Finite Group Databasebeta

← C2×C7×#3392 #13238604

Subgroups — C2×C7×#3392 #13238604

Order 1792.

2238 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×C7×#550C211yesmaximal, characteristic
3C7×#3392C2116yesmaximal, direct product factor
4C2×C7×#492C213yesmaximal, complemented
5C2×C7×#460C213yesmaximal, complemented
6C2×C7×#461C213yesmaximal, complemented
7C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
8C2×C4×C7×SD16C213yesmaximal, complemented
9C22×C7×#41—21no—
10C22×C7×SD16C2213yes—
11C22×C7×SD16C2216yes—
12C7×#557—24no—
13C7×#557C2218yescomplemented
14C7×#550—24no—
15C7×#550C2218yes—
16C2×C7×C4⋊Q8C2213yes—
17C2×C7×#127C2213yes—
18C2×C7×#125—23no—
19C2×C7×#121—23no—
20C2×C4×C7×Q8—21no—
21C2×C4×C7×Q8C2213yes—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×D4C2213yes—
24C7×#492C22124yescomplemented
25C7×#461C22124yescomplemented
26C7×#460C22124yescomplemented
27C7×C8×Q8C2218yescomplemented
28C4×C7×SD16C22124yescomplemented
29C2×C7×#106C2211yescharacteristic
30C2×C7×C4⋊C8C2213yes—
31C2×C7×#103C2213yes—
32C2×C7×#102C2213yes—
33C2×C4×C7×C8C2213yes—
34C2×#3392C711yesmaximal, characteristic, direct product factor, Sylow
35C2×C7×#41—28no—
36C2×C7×#41—48no—
37C22×C7×Q8D411yescharacteristic
38C22×C7×Q8C2313yes—
39C22×C7×D4—23no—
40C22×C7×D4—23no—
41C22×C7×D4C2313yes—
42C23×C4×C7—23no—
43C2×C7×SD16—212no—
44C2×C7×SD16—412no—
45C2×C7×SD16C23112yes—
46C2×C7×SD16C23124yes—
47C22×C7×C8C2313yes—
48C22×C7×C8C2311yescharacteristic
49C7×C4⋊Q8—26no—
50C7×C4⋊Q8C23112yes—
51C7×#127—26no—
52C7×#127C23112yes—
53C7×#125—224no—
54C7×#121—224no—
55C4×C7×Q8—28no—
56C4×C7×Q8—26no—
57C4×C7×Q8D414yes—
58C4×C7×Q8C23112yes—
59C4×C7×Q8C2314yes—
60C4×C7×D4—26no—
61C4×C7×D4C23112yes—
62C2×C7×C4⋊C4—23no—
63C2×C7×C4⋊C4—23no—
64C2×C7×C4⋊C4—23no—
65C2×C7×C4⋊C4D413yes—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C22⋊C4—23no—
69C2×C42×C7—23no—
70C2×C42×C7C2313yes—
71C7×#106C2314yes—
72C7×C4⋊C8C23112yes—
73C7×#103C23112yes—
74C7×#102C23112yes—
75C4×C7×C8C23112yes—
76C2×#557C2×C711yescharacteristic, complemented
77C2×#550C2×C711yescharacteristic
78#3392C2×C7116yesdirect product factor
79C2×#492C2×C713yescomplemented
80C2×#460C2×C713yescomplemented
81C2×#461C2×C713yescomplemented
82C2×C8×Q8C2×C711yescharacteristic, complemented
83C2×C4×SD16C2×C713yescomplemented
84C24×C7—23no—
85C7×#41—432no—
86C2×C7×Q8—23no—
87C2×C7×Q8—26no—
88C2×C7×Q8—26no—
89C2×C7×Q8SD1618yescomplemented
90C2×C7×Q8C2×D414yes—
91C2×C7×Q8C2416yes—
92C2×C7×D4—23no—
93C2×C7×D4—212no—
94C2×C7×D4—212no—
95C2×C7×D4—26no—
96C2×C7×D4—26no—
97C2×C7×D4—412no—
98C2×C7×D4—86no—
99C2×C7×D4C2416yes—
100C22×C4×C7—21no—
101C22×C4×C7—23no—
102C22×C4×C7—21no—
103C22×C4×C7—212no—
104C22×C4×C7—412no—
105C22×C4×C7C2×D413yes—
106C22×C4×C7C2×D413yes—
107C22×C4×C7C2411yescharacteristic
108C7×SD16—224no—
109C7×SD16—424no—
110C2×C7×C8—26no—
111C2×C7×C8#4114yescomplemented
112C2×C7×C8C2416yes—
113C2×C7×C8C2412yes—
114C7×C4⋊C4—212no—
115C7×C4⋊C4—212no—
