Finite Group Databasebeta

← C7×#23989 #13201205

Subgroups — C7×#23989 #13201205

Order 1792.

1210 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3491C211yesmaximal, characteristic
2C7×#2614C212yesmaximal, complemented
3C7×#2310C211yesmaximal, characteristic, complemented
4C7×#2030C212yesmaximal, complemented
5C7×#2022C212yesmaximal
6C7×#1899C212yesmaximal, complemented
7C7×#1890C214yesmaximal, complemented
8C7×#1820C211yesmaximal, characteristic, complemented
9C23×C7×Q8C2211yescharacteristic
10C7×#552—24no—
11C7×#551—24no—
12C7×#543C2211yescharacteristic
13C7×#539—22no—
14C7×#534C2211yescharacteristic
15C7×#529—22no—
16C2×C7×#123C2214yes—
17C2×C7×#123C2212yes—
18C7×#518C2212yes—
19C2×C7×#117C2211yescharacteristic
20C2×C7×#117C2211yescharacteristic
21C2×C7×#99C2214yes—
22C7×#395—24no—
23C7×#393—24no—
24C7×#391—22no—
25C7×#391—22no—
26C7×#381C2212yes—
27C7×#381C2214yes—
28C7×#381C2212yes—
29C7×#381C2212yes—
30C7×#342C2212yes—
31C7×#342C2214yes—
32C7×#342C2212yes—
33#23989C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8—21no—
35C22×C7×Q8—21no—
36C22×C7×Q8—24no—
37C22×C7×Q8—24no—
38C22×C7×Q8—42no—
39C22×C7×D4C2312yes—
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C7×#126—24no—
43C7×#126—24no—
44C7×#125—22no—
45C7×#125—24no—
46C7×#124—22no—
47C7×#123—24no—
48C7×#123—22no—
49C7×#123—22no—
50C7×#123—24no—
51C7×#123—22no—
52C7×#123—28no—
53C7×#123—44no—
54C7×C22⋊Q8—24no—
55C7×C22⋊Q8—22no—
56C7×C22⋊Q8—24no—
57C7×#121—22no—
58C7×#120—22no—
59C4×C7×Q8—22no—
60C4×C7×D4—22no—
61C4×C7×D4—24no—
62C4×C7×D4—24no—
63C7×#117—24no—
64C7×#117—24no—
65C7×#117—22no—
66C7×#117—42no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—24no—
69C2×C7×C4⋊C4D412yes—
70C2×C7×C4⋊C4D414yes—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4D414yes—
74C2×C7×C22⋊C4C2314yes—
75C2×C7×C22⋊C4C2312yes—
76C2×C7×C22⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4C2312yes—
79C2×C42×C7—21no—
80C2×C42×C7D412yes—
81C7×#99—44no—
82C7×#95—24no—
83C7×#95—24no—
84C7×#95—28no—
85C7×#95—24no—
86#3491C2×C711yescharacteristic
87#2614C2×C712yescomplemented
88#2310C2×C711yescharacteristic, complemented
89#2030C2×C712yescomplemented
90#2022C2×C712yes—
91#1899C2×C712yescomplemented
92#1890C2×C714yescomplemented
93#1820C2×C711yescharacteristic, complemented
94C24×C7C2×D412yes—
95C24×C7C2411yescharacteristic
96C2×C7×Q8—24no—
97C2×C7×Q8—24no—
98C2×C7×Q8—22no—
99C2×C7×Q8—416no—
100C2×C7×Q8—42no—
101C2×C7×Q8—42no—
102C2×C7×Q8—42no—
103C2×C7×Q8—42no—
104C2×C7×D4—22no—
105C2×C7×D4—22no—
106C2×C7×D4—24no—
107C2×C7×D4—24no—
108C2×C7×D4—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—24no—
113C22×C4×C7—22no—
114C22×C4×C7—24no—
115C22×C4×C7—24no—
116C22×C4×C7—22no—
117C22×C4×C7—24no—
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2×D414yes—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D414yes—
125C22×C4×C7#4114yes—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—24no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—24no—
131C7×C4⋊C4—24no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—48no—
134C7×C4⋊C4—42no—
135C7×C22⋊C4—24no—
136C7×C22⋊C4—24no—
137C7×C22⋊C4—22no—
138C7×C22⋊C4—22no—
139C7×C22⋊C4—22no—
140C7×C22⋊C4—22no—
141C7×C22⋊C4—24no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—24no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—48no—
147C42×C7—24no—
148C42×C7—22no—
149C42×C7—22no—
150C42×C7—42no—
151C23×Q8C22×C711yescharacteristic
152#552—24no—
153#551—24no—
154#543C22×C711yescharacteristic
155#539—22no—
156#534C22×C711yescharacteristic
157#529—22no—
158C2×#123C22×C714yes—
159C2×#123C22×C712yes—
160#518C22×C712yes—
161C2×#117C22×C711yescharacteristic
162C2×#117C22×C711yescharacteristic
163C2×#99C22×C714yes—
164#395—24no—
165#393—24no—
166#391—22no—
167#391—22no—
168#381C22×C712yes—
169#381C22×C712yes—
170#381C22×C712yes—
171#381C22×C714yes—
172#342C22×C712yes—
173#342C22×C712yes—
174#342C22×C714yes—
175C23×C7—22no—
176C23×C7—24no—
177C23×C7—42no—
178C23×C7—44no—
179C23×C7C22×D411yescharacteristic
