Finite Group Databasebeta

← C4×D7×S4 #13062037

Subgroups — C4×D7×S4 #13062037

Order 1344.

466 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×S4C211yesmaximal, characteristic
2C4×A4×D7C211yesmaximal, characteristic, complemented
3C4×C7⋊S4C211yesmaximal, characteristic, complemented
4C4×C7×S4C211yesmaximal, characteristic, complemented
5D7×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
6(C7⋊S4)⋊C4C211yesmaximal, characteristic, complemented
7S4×Dic7C211yesmaximal, characteristic, complemented
8C4×D4×D7—31nomaximal, Hall
9C2×A4×D7C2211yescharacteristic
10C2×C7⋊S4C2211yescharacteristic
11C2×C7×S4C2211yescharacteristic
12D7×S4C414yesdirect product factor
13C4×C7×A4C2211yescharacteristic, complemented
14C4×D3×D7—41nomaximal
15A4×Dic7C2211yescharacteristic, complemented
16C7⋊SL(2,ℤ/4)C2211yescharacteristic, complemented
17C7×SL(2,ℤ/4)C2211yescharacteristic, complemented
18C2×D4×D7—31no—
19C22×C4×D7—31no—
20C22×C4×D7D311yescharacteristic, complemented
21C4×C7×D4—31no—
22D4×Dic7—31no—
23C4×C22⋊D7—31no—
24C4×C22⋊D7—31no—
25#6456—31no—
26D7×C4⋊C4—31no—
27(C22⋊D7)⋊C4—31no—
28(C22⋊D7)⋊C4—31no—
29D7×C22⋊C4—31no—
30D7×C22⋊C4—31no—
31C4×D28—31no—
32C42×D7—31no—
33C2×C4×S4—71nomaximal, Hall
34C2×C7×A4C2311yescharacteristic
35C2×D3×D7—41no—
36A4×D7C2×C412yescomplemented
37C7⋊S4C2×C412yescomplemented
38C7×S4C2×C412yescomplemented
39C4×D21—41no—
40C4×D3×C7—41no—
41C3×C4×D7—41no—
42D21⋊C4—41no—
43D3×Dic7—41no—
44Dic3×D7—41no—
45C23×D7—31no—
46C23×D7C2×D311yescharacteristic
47C2×C7×D4—31no—
48C22×C4×C7—31no—
49C22×C4×C7C2×D311yescharacteristic, complemented
50C2×C22⋊D7—31no—
51C2×C22⋊D7—31no—
52C22×Dic7—31no—
53C22×Dic7C2×D311yescharacteristic, complemented
54D4×D7—34no—
55D4×D7—34no—
56C2×D28—31no—
57C2×C4×D7—31no—
58C2×C4×D7—31no—
59C2×C4×D7—31no—
60C2×C4×D7—31no—
61C2×C4×D7—31no—
62C2×C4×D7—31no—
63C2×C4×D7—31no—
64C2×C4×D7—31no—
65C2×C4×D7—31no—
66C2×C4×D7—61no—
67C2×C4×D7—61no—
68C2×C4×D7—61no—
69C2×C4×D7—61no—
70C7×C4⋊C4—31no—
71C7×C22⋊C4—31no—
72C7×C22⋊C4—31no—
73C42×C7—31no—
74C22⋊Dic7—31no—
75C22⋊Dic7—31no—
76(C2×D7)⋊C4—31no—
77(C2×D7)⋊C4—31no—
78(C2×D7)⋊C4—31no—
79(C2×D7)⋊C4—31no—
80C4⋊Dic7—31no—
81Dic7⋊C4—31no—
82Dic7⋊C4—31no—
83C4×Dic7—31no—
84C4×Dic7—31no—
85C4×Dic7—31no—
86C22×S4—71no—
87C2×C4×A4—71no—
88C2×SL(2,ℤ/4)—71no—
89C4×S4—71no—
90C4×S4—71no—
91C4×S4—71no—
92C4×S4D711yescharacteristic, direct product factor
93C2×D21—41no—
94C2×D3×C7—41no—
95C2×C3×D7—41no—
96C7×A4C22×C411yescharacteristic, complemented
97D3×D7—44no—
98C3×C4×C7—41no—
99Dic21—41no—
100C3×Dic7—41no—
101C7×Dic3—41no—
102C2×C4×D4—211noSylow
103C23×C7—31no—
104C23×C7C22×D311yescharacteristic
105C22×D7—31no—
106C22×D7—31no—
107C22×D7—32no—
108C22×D7—31no—
109C22×D7—34no—
110C22×D7—61no—
111C22×D7—62no—
