Finite Group Databasebeta

← C7×#19005 #13196221

Subgroups — C7×#19005 #13196221

Order 1792.

820 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3531C211yesmaximal, characteristic
2C7×#3239C212yesmaximal, complemented
3C7×#3139C211yesmaximal, characteristic, complemented
4C7×#1664C214yesmaximal, complemented
5C7×#1663C212yesmaximal, complemented
6C7×#1662C212yesmaximal, complemented
7C7×#1537C211yesmaximal, characteristic, complemented
8C7×#1529C212yesmaximal, complemented
9C7×#564C2211yescharacteristic
10C7×#563—22no—
11C7×#563—22no—
12C7×#563C2212yes—
13C7×#562—22no—
14C7×#562C2211yescharacteristic
15C7×#533—22no—
16C7×#533C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C7×#529—22no—
19C7×#529—22no—
20C7×#529C2212yes—
21C7×#501C2214yes—
22C7×#491C2212yes—
23C7×#491C2212yes—
24C7×#489C2214yes—
25C7×#487—22no—
26C7×#484—22no—
27C7×#481—22no—
28C7×#475—22no—
29C7×#474—22no—
30C7×#432C2212yes—
31C7×#419—22no—
32C4×C7×M16C2211yescharacteristic
33C7×#330C2214yes—
34C7×#330C2212yes—
35C7×#330C2212yes—
36C7×#329C2214yes—
37#19005C711yesmaximal, characteristic, direct product factor, Sylow
38C22×C7×Q8—21no—
39C2×C7×M16—21no—
40C7×C4⋊Q8—22no—
41C7×C4⋊Q8—22no—
42C7×C4⋊Q8—22no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8D412yes—
45C7×C4⋊Q8C2312yes—
46C7×#126—22no—
47C7×#126—22no—
48C7×#126—22no—
49C7×#126—22no—
50C7×#126—22no—
51C7×#125—22no—
52C7×#125—22no—
53C7×#125—22no—
54C7×#125—22no—
55C7×#125C2311yescharacteristic
56C7×#125C2311yescharacteristic
57C7×#125C2312yes—
58C7×#124—22no—
59C7×#124—22no—
60C7×#124C2312yes—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8D412yes—
74C7×#121—22no—
75C7×#121—22no—
76C7×#121D412yes—
77C4×C7×Q8—22no—
78C4×C7×Q8—22no—
79C4×C7×Q8D412yes—
80C4×C7×D4—22no—
81C4×C7×D4—22no—
82C4×C7×D4D412yes—
83C7×#117—22no—
84C7×#117—22no—
85C7×#117D412yes—
86C2×C7×C4⋊C4—22no—
87C2×C42×C7—22no—
88C2×C42×C7C2311yescharacteristic
89C7×#107—24no—
90C7×#106—24no—
91C7×C4⋊C8—22no—
92C7×#103—24no—
93C7×#103—24no—
94C7×#103—24no—
95C7×#102—24no—
96C7×C22⋊C8—22no—
97C7×#97C2314yes—
98C7×#97C2312yes—
99C4×C7×C8—21no—
100#3531C2×C711yescharacteristic
101#3239C2×C712yescomplemented
102#3139C2×C711yescharacteristic, complemented
103#1664C2×C714yescomplemented
104#1663C2×C712yescomplemented
105#1662C2×C712yescomplemented
106#1537C2×C711yescharacteristic, complemented
107#1529C2×C712yescomplemented
108C2×C7×Q8—22no—
109C2×C7×Q8—22no—
110C2×C7×Q8—22no—
111C2×C7×Q8—44no—
112C2×C7×Q8—41no—
113C2×C7×Q8C2×D412yes—
114C2×C7×D4—22no—
115C2×C7×D4—22no—
116C2×C7×D4C2×D412yes—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—41no—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D412yes—
123C7×M16—42no—
124C2×C7×C8—22no—
125C2×C7×C8—24no—
126C7×C4⋊C4—22no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—24no—
140C7×C4⋊C4—42no—
141C7×C4⋊C4C2×D412yes—
142C7×C4⋊C4C2×D412yes—
143C7×C4⋊C4C2×D412yes—
144C7×C4⋊C4C2×D412yes—
145C7×C22⋊C4—22no—
146C7×C22⋊C4—22no—
147C7×C22⋊C4—22no—
148C7×C22⋊C4—22no—
149C7×C22⋊C4—22no—
150C7×C22⋊C4—22no—
151C7×C22⋊C4—22no—
152C7×C22⋊C4—22no—
153C7×C22⋊C4—22no—
154C7×C22⋊C4—22no—
155C42×C7—22no—
156C42×C7—22no—
157C42×C7—22no—
158C42×C7—22no—
159C42×C7C2×D411yescharacteristic
160C42×C7C2×D412yes—
161C42×C7C2411yescharacteristic
162#564C22×C711yescharacteristic
163#563—22no—
164#563—22no—
165#563C22×C712yes—
166#562—22no—
167#562C22×C711yescharacteristic
168#533—22no—
169#533C22×C711yescharacteristic
170C2×C4⋊Q8C22×C711yescharacteristic
171#529—22no—
172#529—22no—
173#529C22×C712yes—
174#501C22×C714yes—
175#491C22×C712yes—
176#491C22×C712yes—
177#489C22×C714yes—
178#487—22no—
179#484—22no—
180#481—22no—
181#475—22no—
182#474—22no—
183#432C22×C712yes—
184#419—22no—
185C4×M16C22×C711yescharacteristic
186#330C22×C712yes—
187#330C22×C712yes—
188#330C22×C714yes—