180C23×C7C22×D412yes—
181C23×C7C2×#4112yes—
182C23×C7H(5,𝔽2)12yes—
183C7×Q8—48no—
184C7×Q8—48no—
185C7×Q8—82no—
186C7×D4—42no—
187C7×D4—42no—
188C7×D4—44no—
189C7×D4—82no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—28no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—24no—
199C2×C4×C7—24no—
200C2×C4×C7—28no—
201C2×C4×C7—24no—
202C2×C4×C7—42no—
203C2×C4×C7—42no—
204C2×C4×C7—44no—
205C2×C4×C7—42no—
206C2×C4×C7—44no—
207C2×C4×C7—42no—
208C2×C4×C7—44no—
209C2×C4×C7—48no—
210C2×C4×C7—42no—
211C2×C4×C7—44no—
212C2×C4×C7—84no—
213C2×C4×C7#12012yes—
214C2×C4×C7#12012yes—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—42no—
220C22×D4C23×C712yes—
221C23×C4C23×C712yes—
222C23×C4C23×C711yescharacteristic
223#126—24no—
224#126—24no—
225#125—22no—
226#125—24no—
227#124—22no—
228#123—24no—
229#123—22no—
230#123—22no—
231#123—24no—
232#123—22no—
233#123—28no—
234#123—44no—
235C22⋊Q8—24no—
236C22⋊Q8—22no—
237C22⋊Q8—24no—
238#121—22no—
239#120—22no—
240C4×Q8—22no—
241C4×D4—22no—
242C4×D4—24no—
243C4×D4—24no—
244#117—24no—
245#117—24no—
246#117—22no—
247#117—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4C7×D412yes—
251C2×C4⋊C4C7×D414yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4C7×D414yes—
255C2×C22⋊C4C23×C712yes—
256C2×C22⋊C4C23×C714yes—
257C2×C22⋊C4C23×C711yescharacteristic
258C2×C22⋊C4C23×C711yescharacteristic
259C2×C22⋊C4C23×C712yes—
260C2×C42—21no—
261C2×C42C7×D412yes—
262#99—44no—
263#95—24no—
264#95—24no—
265#95—28no—
266#95—24no—
267C22×C7—22no—
268C22×C7—24no—
269C22×C7—42no—
270C22×C7—42no—
271C22×C7—42no—
272C22×C7—44no—
273C22×C7—84no—
274C22×C7C2×#12011yescharacteristic
275C22×C7#52511yescharacteristic
276C22×C7#53511yescharacteristic
277C22×C7#54614yes—
278C4×C7—22no—
279C4×C7—22no—
280C4×C7—44no—
281C4×C7—42no—
282C4×C7—42no—
283C4×C7—44no—
284C4×C7—42no—
285C4×C7—82no—
286C4×C7—84no—
287C24C2×C7×D412yes—
288C24C24×C711yescharacteristic
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—416no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4C2×C7×D412yes—
312C22×C4C2×C7×D414yes—
313C22×C4C2×C7×D411yescharacteristic
314C22×C4C2×C7×D411yescharacteristic
315C22×C4C2×C7×D412yes—
316C22×C4C2×C7×D412yes—
317C22×C4C2×C7×D414yes—
318C22×C4C7×#4114yes—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—48no—
327C4⋊C4—42no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—48no—
340C42—24no—
341C42—22no—
342C42—22no—
343C42—42no—
344C2×C7—21no—
345C2×C7—24no—
346C2×C7—21no—
347C2×C7—82no—
348C2×C7#240411yescharacteristic
349C2×C7#302812yes—
350C23—22no—
351C23—24no—
352C23—42no—
353C23—44no—
354C23C22×C7×D411yescharacteristic
355C23C22×C7×D412yes—
356C23C2×C7×#4112yes—
357C23C7×H(5,𝔽2)12yes—
358Q8—48no—
359Q8—48no—
360Q8—82no—
361D4—42no—
362D4—42no—
363D4—44no—
364D4—82no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—28no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—84no—
388C2×C4C7×#12012yes—
389C2×C4C7×#12012yes—
390C7#2398911yescharacteristic, direct product factor, Sylow
391C22—22no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—44no—
397C22—84no—
398C22C2×C7×#12011yescharacteristic
399C22C7×#52511yescharacteristic
400C22C7×#53511yescharacteristic
401C22C7×#54614yes—
402C4—22no—
403C4—22no—
404C4—44no—
405C4—42no—
406C4—42no—
407C4—44no—
408C4—42no—
409C4—82no—
410C4—84no—
411C2—21no—
412C2—24no—
413C2—21no—
414C2—82no—
415C2C7×#240411yescharacteristic
416C2C7×#302812yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#534C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2614C2semidirect
3. C7×#2310C2semidirect
4. C7×#2030C2semidirect
6. C7×#1899C2semidirect
7. C7×#1890C2semidirect
8. C7×#1820C2semidirect
33. #23989C7direct (N × H)
87. #2614C2×C7semidirect
88. #2310C2×C7semidirect
89. #2030C2×C7semidirect
91. #1899C2×C7semidirect
92. #1890C2×C7semidirect
93. #1820C2×C7semidirect
390. C7#23989direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34912
C7×#26142
C7×#23102
C7×#20302
C7×#20222
C7×#18992
C7×#18902
C7×#18202
#239897