Finite Group Databasebeta

← C7×#4956 #13055992

Subgroups — C7×#4956 #13055992

Order 1344.

572 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#67C211yesmaximal, characteristic
2C22×C4×D3×C7C211yesmaximal, characteristic, complemented
3C2×C3×C7×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×C7×#217C211yesmaximal, characteristic, complemented
5C2×C7×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C7×C3⋊#95C211yesmaximal, characteristic, complemented
7C7×C3⋊#95C211yesmaximal, characteristic, complemented
8C7×#386—31nomaximal, Hall
9C24×C3×C7C2211yescharacteristic
10C23×D3×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C2×C7×#67—22no—
14C2×C7×#67—22no—
15C2×C7×#67C414yescomplemented
16C22×C7×Dic3C2211yescharacteristic
17C22×C7×Dic3C2211yescharacteristic
18C22×C7×Dic3C2211yescharacteristic
19C2×C4×D3×C7—22no—
20C2×C4×D3×C7—22no—
21C2×C4×D3×C7—22no—
22C3×C7×C22⋊C4—22no—
23C7×#217—22no—
24C7×(C2×D3)⋊C4—22no—
25C22×C7×D4—31no—
26C23×C4×C7—31no—
27C2×C7×C22⋊C4—31no—
28C2×C7×C22⋊C4—31no—
29C2×C7×C22⋊C4D311yescharacteristic, complemented
30C7×#95—31no—
31C7×#95—31no—
32#4956C711yesmaximal, characteristic, direct product factor, Hall
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7—21no—
36C23×C3×C7—21no—
37C23×C3×C7—21no—
38C23×C3×C7—21no—
39C23×C3×C7C2×C412yescomplemented
40C23×C3×C7C2311yescharacteristic
41C22×D3×C7—21no—
42C22×D3×C7—21no—
43C22×D3×C7—21no—
44C22×D3×C7—21no—
45C22×D3×C7C2×C412yes—
46C22×D3×C7D412yes—
47C22×D3×C7D412yes—
48C2×C3×C4×C7—21no—
49C2×C3×C4×C7—21no—
50C2×C3×C4×C7—21no—
51C2×C3×C4×C7—21no—
52C2×C3×C4×C7—21no—
53C2×C3×C4×C7D412yescomplemented
54C7×#67—28no—
55C2×C7×Dic3—21no—
56C2×C7×Dic3—21no—
57C2×C7×Dic3—21no—
58C2×C7×Dic3—21no—
59C2×C7×Dic3—21no—
60C2×C7×Dic3—21no—
61C2×C7×Dic3—21no—
62C2×C7×Dic3C2×C412yescomplemented
63C2×C7×Dic3D412yescomplemented
64C4×D3×C7—28no—
65C24×C7—31no—
66C24×C7C2×D311yescharacteristic
67C2×C7×D4—34no—
68C2×C7×D4—62no—
69C2×C7×D4—62no—
70C22×C4×C7—31no—
71C22×C4×C7—31no—
72C22×C4×C7—31no—
73C22×C4×C7—62no—
74C22×C4×C7—62no—
75C22×C4×C7—62no—
76C22×C4×C7C2×D311yescharacteristic
77C22×C4×C7C2×D311yescharacteristic
78C7×C22⋊C4—22no—
79C7×C22⋊C4—62no—
80C7×C22⋊C4—62no—
81C22×#67C2×C711yescharacteristic
82C22×C4×D3C2×C711yescharacteristic, complemented
83C2×C3×C22⋊C4C2×C711yescharacteristic, complemented
84C2×#217C2×C711yescharacteristic, complemented
85C2×(C2×D3)⋊C4C2×C711yescharacteristic, complemented
86C3⋊#95C2×C711yescharacteristic, complemented
87C3⋊#95C2×C711yescharacteristic, complemented
88C22×C3×C7—22no—
89C22×C3×C7—22no—
90C22×C3×C7—22no—
91C22×C3×C7—41no—
92C22×C3×C7—41no—
93C22×C3×C7—41no—
94C22×C3×C7—41no—
95C22×C3×C7C22×C411yescharacteristic
96C22×C3×C7C2×D411yescharacteristic
97C22×C3×C7C2×D411yescharacteristic
98C22×C3×C7C2×D411yescharacteristic
99C22×C3×C7C2×D411yescharacteristic
100C22×C3×C7#4111yescharacteristic
101C22×C3×C7#4111yescharacteristic
102C2×D3×C7—22no—
103C2×D3×C7—22no—
104C2×D3×C7—22no—
105C2×D3×C7—22no—
106C2×D3×C7—22no—
107C2×D3×C7—22no—
108C2×D3×C7C22⋊C414yescomplemented
109C3×C4×C7—22no—
110C3×C4×C7—41no—
111C7×Dic3—22no—
112C7×Dic3—22no—
113C7×Dic3—41no—
114#386—31noSylow
