Finite Group Databasebeta

← C2×C7×#3359 #13238512

Subgroups — C2×C7×#3359 #13238512

Order 1792.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×C7×D4×Q8C211yesmaximal, characteristic
3C7×#3359C2116yesmaximal, direct product factor
4C2×C7×#487C213yesmaximal, complemented
5C2×C7×#475C213yesmaximal, complemented
6C2×C7×#461C213yesmaximal
7C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
8C2×C4×C7×SD16C213yesmaximal
9C23×C7×Q8—21no—
10C22×C7×SD16C2213yes—
11C7×#557—24no—
12C7×#557C2218yescomplemented
13C7×D4×Q8—24no—
14C7×D4×Q8C2218yes—
15C2×C7×C4⋊Q8C2213yes—
16C2×C7×C4⋊Q8C2213yes—
17C2×C7×#125—23no—
18C2×C7×C22⋊Q8—23no—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2213yes—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4C2213yes—
23C7×#487C22124yescomplemented
24C7×#475C22124yescomplemented
25C7×#461C22124yes—
26C7×C8×Q8C2218yescomplemented
27C4×C7×SD16C22124yes—
28C2×C7×#106C2213yes—
29C2×C7×C4⋊C8C2213yes—
30C2×C7×#103C2213yes—
31C2×C7×#103C2216yes—
32C2×C7×#102C2211yescharacteristic
33C2×C4×C7×C8C2213yes—
34C2×#3359C711yesmaximal, characteristic, direct product factor, Sylow
35C22×C7×Q8—28no—
36C22×C7×Q8—23no—
37C22×C7×Q8—48no—
38C22×C7×Q8D411yescharacteristic
39C22×C7×Q8C2313yes—
40C22×C7×D4C2311yescharacteristic
41C23×C4×C7—23no—
42C2×C7×SD16—212no—
43C2×C7×SD16C23112yes—
44C22×C7×C8C2311yescharacteristic
45C22×C7×C8C2313yes—
46C7×C4⋊Q8—26no—
47C7×C4⋊Q8—26no—
48C7×C4⋊Q8C23112yes—
49C7×C4⋊Q8C23112yes—
50C7×#125—224no—
51C7×C22⋊Q8—224no—
52C4×C7×Q8—28no—
53C4×C7×Q8—26no—
54C4×C7×Q8D414yes—
55C4×C7×Q8C23112yes—
56C4×C7×Q8C2314yes—
57C4×C7×D4—26no—
58C4×C7×D4C23112yes—
59C2×C7×C4⋊C4—23no—
60C2×C7×C4⋊C4—23no—
61C2×C7×C4⋊C4—23no—
62C2×C7×C4⋊C4—23no—
63C2×C7×C4⋊C4D413yes—
64C2×C7×C4⋊C4C2313yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—23no—
67C2×C42×C7—23no—
68C2×C42×C7C2313yes—
69C7×#106C23112yes—
70C7×C4⋊C8C23112yes—
71C7×#103C23124yes—
72C7×#103C23112yes—
73C7×#102C2314yes—
74C4×C7×C8C23112yes—
75C2×#557C2×C711yescharacteristic, complemented
76C2×D4×Q8C2×C711yescharacteristic
77#3359C2×C7116yesdirect product factor
78C2×#487C2×C713yescomplemented
79C2×#475C2×C713yescomplemented
80C2×#461C2×C713yes—
81C2×C8×Q8C2×C711yescharacteristic, complemented
82C2×C4×SD16C2×C713yes—
83C24×C7—21no—
84C2×C7×Q8—23no—
85C2×C7×Q8—212no—
86C2×C7×Q8—26no—
87C2×C7×Q8—26no—
88C2×C7×Q8—432no—
89C2×C7×Q8—412no—
90C2×C7×Q8SD1618yescomplemented
91C2×C7×Q8C2×D414yes—
92C2×C7×Q8C2416yes—
93C2×C7×D4—21no—
94C2×C7×D4—22no—
95C2×C7×D4#4114yes—
96C2×C7×D4C2412yes—
97C22×C4×C7—21no—
98C22×C4×C7—23no—
99C22×C4×C7—23no—
100C22×C4×C7—212no—
101C22×C4×C7—412no—
102C22×C4×C7C2×D413yes—
103C22×C4×C7C2×D413yes—
104C22×C4×C7C2411yescharacteristic
105C7×SD16—224no—
106C2×C7×C8—26no—
107C2×C7×C8—22no—
108C2×C7×C8C2412yes—
109C2×C7×C8C2416yes—
