Finite Group Databasebeta

← C7×#10267 #13187483

Subgroups — C7×#10267 #13187483

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#486C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×C7×#459C211yesmaximal, characteristic
4C7×#1954C211yesmaximal, characteristic, complemented
5C2×C7×#333C211yesmaximal, characteristic, complemented
6C2×C7×#331C211yesmaximal, characteristic, complemented
7C7×#1277C211yesmaximal, characteristic, complemented
8C22×C7×D8—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C7×#486—22no—
13C7×#486—22no—
14C7×#476—22no—
15C7×#476—22no—
16C7×#459—22no—
17C7×#459—22no—
18C2×C7×#106—21no—
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#102—21no—
23C2×C7×#102—22no—
24C2×C7×#102—21no—
25C2×C7×C22⋊C8—22no—
26C2×C4×C7×C8C2211yescharacteristic
27C7×#333—22no—
28C7×#331—22no—
29#10267C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7—21no—
36C2×C7×D8—24no—
37C2×C7×D8—42no—
38C2×C7×D8—42no—
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C22×C7×C8—21no—
42C22×C7×C8D412yes—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8D412yes—
47C7×#127—21no—
48C7×#127—21no—
49C7×#127—21no—
50C7×#127D412yes—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—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×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8D412yes—
66C7×C4⋊C8D412yes—
67C7×#102—24no—
68C7×#102—28no—
69C7×#102—24no—
70C7×C22⋊C8—44no—
71C4×C7×C8—21no—
72C4×C7×C8—21no—
73C2×#486C2×C711yescharacteristic, complemented
74C2×#476C2×C711yescharacteristic, complemented
75C2×#459C2×C711yescharacteristic
76#1954C2×C711yescharacteristic, complemented
77C2×#333C2×C711yescharacteristic, complemented
78C2×#331C2×C711yescharacteristic, complemented
79#1277C2×C711yescharacteristic, complemented
80C24×C7—21no—
81C24×C7—41no—
82C2×C7×Q8—22no—
83C2×C7×Q8—41no—
84C2×C7×Q8—82no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—42no—
99C2×C7×D4—41no—
100C2×C7×D4—42no—
101C2×C7×D4—41no—
102C2×C7×D4—82no—
103C2×C7×D4SD1612yes—
104C22×C4×C7—21no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—42no—
109C22×C4×C7—42no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7#4111yescharacteristic
113C7×D8—48no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—22no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C2×C7×C8—41no—
122C2×C7×C8—41no—
123C2×C7×C8—41no—
124C2×C7×C8—42no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C22×D8—21no—
138C2×C4⋊Q8C22×C711yescharacteristic
139C2×#127C22×C711yescharacteristic
140C2×C4×D4C22×C711yescharacteristic
141#486—22no—
142#486—22no—
143#476—22no—
144#476—22no—
145#459—22no—
146#459—22no—
147C2×#106—21no—
148C2×C4⋊C8C22×C711yescharacteristic
149C2×C4⋊C8C22×C711yescharacteristic
150C2×C4⋊C8C22×C711yescharacteristic
151C2×#102—21no—
152C2×#102—22no—
153C2×#102—21no—
154C2×C22⋊C8—22no—
155C2×C4×C8C22×C711yescharacteristic
156#333—22no—
157#331—22no—
158C23×C7—22no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—41no—
162C23×C7—42no—
163C23×C7—42no—
164C23×C7—82no—
165C23×C7—81no—
166C23×C7—81no—
167C23×C7—82no—
168C23×C7#12111yescharacteristic
169C7×Q8—82no—
170C7×Q8—82no—
171C7×D4—22no—
172C7×D4—22no—
173C7×D4—41no—
174C7×D4—44no—
175C7×D4—42no—
176C7×D4—41no—
177C7×D4—42no—
178C7×D4—82no—
179C7×D4—82no—
180C7×D4—81no—
181C7×D4—81no—
182C7×D4—81no—
183C7×D4—81no—
184C2×C4×C7—22no—
185C2×C4×C7—41no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—82no—
189C2×C4×C7—82no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12011yescharacteristic
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12711yescharacteristic
198C2×C4×C7C2×SD1611yescharacteristic
199C2×C4×C7#13511yescharacteristic
200C2×C4×C7#13512yes—
201C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C7AGL(1,ℤ/8)12yes—
203C2×C4×C7#13711yescharacteristic
204C7×C8—42no—
205C7×C8—42no—
206C7×C8—42no—
207C7×C8—81no—
208C22×Q8—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C7×D411yescharacteristic