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—32no—
122C23×C7—32no—
123C23×C7—32no—
124C23×C7—61no—
125C23×C7—61no—
126C23×C7—61no—
127C23×C7—61no—
128C23×C7C4×D312yescomplemented
129C23×C7C22×D311yescharacteristic
130C7×D4—68no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—32no—
137C2×C4×C7—32no—
138C2×C4×C7—68no—
139C2×C4×C7—61no—
140C2×C4×C7—61no—
141C2×C4×C7—61no—
142C2×C4×C7—61no—
143C2×C4×C7—61no—
144C2×C4×C7—61no—
145C2×C4×C7—61no—
146C2×C4×C7#6712yescomplemented
147C24×C3C22×C711yescharacteristic
148C23×D3C22×C711yescharacteristic
149C22×C3×C4C22×C711yescharacteristic
150C22×C3×C4C22×C711yescharacteristic
151C2×#67—22no—
152C2×#67—22no—
153C2×#67C4×C714yescomplemented
154C22×Dic3C22×C711yescharacteristic
155C22×Dic3C22×C711yescharacteristic
156C22×Dic3C22×C711yescharacteristic
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C3×C22⋊C4—22no—
161#217—22no—
162(C2×D3)⋊C4—22no—
163C2×C3×C7—42no—
164C2×C3×C7C2×C22⋊C411yescharacteristic
165C2×C3×C7C4×D411yescharacteristic
166C2×C3×C7C4×D411yescharacteristic
167C2×C3×C7#12011yescharacteristic
168C2×C3×C7#12111yescharacteristic
169C2×C3×C7#12111yescharacteristic
170C2×C3×C7#12311yescharacteristic
171D3×C7—24no—
172C22×D4—31no—
173C23×C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4D3×C711yescharacteristic, complemented
177#95—31no—
178#95—31no—
179C22×C7—22no—
180C22×C7—22no—
181C22×C7—22no—
182C22×C7—34no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—41no—
187C22×C7—62no—
188C22×C7—62no—
189C22×C7—62no—
190C22×C7—62no—
191C22×C7—62no—
192C22×C7—62no—
193C22×C7C2×C4×D311yescharacteristic
194C22×C7#23511yescharacteristic
195C22×C7D3×D411yescharacteristic
196C22×C7D3×D411yescharacteristic
197C22×C7D3×D411yescharacteristic
198C22×C7#23711yescharacteristic
199C22×C7C2×#6711yescharacteristic
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—62no—
203C4×C7—62no—
204C4×C7—121no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3C2×C4×C712yescomplemented
212C23×C3C23×C711yescharacteristic
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3C2×C4×C712yes—
218C22×D3C7×D412yes—
219C22×D3C7×D412yes—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4C7×D412yescomplemented
226#67—28no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3C2×C4×C712yescomplemented
235C2×Dic3C7×D412yescomplemented
236C4×D3—28no—
237C3×C7#38611yescharacteristic, complemented, Hall
238C24—31no—
239C24C2×D3×C711yescharacteristic
240C2×D4—34no—
241C2×D4—62no—
242C2×D4—62no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4C2×D3×C711yescharacteristic
250C22×C4C2×D3×C711yescharacteristic
251C22⋊C4—22no—
252C22⋊C4—62no—
253C22⋊C4—62no—
254C2×C7—42no—
255C2×C7—64no—
256C2×C7D3×C22⋊C411yescharacteristic
257C2×C7#67⋊C411yescharacteristic
258C2×C7#88311yescharacteristic
259C2×C7#88411yescharacteristic
260C2×C7C4×#6711yescharacteristic
261C2×C7#93711yescharacteristic
262C2×C7#93811yescharacteristic
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3C22×C4×C711yescharacteristic
271C22×C3C2×C7×D411yescharacteristic
272C22×C3C2×C7×D411yescharacteristic
273C22×C3C2×C7×D411yescharacteristic
274C22×C3C2×C7×D411yescharacteristic
275C22×C3C7×#4111yescharacteristic
