Finite Group Databasebeta

← C7×#23560 #13200776

Subgroups — C7×#23560 #13200776

Order 1792.

1082 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3458C211yesmaximal, characteristic
2C7×#2911C212yesmaximal
3C7×#2285C211yesmaximal, characteristic
4C7×#2061C214yesmaximal, complemented
5C7×#2060C214yesmaximal, complemented
6C7×#1742C211yesmaximal, characteristic
7C2×C7×#339C212yesmaximal
8C22×C7×M16C2212yes—
9C7×#544—24no—
10C7×#544C2214yes—
11C7×#543—22no—
12C7×#533—24no—
13C7×#533—22no—
14C2×C7×#117C2212yes—
15C2×C7×#117C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C7×#428—24no—
19C7×#428C2218yes—
20C7×#427—24no—
21C7×#384C2212yes—
22C7×#382C2212yes—
23C7×#380—22no—
24C4×C7×C4⋊C4—22no—
25C7×#343C2214yes—
26C7×#339C2214yes—
27C7×#339C2214yes—
28#23560C711yesmaximal, characteristic, direct product factor, Sylow
29C23×C4×C7—21no—
30C23×C4×C7C2311yescharacteristic
31C2×C7×M16—22no—
32C2×C7×M16—24no—
33C2×C7×M16Q818yescomplemented
34C2×C7×M16C2314yes—
35C22×C7×C8—22no—
36C7×C4⋊Q8—24no—
37C7×C4⋊Q8—28no—
38C7×#125—24no—
39C7×#125—28no—
40C7×C22⋊Q8—216no—
41C7×C22⋊Q8—28no—
42C4×C7×Q8—28no—
43C7×#117—22no—
44C7×#117—22no—
45C7×#117—22no—
46C7×#117—22no—
47C7×#117—22no—
48C7×#117—22no—
49C7×#117D414yes—
50C7×#117C2314yes—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4D414yes—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2312yes—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4D412yes—
58C2×C42×C7—21no—
59C2×C42×C7—22no—
60C2×C42×C7—22no—
61C2×C42×C7—21no—
62C2×C42×C7D412yes—
63C2×C42×C7Q814yescomplemented
64C2×C42×C7C2311yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C7×#107—28no—
67C7×#106—28no—
68C7×#95—24no—
69#3458C2×C711yescharacteristic
70#2911C2×C712yes—
71#2285C2×C711yescharacteristic
72#2061C2×C714yescomplemented
73#2060C2×C714yescomplemented
74#1742C2×C711yescharacteristic
75C2×#339C2×C712yes—
76C24×C7C2×D411yescharacteristic
77C2×C7×Q8—28no—
78C22×C4×C7—21no—
79C22×C4×C7—24no—
80C22×C4×C7—21no—
81C22×C4×C7—24no—
82C22×C4×C7—22no—
83C22×C4×C7—44no—
84C22×C4×C7C2×D414yes—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D412yes—
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×Q814yes—
89C22×C4×C7C2×Q812yes—
90C22×C4×C7#4112yes—
91C22×C4×C7C2411yescharacteristic
92C7×M16—28no—
93C7×M16—24no—
94C7×M16C2×Q818yes—
95C2×C7×C8—28no—
96C2×C7×C8—24no—
97C7×C4⋊C4—216no—
98C7×C4⋊C4—28no—
99C7×C4⋊C4—24no—
100C7×C4⋊C4—24no—
101C7×C4⋊C4—24no—
102C7×C4⋊C4—48no—
103C7×C4⋊C4C2×D418yes—
104C7×C22⋊C4—24no—
105C7×C22⋊C4—24no—
106C7×C22⋊C4—24no—
107C7×C22⋊C4—42no—
108C42×C7—22no—
109C42×C7—24no—
110C42×C7—22no—
111C42×C7—24no—
112C42×C7—24no—
113C42×C7—21no—
114C42×C7—21no—
115C42×C7—41no—
116C42×C7—41no—
117C42×C7C2×Q812yes—
118C42×C7C2×Q812yes—
119C22×M16C22×C712yes—
120#544—24no—
121#544C22×C714yes—
122#543—22no—
123#533—24no—
124#533—22no—
125C2×#117C22×C712yes—
126C2×#117C22×C711yescharacteristic
127C2×#117C22×C711yescharacteristic
128C22×C42C22×C711yescharacteristic
129#428—24no—
130#428C22×C718yes—
131#427—24no—
132#384C22×C712yes—
133#382C22×C712yes—
134#380—22no—
135C4×C4⋊C4—22no—
136#343C22×C714yes—
137#339C22×C714yes—
138#339C22×C714yes—
139C23×C7—42no—
140C23×C7#12012yes—
141C23×C7C22⋊Q812yes—
142C23×C7C4⋊Q812yes—
143C23×C7C22×D411yescharacteristic
144C7×Q8—88no—
145C2×C4×C7—24no—
146C2×C4×C7—22no—
147C2×C4×C7—24no—
148C2×C4×C7—28no—
149C2×C4×C7—22no—
150C2×C4×C7—48no—
151C2×C4×C7—42no—
152C2×C4×C7—44no—
153C2×C4×C7—44no—
154C2×C4×C7—42no—
155C2×C4×C7—84no—
156C2×C4×C7#12011yescharacteristic
157C2×C4×C7#12011yescharacteristic
158C2×C4×C7C22⋊Q814yes—
159C2×C4×C7C22⋊Q814yes—
160C2×C4×C7C22⋊Q812yes—
