Finite Group Databasebeta

← C7×#60492 #13237708

Subgroups — C7×#60492 #13237708

Order 1792.

2038 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×#3458C211yesmaximal, characteristic, complemented
2C2×C7×#520C211yesmaximal, characteristic
3C7×#3084C218yesmaximal, complemented
4C7×#3083C218yesmaximal, complemented
5C7×#3078C214yesmaximal, complemented
6C7×#3077C214yesmaximal, complemented
7C2×C7×#428C212yesmaximal, complemented
8C7×#2905C211yesmaximal, characteristic, complemented
9C7×#2896C212yesmaximal
10C22×C7×#41C2211yescharacteristic
11C22×C7×M16C2212yes—
12C7×#544—21no—
13C7×#544—21no—
14C7×#544C2214yescomplemented
15C7×#544C2214yescomplemented
16C7×#543—22no—
17C7×#533—24no—
18C7×#533C2214yes—
19C7×#520—24no—
20C7×#520C2218yes—
21C2×C4×C7×Q8C2212yes—
22C2×C4×C7×D4—22no—
23C2×C4×C7×D4C2212yes—
24C2×C7×#117—21no—
25C2×C7×#117C2212yes—
26C2×C7×#117C2211yescharacteristic
27C22×C7×C4⋊C4—21no—
28C22×C7×C4⋊C4C2211yescharacteristic
29C7×#479C22116yescomplemented
30C7×#478C22116yescomplemented
31C7×#477C2218yescomplemented
32C7×#476C2218yescomplemented
33C7×#475C2218yescomplemented
34C7×#474C2218yescomplemented
35C7×#428C2218yes—
36C7×#428C2218yescomplemented
37C2×C7×#107C2214yescomplemented
38C2×C7×#106C2214yescomplemented
39C7×#424C2218yescomplemented
40C7×#423C2214yes—
41C7×#419C2214yes—
42C7×#418C2214yes—
43C7×#417C2214yes—
44C7×#417C2214yes—
45C7×#416C2218yes—
46#60492C711yesmaximal, characteristic, direct product factor, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—24no—
49C2×C7×#41Q818yescomplemented
50C2×C7×#41C2314yes—
51C22×C7×Q8C2311yescharacteristic
52C22×C7×D4—21no—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7—22no—
57C23×C4×C7C2311yescharacteristic
58C2×C7×M16—28no—
59C2×C7×M16C2314yes—
60C2×C7×M16C2314yes—
61C22×C7×C8C2314yes—
62C7×C4⋊Q8—24no—
63C7×C4⋊Q8—24no—
64C7×C4⋊Q8C2314yes—
65C7×C4⋊Q8C2314yes—
66C7×#125—24no—
67C7×#125—24no—
68C7×#125C2318yes—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8—24no—
71C7×C22⋊Q8—28no—
72C7×C22⋊Q8—28no—
73C4×C7×Q8—24no—
74C4×C7×Q8—24no—
75C4×C7×Q8—24no—
76C4×C7×Q8C2318yes—
77C4×C7×D4—216no—
78C4×C7×D4—24no—
79C4×C7×D4C2318yes—
80C7×#117—22no—
81C7×#117—22no—
82C7×#117—28no—
83C7×#117D414yes—
84C7×#117C2314yes—
85C7×#117C2314yes—
86C7×#117C2314yes—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—28no—
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2314yes—
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C4⋊C4C2312yes—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4D412yes—
100C2×C42×C7—22no—
101C2×C42×C7—22no—
102C2×C42×C7C2312yes—
103C7×#107C23116yes—
104C7×#106C23116yes—
105C7×C4⋊C8C2318yes—
106C7×C4⋊C8C2318yes—
107C7×#103C2318yes—
108C7×#103C2318yes—
109C7×#102C2318yes—
110C7×#102C2318yes—
111#3458C2×C711yescharacteristic, complemented