112C22×D7—61no—
113C22×D7C4×D312yescomplemented
114C7×D4—32no—
115C7×D4—32no—
116C2×C4×C7—31no—
117C2×C4×C7—31no—
118C2×C4×C7—31no—
119C2×C4×C7—31no—
120C2×C4×C7—31no—
121C2×C4×C7—61no—
122C2×C4×C7—61no—
123C22⋊D7—34no—
124C22⋊D7—32no—
125C22⋊D7—32no—
126C2×Dic7—31no—
127C2×Dic7—31no—
128C2×Dic7—31no—
129C2×Dic7—31no—
130C2×Dic7—31no—
131C2×Dic7—61no—
132C2×Dic7—61no—
133D28—34no—
134C4×D7—34no—
135C4×D7—31no—
136C4×D7—31no—
137C4×D7—31no—
138C4×D7—61no—
139C4×D7—61no—
140C4×D7—61no—
141C4×D7—62no—
142C4×D7—61no—
143C4×D7—61no—
144C4×D7—61no—
145C4×D7—61no—
146C4×D7S411yescharacteristic, direct product factor
147C22×A4—71no—
148C2×S4—71no—
149C2×S4—74no—
150C2×S4C2×D711yescharacteristic
151C2×C4×D3—281no—
152C4×A4—71no—
153C4×A4C2×D711yescharacteristic, complemented
154SL(2,ℤ/4)—71no—
155SL(2,ℤ/4)C2×D711yescharacteristic, complemented
156C2×C3×C7—41no—
157D21—42no—
158C3×D7—42no—
159D3×C7—42no—
160C22×D4—211no—
161C23×C4—71no—
162C23×C4—211no—
163C4×D4—31no—
164C4×D4—211no—
165C4×D4—211no—
166C4×D4—211no—
167C4×D4—211no—
168C4×D4—211no—
169C4×D4—211no—
170C4×D4—211no—
171C2×C4⋊C4—211no—
172C2×C22⋊C4—211no—
173C2×C22⋊C4—211no—
174C2×C42—211no—
175C22×C7—31no—
176C22×C7—31no—
177C22×C7—32no—
178C22×C7—61no—
179C22×C7—61no—
180C22×C7C2×C4×D311yescharacteristic, complemented
181C2×D7—31no—
182C2×D7—32no—
183C2×D7—32no—
184C2×D7—32no—
185C2×D7—32no—
186C2×D7—61no—
187C2×D7—64no—
188C2×D7—61no—
189C2×D7—61no—
190C2×D7—64no—
191C2×D7C2×S411yescharacteristic
192C4×C7—32no—
193C4×C7—31no—
194C4×C7—61no—
195C4×C7—61no—
196C4×C7C2×S411yescharacteristic, complemented
197Dic7—32no—
198Dic7—31no—
199Dic7—61no—
200Dic7—61no—
201Dic7C2×S411yescharacteristic, complemented
202C22×D3—281no—
203C2×A4—72no—
204C2×A4C22×D711yescharacteristic
205S4—72no—
206S4C4×D712yesdirect product factor
207C2×C3×C4—281no—
208C2×Dic3—281no—
209C4×D3—41no—
210C4×D3—281no—
211C4×D3—281no—
212C4×D3—281no—
213C3×C7—41noHall
214C24—71no—
215C24—211no—
216C2×D4—31no—
217C2×D4—211no—
218C2×D4—214no—
219C2×D4—214no—
220C2×D4—211no—
221C2×D4—211no—
222C22×C4—31no—
223C22×C4—71no—
224C22×C4—211no—
225C22×C4—211no—
226C22×C4—211no—
227C22×C4—211no—
228C22×C4—211no—
229C22×C4—211no—
230C22×C4—211no—
231C22×C4—211no—
232C22×C4—211no—
233C22×C4—211no—
234C22×C4—421no—
235C22×C4—421no—
236C22×C4—421no—
237C22×C4—421no—
238C22×C4D3×D711yescharacteristic, complemented
239C4⋊C4—31no—
240C4⋊C4—211no—
241C4⋊C4—211no—
242C4⋊C4—211no—
243C22⋊C4—31no—
244C22⋊C4—31no—
245C22⋊C4—211no—
246C22⋊C4—211no—
247C22⋊C4—211no—
248C22⋊C4—211no—
249C22⋊C4—211no—
250C22⋊C4—211no—
251C42—31no—
252C42—211no—
253C42—211no—
254C42—211no—
255C2×C7—31no—
256C2×C7—31no—
257C2×C7—62no—
258C2×C7C22×S411yescharacteristic
259D7—32no—
260D7—62no—