161C2×C4×C7C4⋊Q814yes—
162C2×C4×C7C4⋊Q814yes—
163C2×C4×C7C4⋊Q812yes—
164C2×C4×C7C22×D412yes—
165C2×C4×C7C22×Q812yes—
166C2×C4×C7C22×Q811yescharacteristic
167C2×C4×C7C2×#4111yescharacteristic
168C7×C8—28no—
169C23×C4—21no—
170C23×C4C23×C711yescharacteristic
171C2×M16—22no—
172C2×M16—24no—
173C2×M16C7×Q818yescomplemented
174C2×M16C23×C714yes—
175C22×C8—22no—
176C4⋊Q8—24no—
177C4⋊Q8—28no—
178#125—24no—
179#125—28no—
180C22⋊Q8—216no—
181C22⋊Q8—28no—
182C4×Q8—28no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117C7×D414yes—
190#117C23×C714yes—
191C2×C4⋊C4—24no—
192C2×C4⋊C4C7×D414yes—
193C2×C4⋊C4C23×C712yes—
194C2×C4⋊C4C23×C712yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4C7×D412yes—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42C7×D412yes—
203C2×C42C7×Q814yescomplemented
204C2×C42C23×C711yescharacteristic
205C2×C42C23×C711yescharacteristic
206#107—28no—
207#106—28no—
208#95—24no—
209C22×C7—22no—
210C22×C7—24no—
211C22×C7—42no—
212C22×C7—42no—
213C22×C7#39514yes—
214C22×C7C2×#12011yescharacteristic
215C22×C7C2×C22⋊Q811yescharacteristic
216C22×C7C2×C4⋊Q811yescharacteristic
217C4×C7—24no—
218C4×C7—42no—
219C4×C7—44no—
220C4×C7—48no—
221C4×C7—82no—
222C4×C7#39512yes—
223C4×C7#39512yes—
224C4×C7C2×C22⋊Q812yes—
225C4×C7C2×C4⋊Q812yes—
226C24C2×C7×D411yescharacteristic
227C2×Q8—28no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4C2×C7×D414yes—
235C22×C4C2×C7×D411yescharacteristic
236C22×C4C2×C7×D412yes—
237C22×C4C2×C7×D412yes—
238C22×C4C2×C7×Q814yes—
239C22×C4C2×C7×Q812yes—
240C22×C4C7×#4112yes—
241C22×C4C24×C711yescharacteristic
242M16—28no—
243M16—24no—
244M16C2×C7×Q818yes—
245C2×C8—28no—
246C2×C8—24no—
247C4⋊C4—216no—
248C4⋊C4—28no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—48no—
253C4⋊C4C2×C7×D418yes—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—42no—
258C42—24no—
259C42—22no—
260C42—22no—
261C42—24no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—41no—
266C42—41no—
267C42C2×C7×Q812yes—
268C42C2×C7×Q812yes—
269C2×C7—21no—
270C2×C7—21no—
271C2×C7—24no—
272C2×C7C2×#39511yescharacteristic
273C2×C7#301812yes—
274C23—42no—
275C23C7×#12012yes—
276C23C7×C22⋊Q812yes—
277C23C7×C4⋊Q812yes—
278C23C22×C7×D411yescharacteristic
279Q8—88no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4C7×#12011yescharacteristic
292C2×C4C7×#12011yescharacteristic
293C2×C4C7×C22⋊Q814yes—
294C2×C4C7×C22⋊Q814yes—
295C2×C4C7×C22⋊Q812yes—
296C2×C4C7×C4⋊Q814yes—
297C2×C4C7×C4⋊Q814yes—
298C2×C4C7×C4⋊Q812yes—
299C2×C4C22×C7×D412yes—
300C2×C4C22×C7×Q812yes—
301C2×C4C22×C7×Q811yescharacteristic
302C2×C4C2×C7×#4111yescharacteristic
303C8—28no—
304C7#2356011yescharacteristic, direct product factor, Sylow
305C22—22no—
306C22—24no—
307C22—42no—
308C22—42no—
309C22C7×#39514yes—
310C22C2×C7×#12011yescharacteristic
311C22C2×C7×C22⋊Q811yescharacteristic
312C22C2×C7×C4⋊Q811yescharacteristic
313C4—24no—
314C4—42no—
315C4—44no—
316C4—48no—
317C4—82no—
318C4C7×#39512yes—
319C4C7×#39512yes—
320C4C2×C7×C22⋊Q812yes—
321C4C2×C7×C4⋊Q812yes—
322C2—21no—
323C2—21no—
324C2—24no—
325C2C2×C7×#39511yescharacteristic
326C2C7×#301812yes—
subgroupG/H
centerC2×C4×C7#120
commutator, Frattini, cosocleC22×C4C24×C7
socleC22×C7C2×#120
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#2061C2semidirect
5. C7×#2060C2semidirect
28. #23560C7direct (N × H)
33. C2×C7×M16Q8semidirect
63. C2×C42×C7Q8semidirect
72. #2061C2×C7semidirect
73. #2060C2×C7semidirect
173. C2×M16C7×Q8semidirect
203. C2×C42C7×Q8semidirect
304. C7#23560direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34582
C7×#29112
C7×#22852
C7×#20612
C7×#20602
C7×#17422
C2×C7×#3392
#235607