Finite Group Databasebeta

← C4×C7×#324 #13185367

Subgroups — C4×C7×#324 #13185367

Order 1792.

730 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×C4×C7×C4⋊C4C211yesmaximal, characteristic
2C4×C7×C22⋊C8C212yesmaximal
3C2×C7×#324C212yesmaximal
4C7×#1280C212yesmaximal
5C22×C7×C4⋊C4C411yescharacteristic
6C22×C7×C4⋊C4C2211yescharacteristic
7C22×C42×C7C411yescharacteristic
8C22×C42×C7C411yescharacteristic
9C22×C42×C7C2211yescharacteristic
10C2×C7×C22⋊C8C2214yes—
11C2×C4×C7×C8—22no—
12C4×C7×C4⋊C4—22no—
13C4×C7×C4⋊C4—22no—
14C2×C7×#95C411yescharacteristic
15C2×C7×#95C2211yescharacteristic
16C7×#336—22no—
17C7×#324C418yesdirect product factor
18C4×#324C711yesmaximal, characteristic, direct product factor, Sylow
19C23×C4×C7C2×C411yescharacteristic
20C23×C4×C7C2×C411yescharacteristic
21C23×C4×C7C2×C411yescharacteristic
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×C7×C8—24no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4C814yescomplemented
32C2×C7×C4⋊C4C2×C414yes—
33C2×C42×C7—21no—
34C2×C42×C7—21no—
35C2×C42×C7—22no—
36C2×C42×C7—21no—
37C2×C42×C7—21no—
38C2×C42×C7—42no—
39C2×C42×C7—42no—
40C2×C42×C7C814yescomplemented
41C2×C42×C7D412yes—
42C2×C42×C7D412yes—
43C7×C22⋊C8—24no—
44C7×C22⋊C8C2×C418yes—
45C4×C7×C8—44no—
46C7×#95—22no—
47C7×#95—22no—
48C2×C4×C4⋊C4C2×C711yescharacteristic
49C4×C22⋊C8C2×C712yes—
50C2×#324C2×C712yes—
51#1280C2×C712yes—
52C24×C7C22×C411yescharacteristic
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—41no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7—41no—
69C22×C4×C7—41no—
70C22×C4×C7—41no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7C4212yes—
76C22×C4×C7C4212yes—
77C22×C4×C7C22⋊C412yes—
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C2×C812yes—
81C22×C4×C7C2×C812yes—
82C22×C4×C7C2×C812yes—
83C22×C4×C7C2×C812yes—
84C22×C4×C7C2×C812yes—
85C22×C4×C7C2×C812yes—
86C22×C4×C7M1612yes—
87C22×C4×C7M1612yes—
88C22×C4×C7C22×C412yes—
89C22×C4×C7C2×D412yes—
90C22×C4×C7#4112yes—
91C2×C7×C8—28no—
92C2×C7×C8—44no—
93C2×C7×C8—44no—
94C7×C4⋊C4—28no—
95C7×C4⋊C4—28no—
96C42×C7—24no—
97C42×C7—24no—
98C42×C7—24no—
99C42×C7—42no—
100C42×C7—42no—
101C42×C7—42no—
102C22×C4⋊C4C4×C711yescharacteristic
103C22×C4⋊C4C22×C711yescharacteristic
104C22×C42C4×C711yescharacteristic
105C22×C42C4×C711yescharacteristic
106C22×C42C22×C711yescharacteristic
107C2×C22⋊C8C22×C714yes—
108C2×C4×C8—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C2×#95C4×C711yescharacteristic
112C2×#95C22×C711yescharacteristic
113#336—22no—
114#324C4×C718yesdirect product factor
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×C4211yescharacteristic
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#11711yescharacteristic
122C23×C7C22×C811yescharacteristic
123C23×C7C22×C811yescharacteristic
124C23×C7C2×M1611yescharacteristic
125C23×C7C2×M1611yescharacteristic
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—24no—
139C2×C4×C7—24no—
140C2×C4×C7—22no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7C4×C814yescomplemented
160C2×C4×C7C22⋊C812yes—
161C2×C4×C7C22⋊C814yes—
162C2×C4×C7C22⋊C812yes—
163C2×C4×C7#9912yes—
164C2×C4×C7#10112yes—
165C2×C4×C7C4×D414yes—
166C7×C8—48no—
167C23×C4C2×C4×C711yescharacteristic
168C23×C4C2×C4×C711yescharacteristic
169C23×C4C2×C4×C711yescharacteristic
170C23×C4C2×C4×C711yescharacteristic
171C23×C4C2×C4×C711yescharacteristic
172C23×C4C2×C4×C711yescharacteristic
173C23×C4C23×C711yescharacteristic
174C22×C8—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C7×C814yescomplemented
180C2×C4⋊C4C2×C4×C714yes—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42C7×C814yescomplemented
