Finite Group Databasebeta

← C7×#10477 #13187693

Subgroups — C7×#10477 #13187693

Order 1792.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#461C211yesmaximal, characteristic
2C7×#1990C211yesmaximal, characteristic, complemented
3C7×#1954C211yesmaximal, characteristic
4C7×#1802C211yesmaximal, characteristic
5C2×C7×#334C211yesmaximal, characteristic, complemented
6C2×C7×#332C211yesmaximal, characteristic, complemented
7C7×#1296C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#461—22no—
12C7×#461—22no—
13C2×C7×C4⋊C8C2211yescharacteristic
14C2×C7×C4⋊C8C2211yescharacteristic
15C2×C7×C4⋊C8C2211yescharacteristic
16C2×C7×#103—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×#332—22no—
27#10477C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8—21no—
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C2×C7×SD16—24no—
32C2×C7×SD16—42no—
33C2×C7×SD16—42no—
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C22×C7×C8—21no—
37C22×C7×C8D411yescharacteristic
38C22×C7×C8D411yescharacteristic
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8D412yes—
43C4×C7×D4—21no—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C22⋊C4—21no—
54C2×C42×C7C2311yescharacteristic
55C7×C4⋊C8—21no—
56C7×C4⋊C8—21no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8D412yes—
61C7×#103—24no—
62C7×#102—42no—
63C7×C22⋊C8—42no—
64C7×C22⋊C8—42no—
65C4×C7×C8—21no—
66C4×C7×C8—21no—
67C2×#461C2×C711yescharacteristic
68#1990C2×C711yescharacteristic, complemented
69#1954C2×C711yescharacteristic
70#1802C2×C711yescharacteristic
71C2×#334C2×C711yescharacteristic, complemented
72C2×#332C2×C711yescharacteristic, complemented
73#1296C2×C711yescharacteristic, complemented
74C24×C7—21no—
75C2×C7×Q8—22no—
76C2×C7×Q8—41no—
77C2×C7×Q8—41no—
78C2×C7×Q8—41no—
79C2×C7×Q8—41no—
80C2×C7×Q8—41no—
81C2×C7×D4—21no—
82C2×C7×D4—21no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4SD1612yes—
87C22×C4×C7—21no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7#4111yescharacteristic
99C7×SD16—48no—
100C2×C7×C8—21no—
101C2×C7×C8—21no—
102C2×C7×C8—22no—
103C2×C7×C8—21no—
104C2×C7×C8—21no—
105C2×C7×C8—21no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8—41no—
114C2×C7×C8SD1612yescomplemented
115C7×C4⋊C4—21no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—81no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C42×C7#4111yescharacteristic
132C22×SD16—21no—
133C2×C4⋊Q8C22×C711yescharacteristic
134C2×C4×D4C22×C711yescharacteristic
135#461—22no—
136#461—22no—
137C2×C4⋊C8C22×C711yescharacteristic
138C2×C4⋊C8C22×C711yescharacteristic
139C2×C4⋊C8C22×C711yescharacteristic
140C2×#103—21no—
141C2×#102—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8—21no—
144C2×C4×C8C22×C711yescharacteristic
145#384C22×C711yescharacteristic
146#340—21no—
147#340—21no—
148#340—21no—
149#334—22no—
150#332—22no—
151C23×C7—22no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7#12111yescharacteristic
159C7×Q8—42no—
160C7×Q8—42no—
161C7×Q8—81no—
162C7×Q8—81no—
163C7×D4—22no—
164C7×D4—22no—
165C7×D4—41no—
166C7×D4—41no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—81no—
174C2×C4×C7—81no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7—81no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12311yescharacteristic
186C2×C4×C7#12311yescharacteristic
187C2×C4×C7#12411yescharacteristic
188C2×C4×C7C2×SD1611yescharacteristic
189C2×C4×C7C2×SD1611yescharacteristic
190C2×C4×C7#13511yescharacteristic
191C2×C4×C7#13511yescharacteristic
192C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C7#13711yescharacteristic
196C7×C8—22no—
197C7×C8—42no—
198C7×C8—41no—
199C7×C8—81no—
200C7×C8—81no—
201C22×Q8—21no—
202C22×D4C7×D411yescharacteristic
203C23×C4—21no—
204C2×SD16—24no—
205C2×SD16—42no—
206C2×SD16—42no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8C7×D411yescharacteristic
