Finite Group Databasebeta

← C7×#18294 #13195510

Subgroups — C7×#18294 #13195510

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#544C211yesmaximal, characteristic, complemented
2C7×#3187C212yesmaximal
3C7×#2618C211yesmaximal, characteristic
4C7×#1611C214yesmaximal, complemented
5C7×#1599C214yesmaximal, complemented
6C7×#1520C212yesmaximal, complemented
7C2×C7×#324C211yesmaximal, characteristic, complemented
8C7×#544—24no—
9C7×#544—22no—
10C7×#544C2214yes—
11C2×C7×C4⋊Q8—22no—
12C2×C7×#125—22no—
13C2×C7×#123—21no—
14C2×C7×C22⋊Q8—21no—
15C2×C7×C22⋊Q8C2212yes—
16C2×C7×#121C2211yescharacteristic
17C2×C7×#117—21no—
18C22×C7×C4⋊C4C2211yescharacteristic
19C7×#483C2214yes—
20C7×#481—22no—
21C7×#465—24no—
22C7×#450C2214yes—
23C7×#448—22no—
24C2×C7×C22⋊C8C2212yes—
25C7×#398C2212yes—
26C7×#396—22no—
27C7×#394C2212yes—
28C7×#386—21no—
29C7×#380C2211yescharacteristic
30C7×#328C2218yes—
31C7×#324C2214yes—
32#18294C711yesmaximal, characteristic, direct product factor, Sylow
33C22×C7×Q8—21no—
34C22×C7×Q8D412yes—
35C22×C7×D4—21no—
36C22×C7×D4C2311yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C2×C7×SD16—24no—
41C22×C7×C8—22no—
42C7×C4⋊Q8—28no—
43C7×C4⋊Q8—28no—
44C7×#125—28no—
45C7×#125—28no—
46C7×#123—44no—
47C7×C22⋊Q8—24no—
48C7×C22⋊Q8—24no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8D414yes—
52C7×C22⋊Q8C2314yes—
53C7×#121—22no—
54C7×#121—22no—
55C7×#117—24no—
56C7×#117—24no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4D412yes—
67C2×C7×C4⋊C4D412yes—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C42×C7—22no—
76C7×#106—24no—
77C7×#103—24no—
78C7×#102—24no—
79C7×C22⋊C8—22no—
80C7×C22⋊C8C2314yes—
81C7×#95—22no—
82C7×#95C2312yes—
83C2×#544C2×C711yescharacteristic, complemented
84#3187C2×C712yes—
85#2618C2×C711yescharacteristic
86#1611C2×C714yescomplemented
87#1599C2×C714yescomplemented
88#1520C2×C712yescomplemented
89C2×#324C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C24×C7C2×D411yescharacteristic
92C2×C7×Q8—24no—
93C2×C7×Q8—22no—
94C2×C7×Q8—82no—
95C2×C7×Q8—82no—
96C2×C7×Q8C2×D414yes—
97C2×C7×D4—22no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—42no—
101C2×C7×D4—42no—
102C2×C7×D4—81no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—22no—
112C22×C4×C7—81no—
113C22×C4×C7—81no—
114C22×C4×C7SD1614yescomplemented
115C22×C4×C7C2×D412yes—
116C22×C4×C7C2×D412yes—
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2×D412yes—
120C22×C4×C7C2411yescharacteristic
121C7×SD16—84no—
122C2×C7×C8—24no—
123C2×C7×C8—82no—
124C7×C4⋊C4—24no—
125C7×C4⋊C4—24no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—28no—
128C7×C4⋊C4—24no—
129C7×C4⋊C4—24no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—24no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—44no—
135C7×C4⋊C4—81no—
136C7×C4⋊C4C2×D414yes—
137C7×C22⋊C4—24no—
138C7×C22⋊C4—24no—
139C7×C22⋊C4—42no—
140C7×C22⋊C4—42no—
141C7×C22⋊C4—44no—
142C7×C22⋊C4—82no—
143C42×C7—28no—
144#544—24no—
145#544—22no—
146#544C22×C714yes—
147C2×C4⋊Q8—22no—
148C2×#125—22no—
149C2×#123—21no—
150C2×C22⋊Q8—21no—
151C2×C22⋊Q8C22×C712yes—
152C2×#121C22×C711yescharacteristic
153C2×#117—21no—
154C22×C4⋊C4C22×C711yescharacteristic
155#483C22×C714yes—
156#481—22no—
157#465—24no—
158#450C22×C714yes—
159#448—22no—
160C2×C22⋊C8C22×C712yes—
161#398C22×C712yes—
162#396—22no—
163#394C22×C712yes—
164#386—21no—
165#380C22×C711yescharacteristic
166#328C22×C718yes—
167#324C22×C714yes—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—42no—
171C23×C7—42no—
172C23×C7—41no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7#12011yescharacteristic
176C23×C7#12011yescharacteristic
