Finite Group Databasebeta

← C7×#10245 #13187461

Subgroups — C7×#10245 #13187461

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3227C211yesmaximal, characteristic
2C7×#3037C211yesmaximal, characteristic
3C7×#2567C211yesmaximal, characteristic, complemented
4C7×#1549C211yesmaximal, characteristic, complemented
5C7×#1502C211yesmaximal, characteristic
6C7×#1479C211yesmaximal, characteristic, complemented
7C7×#1303C211yesmaximal, characteristic, complemented
8C2×C7×#137—21no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#498—21no—
13C7×#486—21no—
14C7×#481—22no—
15C7×#476—22no—
16C7×#464—22no—
17C7×#462—21no—
18C7×#460—21no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#423C2211yescharacteristic
22C7×#417—21no—
23C7×#395—22no—
24C7×#391—21no—
25C7×#389—22no—
26C7×#384—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C7×#332—22no—
29C7×#327—22no—
30C7×Q8⋊C8—22no—
31#10245C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C22×C7×Q8—21no—
34C22×C7×Q8D411yescharacteristic
35C7×#137—42no—
36C7×#137—42no—
37C2×C7×Q16—22no—
38C2×C7×SD16—22no—
39C2×C7×M16—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8D411yescharacteristic
45C7×#127D411yescharacteristic
46C7×#124—21no—
47C7×#121—21no—
48C7×#121—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C42×C7—21no—
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64C7×#106—24no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8D412yes—
68C7×C4⋊C8D412yes—
69C7×#103—22no—
70C7×#102—24no—
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#3227C2×C711yescharacteristic
79#3037C2×C711yescharacteristic
80#2567C2×C711yescharacteristic, complemented
81#1549C2×C711yescharacteristic, complemented
82#1502C2×C711yescharacteristic
83#1479C2×C711yescharacteristic, complemented
84#1303C2×C711yescharacteristic, complemented
85C7×#41—41no—
86C7×#41—41no—
87C7×#41—41no—
88C7×#41—41no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—41no—
96C2×C7×Q8—41no—
97C2×C7×Q8—82no—
98C2×C7×Q8SD1612yes—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—41no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—41no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C7×Q16—44no—
112C7×SD16—44no—
113C7×M16—41no—
114C7×M16—41no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—41no—
118C7×C4⋊C4—24no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—84no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7—41no—
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C42×C7#4111yescharacteristic
140C2×#137—21no—
141#532C22×C711yescharacteristic
142C2×C4⋊Q8C22×C711yescharacteristic
143C2×C4×Q8C22×C711yescharacteristic
144#498—21no—
145#486—21no—
146#481—22no—
147#476—22no—
148#464—22no—
149#462—21no—
150#460—21no—
151#431C22×C711yescharacteristic
152#431C22×C711yescharacteristic
153#423C22×C711yescharacteristic
154#417—21no—
155#395—22no—
156#391—21no—
157#389—22no—
158#384—21no—
159C4×C4⋊C4C22×C711yescharacteristic
160#332—22no—
161#327—22no—
162Q8⋊C8—22no—
163C23×C7—41no—
164C23×C7#12011yescharacteristic
165C7×Q8—22no—
166C7×Q8—22no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—41no—
170C7×Q8—41no—
171C7×Q8—82no—
172C7×Q8—82no—
173C7×D4—41no—
174C7×D4—41no—
175C7×D4—42no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—81no—
179C2×C4×C7—22no—
180C2×C4×C7—24no—
181C2×C4×C7—22no—
182C2×C4×C7—41no—
183C2×C4×C7—42no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—44no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12711yescharacteristic
200C2×C4×C7C2×SD1611yescharacteristic
201C2×C4×C7#13511yescharacteristic
202C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C7#13711yescharacteristic
204C7×C8—42no—
205C7×C8—42no—
206C7×C8—81no—
207C2×#41—21no—
208C22×Q8—21no—
209C22×Q8C7×D411yescharacteristic
210#137—42no—
211#137—42no—
212C2×Q16—22no—
213C2×SD16—22no—
214C2×M16—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8C7×D411yescharacteristic
220#127C7×D411yescharacteristic
221#124—21no—
222#121—21no—
223#121—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×D4—21no—
229C4×D4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4C7×D411yescharacteristic
235C2×C4⋊C4C7×D412yes—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C23×C711yescharacteristic
239#106—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C7×D412yes—
243C4⋊C8C7×D412yes—
244#103—22no—
245#102—22no—
246#102—24no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—24no—
252#95C7×D412yes—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7#39211yescharacteristic
259C22×C7#44811yescharacteristic
260C22×C7#44911yescharacteristic
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—44no—
265C4×C7—42no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7—82no—
269C4×C7#45011yescharacteristic
270C4×C7#45211yescharacteristic
271C4×C7#45612yes—
272C4×C7#46511yescharacteristic
273C4×C7#46811yescharacteristic
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—82no—
287C2×Q8C7×SD1612yes—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4C2×C7×D411yescharacteristic
298C22×C4C2×C7×D411yescharacteristic
299C22×C4C2×C7×D411yescharacteristic
300Q16—44no—
301SD16—44no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C4⋊C4—24no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—84no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42C2×C7×D411yescharacteristic
326C42C2×C7×D411yescharacteristic
327C42C2×C7×D411yescharacteristic
328C42C7×#4111yescharacteristic
329C2×C7—21no—
330C2×C7—21no—
331C2×C7—161no—
332C2×C7#200011yescharacteristic
333C2×C7#200511yescharacteristic
334C2×C7#200811yescharacteristic
335C23—41no—
336C23C7×#12011yescharacteristic
337Q8—22no—
338Q8—22no—
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343Q8—82no—
344Q8—82no—
345D4—41no—
346D4—41no—
347D4—42no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4C7×#12011yescharacteristic
367C2×C4C7×#12011yescharacteristic
368C2×C4C7×#12111yescharacteristic
369C2×C4C7×#12111yescharacteristic
370C2×C4C7×#12111yescharacteristic
371C2×C4C7×#12711yescharacteristic
372C2×C4C2×C7×SD1611yescharacteristic
373C2×C4C7×#13511yescharacteristic
374C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
375C2×C4C7×#13711yescharacteristic
376C8—42no—
377C8—42no—
378C8—81no—
379C7#1024511yescharacteristic, direct product factor, Sylow
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C7×#39211yescharacteristic
386C22C7×#44811yescharacteristic
387C22C7×#44911yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—44no—
392C4—42no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4C7×#45011yescharacteristic
397C4C7×#45211yescharacteristic
398C4C7×#45612yes—
399C4C7×#46511yescharacteristic
400C4C7×#46811yescharacteristic
401C2—21no—
402C2—21no—
403C2—161no—
404C2C7×#200011yescharacteristic
405C2C7×#200511yescharacteristic
406C2C7×#200811yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
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
3. C7×#2567C2semidirect
4. C7×#1549C2semidirect
6. C7×#1479C2semidirect
7. C7×#1303C2semidirect
31. #10245C7direct (N × H)
80. #2567C2×C7semidirect
81. #1549C2×C7semidirect
83. #1479C2×C7semidirect
84. #1303C2×C7semidirect
379. C7#10245direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32272
C7×#30372
C7×#25672
C7×#15492
C7×#15022
C7×#14792
C7×#13032
#102457