110C7×C4⋊C4—212no—
111C7×C4⋊C4—212no—
112C7×C4⋊C4—212no—
113C7×C4⋊C4—212no—
114C7×C4⋊C4—23no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4C2×D4112yes—
117C7×C4⋊C4C2412yes—
118C7×C4⋊C4C2416yes—
119C7×C22⋊C4—212no—
120C42×C7—212no—
121C42×C7C2×D416yes—
122C42×C7C2416yes—
123C23×Q8—21no—
124C22×SD16C22×C713yes—
125#557—24no—
126#557C22×C718yescomplemented
127D4×Q8—24no—
128D4×Q8C22×C718yes—
129C2×C4⋊Q8C22×C713yes—
130C2×C4⋊Q8C22×C713yes—
131C2×#125—23no—
132C2×C22⋊Q8—23no—
133C2×C4×Q8—21no—
134C2×C4×Q8C22×C713yes—
135C2×C4×Q8C22×C711yescharacteristic
136C2×C4×D4C22×C713yes—
137#487C22×C7124yescomplemented
138#475C22×C7124yescomplemented
139#461C22×C7124yes—
140C8×Q8C22×C718yescomplemented
141C4×SD16C22×C7124yes—
142C2×#106C22×C713yes—
143C2×C4⋊C8C22×C713yes—
144C2×#103C22×C716yes—
145C2×#103C22×C713yes—
146C2×#102C22×C711yescharacteristic
147C2×C4×C8C22×C713yes—
148C23×C7—24no—
149C23×C7—22no—
150C23×C7—42no—
151C23×C7—42no—
152C23×C7C22×D411yescharacteristic
153C7×Q8—26no—
154C7×Q8—26no—
155C7×Q8—46no—
156C7×Q8—424no—
157C7×Q8C2×SD16116yescomplemented
158C7×D4—24no—
159C7×D4—22no—
160C7×D4C2×#4114yes—
161C2×C4×C7—26no—
162C2×C4×C7—22no—
163C2×C4×C7—26no—
164C2×C4×C7—26no—
165C2×C4×C7—46no—
166C2×C4×C7—46no—
167C2×C4×C7—42no—
168C2×C4×C7—424no—
169C2×C4×C7C2×SD16112yes—
170C2×C4×C7C22×D416yes—
171C2×C4×C7C22×D411yescharacteristic
172C2×C4×C7C22×D416yes—
173C2×C4×C7C2×#4112yes—
174C2×C4×C72−1+412yes—
175C2×C4×C7C2511yescharacteristic
176C7×C8—26no—
177C7×C8—22no—
178C22×Q8—28no—
179C22×Q8—23no—
180C22×Q8—48no—
181C22×Q8C7×D411yescharacteristic
182C22×Q8C23×C713yes—
183C22×D4C23×C711yescharacteristic
184C23×C4—23no—
185C2×SD16—212no—
186C2×SD16C23×C7112yes—
187C22×C8C23×C713yes—
188C22×C8C23×C711yescharacteristic
189C4⋊Q8—26no—
190C4⋊Q8—26no—
191C4⋊Q8C23×C7112yes—
192C4⋊Q8C23×C7112yes—
193#125—224no—
194C22⋊Q8—224no—
195C4×Q8—28no—
196C4×Q8—26no—
197C4×Q8C7×D414yes—
198C4×Q8C23×C7112yes—
199C4×Q8C23×C714yes—
200C4×D4—26no—
201C4×D4C23×C7112yes—
202C2×C4⋊C4—23no—
203C2×C4⋊C4—23no—
204C2×C4⋊C4—23no—
205C2×C4⋊C4—23no—
206C2×C4⋊C4C7×D413yes—
207C2×C4⋊C4C23×C713yes—
208C2×C4⋊C4C23×C711yescharacteristic
209C2×C22⋊C4—23no—
210C2×C42—23no—
211C2×C42C23×C713yes—
212#106C23×C7112yes—
213C4⋊C8C23×C7112yes—
214#103C23×C7112yes—
215#103C23×C7124yes—
216#102C23×C714yes—
217C4×C8C23×C7112yes—
218C22×C7—24no—
219C22×C7—48no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7#54812yes—
223C22×C7C22×SD1612yes—
224C22×C7#57812yes—
225C22×C7C23×D411yescharacteristic
226C4×C7—26no—
227C4×C7—46no—
228C4×C7—46no—
229C4×C7—42no—
230C4×C7#54812yes—
231C4×C7C22×SD16112yes—
232C4×C7C22×#4111yescharacteristic
233C4×C7C2×2−1+411yescharacteristic
234C24—21no—
235C2×Q8—23no—
236C2×Q8—212no—
237C2×Q8—26no—
238C2×Q8—26no—
239C2×Q8—432no—
240C2×Q8—412no—