189#329C22×C714yes—
190C23×C7—22no—
191C23×C7#12011yescharacteristic
192C7×Q8—42no—
193C7×Q8—44no—
194C7×Q8—82no—
195C7×Q8—82no—
196C7×Q8—82no—
197C7×D4—42no—
198C7×D4—82no—
199C7×D4—82no—
200C2×C4×C7—22no—
201C2×C4×C7—22no—
202C2×C4×C7—22no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—22no—
206C2×C4×C7—42no—
207C2×C4×C7—44no—
208C2×C4×C7—42no—
209C2×C4×C7—82no—
210C2×C4×C7—82no—
211C2×C4×C7—82no—
212C2×C4×C7#12012yes—
213C2×C4×C7#12011yescharacteristic
214C2×C4×C7C22×D411yescharacteristic
215C2×C4×C7C22×D412yes—
216C7×C8—44no—
217C7×C8—42no—
218C22×Q8—21no—
219C2×M16—21no—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—21no—
224C4⋊Q8C7×D412yes—
225C4⋊Q8C23×C712yes—
226#126—22no—
227#126—22no—
228#126—22no—
229#126—22no—
230#126—22no—
231#125—22no—
232#125—22no—
233#125—22no—
234#125—22no—
235#125C23×C712yes—
236#125C23×C711yescharacteristic
237#125C23×C711yescharacteristic
238#124—22no—
239#124—22no—
240#124C23×C712yes—
241#123—22no—
242#123—22no—
243#123—22no—
244#123—22no—
245#123—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8C7×D412yes—
254#121—22no—
255#121—22no—
256#121C7×D412yes—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8C7×D412yes—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4C7×D412yes—
263#117—22no—
264#117—22no—
265#117C7×D412yes—
266C2×C4⋊C4—22no—
267C2×C42—22no—
268C2×C42C23×C711yescharacteristic
269#107—24no—
270#106—24no—
271C4⋊C8—22no—
272#103—24no—
273#103—24no—
274#103—24no—
275#102—24no—
276C22⋊C8—22no—
277#97C23×C712yes—
278#97C23×C714yes—
279C4×C8—21no—
280C22×C7—21no—
281C22×C7—22no—
282C22×C7—42no—
283C22×C7—42no—
284C22×C7—42no—
285C22×C7C2×#12011yescharacteristic
286C4×C7—21no—
287C4×C7—22no—
288C4×C7—22no—
289C4×C7—42no—
290C4×C7—42no—
291C4×C7—42no—
292C4×C7—42no—
293C4×C7—42no—
294C4×C7—82no—
295C4×C7#45412yes—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—44no—
300C2×Q8—41no—
301C2×Q8C2×C7×D412yes—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4C2×C7×D412yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4C2×C7×D411yescharacteristic
310C22×C4C2×C7×D412yes—
311M16—42no—
312C2×C8—22no—
313C2×C8—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4C2×C7×D412yes—
330C4⋊C4C2×C7×D412yes—
331C4⋊C4C2×C7×D412yes—
332C4⋊C4C2×C7×D412yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42C2×C7×D411yescharacteristic
348C42C2×C7×D412yes—
349C42C24×C711yescharacteristic
350C2×C7—41no—
351C2×C7—82no—
352C2×C7C2×#45411yescharacteristic
353C2×C7#302212yes—
354C23—22no—
355C23C7×#12011yescharacteristic
356Q8—42no—
357Q8—44no—
358Q8—82no—
359Q8—82no—
360Q8—82no—
361D4—42no—
362D4—82no—
363D4—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4C7×#12012yes—
377C2×C4C7×#12011yescharacteristic
378C2×C4C22×C7×D411yescharacteristic
379C2×C4C22×C7×D412yes—
380C8—44no—
381C8—42no—
382C7#1900511yescharacteristic, direct product factor, Sylow
383C22—21no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22C2×C7×#12011yescharacteristic
389C4—21no—
390C4—22no—
391C4—22no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—82no—
398C4C7×#45412yes—
399C2—41no—
400C2—82no—
401C2C2×C7×#45411yescharacteristic
402C2C7×#302212yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3531C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3239C2semidirect
3. C7×#3139C2semidirect
4. C7×#1664C2semidirect
5. C7×#1663C2semidirect
6. C7×#1662C2semidirect
7. C7×#1537C2semidirect
8. C7×#1529C2semidirect
37. #19005C7direct (N × H)
101. #3239C2×C7semidirect
102. #3139C2×C7semidirect
103. #1664C2×C7semidirect
104. #1663C2×C7semidirect
105. #1662C2×C7semidirect
106. #1537C2×C7semidirect
107. #1529C2×C7semidirect
382. C7#19005direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35312
C7×#32392
C7×#31392
C7×#16642
C7×#16632
C7×#16622
C7×#15372
C7×#15292
#190057