112C2×#520C2×C711yescharacteristic
113#3084C2×C718yescomplemented
114#3083C2×C718yescomplemented
115#3078C2×C714yescomplemented
116#3077C2×C714yescomplemented
117C2×#428C2×C712yescomplemented
118#2905C2×C711yescharacteristic, complemented
119#2896C2×C712yes—
120C24×C7—21no—
121C24×C7C2×D411yescharacteristic
122C7×#41—216no—
123C7×#41—28no—
124C7×#41C2×Q8116yes—
125C2×C7×Q8—21no—
126C2×C7×Q8—24no—
127C2×C7×Q8—24no—
128C2×C7×Q8—22no—
129C2×C7×Q8C2×Q814yes—
130C2×C7×Q8C2412yes—
131C2×C7×D4—28no—
132C2×C7×D4—24no—
133C2×C7×D4—21no—
134C2×C7×D4—22no—
135C2×C7×D4C2×Q814yes—
136C2×C7×D4C2412yes—
137C22×C4×C7—21no—
138C22×C4×C7—28no—
139C22×C4×C7—24no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—24no—
144C22×C4×C7—28no—
145C22×C4×C7—84no—
146C22×C4×C7—82no—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×Q814yes—
152C22×C4×C7#4114yes—
153C22×C4×C7C2411yescharacteristic
154C22×C4×C7C2412yes—
155C7×M16—28no—
156C7×M16—28no—
157C2×C7×C8—28no—
158C2×C7×C8C2418yes—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—28no—
168C7×C4⋊C4C2×D414yes—
169C7×C4⋊C4C2×D414yes—
170C7×C4⋊C4C2414yes—
171C7×C4⋊C4C2414yes—
172C7×C4⋊C4C2414yes—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—28no—
176C7×C22⋊C4C2×D418yes—
177C42×C7—28no—
178C42×C7—24no—
179C42×C7—24no—
180C42×C7C2×D414yes—
181C42×C7C2414yes—
182C22×#41C22×C711yescharacteristic
183C22×M16C22×C712yes—
184#544—21no—
185#544—21no—
186#544C22×C714yescomplemented
187#544C22×C714yescomplemented
188#543—22no—
189#533—24no—
190#533C22×C714yes—
191#520—24no—
192#520C22×C718yes—
193C2×C4×Q8C22×C712yes—
194C2×C4×D4—22no—
195C2×C4×D4C22×C712yes—
196C2×#117—21no—
197C2×#117C22×C712yes—
198C2×#117C22×C711yescharacteristic
199C22×C4⋊C4—21no—
200C22×C4⋊C4C22×C711yescharacteristic
201#479C22×C7116yescomplemented
202#478C22×C7116yescomplemented
203#477C22×C718yescomplemented
204#476C22×C718yescomplemented
205#475C22×C718yescomplemented
206#474C22×C718yescomplemented
207#428C22×C718yes—
208#428C22×C718yescomplemented
209C2×#107C22×C714yescomplemented
210C2×#106C22×C714yescomplemented
211#424C22×C718yescomplemented
212#423C22×C714yes—
213#419C22×C714yes—
214#418C22×C714yes—
215#417C22×C714yes—
216#417C22×C714yes—
217#416C22×C718yes—
218C23×C7—24no—
219C23×C7—22no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7C22⋊Q814yes—
225C23×C7C22×D411yescharacteristic
226C23×C7C22×D412yes—
227C7×Q8—24no—
228C7×Q8—22no—
229C7×Q8—84no—
230C7×Q8—84no—
231C7×Q8C22×Q814yes—
232C7×D4—216no—
233C7×D4—24no—
234C7×D4—22no—
235C7×D4C22×Q814yes—
236C2×C4×C7—24no—
237C2×C4×C7—22no—
238C2×C4×C7—216no—
239C2×C4×C7—24no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—44no—
245C2×C4×C7—88no—
246C2×C4×C7—82no—
247C2×C4×C7—82no—
248C2×C4×C7—84no—
249C2×C4×C7C22⋊Q814yes—
250C2×C4×C7C22⋊Q818yes—
251C2×C4×C7C22×D414yes—
252C2×C4×C7C22×D414yes—