213C23×C4—21no—
214C2×D8—24no—
215C2×D8—42no—
216C2×D8—42no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8C7×D412yes—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8C7×D412yes—
225#127—21no—
226#127—21no—
227#127—21no—
228#127C7×D412yes—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C42C23×C711yescharacteristic
238#106—24no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C7×D412yes—
244C4⋊C8C7×D412yes—
245#102—24no—
246#102—24no—
247#102—28no—
248C22⋊C8—44no—
249C4×C8—21no—
250C4×C8—21no—
251C22×C7—44no—
252C22×C7—42no—
253C22×C7—82no—
254C22×C7—81no—
255C22×C7—81no—
256C22×C7—82no—
257C22×C7—82no—
258C22×C7—82no—
259C22×C7—162no—
260C22×C7—162no—
261C22×C7#39211yescharacteristic
262C22×C7#46312yes—
263C22×C7#46511yescharacteristic
264C22×C7#46811yescharacteristic
265C22×C7#47212yes—
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—81no—
269C4×C7—82no—
270C4×C7#44811yescharacteristic
271C4×C7#44911yescharacteristic
272C4×C7#44912yes—
273C4×C7#45011yescharacteristic
274C4×C7#45211yescharacteristic
275C4×C7#49511yescharacteristic
276C4×C7#49611yescharacteristic
277C24—21no—
278C24—41no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—82no—
300C2×D4C7×SD1612yes—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C2×C7×D411yescharacteristic
308C22×C4C2×C7×D411yescharacteristic
309C22×C4C7×#4111yescharacteristic
310D8—48no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42C2×C7×D411yescharacteristic
331C42C2×C7×D411yescharacteristic
332C42C2×C7×D411yescharacteristic
333C42C2×C7×D411yescharacteristic
334C2×C7—82no—
335C2×C7—162no—
336C2×C7#167712yes—
337C2×C7#169712yes—
338C2×C7#200011yescharacteristic
339C2×C7#201211yescharacteristic
340C2×C7#201811yescharacteristic
341C23—22no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—82no—
349C23—82no—
350C23—81no—
351C23C7×#12111yescharacteristic
352Q8—82no—
353Q8—82no—
354D4—22no—
355D4—22no—
356D4—41no—
357D4—44no—
358D4—42no—
359D4—41no—
360D4—42no—
361D4—82no—
362D4—82no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4C7×#12011yescharacteristic
376C2×C4C7×#12011yescharacteristic
377C2×C4C7×#12011yescharacteristic
378C2×C4C7×#12111yescharacteristic
379C2×C4C7×#12111yescharacteristic
380C2×C4C7×#12711yescharacteristic
381C2×C4C2×C7×SD1611yescharacteristic
382C2×C4C7×#13512yes—
383C2×C4C7×#13511yescharacteristic
384C2×C4C7×AGL(1,ℤ/8)12yes—
385C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C7×#13711yescharacteristic
387C8—42no—
388C8—42no—
389C8—42no—
390C8—81no—
391C7#1026711yescharacteristic, direct product factor, Sylow
392C22—42no—
393C22—44no—
394C22—81no—
395C22—81no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—162no—
401C22—162no—
402C22C7×#39211yescharacteristic
403C22C7×#46312yes—
404C22C7×#46511yescharacteristic
405C22C7×#46811yescharacteristic
406C22C7×#47212yes—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—82no—
411C4C7×#44811yescharacteristic
412C4C7×#44911yescharacteristic
413C4C7×#44912yes—
414C4C7×#45011yescharacteristic
415C4C7×#45211yescharacteristic
416C4C7×#49511yescharacteristic
417C4C7×#49611yescharacteristic
418C2—82no—
419C2—162no—
420C2C7×#167712yes—
421C2C7×#169712yes—
422C2C7×#200011yescharacteristic
423C2C7×#201211yescharacteristic
424C2C7×#201811yescharacteristic
subgroupG/H
center, socleC23×C7#121
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#459C2
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×#486C2semidirect
2. C2×C7×#476C2semidirect
4. C7×#1954C2semidirect
5. C2×C7×#333C2semidirect
6. C2×C7×#331C2semidirect
7. C7×#1277C2semidirect
29. #10267C7direct (N × H)
73. C2×#486C2×C7semidirect
74. C2×#476C2×C7semidirect
76. #1954C2×C7semidirect
77. C2×#333C2×C7semidirect
78. C2×#331C2×C7semidirect
79. #1277C2×C7semidirect
391. C7#10267direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4862
C2×C7×#4762
C2×C7×#4592
C7×#19542
C2×C7×#3332
C2×C7×#3312
C7×#12772
#102677