Finite Group Databasebeta

← C2×C7×#1683 #13196325

Subgroups — C2×C7×#1683 #13196325

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#487C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic
3C2×C7×#474C211yesmaximal, characteristic
4C2×C7×C8×D4C211yesmaximal, characteristic
5C7×#1683C218yesmaximal, direct product factor
6C2×C7×#335C211yesmaximal, characteristic, complemented
7C2×C7×#333C211yesmaximal, characteristic, complemented
8C2×C7×#332C211yesmaximal, characteristic, complemented
9C23×C7×C8—21no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C7×#487—22no—
14C7×#487C2214yescomplemented
15C7×#476—22no—
16C7×#476C2214yes—
17C7×#474—22no—
18C7×#474C2214yes—
19C7×C8×D4—22no—
20C7×C8×D4C2214yes—
21C2×C7×#107—21no—
22C2×C7×#106—21no—
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#103—21no—
27C2×C7×#103—21no—
28C2×C7×#102—21no—
29C2×C7×#102—21no—
30C2×C7×C22⋊C8—21no—
31C2×C4×C7×C8C2211yescharacteristic
32C7×#335C2214yescomplemented
33C7×#333C2214yescomplemented
34C7×#332C2214yescomplemented
35C2×#1683C711yesmaximal, characteristic, direct product factor, Sylow
36C22×C7×Q8—21no—
37C22×C7×Q8—21no—
38C22×C7×D4D411yescharacteristic
39C23×C4×C7—21no—
40C22×C7×C8—21no—
41C22×C7×C8—21no—
42C22×C7×C8—21no—
43C22×C7×C8—24no—
44C22×C7×C8—44no—
45C22×C7×C8D411yescharacteristic
46C22×C7×C8D411yescharacteristic
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8—21no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8C2312yes—
54C7×C4⋊Q8C2312yes—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C4×C7×D4D412yes—
58C4×C7×D4C2312yes—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C22⋊C4—21no—
65C2×C42×C7C2311yescharacteristic
66C7×#107—24no—
67C7×#106—24no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8C2312yes—
72C7×C4⋊C8C2312yes—
73C7×C4⋊C8C2312yes—
74C7×#103—24no—
75C7×#103—24no—
76C7×#102—24no—
77C7×#102—24no—
78C7×C22⋊C8—24no—
79C4×C7×C8D412yes—
80C4×C7×C8C2312yes—
81C2×#487C2×C711yescharacteristic, complemented
82C2×#476C2×C711yescharacteristic
83C2×#474C2×C711yescharacteristic
84C2×C8×D4C2×C711yescharacteristic
85#1683C2×C718yesdirect product factor
86C2×#335C2×C711yescharacteristic, complemented
87C2×#333C2×C711yescharacteristic, complemented
88C2×#332C2×C711yescharacteristic, complemented
89C24×C7—21no—
90C2×C7×Q8—22no—
91C2×C7×Q8—22no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C2×C7×Q8—42no—
95C2×C7×Q8—42no—
96C2×C7×Q8—82no—
97C2×C7×D4—21no—
98C2×C7×D4—22no—
99C2×C7×D4Q1614yes—
100C2×C7×D4C2×D412yes—
101C22×C4×C7—21no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—42no—
105C22×C4×C7—42no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4111yescharacteristic
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—48no—
117C2×C7×C8—42no—
118C2×C7×C8—42no—
119C2×C7×C8—42no—
120C2×C7×C8C2×D412yes—
121C2×C7×C8C2×D412yes—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4C2×D412yes—
132C7×C22⋊C4—22no—
133C7×C22⋊C4—22no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7#4111yescharacteristic
137C42×C7C2411yescharacteristic
138C23×C8—21no—
139C2×C4⋊Q8C22×C711yescharacteristic
140C2×C4⋊Q8C22×C711yescharacteristic
141C2×C4×D4C22×C711yescharacteristic
142#487—22no—
143#487C22×C714yescomplemented
144#476—22no—
145#476C22×C714yes—
146#474—22no—
147#474C22×C714yes—
148C8×D4—22no—
149C8×D4C22×C714yes—
150C2×#107—21no—
151C2×#106—21no—
152C2×C4⋊C8C22×C711yescharacteristic
153C2×C4⋊C8C22×C711yescharacteristic
154C2×C4⋊C8C22×C711yescharacteristic
155C2×#103—21no—
156C2×#103—21no—
157C2×#102—21no—
158C2×#102—21no—
159C2×C22⋊C8—21no—
160C2×C4×C8C22×C711yescharacteristic
161#335C22×C714yescomplemented
162#333C22×C714yescomplemented
163#332C22×C714yescomplemented
164C23×C7—22no—
165C23×C7—24no—
166C23×C7—42no—
167C23×C7—42no—
168C23×C7#12111yescharacteristic
169C7×Q8—42no—
170C7×Q8—42no—
171C7×Q8—82no—
172C7×Q8—82no—
173C7×Q8—82no—
174C7×D4—22no—
175C7×D4—24no—
176C7×D4C2×Q1614yes—
177C2×C4×C7—22no—
178C2×C4×C7—22no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—44no—
183C2×C4×C7—42no—
184C2×C4×C7—42no—
185C2×C4×C7—82no—
186C2×C4×C7—82no—
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7#12111yescharacteristic
190C2×C4×C7C2×Q1612yes—
191C2×C4×C7#13512yes—
192C2×C4×C7#13512yes—
193C2×C4×C7AGL(1,ℤ/8)12yes—
194C2×C4×C7C22×D411yescharacteristic
195C2×C4×C7C22×D411yescharacteristic