276C22×C3C7×#4111yescharacteristic
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—22no—
282C2×D3—22no—
283C2×D3C7×C22⋊C414yescomplemented
284C3×C4—22no—
285C3×C4—41no—
286Dic3—22no—
287Dic3—22no—
288Dic3—41no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—32no—
296C23—32no—
297C23—32no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23C4×D3×C712yescomplemented
303C23C22×D3×C711yescharacteristic
304D4—68no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—32no—
311C2×C4—32no—
312C2×C4—68no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4C7×#6712yescomplemented
321C7#495611yescharacteristic, direct product factor, Sylow
322C2×C3—42no—
323C2×C3C2×C7×C22⋊C411yescharacteristic
324C2×C3C4×C7×D411yescharacteristic
325C2×C3C4×C7×D411yescharacteristic
326C2×C3C7×#12011yescharacteristic
327C2×C3C7×#12111yescharacteristic
328C2×C3C7×#12111yescharacteristic
329C2×C3C7×#12311yescharacteristic
330D3—24no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—34no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—62no—
340C22—62no—
341C22—62no—
342C22—62no—
343C22—62no—
344C22—62no—
345C22C2×C4×D3×C711yescharacteristic
346C22C7×#23511yescharacteristic
347C22D3×C7×D411yescharacteristic
348C22D3×C7×D411yescharacteristic
349C22D3×C7×D411yescharacteristic
350C22C7×#23711yescharacteristic
351C22C2×C7×#6711yescharacteristic
352C4—22no—
353C4—41no—
354C4—62no—
355C4—62no—
356C4—121no—
357C3C7×#38611yescharacteristic, complemented, Sylow
358C2—42no—
359C2—64no—
360C2D3×C7×C22⋊C411yescharacteristic
361C2C7×#67⋊C411yescharacteristic
362C2C7×#88311yescharacteristic
363C2C7×#88411yescharacteristic
364C2C4×C7×#6711yescharacteristic
365C2C7×#93711yescharacteristic
366C2C7×#93811yescharacteristic
subgroupG/H
centerC23×C7C22×D3
commutatorC22×C3C22×C4×C7
FrattiniC23C22×D3×C7
FittingC2×C3×C7×C22⋊C4C2
socleC23×C3×C7C23
prime radicalC22×C7×#67C2
cosocleC23×C3C23×C7
hypercenterC2×C7×C22⋊C4D3
hypocenterC3C7×#386
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C7 ◁ C2×C7×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×D3×C7C2semidirect
3. C2×C3×C7×C22⋊C4C2semidirect
4. C2×C7×#217C2semidirect
5. C2×C7×(C2×D3)⋊C4C2semidirect
6. C7×C3⋊#95C2semidirect
7. C7×C3⋊#95C2semidirect
15. C2×C7×#67C4semidirect
29. C2×C7×C22⋊C4D3semidirect
32. #4956C7direct (N × H)
39. C23×C3×C7C2×C4semidirect
53. C2×C3×C4×C7D4semidirect
62. C2×C7×Dic3C2×C4semidirect
63. C2×C7×Dic3D4semidirect
82. C22×C4×D3C2×C7semidirect
83. C2×C3×C22⋊C4C2×C7semidirect
84. C2×#217C2×C7semidirect
85. C2×(C2×D3)⋊C4C2×C7semidirect
86. C3⋊#95C2×C7semidirect
87. C3⋊#95C2×C7semidirect
108. C2×D3×C7C22⋊C4semidirect
128. C23×C7C4×D3semidirect
146. C2×C4×C7#67semidirect
153. C2×#67C4×C7semidirect
176. C2×C22⋊C4D3×C7semidirect
211. C23×C3C2×C4×C7semidirect
225. C2×C3×C4C7×D4semidirect
234. C2×Dic3C2×C4×C7semidirect
235. C2×Dic3C7×D4semidirect
237. C3×C7#386semidirect
283. C2×D3C7×C22⋊C4semidirect
302. C23C4×D3×C7semidirect
320. C2×C4C7×#67semidirect
321. C7#4956direct (N × H)
357. C3C7×#386semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#672
C22×C4×D3×C72
C2×C3×C7×C22⋊C42
C2×C7×#2172
C2×C7×(C2×D3)⋊C42
C7×C3⋊#952
C7×#3863
#49567