Finite Group Databasebeta

← C7×#10247 #13187463

Subgroups — C7×#10247 #13187463

Order 1792.

556 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3229C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic
3C7×#2547C211yesmaximal, characteristic
4C7×#1546C211yesmaximal, characteristic, complemented
5C7×#1503C211yesmaximal, characteristic, complemented
6C7×#1478C211yesmaximal, characteristic
7C7×#1303C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#124—21no—
11C2×C4×C7×D4C2211yescharacteristic
12C7×#498—21no—
13C7×#487—21no—
14C7×#483—22no—
15C7×#478—22no—
16C7×#476—21no—
17C7×#475—22no—
18C7×#474—21no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#428—21no—
22C7×#423C2211yescharacteristic
23C7×#418—21no—
24C7×#397—22no—
25C7×#391—21no—
26C7×#380—22no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C7×#332—22no—
29C7×#328—22no—
30C7×D4⋊C8—22no—
31#10247C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8—21no—
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C2×C7×M16—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×#125—21no—
43C7×#124—42no—
44C7×#124—42no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4D412yes—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—22no—
63C7×#106—24no—
64C7×#106—22no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8D412yes—
68C7×C4⋊C8D412yes—
69C7×#103—24no—
70C7×#102—22no—
71C7×#102—22no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C4×C7×C8—21no—
75C4×C7×C8—21no—
76C7×#95—24no—
77C7×#95D412yes—
78#3229C2×C711yescharacteristic, complemented
79#3077C2×C711yescharacteristic
80#2547C2×C711yescharacteristic
81#1546C2×C711yescharacteristic, complemented
82#1503C2×C711yescharacteristic, complemented
83#1478C2×C711yescharacteristic
84#1303C2×C711yescharacteristic, complemented
85C24×C7—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4SD1612yes—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—44no—
104C22×C4×C7—41no—
105C22×C4×C7—41no—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C7×M16—41no—
110C7×M16—41no—
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—41no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—42no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C22⋊C4—48no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C42×C7—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7#4111yescharacteristic
138#533C22×C711yescharacteristic
139C2×C4⋊Q8C22×C711yescharacteristic
140C2×#124—21no—
141C2×C4×D4C22×C711yescharacteristic
142#498—21no—
143#487—21no—
144#483—22no—
145#478—22no—
146#476—21no—
147#475—22no—
148#474—21no—
149#431C22×C711yescharacteristic
150#431C22×C711yescharacteristic
151#428—21no—
152#423C22×C711yescharacteristic
153#418—21no—
154#397—22no—
155#391—21no—
156#380—22no—
157C4×C4⋊C4C22×C711yescharacteristic
158#332—22no—
159#328—22no—
160D4⋊C8—22no—
161C23×C7—22no—
162C23×C7—42no—
163C23×C7—42no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7#12011yescharacteristic
167C7×Q8—42no—
168C7×Q8—81no—
169C7×Q8—81no—
170C7×Q8—81no—
171C7×Q8—81no—
172C7×Q8—81no—
173C7×D4—22no—
174C7×D4—22no—
175C7×D4—41no—
176C7×D4—41no—
177C2×C4×C7—22no—
178C2×C4×C7—41no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—82no—
190C2×C4×C7—82no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12711yescharacteristic
197C2×C4×C7C2×SD1611yescharacteristic
198C2×C4×C7#13511yescharacteristic
199C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
200C2×C4×C7#13711yescharacteristic
201C7×C8—42no—
202C7×C8—42no—
203C7×C8—81no—
204C22×Q8—21no—
205C22×D4C7×D411yescharacteristic
206C23×C4—21no—
207C2×M16—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8C7×D411yescharacteristic
213C4⋊Q8C7×D411yescharacteristic
214#125—21no—
215#124—42no—
216#124—42no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225#117—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C7×D411yescharacteristic
229C2×C4⋊C4C7×D412yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42C23×C711yescharacteristic
234#107—22no—
235#106—22no—
236#106—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8C7×D412yes—
240C4⋊C8C7×D412yes—
241#103—24no—
242#102—22no—
243#102—22no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248#95—24no—
249#95C7×D412yes—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—44no—
253C22×C7—42no—
254C22×C7—82no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7#39211yescharacteristic
258C22×C7#45011yescharacteristic
259C22×C7#45211yescharacteristic
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—82no—
264C4×C7—82no—
265C4×C7—81no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7#44811yescharacteristic
269C4×C7#44911yescharacteristic
270C4×C7#45412yes—
271C4×C7#46511yescharacteristic
272C4×C7#46811yescharacteristic
273C24—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4C7×SD1612yes—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—44no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×C7×D411yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297M16—41no—
298M16—41no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—48no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42C2×C7×D411yescharacteristic
323C42C2×C7×D411yescharacteristic
324C42C2×C7×D411yescharacteristic
325C42C7×#4111yescharacteristic
326C2×C7—21no—
327C2×C7—21no—
328C2×C7—82no—
329C2×C7#200011yescharacteristic
330C2×C7#200311yescharacteristic
331C2×C7#200911yescharacteristic
332C23—22no—
333C23—42no—
334C23—42no—
335C23—41no—
336C23—41no—
337C23C7×#12011yescharacteristic
338Q8—42no—
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343Q8—81no—
344D4—22no—
345D4—22no—
346D4—41no—
347D4—41no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4C7×#12011yescharacteristic
363C2×C4C7×#12011yescharacteristic
364C2×C4C7×#12111yescharacteristic
365C2×C4C7×#12111yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12711yescharacteristic
368C2×C4C2×C7×SD1611yescharacteristic
369C2×C4C7×#13511yescharacteristic
370C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C7×#13711yescharacteristic
372C8—42no—
373C8—42no—
374C8—81no—
375C7#1024711yescharacteristic, direct product factor, Sylow
376C22—21no—
377C22—21no—
378C22—44no—
379C22—42no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22C7×#39211yescharacteristic
384C22C7×#45011yescharacteristic
385C22C7×#45211yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—82no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4C7×#44811yescharacteristic
395C4C7×#44911yescharacteristic
396C4C7×#45412yes—
397C4C7×#46511yescharacteristic
398C4C7×#46811yescharacteristic
399C2—21no—
400C2—21no—
401C2—82no—
402C2C7×#200011yescharacteristic
403C2C7×#200311yescharacteristic
404C2C7×#200911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
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
1. C7×#3229C2semidirect
4. C7×#1546C2semidirect
5. C7×#1503C2semidirect
7. C7×#1303C2semidirect
31. #10247C7direct (N × H)
78. #3229C2×C7semidirect
81. #1546C2×C7semidirect
82. #1503C2×C7semidirect
84. #1303C2×C7semidirect
375. C7#10247direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32292
C7×#30772
C7×#25472
C7×#15462
C7×#15032
C7×#14782
C7×#13032
#102477