196C2×C4×C7C2×#4111yescharacteristic
197C7×C8—22no—
198C7×C8—22no—
199C7×C8—42no—
200C7×C8—42no—
201C7×C8—42no—
202C22×Q8—21no—
203C22×Q8—21no—
204C22×D4C7×D411yescharacteristic
205C23×C4—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—24no—
210C22×C8—44no—
211C22×C8C7×D411yescharacteristic
212C22×C8C7×D411yescharacteristic
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8C23×C712yes—
220C4⋊Q8C23×C712yes—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4C7×D412yes—
224C4×D4C23×C712yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C7×D411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C42C23×C711yescharacteristic
232#107—24no—
233#106—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C23×C712yes—
238C4⋊C8C23×C712yes—
239C4⋊C8C23×C712yes—
240#103—24no—
241#103—24no—
242#102—24no—
243#102—24no—
244C22⋊C8—24no—
245C4×C8C7×D412yes—
246C4×C8C23×C712yes—
247C22×C7—24no—
248C22×C7—48no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7#46312yes—
252C22×C7#46712yes—
253C22×C7#47112yes—
254C22×C7C2×#12111yescharacteristic
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—42no—
258C4×C7—82no—
259C4×C7—82no—
260C4×C7#46312yes—
261C4×C7#46712yes—
262C4×C7C22×Q1611yescharacteristic
263C4×C7C2×#13511yescharacteristic
264C4×C7C2×#13511yescharacteristic
265C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
266C24—21no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—82no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4C7×Q1614yes—
277C2×D4C2×C7×D412yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C2×C7×D411yescharacteristic
286C22×C4C2×C7×D411yescharacteristic
287C22×C4C7×#4111yescharacteristic
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—48no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8C2×C7×D412yes—
298C2×C8C2×C7×D412yes—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4C2×C7×D412yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42C2×C7×D411yescharacteristic
312C42C2×C7×D411yescharacteristic
313C42C7×#4111yescharacteristic
314C42C24×C711yescharacteristic
315C2×C7—44no—
316C2×C7#168314yesdirect product factor
317C2×C7C2×#46311yescharacteristic
318C2×C7C2×#46711yescharacteristic
319C2×C7C2×#47111yescharacteristic
320C23—24no—
321C23—22no—
322C23—42no—
323C23—42no—
324C23C7×#12111yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—82no—
328Q8—82no—
329Q8—82no—
330D4—24no—
331D4—22no—
332D4C2×C7×Q1614yes—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4C7×#12111yescharacteristic
344C2×C4C7×#12111yescharacteristic
345C2×C4C7×#12111yescharacteristic
346C2×C4C2×C7×Q1612yes—
347C2×C4C7×#13512yes—
348C2×C4C7×#13512yes—
349C2×C4C7×AGL(1,ℤ/8)12yes—
350C2×C4C22×C7×D411yescharacteristic
351C2×C4C22×C7×D411yescharacteristic
352C2×C4C2×C7×#4111yescharacteristic
353C8—22no—
354C8—22no—
355C8—42no—
356C8—42no—
357C8—42no—
358C7C2×#168311yescharacteristic, direct product factor, Sylow
359C22—24no—
360C22—42no—
361C22—42no—
362C22—48no—
363C22C7×#46312yes—
364C22C7×#46712yes—
365C22C7×#47112yes—
366C22C2×C7×#12111yescharacteristic
367C4—21no—
368C4—21no—
369C4—42no—
370C4—82no—
371C4—82no—
372C4C7×#46312yes—
373C4C7×#46712yes—
374C4C22×C7×Q1611yescharacteristic
375C4C2×C7×#13511yescharacteristic
376C4C2×C7×#13511yescharacteristic
377C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
378C2—44no—
379C2C7×#168314yesdirect product factor
380C2C2×C7×#46311yescharacteristic
381C2C2×C7×#46711yescharacteristic
382C2C2×C7×#47111yescharacteristic
subgroupG/H
center, socleC23×C7#121
commutator, Frattini, cosocleC42C24×C7
prime radicalC22×C7×D4D4
LCSG ▷ 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×#487C2semidirect
5. C7×#1683C2direct (N × H)
6. C2×C7×#335C2semidirect
7. C2×C7×#333C2semidirect
8. C2×C7×#332C2semidirect
14. C7×#487C22semidirect
32. C7×#335C22semidirect
33. C7×#333C22semidirect
34. C7×#332C22semidirect
35. C2×#1683C7direct (N × H)
81. C2×#487C2×C7semidirect
85. #1683C2×C7direct (N × H)
86. C2×#335C2×C7semidirect
87. C2×#333C2×C7semidirect
88. C2×#332C2×C7semidirect
143. #487C22×C7semidirect
161. #335C22×C7semidirect
162. #333C22×C7semidirect
163. #332C22×C7semidirect
316. C2×C7#1683direct (N × H)
358. C7C2×#1683direct (N × H)
379. C2C7×#1683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4872
C2×C7×#4762
C2×C7×#4742
C2×C7×C8×D42
C7×#16832
C2×C7×#3352
C2×C7×#3332
C2×C7×#3322
C2×#16837