241C2×Q8C7×SD1618yescomplemented
242C2×Q8C2×C7×D414yes—
243C2×Q8C24×C716yes—
244C2×D4—21no—
245C2×D4—22no—
246C2×D4C7×#4114yes—
247C2×D4C24×C712yes—
248C22×C4—21no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—212no—
252C22×C4—412no—
253C22×C4C2×C7×D413yes—
254C22×C4C2×C7×D413yes—
255C22×C4C24×C711yescharacteristic
256SD16—224no—
257C2×C8—26no—
258C2×C8—22no—
259C2×C8C24×C712yes—
260C2×C8C24×C716yes—
261C4⋊C4—212no—
262C4⋊C4—212no—
263C4⋊C4—212no—
264C4⋊C4—212no—
265C4⋊C4—23no—
266C4⋊C4—21no—
267C4⋊C4C2×C7×D4112yes—
268C4⋊C4C24×C712yes—
269C4⋊C4C24×C716yes—
270C22⋊C4—212no—
271C42—212no—
272C42C2×C7×D416yes—
273C42C24×C716yes—
274C2×C7—44no—
275C2×C7#335914yesdirect product factor
276C2×C7C2×#54811yescharacteristic
277C2×C7C23×SD1611yescharacteristic
278C2×C7C2×#57811yescharacteristic
279C23—22no—
280C23—24no—
281C23—42no—
282C23—42no—
283C23C22×C7×D411yescharacteristic
284Q8—26no—
285Q8—26no—
286Q8—46no—
287Q8—424no—
288Q8C2×C7×SD16116yescomplemented
289D4—22no—
290D4—24no—
291D4C2×C7×#4114yes—
292C2×C4—26no—
293C2×C4—22no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—42no—
299C2×C4—424no—
300C2×C4C2×C7×SD16112yes—
301C2×C4C22×C7×D416yes—
302C2×C4C22×C7×D411yescharacteristic
303C2×C4C22×C7×D416yes—
304C2×C4C2×C7×#4112yes—
305C2×C4C7×2−1+412yes—
306C2×C4C25×C711yescharacteristic
307C8—26no—
308C8—22no—
309C7C2×#335911yescharacteristic, direct product factor, Sylow
310C22—24no—
311C22—42no—
312C22—42no—
313C22—48no—
314C22C7×#54812yes—
315C22C22×C7×SD1612yes—
316C22C7×#57812yes—
317C22C23×C7×D411yescharacteristic
318C4—26no—
319C4—46no—
320C4—42no—
321C4—46no—
322C4C7×#54812yes—
323C4C22×C7×SD16112yes—
324C4C22×C7×#4111yescharacteristic
325C4C2×C7×2−1+411yescharacteristic
326C2—44no—
327C2C7×#335914yesdirect product factor
328C2C2×C7×#54811yescharacteristic
329C2C23×C7×SD1611yescharacteristic
330C2C2×C7×#57811yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC22×C7×D4C23
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×#3359C2direct (N × H)
4. C2×C7×#487C2semidirect
5. C2×C7×#475C2semidirect
7. C2×C7×C8×Q8C2semidirect
12. C7×#557C22semidirect
23. C7×#487C22semidirect
24. C7×#475C22semidirect
26. C7×C8×Q8C22semidirect
34. C2×#3359C7direct (N × H)
75. C2×#557C2×C7semidirect
77. #3359C2×C7direct (N × H)
78. C2×#487C2×C7semidirect
79. C2×#475C2×C7semidirect
81. C2×C8×Q8C2×C7semidirect
90. C2×C7×Q8SD16semidirect
126. #557C22×C7semidirect
137. #487C22×C7semidirect
138. #475C22×C7semidirect
140. C8×Q8C22×C7semidirect
157. C7×Q8C2×SD16semidirect
241. C2×Q8C7×SD16semidirect
275. C2×C7#3359direct (N × H)
288. Q8C2×C7×SD16semidirect
309. C7C2×#3359direct (N × H)
327. C2C7×#3359direct (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×D4×Q82
C7×#33592
C2×C7×#4872
C2×C7×#4752
C2×C7×#4612
C2×C7×C8×Q82
C2×C4×C7×SD162
C2×#33597