189C2×C42C7×D412yes—
190C2×C42C7×D412yes—
191C22⋊C8—24no—
192C22⋊C8C2×C4×C718yes—
193C4×C8—44no—
194#95—22no—
195#95—22no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7C4×C22⋊C411yescharacteristic
209C22×C7C2×C4×C811yescharacteristic
210C22×C7C4×M1611yescharacteristic
211C22×C7C2×C22⋊C811yescharacteristic
212C22×C7C2×C22⋊C811yescharacteristic
213C22×C7C2×#9911yescharacteristic
214C22×C7#41011yescharacteristic
215C22×C7C2×#10111yescharacteristic
216C22×C7#41311yescharacteristic
217C22×C7#43111yescharacteristic
218C22×C7#43111yescharacteristic
219C4×C7—22no—
220C4×C7—24no—
221C4×C7—24no—
222C4×C7—24no—
223C4×C7—42no—
224C4×C7—42no—
225C4×C7—42no—
226C4×C7#32414yesdirect product factor
227C24C22×C4×C711yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C42×C712yes—
251C22×C4C42×C712yes—
252C22×C4C7×C22⋊C412yes—
253C22×C4C7×C22⋊C411yescharacteristic
254C22×C4C7×C22⋊C411yescharacteristic
255C22×C4C2×C7×C812yes—
256C22×C4C2×C7×C812yes—
257C22×C4C2×C7×C812yes—
258C22×C4C2×C7×C812yes—
259C22×C4C2×C7×C812yes—
260C22×C4C2×C7×C812yes—
261C22×C4C7×M1612yes—
262C22×C4C7×M1612yes—
263C22×C4C22×C4×C712yes—
264C22×C4C2×C7×D412yes—
265C22×C4C7×#4112yes—
266C2×C8—28no—
267C2×C8—44no—
268C2×C8—44no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—42no—
275C42—42no—
276C42—42no—
277C2×C7—21no—
278C2×C7—21no—
279C2×C7—21no—
280C2×C7—21no—
281C2×C7C2×#32411yescharacteristic
282C2×C7#145811yescharacteristic
283C2×C7#145911yescharacteristic
284C2×C7#146111yescharacteristic
285C2×C7C4×C22⋊C811yescharacteristic
286C2×C7C4×#9911yescharacteristic
287C2×C7C4×#10111yescharacteristic
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23C2×C42×C711yescharacteristic
293C23C2×C7×C22⋊C411yescharacteristic
294C23C7×#11711yescharacteristic
295C23C22×C7×C811yescharacteristic
296C23C22×C7×C811yescharacteristic
297C23C2×C7×M1611yescharacteristic
298C23C2×C7×M1611yescharacteristic
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4C4×C7×C814yescomplemented
333C2×C4C7×C22⋊C812yes—
334C2×C4C7×C22⋊C814yes—
335C2×C4C7×C22⋊C812yes—
336C2×C4C7×#9912yes—
337C2×C4C7×#10112yes—
338C2×C4C4×C7×D414yes—
339C8—48no—
340C7C4×#32411yescharacteristic, direct product factor, Sylow
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22C4×C7×C22⋊C411yescharacteristic
354C22C2×C4×C7×C811yescharacteristic
355C22C4×C7×M1611yescharacteristic
356C22C2×C7×C22⋊C811yescharacteristic
357C22C2×C7×C22⋊C811yescharacteristic
358C22C2×C7×#9911yescharacteristic
359C22C7×#41011yescharacteristic
360C22C2×C7×#10111yescharacteristic
361C22C7×#41311yescharacteristic
362C22C7×#43111yescharacteristic
363C22C7×#43111yescharacteristic
364C4—22no—
365C4—24no—
366C4—24no—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4C7×#32414yesdirect product factor
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2C2×C7×#32411yescharacteristic
377C2C7×#145811yescharacteristic
378C2C7×#145911yescharacteristic
379C2C7×#146111yescharacteristic
380C2C4×C7×C22⋊C811yescharacteristic
381C2C4×C7×#9911yescharacteristic
382C2C4×C7×#10111yescharacteristic
subgroupG/H
centerC22×C4×C7C22⋊C4
commutatorC22C2×C4×C7×C8
Frattini, cosocleC23×C4C23×C7
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
17. C7×#324C4direct (N × H)
18. C4×#324C7direct (N × H)
31. C2×C7×C4⋊C4C8semidirect
40. C2×C42×C7C8semidirect
114. #324C4×C7direct (N × H)
159. C2×C4×C7C4×C8semidirect
179. C2×C4⋊C4C7×C8semidirect
188. C2×C42C7×C8semidirect
226. C4×C7#324direct (N × H)
332. C2×C4C4×C7×C8semidirect
340. C7C4×#324direct (N × H)
371. C4C7×#324direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×C4⋊C42
C4×C7×C22⋊C82
C2×C7×#3242
C7×#12802
C4×#3247