Finite Group Databasebeta

← C2×C7×#3358 #13238511

Subgroups — C2×C7×#3358 #13238511

Order 1792.

2402 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic, complemented
2C2×C7×D4×Q8C211yesmaximal, characteristic, complemented
3C7×#3358C2116yesmaximal, direct product factor
4C2×C7×#486C213yesmaximal, complemented
5C2×C7×#474C213yesmaximal, complemented
6C2×C7×#459C213yesmaximal, complemented
7C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
8C2×C4×C7×D8C213yesmaximal, complemented
9C22×C7×#41—21no—
10C23×C7×Q8—21no—
11C22×C7×D8C2213yes—
12C7×#550—24no—
13C7×#550C2218yescomplemented
14C7×D4×Q8—24no—
15C7×D4×Q8C2218yescomplemented
16C2×C7×C4⋊Q8C2213yes—
17C2×C7×#127C2213yes—
18C2×C7×C22⋊Q8—23no—
19C2×C7×#121—23no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2213yes—
22C2×C4×C7×D4C2213yes—
23C7×#486C22124yescomplemented
24C7×#474C22124yescomplemented
25C7×#459C22124yescomplemented
26C7×C8×Q8C2218yescomplemented
27C4×C7×D8C22124yescomplemented
28C2×C7×#107C2213yes—
29C2×C7×C4⋊C8C2213yes—
30C2×C7×#102C2211yescharacteristic
31C2×C7×#102C2213yes—
32C2×C7×#102C2216yes—
33C2×C4×C7×C8C2213yes—
34C2×#3358C711yesmaximal, characteristic, direct product factor, Sylow
35C2×C7×#41—28no—
36C2×C7×#41—48no—
37C22×C7×Q8—28no—
38C22×C7×Q8—23no—
39C22×C7×Q8—48no—
40C22×C7×Q8D411yescharacteristic
41C22×C7×D4—23no—
42C22×C7×D4—23no—
43C22×C7×D4C2313yes—
44C22×C7×D4C2311yescharacteristic
45C23×C4×C7—23no—
46C23×C4×C7—23no—
47C2×C7×D8—212no—
48C2×C7×D8C23112yes—
49C22×C7×C8C2311yescharacteristic
50C22×C7×C8C2313yes—
51C7×C4⋊Q8—26no—
52C7×C4⋊Q8C23112yes—
53C7×#127—26no—
54C7×#127C23112yes—
55C7×C22⋊Q8—224no—
56C7×#121—224no—
57C4×C7×Q8D414yes—
58C4×C7×Q8C2314yes—
59C4×C7×D4—26no—
60C4×C7×D4—26no—
61C4×C7×D4C23112yes—
62C4×C7×D4C23112yes—
63C2×C7×C4⋊C4—23no—
64C2×C7×C4⋊C4D413yes—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2313yes—
67C2×C7×C22⋊C4—23no—
68C2×C7×C22⋊C4—23no—
69C2×C42×C7C2313yes—
70C7×#107C23112yes—
71C7×C4⋊C8C23112yes—
72C7×#102C23124yes—
73C7×#102C23112yes—
74C7×#102C2314yes—
75C4×C7×C8C23112yes—
76C2×#550C2×C711yescharacteristic, complemented
77C2×D4×Q8C2×C711yescharacteristic, complemented
78#3358C2×C7116yesdirect product factor
79C2×#486C2×C713yescomplemented
80C2×#474C2×C713yescomplemented
81C2×#459C2×C713yescomplemented
82C2×C8×Q8C2×C711yescharacteristic, complemented
83C2×C4×D8C2×C713yescomplemented
84C24×C7—21no—
85C24×C7—23no—
86C7×#41—432no—
87C2×C7×Q8—212no—
88C2×C7×Q8—432no—
89C2×C7×Q8—412no—
90C2×C7×Q8D818yescomplemented
91C2×C7×Q8C2×D414yes—
92C2×C7×D4—21no—
93C2×C7×D4—23no—
94C2×C7×D4—212no—
95C2×C7×D4—212no—
96C2×C7×D4—26no—
97C2×C7×D4—22no—
98C2×C7×D4—26no—
99C2×C7×D4—412no—
100C2×C7×D4—86no—
101C2×C7×D4#4114yescomplemented
102C2×C7×D4C2416yes—
103C2×C7×D4C2412yes—
104C22×C4×C7—21no—
105C22×C4×C7—23no—
106C22×C4×C7—212no—
107C22×C4×C7—212no—
108C22×C4×C7—412no—
109C22×C4×C7—412no—
110C22×C4×C7C2×D413yes—
111C22×C4×C7C2×D413yes—
112C22×C4×C7C2411yescharacteristic
113C7×D8—224no—
114C2×C7×C8—26no—
115C2×C7×C8—22no—
116C2×C7×C8C2412yes—
117C2×C7×C8C2416yes—
118C7×C4⋊C4—212no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—23no—