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C22×D412yes—
255C2×C4×C7C22×Q812yes—
256C2×C4×C7C22×Q814yes—
257C2×C4×C7C2×#4114yes—
258C2×C4×C7C2×#4112yes—
259C2×C4×C7C2511yescharacteristic
260C7×C8—28no—
261C2×#41—22no—
262C2×#41—24no—
263C2×#41C7×Q818yescomplemented
264C2×#41C23×C714yes—
265C22×Q8C23×C711yescharacteristic
266C22×D4—21no—
267C22×D4C23×C711yescharacteristic
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4C23×C711yescharacteristic
272C2×M16—28no—
273C2×M16C23×C714yes—
274C2×M16C23×C714yes—
275C22×C8C23×C714yes—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8C23×C714yes—
279C4⋊Q8C23×C714yes—
280#125—24no—
281#125—24no—
282#125C23×C718yes—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8C23×C718yes—
291C4×D4—216no—
292C4×D4—24no—
293C4×D4C23×C718yes—
294#117—22no—
295#117—22no—
296#117—28no—
297#117C7×D414yes—
298#117C23×C714yes—
299#117C23×C714yes—
300#117C23×C714yes—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4C7×D411yescharacteristic
306C2×C4⋊C4C7×D411yescharacteristic
307C2×C4⋊C4C23×C712yes—
308C2×C4⋊C4C23×C714yes—
309C2×C4⋊C4C23×C712yes—
310C2×C4⋊C4C23×C712yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4C7×D412yes—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C23×C712yes—
317#107C23×C7116yes—
318#106C23×C7116yes—
319C4⋊C8C23×C718yes—
320C4⋊C8C23×C718yes—
321#103C23×C718yes—
322#103C23×C718yes—
323#102C23×C718yes—
324#102C23×C718yes—
325C22×C7—21no—
326C22×C7—22no—
327C22×C7—21no—
328C22×C7—22no—
329C22×C7—24no—
330C22×C7—44no—
331C22×C7—48no—
332C22×C7—42no—
333C22×C7—42no—
334C22×C7C2×C22⋊Q812yes—
335C22×C7C2×C22⋊Q814yes—
336C22×C7C23×D411yescharacteristic
337C4×C7—24no—
338C4×C7—24no—
339C4×C7—24no—
340C4×C7—44no—
341C4×C7—44no—
342C4×C7—44no—
343C4×C7C2×C22⋊Q814yes—
344C4×C7C2×C22⋊Q812yes—
345C4×C7C23×Q811yescharacteristic
346C4×C7C22×#4111yescharacteristic
347C24—21no—
348C24C2×C7×D411yescharacteristic
349#41—216no—
350#41—28no—
351#41C2×C7×Q8116yes—
352C2×Q8—21no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8C2×C7×Q814yes—
357C2×Q8C24×C712yes—
358C2×D4—28no—
359C2×D4—24no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4C2×C7×Q814yes—
363C2×D4C24×C712yes—
364C22×C4—21no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—84no—
373C22×C4—82no—
374C22×C4C2×C7×D412yes—
375C22×C4C2×C7×D412yes—
376C22×C4C2×C7×D412yes—
377C22×C4C2×C7×D411yescharacteristic
378C22×C4C2×C7×Q814yes—
379C22×C4C7×#4114yes—
380C22×C4C24×C711yescharacteristic
381C22×C4C24×C712yes—
382M16—28no—
383M16—28no—
384C2×C8—28no—
385C2×C8C24×C718yes—
386C4⋊C4—24no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4C2×C7×D414yes—
396C4⋊C4C2×C7×D414yes—
397C4⋊C4C24×C714yes—
398C4⋊C4C24×C714yes—
399C4⋊C4C24×C714yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—28no—
403C22⋊C4C2×C7×D418yes—
404C42—28no—
