Finite Group Databasebeta

← C7×#10480 #13187696

Subgroups — C7×#10480 #13187696

Order 1792.

474 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
1C2×C7×#476C211yesmaximal, characteristic
2C7×#1990C211yesmaximal, characteristic, complemented
3C7×#1954C211yesmaximal, characteristic
4C7×#1802C211yesmaximal, characteristic
5C2×C7×#334C211yesmaximal, characteristic, complemented
6C2×C7×#333C211yesmaximal, characteristic, complemented
7C7×#1296C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#476—22no—
11C7×#476—22no—
12C2×C7×#106—21no—
13C2×C7×C4⋊C8C2211yescharacteristic
14C2×C7×C4⋊C8C2211yescharacteristic
15C2×C7×C4⋊C8C2211yescharacteristic
16C2×C7×#102—21no—
17C2×C7×#102—21no—
18C2×C7×C22⋊C8—21no—
19C2×C7×C22⋊C8—21no—
20C2×C4×C7×C8C2211yescharacteristic
21C7×#384C2211yescharacteristic
22C7×#340—21no—
23C7×#340—21no—
24C7×#340—21no—
25C7×#334—22no—
26C7×#333—22no—
27#10480C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8—21no—
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C22×C7×C8—21no—
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C22×C7×C8D411yescharacteristic
35C22×C7×C8D411yescharacteristic
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8D412yes—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C4×C7×D4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C42×C7C2311yescharacteristic
52C7×#106—24no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C4⋊C8—21no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8D412yes—
59C7×#102—24no—
60C7×#102—42no—
61C7×C22⋊C8—42no—
62C7×C22⋊C8—42no—
63C4×C7×C8—21no—
64C4×C7×C8—21no—
65C2×#476C2×C711yescharacteristic
66#1990C2×C711yescharacteristic, complemented
67#1954C2×C711yescharacteristic
68#1802C2×C711yescharacteristic
69C2×#334C2×C711yescharacteristic, complemented
70C2×#333C2×C711yescharacteristic, complemented
71#1296C2×C711yescharacteristic, complemented
72C24×C7—21no—
73C2×C7×Q8—22no—
74C2×C7×Q8—41no—
75C2×C7×Q8—81no—
76C2×C7×Q8—81no—
77C2×C7×D4—21no—
78C2×C7×D4—21no—
79C2×C7×D4—21no—
80C2×C7×D4—21no—
81C2×C7×D4—21no—
82C2×C7×D4SD1612yes—
83C22×C4×C7—21no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7—41no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C2×C7×C8—21no—
96C2×C7×C8—21no—
97C2×C7×C8—22no—
98C2×C7×C8—21no—
99C2×C7×C8—21no—
100C2×C7×C8—21no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8SD1612yes—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—81no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C42×C7C2×D411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7#4111yescharacteristic
126C42×C7#4111yescharacteristic
127C2×C4⋊Q8C22×C711yescharacteristic
128C2×C4×D4C22×C711yescharacteristic
129#476—22no—
130#476—22no—
131C2×#106—21no—
132C2×C4⋊C8C22×C711yescharacteristic
133C2×C4⋊C8C22×C711yescharacteristic
134C2×C4⋊C8C22×C711yescharacteristic
135C2×#102—21no—
136C2×#102—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C4×C8C22×C711yescharacteristic
140#384C22×C711yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#334—22no—
145#333—22no—
146C23×C7—22no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7#12111yescharacteristic
154C7×Q8—82no—
155C7×Q8—82no—
156C7×D4—22no—
157C7×D4—22no—
158C7×D4—41no—
159C7×D4—41no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—41no—
163C2×C4×C7—42no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7—81no—
170C2×C4×C7—81no—
171C2×C4×C7—81no—
172C2×C4×C7—81no—
173C2×C4×C7—81no—
174C2×C4×C7—81no—
175C2×C4×C7#12011yescharacteristic
176C2×C4×C7#12111yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12311yescharacteristic
179C2×C4×C7#12311yescharacteristic
180C2×C4×C7#12411yescharacteristic
181C2×C4×C7C2×SD1611yescharacteristic
182C2×C4×C7C2×SD1611yescharacteristic
183C2×C4×C7#13511yescharacteristic
184C2×C4×C7#13511yescharacteristic
185C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C7#13711yescharacteristic
187C2×C4×C7#13711yescharacteristic
188C2×C4×C7#13711yescharacteristic
189C7×C8—22no—
190C7×C8—42no—
191C7×C8—41no—
192C7×C8—81no—
193C7×C8—81no—
194C22×Q8—21no—
195C22×D4C7×D411yescharacteristic