121C7×C4⋊C4C2×D4112yes—
122C7×C4⋊C4C2416yes—
123C7×C4⋊C4C2412yes—
124C7×C22⋊C4—212no—
125C7×C22⋊C4—212no—
126C42×C7C2×D416yes—
127C42×C7C2416yes—
128C22×#41—21no—
129C23×Q8—21no—
130C22×D8C22×C713yes—
131#550—24no—
132#550C22×C718yescomplemented
133D4×Q8—24no—
134D4×Q8C22×C718yescomplemented
135C2×C4⋊Q8C22×C713yes—
136C2×#127C22×C713yes—
137C2×C22⋊Q8—23no—
138C2×#121—23no—
139C2×C4×Q8C22×C711yescharacteristic
140C2×C4×D4C22×C713yes—
141C2×C4×D4C22×C713yes—
142#486C22×C7124yescomplemented
143#474C22×C7124yescomplemented
144#459C22×C7124yescomplemented
145C8×Q8C22×C718yescomplemented
146C4×D8C22×C7124yescomplemented
147C2×#107C22×C713yes—
148C2×C4⋊C8C22×C713yes—
149C2×#102C22×C713yes—
150C2×#102C22×C716yes—
151C2×#102C22×C711yescharacteristic
152C2×C4×C8C22×C713yes—
153C23×C7—24no—
154C23×C7—22no—
155C23×C7—26no—
156C23×C7—42no—
157C23×C7—42no—
158C23×C7—46no—
159C23×C7—46no—
160C23×C7—46no—
161C23×C7C22×D411yescharacteristic
162C7×Q8—424no—
163C7×Q8C2×D8116yescomplemented
164C7×D4—24no—
165C7×D4—26no—
166C7×D4—22no—
167C7×D4—26no—
168C7×D4—46no—
169C7×D4—424no—
170C7×D4—812no—
171C7×D4C2×#4114yescomplemented
172C2×C4×C7—26no—
173C2×C4×C7—22no—
174C2×C4×C7—26no—
175C2×C4×C7—42no—
176C2×C4×C7—46no—
177C2×C4×C7—424no—
178C2×C4×C7—424no—
179C2×C4×C7C2×D8112yes—
180C2×C4×C7C22×D416yes—
181C2×C4×C7C22×D411yescharacteristic
182C2×C4×C7C22×D416yes—
183C2×C4×C7C2×#4112yes—
184C2×C4×C72−1+412yes—
185C2×C4×C7C2511yescharacteristic
186C7×C8—26no—
187C7×C8—22no—
188C2×#41—28no—
189C2×#41—48no—
190C22×Q8—28no—
191C22×Q8—23no—
192C22×Q8—48no—
193C22×Q8C7×D411yescharacteristic
194C22×D4—23no—
195C22×D4—23no—
196C22×D4C23×C713yes—
197C22×D4C23×C711yescharacteristic
198C23×C4—23no—
199C23×C4—23no—
200C2×D8—212no—
201C2×D8C23×C7112yes—
202C22×C8C23×C713yes—
203C22×C8C23×C711yescharacteristic
204C4⋊Q8—26no—
205C4⋊Q8C23×C7112yes—
206#127—26no—
207#127C23×C7112yes—
208C22⋊Q8—224no—
209#121—224no—
210C4×Q8C7×D414yes—
211C4×Q8C23×C714yes—
212C4×D4—26no—
213C4×D4—26no—
214C4×D4C23×C7112yes—
215C4×D4C23×C7112yes—
216C2×C4⋊C4—23no—
217C2×C4⋊C4C7×D413yes—
218C2×C4⋊C4C23×C711yescharacteristic
219C2×C4⋊C4C23×C713yes—
220C2×C22⋊C4—23no—
221C2×C22⋊C4—23no—
222C2×C42C23×C713yes—
223#107C23×C7112yes—
224C4⋊C8C23×C7112yes—
225#102C23×C714yes—
226#102C23×C7112yes—
227#102C23×C7124yes—
228C4×C8C23×C7112yes—
229C22×C7—24no—
230C22×C7—48no—
231C22×C7—42no—
232C22×C7—42no—
233C22×C7—46no—
234C22×C7—46no—
235C22×C7—46no—
236C22×C7—812no—
237C22×C7#54812yes—
238C22×C7C22×D812yes—
239C22×C7#57712yes—
240C22×C7C23×D411yescharacteristic
241C4×C7—26no—
242C4×C7—42no—
243C4×C7—46no—
244C4×C7#54812yes—
245C4×C7C22×D8112yes—
246C4×C7C22×#4111yescharacteristic
247C4×C7C2×2−1+411yescharacteristic
248C24—21no—
249C24—23no—
250#41—432no—
251C2×Q8—212no—
252C2×Q8—432no—
253C2×Q8—412no—
254C2×Q8C7×D818yescomplemented
255C2×Q8C2×C7×D414yes—
256C2×D4—21no—
257C2×D4—23no—
258C2×D4—212no—
259C2×D4—212no—
260C2×D4—26no—
261C2×D4—22no—
262C2×D4—26no—
263C2×D4—412no—
264C2×D4—86no—
265C2×D4C7×#4114yescomplemented
266C2×D4C24×C716yes—
267C2×D4C24×C712yes—
268C22×C4—21no—
269C22×C4—23no—
270C22×C4—212no—
271C22×C4—212no—