116C7×C4⋊C4—212no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4C2×D4112yes—
120C7×C4⋊C4C2412yes—
121C7×C4⋊C4C2412yes—
122C7×C22⋊C4—212no—
123C42×C7—212no—
124C42×C7C2×D416yes—
125C42×C7C2416yes—
126C22×#41—21no—
127C22×SD16C22×C713yes—
128C22×SD16C22×C716yes—
129#557—24no—
130#557C22×C718yescomplemented
131#550—24no—
132#550C22×C718yes—
133C2×C4⋊Q8C22×C713yes—
134C2×#127C22×C713yes—
135C2×#125—23no—
136C2×#121—23no—
137C2×C4×Q8—21no—
138C2×C4×Q8C22×C713yes—
139C2×C4×Q8C22×C711yescharacteristic
140C2×C4×D4C22×C713yes—
141#492C22×C7124yescomplemented
142#461C22×C7124yescomplemented
143#460C22×C7124yescomplemented
144C8×Q8C22×C718yescomplemented
145C4×SD16C22×C7124yescomplemented
146C2×#106C22×C711yescharacteristic
147C2×C4⋊C8C22×C713yes—
148C2×#103C22×C713yes—
149C2×#102C22×C713yes—
150C2×C4×C8C22×C713yes—
151C23×C7—26no—
152C23×C7—46no—
153C23×C7—46no—
154C23×C7—46no—
155C23×C7C22×D411yescharacteristic
156C7×Q8—26no—
157C7×Q8—26no—
158C7×Q8—46no—
159C7×Q8C2×SD16116yescomplemented
160C7×D4—26no—
161C7×D4—26no—
162C7×D4—46no—
163C7×D4—424no—
164C7×D4—812no—
165C2×C4×C7—26no—
166C2×C4×C7—22no—
167C2×C4×C7—26no—
168C2×C4×C7—22no—
169C2×C4×C7—46no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—424no—
173C2×C4×C7C2×SD16112yes—
174C2×C4×C7C22×D416yes—
175C2×C4×C7C22×D411yescharacteristic
176C2×C4×C7C22×D416yes—
177C2×C4×C7C2×#4112yes—
178C2×C4×C7H(5,𝔽2)12yes—
179C2×C4×C7C2511yescharacteristic
180C7×C8—26no—
181C7×C8C2×#4114yescomplemented
182C2×#41—28no—
183C2×#41—48no—
184C22×Q8C7×D411yescharacteristic
185C22×Q8C23×C713yes—
186C22×D4—23no—
187C22×D4—23no—
188C22×D4C23×C713yes—
189C23×C4—23no—
190C2×SD16—212no—
191C2×SD16—412no—
192C2×SD16C23×C7112yes—
193C2×SD16C23×C7124yes—
194C22×C8C23×C713yes—
195C22×C8C23×C711yescharacteristic
196C4⋊Q8—26no—
197C4⋊Q8C23×C7112yes—
198#127—26no—
199#127C23×C7112yes—
200#125—224no—
201#121—224no—
202C4×Q8—28no—
203C4×Q8—26no—
204C4×Q8C7×D414yes—
205C4×Q8C23×C7112yes—
206C4×Q8C23×C714yes—
207C4×D4—26no—
208C4×D4C23×C7112yes—
209C2×C4⋊C4—23no—
210C2×C4⋊C4—23no—
211C2×C4⋊C4—23no—
212C2×C4⋊C4C7×D413yes—
213C2×C4⋊C4C23×C711yescharacteristic
214C2×C4⋊C4C23×C711yescharacteristic
215C2×C22⋊C4—23no—
216C2×C42—23no—
217C2×C42C23×C713yes—
218#106C23×C714yes—
219C4⋊C8C23×C7112yes—
220#103C23×C7112yes—
221#102C23×C7112yes—
222C4×C8C23×C7112yes—
223C22×C7—46no—
224C22×C7—46no—
225C22×C7—46no—
226C22×C7—812no—
227C22×C7#55012yes—
228C22×C7C22×SD1612yes—
229C22×C7#57712yes—
230C22×C7C23×D411yescharacteristic
231C4×C7—26no—
232C4×C7—46no—
233C4×C7—42no—
234C4×C7—42no—
235C4×C7#55012yes—
236C4×C7C22×SD16112yes—
237C4×C7C22×#4111yescharacteristic
238C4×C7C2×H(5,𝔽2)11yescharacteristic
239C24—23no—
240#41—432no—
241C2×Q8—23no—
242C2×Q8—26no—
243C2×Q8—26no—
244C2×Q8C7×SD1618yescomplemented
245C2×Q8C2×C7×D414yes—
246C2×Q8C24×C716yes—
247C2×D4—23no—
248C2×D4—212no—
249C2×D4—212no—
250C2×D4—26no—
251C2×D4—26no—
252C2×D4—412no—
253C2×D4—86no—
254C2×D4C24×C716yes—
255C22×C4—21no—
256C22×C4—23no—
257C22×C4—21no—
258C22×C4—212no—
