Finite Group Databasebeta

← C7×#10271 #13187487

Subgroups — C7×#10271 #13187487

Order 1792.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#498C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic
3C2×C7×#461C211yesmaximal, characteristic
4C7×#1954C211yesmaximal, characteristic
5C2×C7×#333C211yesmaximal, characteristic, complemented
6C2×C7×#332C211yesmaximal, characteristic, complemented
7C7×#1277C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#498—22no—
13C7×#498—22no—
14C7×#476—22no—
15C7×#476—22no—
16C7×#461—22no—
17C7×#461—22no—
18C2×C7×#106—22no—
19C2×C7×#106—21no—
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×#103—21no—
24C2×C7×#102—21no—
25C2×C7×C22⋊C8—22no—
26C2×C4×C7×C8C2211yescharacteristic
27C7×#333—22no—
28C7×#332—22no—
29#10271C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8—21no—
32C22×C7×D4D411yescharacteristic
33C23×C4×C7—21no—
34C2×C7×SD16—24no—
35C2×C7×SD16—42no—
36C2×C7×SD16—42no—
37C22×C7×C8—21no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C22×C7×C8D412yes—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8D412yes—
48C7×C4⋊Q8D412yes—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C42×C7C2311yescharacteristic
60C7×#106—24no—
61C7×#106—28no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8D412yes—
67C7×C4⋊C8D412yes—
68C7×#103—24no—
69C7×#102—24no—
70C7×C22⋊C8—44no—
71C4×C7×C8—21no—
72C4×C7×C8—21no—
73C2×#498C2×C711yescharacteristic, complemented
74C2×#476C2×C711yescharacteristic
75C2×#461C2×C711yescharacteristic
76#1954C2×C711yescharacteristic
77C2×#333C2×C711yescharacteristic, complemented
78C2×#332C2×C711yescharacteristic, complemented
79#1277C2×C711yescharacteristic, complemented
80C24×C7—21no—
81C2×C7×Q8—22no—
82C2×C7×Q8—22no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—41no—
87C2×C7×Q8—41no—
88C2×C7×Q8—41no—
89C2×C7×Q8—82no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—21no—
95C2×C7×D4SD1612yes—
96C22×C4×C7—21no—
97C22×C4×C7—41no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7—41no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4111yescharacteristic
106C7×SD16—48no—
107C2×C7×C8—22no—
108C2×C7×C8—22no—
109C2×C7×C8—22no—
110C2×C7×C8—22no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8—41no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—42no—
117C2×C7×C8SD1614yescomplemented
118C7×C4⋊C4—22no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C42×C7C2×D411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C22×SD16—21no—
135C2×C4⋊Q8C22×C711yescharacteristic
136C2×C4⋊Q8C22×C711yescharacteristic
137C2×C4×D4C22×C711yescharacteristic
138#498—22no—
139#498—22no—
140#476—22no—
141#476—22no—
142#461—22no—
143#461—22no—
144C2×#106—22no—
145C2×#106—21no—
146C2×C4⋊C8C22×C711yescharacteristic
147C2×C4⋊C8C22×C711yescharacteristic
148C2×C4⋊C8C22×C711yescharacteristic
149C2×#103—21no—
150C2×#102—21no—
151C2×C22⋊C8—22no—
152C2×C4×C8C22×C711yescharacteristic
153#333—22no—
154#332—22no—
155C23×C7—22no—
156C23×C7—41no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7#12111yescharacteristic
161C7×Q8—42no—
162C7×Q8—42no—
163C7×Q8—82no—
164C7×Q8—82no—
165C7×Q8—81no—
166C7×Q8—81no—
167C7×D4—22no—
168C7×D4—22no—
169C7×D4—41no—
170C7×D4—41no—
171C2×C4×C7—22no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—81no—
177C2×C4×C7—81no—
178C2×C4×C7—82no—
179C2×C4×C7—82no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12011yescharacteristic
184C2×C4×C7#12011yescharacteristic
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12711yescharacteristic
188C2×C4×C7C2×SD1611yescharacteristic
189C2×C4×C7C2×SD1612yes—
190C2×C4×C7#13511yescharacteristic
191C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C7#13711yescharacteristic
193C2×C4×C7#13712yes—
194C7×C8—24no—
195C7×C8—42no—
196C7×C8—42no—