405C42—24no—
406C42—24no—
407C42C2×C7×D414yes—
408C42C24×C714yes—
409C2×C7—22no—
410C2×C7—24no—
411C2×C7—44no—
412C2×C7C22×C22⋊Q811yescharacteristic
413C2×C7#358511yescharacteristic
414C2×C7#358611yescharacteristic
415C23—22no—
416C23—24no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23C7×C22⋊Q814yes—
422C23C22×C7×D411yescharacteristic
423C23C22×C7×D412yes—
424Q8—22no—
425Q8—24no—
426Q8—84no—
427Q8—84no—
428Q8C22×C7×Q814yes—
429D4—216no—
430D4—22no—
431D4—24no—
432D4C22×C7×Q814yes—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—216no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—88no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4C7×C22⋊Q814yes—
447C2×C4C7×C22⋊Q818yes—
448C2×C4C22×C7×D414yes—
449C2×C4C22×C7×D414yes—
450C2×C4C22×C7×D411yescharacteristic
451C2×C4C22×C7×D412yes—
452C2×C4C22×C7×Q812yes—
453C2×C4C22×C7×Q814yes—
454C2×C4C2×C7×#4114yes—
455C2×C4C2×C7×#4112yes—
456C2×C4C25×C711yescharacteristic
457C8—28no—
458C7#6049211yescharacteristic, direct product factor, Sylow
459C22—21no—
460C22—22no—
461C22—21no—
462C22—22no—
463C22—24no—
464C22—44no—
465C22—48no—
466C22—42no—
467C22—42no—
468C22C2×C7×C22⋊Q812yes—
469C22C2×C7×C22⋊Q814yes—
470C22C23×C7×D411yescharacteristic
471C4—24no—
472C4—24no—
473C4—24no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4C2×C7×C22⋊Q814yes—
478C4C2×C7×C22⋊Q812yes—
479C4C23×C7×Q811yescharacteristic
480C4C22×C7×#4111yescharacteristic
481C2—22no—
482C2—24no—
483C2—44no—
484C2C22×C7×C22⋊Q811yescharacteristic
485C2C7×#358511yescharacteristic
486C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3458C2semidirect
3. C7×#3084C2semidirect
4. C7×#3083C2semidirect
5. C7×#3078C2semidirect
6. C7×#3077C2semidirect
7. C2×C7×#428C2semidirect
8. C7×#2905C2semidirect
14. C7×#544C22semidirect
15. C7×#544C22semidirect
29. C7×#479C22semidirect
30. C7×#478C22semidirect
31. C7×#477C22semidirect
32. C7×#476C22semidirect
33. C7×#475C22semidirect
34. C7×#474C22semidirect
36. C7×#428C22semidirect
37. C2×C7×#107C22semidirect
38. C2×C7×#106C22semidirect
39. C7×#424C22semidirect
46. #60492C7direct (N × H)
49. C2×C7×#41Q8semidirect
111. #3458C2×C7semidirect
113. #3084C2×C7semidirect
114. #3083C2×C7semidirect
115. #3078C2×C7semidirect
116. #3077C2×C7semidirect
117. C2×#428C2×C7semidirect
118. #2905C2×C7semidirect
186. #544C22×C7semidirect
187. #544C22×C7semidirect
201. #479C22×C7semidirect
202. #478C22×C7semidirect
203. #477C22×C7semidirect
204. #476C22×C7semidirect
205. #475C22×C7semidirect
206. #474C22×C7semidirect
208. #428C22×C7semidirect
209. C2×#107C22×C7semidirect
210. C2×#106C22×C7semidirect
211. #424C22×C7semidirect
263. C2×#41C7×Q8semidirect
458. C7#60492direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34582
C2×C7×#5202
C7×#30842
C7×#30832
C7×#30782
C7×#30772
C2×C7×#4282
C7×#29052
C7×#28962
#604927