259C22×C4—412no—
260C22×C4C2×C7×D413yes—
261C22×C4C2×C7×D413yes—
262C22×C4C24×C711yescharacteristic
263SD16—224no—
264SD16—424no—
265C2×C8—26no—
266C2×C8C7×#4114yescomplemented
267C2×C8C24×C716yes—
268C2×C8C24×C712yes—
269C4⋊C4—212no—
270C4⋊C4—212no—
271C4⋊C4—212no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4C2×C7×D4112yes—
275C4⋊C4C24×C712yes—
276C4⋊C4C24×C712yes—
277C22⋊C4—212no—
278C42—212no—
279C42C2×C7×D416yes—
280C42C24×C716yes—
281C2×C7—86no—
282C2×C7#339214yesdirect product factor
283C2×C7C2×#55011yescharacteristic
284C2×C7C23×SD1611yescharacteristic
285C2×C7C2×#57711yescharacteristic
286C23—26no—
287C23—46no—
288C23—46no—
289C23—46no—
290C23C22×C7×D411yescharacteristic
291Q8—26no—
292Q8—26no—
293Q8—46no—
294Q8C2×C7×SD16116yescomplemented
295D4—26no—
296D4—26no—
297D4—46no—
298D4—424no—
299D4—812no—
300C2×C4—26no—
301C2×C4—22no—
302C2×C4—26no—
303C2×C4—22no—
304C2×C4—46no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—424no—
308C2×C4C2×C7×SD16112yes—
309C2×C4C22×C7×D416yes—
310C2×C4C22×C7×D411yescharacteristic
311C2×C4C22×C7×D416yes—
312C2×C4C2×C7×#4112yes—
313C2×C4C7×H(5,𝔽2)12yes—
314C2×C4C25×C711yescharacteristic
315C8—26no—
316C8C2×C7×#4114yescomplemented
317C7C2×#339211yescharacteristic, direct product factor, Sylow
318C22—46no—
319C22—46no—
320C22—46no—
321C22—812no—
322C22C7×#55012yes—
323C22C22×C7×SD1612yes—
324C22C7×#57712yes—
325C22C23×C7×D411yescharacteristic
326C4—26no—
327C4—42no—
328C4—42no—
329C4—46no—
330C4C7×#55012yes—
331C4C22×C7×SD16112yes—
332C4C22×C7×#4111yescharacteristic
333C4C2×C7×H(5,𝔽2)11yescharacteristic
334C2—86no—
335C2C7×#339214yesdirect product factor
336C2C2×C7×#55011yescharacteristic
337C2C23×C7×SD1611yescharacteristic
338C2C2×C7×#57711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#550C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
3. C7×#3392C2direct (N × H)
4. C2×C7×#492C2semidirect
5. C2×C7×#460C2semidirect
6. C2×C7×#461C2semidirect
7. C2×C7×C8×Q8C2semidirect
8. C2×C4×C7×SD16C2semidirect
13. C7×#557C22semidirect
24. C7×#492C22semidirect
25. C7×#461C22semidirect
26. C7×#460C22semidirect
27. C7×C8×Q8C22semidirect
28. C4×C7×SD16C22semidirect
34. C2×#3392C7direct (N × H)
76. C2×#557C2×C7semidirect
78. #3392C2×C7direct (N × H)
79. C2×#492C2×C7semidirect
80. C2×#460C2×C7semidirect
81. C2×#461C2×C7semidirect
82. C2×C8×Q8C2×C7semidirect
83. C2×C4×SD16C2×C7semidirect
89. C2×C7×Q8SD16semidirect
111. C2×C7×C8#41semidirect
130. #557C22×C7semidirect
141. #492C22×C7semidirect
142. #461C22×C7semidirect
143. #460C22×C7semidirect
144. C8×Q8C22×C7semidirect
145. C4×SD16C22×C7semidirect
159. C7×Q8C2×SD16semidirect
181. C7×C8C2×#41semidirect
244. C2×Q8C7×SD16semidirect
266. C2×C8C7×#41semidirect
282. C2×C7#3392direct (N × H)
294. Q8C2×C7×SD16semidirect
316. C8C2×C7×#41semidirect
317. C7C2×#3392direct (N × H)
335. C2C7×#3392direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×C7×#5502
C7×#33922
C2×C7×#4922
C2×C7×#4602
C2×C7×#4612
C2×C7×C8×Q82
C2×C4×C7×SD162
C2×#33927