211C22×C8C7×D411yescharacteristic
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C7×D412yes—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C42C23×C711yescharacteristic
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8C7×D412yes—
234#103—24no—
235#102—42no—
236C22⋊C8—42no—
237C22⋊C8—42no—
238C4×C8—21no—
239C4×C8—21no—
240C22×C7—42no—
241C22×C7—42no—
242C22×C7—42no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7#39411yescharacteristic
248C22×C7#46111yescharacteristic
249C22×C7#46311yescharacteristic
250C22×C7#46511yescharacteristic
251C22×C7#46911yescharacteristic
252C22×C7#47111yescharacteristic
253C22×C7#47311yescharacteristic
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—81no—
257C4×C7—82no—
258C4×C7—161no—
259C4×C7#44911yescharacteristic
260C4×C7#45011yescharacteristic
261C4×C7#46011yescharacteristic
262C4×C7#46411yescharacteristic
263C4×C7#46511yescharacteristic
264C4×C7#46811yescharacteristic
265C4×C7#48111yescharacteristic
266C4×C7#48211yescharacteristic
267C24—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4C7×SD1612yes—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291C22×C4C7×#4111yescharacteristic
292SD16—48no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8C7×SD1612yescomplemented
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—81no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42C2×C7×D411yescharacteristic
322C42C2×C7×D411yescharacteristic
323C42C7×#4111yescharacteristic
324C42C7×#4111yescharacteristic
325C2×C7—82no—
326C2×C7#167111yescharacteristic
327C2×C7#167911yescharacteristic
328C2×C7#169311yescharacteristic
329C2×C7#170111yescharacteristic
330C2×C7#203411yescharacteristic
331C2×C7#205211yescharacteristic
332C2×C7#205511yescharacteristic
333C23—22no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23C7×#12111yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4C7×#12011yescharacteristic
365C2×C4C7×#12111yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12311yescharacteristic
368C2×C4C7×#12311yescharacteristic
369C2×C4C7×#12411yescharacteristic
370C2×C4C2×C7×SD1611yescharacteristic
371C2×C4C2×C7×SD1611yescharacteristic
372C2×C4C7×#13511yescharacteristic
373C2×C4C7×#13511yescharacteristic
374C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
375C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
376C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
377C2×C4C7×#13711yescharacteristic
378C8—22no—
379C8—41no—
380C8—42no—
381C8—81no—
382C8—81no—
383C7#1047711yescharacteristic, direct product factor, Sylow
384C22—42no—
385C22—42no—
386C22—42no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22C7×#39411yescharacteristic
392C22C7×#46111yescharacteristic
393C22C7×#46311yescharacteristic
394C22C7×#46511yescharacteristic
395C22C7×#46911yescharacteristic
396C22C7×#47111yescharacteristic
397C22C7×#47311yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—81no—
402C4—161no—
403C4C7×#44911yescharacteristic
404C4C7×#45011yescharacteristic
405C4C7×#46011yescharacteristic
406C4C7×#46411yescharacteristic
407C4C7×#46511yescharacteristic
408C4C7×#46811yescharacteristic
409C4C7×#48111yescharacteristic
410C4C7×#48211yescharacteristic
411C2—82no—
412C2C7×#167111yescharacteristic
413C2C7×#167911yescharacteristic
414C2C7×#169311yescharacteristic
415C2C7×#170111yescharacteristic
416C2C7×#203411yescharacteristic
417C2C7×#205211yescharacteristic
418C2C7×#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×#332C2semidirect
7. C7×#1296C2semidirect
27. #10477C7direct (N × H)
68. #1990C2×C7semidirect
71. C2×#334C2×C7semidirect
72. C2×#332C2×C7semidirect
73. #1296C2×C7semidirect
114. C2×C7×C8SD16semidirect
307. C2×C8C7×SD16semidirect
383. C7#10477direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4612
C7×#19902
C7×#19542
C7×#18022
C2×C7×#3342
C2×C7×#3322
C7×#12962
#104777