261D7C4×S412yesdirect product factor
262C22×C3—281no—
263C2×D3—41no—
264C2×D3—284no—
265C2×D3—281no—
266A4C2×C4×D711yescharacteristic, complemented
267C3×C4—41no—
268C3×C4—281no—
269Dic3—41no—
270Dic3—281no—
271C23—31no—
272C23—72no—
273C23—211no—
274C23—211no—
275C23—212no—
276C23—211no—
277C23—214no—
278C23—421no—
279C23—421no—
280C23—422no—
281C23C2×D3×D711yescharacteristic
282D4—32no—
283D4—32no—
284D4—214no—
285D4—212no—
286D4—214no—
287D4—212no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—71no—
296C2×C4—211no—
297C2×C4—211no—
298C2×C4—214no—
299C2×C4—211no—
300C2×C4—211no—
301C2×C4—211no—
302C2×C4—211no—
303C2×C4—211no—
304C2×C4—211no—
305C2×C4—421no—
306C2×C4—421no—
307C2×C4—421no—
308C2×C4—421no—
309C2×C4—421no—
310C2×C4—422no—
311C2×C4—421no—
312C2×C4—421no—
313C2×C4—421no—
314C2×C4—421no—
315C7C2×C4×S411yescharacteristic, complemented, Sylow
316C2×C3—41no—
317C2×C3—282no—
318D3—42no—
319D3—282no—
320C22—31no—
321C22—31no—
322C22—32no—
323C22—61no—
324C22—61no—
325C22—71no—
326C22—212no—
327C22—212no—
328C22—212no—
329C22—212no—
330C22—211no—
331C22—421no—
332C22—421no—
333C22—424no—
334C22—424no—
335C22—421no—
336C22C4×D3×D711yescharacteristic, complemented
337C4—32no—
338C4—31no—
339C4—61no—
340C4—61no—
341C4—71no—
342C4—212no—
343C4—211no—
344C4—421no—
345C4—421no—
346C4D7×S411yescharacteristic, direct product factor
347C3—41noSylow
348C2—31no—
349C2—31no—
350C2—62no—
351C2—72no—
352C2—212no—
353C2—422no—
354C2C2×D7×S411yescharacteristic
subgroupG/H
center, hypercenterC4D7×S4
commutator, hypocenterC7×A4C22×C4
FrattiniC2C2×D7×S4
FittingC22×C4×C7C2×D3
socleC23×C7C22×D3
prime radicalC2×D7×S4C2
cosocleC2×C7×A4C23
LCSG ▷ C7×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4×A4×D7C2semidirect
3. C4×C7⋊S4C2semidirect
4. C4×C7×S4C2semidirect
5. D7×SL(2,ℤ/4)C2semidirect
6. (C7⋊S4)⋊C4C2semidirect
7. S4×Dic7C2semidirect
12. D7×S4C4direct (N × H)
13. C4×C7×A4C22semidirect
15. A4×Dic7C22semidirect
16. C7⋊SL(2,ℤ/4)C22semidirect
17. C7×SL(2,ℤ/4)C22semidirect
20. C22×C4×D7D3semidirect
36. A4×D7C2×C4semidirect
37. C7⋊S4C2×C4semidirect
38. C7×S4C2×C4semidirect
49. C22×C4×C7C2×D3semidirect
53. C22×Dic7C2×D3semidirect
92. C4×S4D7direct (N × H)
96. C7×A4C22×C4semidirect
113. C22×D7C4×D3semidirect
146. C4×D7S4direct (N × H)
153. C4×A4C2×D7semidirect
155. SL(2,ℤ/4)C2×D7semidirect
180. C22×C7C2×C4×D3semidirect
196. C4×C7C2×S4semidirect
201. Dic7C2×S4semidirect
206. S4C4×D7direct (N × H)
238. C22×C4D3×D7semidirect
261. D7C4×S4direct (N × H)
266. A4C2×C4×D7semidirect
315. C7C2×C4×S4semidirect
336. C22C4×D3×D7semidirect
346. C4D7×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D7×S42
C4×A4×D72
C4×C7⋊S42
C4×C7×S42
D7×SL(2,ℤ/4)2
(C7⋊S4)⋊C42
S4×Dic72
C4×D4×D73
C4×D3×D74
C2×C4×S47