196C23×C4—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8C7×D411yescharacteristic
201C22×C8C7×D411yescharacteristic
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8C7×D412yes—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C7×D411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C42C23×C711yescharacteristic
218#106—24no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8C7×D412yes—
225#102—24no—
226#102—42no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229C4×C8—21no—
230C4×C8—21no—
231C22×C7—42no—
232C22×C7—42no—
233C22×C7—42no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7#39411yescharacteristic
239C22×C7#46111yescharacteristic
240C22×C7#46311yescharacteristic
241C22×C7#46511yescharacteristic
242C22×C7#47011yescharacteristic
243C22×C7#47111yescharacteristic
244C22×C7#47211yescharacteristic
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—82no—
248C4×C7—81no—
249C4×C7—161no—
250C4×C7#44811yescharacteristic
251C4×C7#45211yescharacteristic
252C4×C7#46011yescharacteristic
253C4×C7#46411yescharacteristic
254C4×C7#46511yescharacteristic
255C4×C7#46811yescharacteristic
256C4×C7#48311yescharacteristic
257C4×C7#48511yescharacteristic
258C24—21no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4C7×SD1612yes—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×C7×D411yescharacteristic
279C22×C4C2×C7×D411yescharacteristic
280C22×C4C7×#4111yescharacteristic
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8C7×SD1612yes—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42C2×C7×D411yescharacteristic
310C42C2×C7×D411yescharacteristic
311C42C7×#4111yescharacteristic
312C42C7×#4111yescharacteristic
313C2×C7—82no—
314C2×C7#167111yescharacteristic
315C2×C7#167711yescharacteristic
316C2×C7#169511yescharacteristic
317C2×C7#170311yescharacteristic
318C2×C7#203511yescharacteristic
319C2×C7#205111yescharacteristic
320C2×C7#205511yescharacteristic
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23C7×#12111yescharacteristic
329Q8—82no—
330Q8—82no—
331D4—22no—
332D4—22no—
333D4—41no—
334D4—41no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4C7×#12011yescharacteristic
351C2×C4C7×#12111yescharacteristic
352C2×C4C7×#12111yescharacteristic
353C2×C4C7×#12311yescharacteristic
354C2×C4C7×#12311yescharacteristic
355C2×C4C7×#12411yescharacteristic
356C2×C4C2×C7×SD1611yescharacteristic
357C2×C4C2×C7×SD1611yescharacteristic
358C2×C4C7×#13511yescharacteristic
359C2×C4C7×#13511yescharacteristic
360C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
361C2×C4C7×#13711yescharacteristic
362C2×C4C7×#13711yescharacteristic
363C2×C4C7×#13711yescharacteristic
364C8—22no—
365C8—41no—
366C8—42no—
367C8—81no—
368C8—81no—
369C7#1048011yescharacteristic, direct product factor, Sylow
370C22—42no—
371C22—42no—
372C22—42no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22C7×#39411yescharacteristic
378C22C7×#46111yescharacteristic
379C22C7×#46311yescharacteristic
380C22C7×#46511yescharacteristic
381C22C7×#47011yescharacteristic
382C22C7×#47111yescharacteristic
383C22C7×#47211yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—81no—
388C4—161no—
389C4C7×#44811yescharacteristic
390C4C7×#45211yescharacteristic
391C4C7×#46011yescharacteristic
392C4C7×#46411yescharacteristic
393C4C7×#46511yescharacteristic
394C4C7×#46811yescharacteristic
395C4C7×#48311yescharacteristic
396C4C7×#48511yescharacteristic
397C2—82no—
398C2C7×#167111yescharacteristic
399C2C7×#167711yescharacteristic
400C2C7×#169511yescharacteristic
401C2C7×#170311yescharacteristic
402C2C7×#203511yescharacteristic
403C2C7×#205111yescharacteristic
404C2C7×#205511yescharacteristic
subgroupG/H
center, socleC23×C7#121
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1990C2semidirect
5. C2×C7×#334C2semidirect
6. C2×C7×#333C2semidirect
7. C7×#1296C2semidirect
27. #10480C7direct (N × H)
66. #1990C2×C7semidirect
69. C2×#334C2×C7semidirect
70. C2×#333C2×C7semidirect
71. #1296C2×C7semidirect
369. C7#10480direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4762
C7×#19902
C7×#19542
C7×#18022
C2×C7×#3342
C2×C7×#3332
C7×#12962
#104807