
301C4×Q8—23no—
302C4×Q8—23no—
303C4×Q8—23no—
304C4×Q8—23no—
305C4×Q8—23no—
306C4×Q8—23no—
307C4×Q8C7×D411yescharacteristic
308C4×Q8C7×D411yescharacteristic
309C4×Q8C23×C713yes—
310C4×Q8C23×C716yes—
311C4×Q8C23×C713yes—
312C4×D4—23no—
313C4×D4—26no—
314C4×D4—23no—
315C4×D4—23no—
316C4×D4—26no—
317C4×D4C7×D413yes—
318C4×D4C7×D413yes—
319C4×D4C23×C711yescharacteristic
320C4×D4C23×C713yes—
321#117—23no—
322#117—26no—
323#117—23no—
324#117C23×C713yes—
325#117C23×C716yes—
326#117C23×C713yes—
327C2×C4⋊C4—23no—
328C2×C4⋊C4—23no—
329C2×C4⋊C4—23no—
330C2×C4⋊C4C23×C713yes—
331C2×C4⋊C4C23×C716yes—
332C2×C42—23no—
333C2×C42—23no—
334C2×C42C23×C713yes—
335#107C23×C716yes—
336#107C23×C716yes—
337#106C23×C716yes—
338#106C23×C7112yes—
339#106C23×C712yes—
340C4⋊C8C23×C711yescharacteristic
341C4⋊C8C23×C713yes—
342C4⋊C8C23×C716yes—
343C4⋊C8C23×C716yes—
344#103C23×C716yes—
345#103C23×C7112yes—
346#103C23×C716yes—
347#102C23×C712yes—
348#102C23×C716yes—
349C22×C7—23no—
350C22×C7—23no—
351C22×C7—22no—
352C22×C7—41no—
353C22×C7—41no—
354C22×C7C2×C22⋊Q816yes—
355C22×C7C23×D411yescharacteristic
356C4×C7—21no—
357C4×C7—23no—
358C4×C7—26no—
359C4×C7—23no—
360C4×C7—43no—
361C4×C7—42no—
362C4×C7—41no—
363C4×C7—46no—
364C4×C7C2×C22⋊Q816yes—
365C4×C7#57712yes—
366C4×C7C23×Q811yescharacteristic
367C4×C7C22×#4111yescharacteristic
368#41—224no—
369#41—26no—
370#41C2×C7×Q814yes—
371#41C2×C7×Q8112yescomplemented
372#41C7×#4114yes—
373C2×Q8—28no—
374C2×Q8—23no—
375C2×Q8—21no—
376C2×Q8—23no—
377C2×Q8—26no—
378C2×Q8—26no—
379C2×Q8—86no—
380C2×Q8C2×C7×D411yescharacteristic
381C2×Q8C2×C7×Q8112yescomplemented
382C2×Q8C24×C713yes—
383C2×D4—23no—
384C2×D4C2×C7×D413yes—
385C2×D4C24×C711yescharacteristic
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—26no—
390C22×C4—21no—
391C22×C4—23no—
392C22×C4C2×C7×D413yes—
393C22×C4C2×C7×D413yes—
394C22×C4C24×C713yes—
395M16—212no—
396C2×C8—212no—
397C2×C8C24×C712yes—
398C2×C8C24×C716yes—
399C4⋊C4—26no—
400C4⋊C4—23no—
401C4⋊C4—23no—
402C4⋊C4—23no—
403C4⋊C4—26no—
404C4⋊C4—26no—
405C4⋊C4—23no—
406C4⋊C4—21no—
407C4⋊C4—26no—
408C4⋊C4—26no—
409C4⋊C4C2×C7×D413yes—
410C4⋊C4C2×C7×D413yes—
411C4⋊C4C24×C716yes—
412C4⋊C4C24×C712yes—
413C4⋊C4C24×C713yes—
414C4⋊C4C24×C711yescharacteristic
415C22⋊C4—23no—
416C22⋊C4—26no—
417C22⋊C4—21no—
418C22⋊C4—23no—
419C22⋊C4C2×C7×D416yes—
420C42—23no—
421C42—23no—
422C42—26no—
423C42—23no—
424C42—26no—
425C42C2×C7×D413yes—
426C42C2×C7×D413yes—
427C42C24×C711yescharacteristic
428C42C24×C713yes—
429C2×C7—26no—
430C2×C7—42no—
431C2×C7C22×C22⋊Q811yescharacteristic
432C2×C7C2×#57711yescharacteristic
433C2×C7#358611yescharacteristic
434C23—21no—
435C23C22×C7×D413yes—
436Q8—23no—
437Q8—212no—
438Q8—42no—
439Q8—46no—
440Q8—86no—
441Q8C7×C22⋊Q814yes—
442Q8C22×C7×Q816yes—
443D4—212no—
444D4—21no—
445D4C7×C22⋊Q8112yes—
446D4C22×C7×Q812yes—
447C2×C4—23no—
448C2×C4—212no—
449C2×C4—212no—
450C2×C4—26no—
451C2×C4—21no—
452C2×C4—23no—
453C2×C4—22no—
454C2×C4—26no—
455C2×C4—43no—
456C2×C4—42no—
457C2×C4—83no—
458C2×C4—83no—
459C2×C4—86no—
460C2×C4C22×C7×D413yes—
461C2×C4C22×C7×D413yes—
462C2×C4C22×C7×D411yescharacteristic
463C2×C4C22×C7×D413yes—
464C2×C4C22×C7×D411yescharacteristic
465C2×C4C22×C7×Q816yes—
466C2×C4C2×C7×#4116yes—
467C2×C4C25×C711yescharacteristic
468C8—22no—
469C8—26no—
470C7#6049711yescharacteristic, direct product factor, Sylow
471C22—23no—
472C22—23no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22C2×C7×C22⋊Q816yes—
477C22C23×C7×D411yescharacteristic
478C4—23no—
479C4—21no—
480C4—23no—
481C4—26no—
482C4—41no—
483C4—46no—
484C4—43no—
485C4—42no—
486C4C2×C7×C22⋊Q816yes—
487C4C7×#57712yes—
488C4C23×C7×Q811yescharacteristic
489C4C22×C7×#4111yescharacteristic
490C2—26no—
491C2—42no—
492C2C22×C7×C22⋊Q811yescharacteristic
493C2C2×C7×#57711yescharacteristic
494C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×2−1+4C22
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
1. C7×#3503C2semidirect
3. C7×#3084C2semidirect
4. C7×#3082C2semidirect
5. C7×#3079C2semidirect
6. C7×#3078C2semidirect
7. C7×#3076C2semidirect
8. C2×C7×#475C2semidirect
9. C7×#2912C2semidirect
10. C7×#2906C2semidirect
54. #60497C7direct (N × H)
56. C7×2−1+4Q8semidirect
127. #3503C2×C7semidirect
129. #3084C2×C7semidirect
130. #3082C2×C7semidirect
131. #3079C2×C7semidirect
132. #3078C2×C7semidirect
133. #3076C2×C7semidirect
134. C2×#475C2×C7semidirect
135. #2912C2×C7semidirect
136. #2906C2×C7semidirect
141. C7×#41C2×Q8semidirect
151. C2×C7×Q8C2×Q8semidirect
278. 2−1+4C7×Q8semidirect
371. #41C2×C7×Q8semidirect
381. C2×Q8C2×C7×Q8semidirect
470. C7#60497direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35032
C4×C7×2−1+42
C7×#30842
C7×#30822
C7×#30792
C7×#30782
C7×#30762
C2×C7×#4752
C7×#29122
C7×#29062
C7×#28982
#604977