272C22×C4—412no—
273C22×C4—412no—
274C22×C4C2×C7×D413yes—
275C22×C4C2×C7×D413yes—
276C22×C4C24×C711yescharacteristic
277D8—224no—
278C2×C8—26no—
279C2×C8—22no—
280C2×C8C24×C712yes—
281C2×C8C24×C716yes—
282C4⋊C4—212no—
283C4⋊C4—21no—
284C4⋊C4—23no—
285C4⋊C4C2×C7×D4112yes—
286C4⋊C4C24×C716yes—
287C4⋊C4C24×C712yes—
288C22⋊C4—212no—
289C22⋊C4—212no—
290C42C2×C7×D416yes—
291C42C24×C716yes—
292C2×C7—44no—
293C2×C7—86no—
294C2×C7#335814yesdirect product factor
295C2×C7C2×#54811yescharacteristic
296C2×C7C23×D811yescharacteristic
297C2×C7C2×#57711yescharacteristic
298C23—22no—
299C23—24no—
300C23—26no—
301C23—42no—
302C23—42no—
303C23—46no—
304C23—46no—
305C23—46no—
306C23C22×C7×D411yescharacteristic
307Q8—424no—
308Q8C2×C7×D8116yescomplemented
309D4—26no—
310D4—22no—
311D4—26no—
312D4—24no—
313D4—46no—
314D4—424no—
315D4—812no—
316D4C2×C7×#4114yescomplemented
317C2×C4—26no—
318C2×C4—22no—
319C2×C4—26no—
320C2×C4—42no—
321C2×C4—46no—
322C2×C4—424no—
323C2×C4—424no—
324C2×C4C2×C7×D8112yes—
325C2×C4C22×C7×D416yes—
326C2×C4C22×C7×D411yescharacteristic
327C2×C4C22×C7×D416yes—
328C2×C4C2×C7×#4112yes—
329C2×C4C7×2−1+412yes—
330C2×C4C25×C711yescharacteristic
331C8—26no—
332C8—22no—
333C7C2×#335811yescharacteristic, direct product factor, Sylow
334C22—24no—
335C22—42no—
336C22—42no—
337C22—48no—
338C22—46no—
339C22—46no—
340C22—46no—
341C22—812no—
342C22C7×#54812yes—
343C22C22×C7×D812yes—
344C22C7×#57712yes—
345C22C23×C7×D411yescharacteristic
346C4—26no—
347C4—42no—
348C4—46no—
349C4C7×#54812yes—
350C4C22×C7×D8112yes—
351C4C22×C7×#4111yescharacteristic
352C4C2×C7×2−1+411yescharacteristic
353C2—44no—
354C2—86no—
355C2C7×#335814yesdirect product factor
356C2C2×C7×#54811yescharacteristic
357C2C23×C7×D811yescharacteristic
358C2C2×C7×#57711yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC2×C4C25×C7
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×#550C2semidirect
2. C2×C7×D4×Q8C2semidirect
3. C7×#3358C2direct (N × H)
4. C2×C7×#486C2semidirect
5. C2×C7×#474C2semidirect
6. C2×C7×#459C2semidirect
7. C2×C7×C8×Q8C2semidirect
8. C2×C4×C7×D8C2semidirect
13. C7×#550C22semidirect
15. C7×D4×Q8C22semidirect
23. C7×#486C22semidirect
24. C7×#474C22semidirect
25. C7×#459C22semidirect
26. C7×C8×Q8C22semidirect
27. C4×C7×D8C22semidirect
34. C2×#3358C7direct (N × H)
76. C2×#550C2×C7semidirect
77. C2×D4×Q8C2×C7semidirect
78. #3358C2×C7direct (N × H)
79. C2×#486C2×C7semidirect
80. C2×#474C2×C7semidirect
81. C2×#459C2×C7semidirect
82. C2×C8×Q8C2×C7semidirect
83. C2×C4×D8C2×C7semidirect
90. C2×C7×Q8D8semidirect
101. C2×C7×D4#41semidirect
132. #550C22×C7semidirect
134. D4×Q8C22×C7semidirect
142. #486C22×C7semidirect
143. #474C22×C7semidirect
144. #459C22×C7semidirect
145. C8×Q8C22×C7semidirect
146. C4×D8C22×C7semidirect
163. C7×Q8C2×D8semidirect
171. C7×D4C2×#41semidirect
254. C2×Q8C7×D8semidirect
265. C2×D4C7×#41semidirect
294. C2×C7#3358direct (N × H)
308. Q8C2×C7×D8semidirect
316. D4C2×C7×#41semidirect
333. C7C2×#3358direct (N × H)
355. C2C7×#3358direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5502
C2×C7×D4×Q82
C7×#33582
C2×C7×#4862
C2×C7×#4742
C2×C7×#4592
C2×C7×C8×Q82
C2×C4×C7×D82
C2×#33587