197C7×C8—81no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×D4C7×D411yescharacteristic
201C23×C4—21no—
202C2×SD16—24no—
203C2×SD16—42no—
204C2×SD16—42no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C7×D412yes—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C7×D412yes—
216C4⋊Q8C7×D412yes—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—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
228#106—24no—
229#106—28no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8C7×D412yes—
235C4⋊C8C7×D412yes—
236#103—24no—
237#102—24no—
238C22⋊C8—44no—
239C4×C8—21no—
240C4×C8—21no—
241C22×C7—44no—
242C22×C7—42no—
243C22×C7—82no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7#39211yescharacteristic
247C22×C7#46112yes—
248C22×C7#46511yescharacteristic
249C22×C7#46811yescharacteristic
250C22×C7#47112yes—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—82no—
254C4×C7—81no—
255C4×C7—82no—
256C4×C7#44811yescharacteristic
257C4×C7#44812yes—
258C4×C7#44911yescharacteristic
259C4×C7#45011yescharacteristic
260C4×C7#45211yescharacteristic
261C4×C7#49211yescharacteristic
262C4×C7#49711yescharacteristic
263C24—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—82no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4C7×SD1612yes—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C7×D411yescharacteristic
287C22×C4C2×C7×D411yescharacteristic
288C22×C4C7×#4111yescharacteristic
289SD16—48no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8C7×SD1614yescomplemented
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42C2×C7×D411yescharacteristic
314C42C2×C7×D411yescharacteristic
315C42C2×C7×D411yescharacteristic
316C42C2×C7×D411yescharacteristic
317C2×C7—82no—
318C2×C7#167112yes—
319C2×C7#169112yes—
320C2×C7#200011yescharacteristic
321C2×C7#201311yescharacteristic
322C2×C7#201911yescharacteristic
323C23—22no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—42no—
328C23C7×#12111yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—82no—
332Q8—82no—
333Q8—81no—
334Q8—81no—
335D4—22no—
336D4—22no—
337D4—41no—
338D4—41no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4C7×#12011yescharacteristic
351C2×C4C7×#12011yescharacteristic
352C2×C4C7×#12011yescharacteristic
353C2×C4C7×#12111yescharacteristic
354C2×C4C7×#12111yescharacteristic
355C2×C4C7×#12711yescharacteristic
356C2×C4C2×C7×SD1612yes—
357C2×C4C2×C7×SD1611yescharacteristic
358C2×C4C7×#13511yescharacteristic
359C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C7×#13712yes—
361C2×C4C7×#13711yescharacteristic
362C8—24no—
363C8—42no—
364C8—42no—
365C8—81no—
366C7#1027111yescharacteristic, direct product factor, Sylow
367C22—42no—
368C22—44no—
369C22—81no—
370C22—81no—
371C22—82no—
372C22C7×#39211yescharacteristic
373C22C7×#46112yes—
374C22C7×#46511yescharacteristic
375C22C7×#46811yescharacteristic
376C22C7×#47112yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—81no—
381C4—82no—
382C4C7×#44811yescharacteristic
383C4C7×#44812yes—
384C4C7×#44911yescharacteristic
385C4C7×#45011yescharacteristic
386C4C7×#45211yescharacteristic
387C4C7×#49211yescharacteristic
388C4C7×#49711yescharacteristic
389C2—82no—
390C2C7×#167112yes—
391C2C7×#169112yes—
392C2C7×#200011yescharacteristic
393C2C7×#201311yescharacteristic
394C2C7×#201911yescharacteristic
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
1. C2×C7×#498C2semidirect
5. C2×C7×#333C2semidirect
6. C2×C7×#332C2semidirect
7. C7×#1277C2semidirect
29. #10271C7direct (N × H)
73. C2×#498C2×C7semidirect
77. C2×#333C2×C7semidirect
78. C2×#332C2×C7semidirect
79. #1277C2×C7semidirect
117. C2×C7×C8SD16semidirect
300. C2×C8C7×SD16semidirect
366. C7#10271direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4982
C2×C7×#4762
C2×C7×#4612
C7×#19542
C2×C7×#3332
C2×C7×#3322
C7×#12772
#102717