177C23×C7C2×SD1612yes—
178C23×C7#13712yes—
179C23×C7C22×D411yescharacteristic
180C7×Q8—44no—
181C7×Q8—82no—
182C7×Q8—84no—
183C7×D4—44no—
184C7×D4—82no—
185C7×D4—82no—
186C2×C4×C7—22no—
187C2×C4×C7—22no—
188C2×C4×C7—24no—
189C2×C4×C7—24no—
190C2×C4×C7—24no—
191C2×C4×C7—24no—
192C2×C4×C7—22no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—41no—
197C2×C4×C7—44no—
198C2×C4×C7—42no—
199C2×C4×C7—44no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—42no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—81no—
208C2×C4×C7#12012yes—
209C2×C4×C7C2×SD1614yes—
210C2×C4×C7C22×D412yes—
211C7×C8—44no—
212C22×Q8—21no—
213C22×Q8C7×D412yes—
214C22×D4—21no—
215C22×D4C23×C711yescharacteristic
216C23×C4C7×D411yescharacteristic
217C23×C4C7×D411yescharacteristic
218C23×C4C23×C711yescharacteristic
219C2×SD16—24no—
220C22×C8—22no—
221C4⋊Q8—28no—
222C4⋊Q8—28no—
223#125—28no—
224#125—28no—
225#123—44no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8C7×D414yes—
231C22⋊Q8C23×C714yes—
232#121—22no—
233#121—22no—
234#117—24no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C7×D412yes—
246C2×C4⋊C4C7×D412yes—
247C2×C4⋊C4C23×C711yescharacteristic
248C2×C4⋊C4C23×C712yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—22no—
255#106—24no—
256#103—24no—
257#102—24no—
258C22⋊C8—22no—
259C22⋊C8C23×C714yes—
260#95—22no—
261#95C23×C712yes—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—42no—
269C22×C7—42no—
270C22×C7—82no—
271C22×C7—82no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7—81no—
275C22×C7#44812yes—
276C22×C7#44812yes—
277C22×C7C2×#12011yescharacteristic
278C22×C7C22×SD1611yescharacteristic
279C22×C7C2×#13711yescharacteristic
280C4×C7—24no—
281C4×C7—22no—
282C4×C7—22no—
283C4×C7—44no—
284C4×C7—42no—
285C4×C7—44no—
286C4×C7—82no—
287C4×C7—81no—
288C24—21no—
289C24C2×C7×D411yescharacteristic
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—82no—
293C2×Q8—82no—
294C2×Q8C2×C7×D414yes—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—81no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4C7×SD1614yescomplemented
313C22×C4C2×C7×D412yes—
314C22×C4C2×C7×D412yes—
315C22×C4C2×C7×D411yescharacteristic
316C22×C4C2×C7×D412yes—
317C22×C4C2×C7×D412yes—
318C22×C4C24×C711yescharacteristic
319SD16—84no—
320C2×C8—24no—
321C2×C8—82no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—81no—
334C4⋊C4C2×C7×D414yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C42—28no—
342C2×C7—21no—
343C2×C7—21no—
344C2×C7—22no—
345C2×C7—22no—
346C2×C7—161no—
347C2×C7C2×#44811yescharacteristic
348C2×C7#302011yescharacteristic
349C2×C7#302611yescharacteristic
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C7×#12011yescharacteristic
358C23C7×#12011yescharacteristic
359C23C2×C7×SD1612yes—
360C23C7×#13712yes—
361C23C22×C7×D411yescharacteristic
362Q8—44no—
363Q8—82no—
364Q8—84no—
365D4—44no—
366D4—82no—
367D4—82no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4C7×#12012yes—
391C2×C4C2×C7×SD1614yes—
392C2×C4C22×C7×D412yes—
393C8—44no—
394C7#1829411yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C7×#44812yes—
409C22C7×#44812yes—
410C22C2×C7×#12011yescharacteristic
411C22C22×C7×SD1611yescharacteristic
412C22C2×C7×#13711yescharacteristic
413C4—24no—
414C4—22no—
415C4—22no—
416C4—44no—
417C4—42no—
418C4—44no—
419C4—82no—
420C4—81no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—22no—
425C2—161no—
426C2C2×C7×#44811yescharacteristic
427C2C7×#302011yescharacteristic
428C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#544C2semidirect
4. C7×#1611C2semidirect
5. C7×#1599C2semidirect
6. C7×#1520C2semidirect
7. C2×C7×#324C2semidirect
32. #18294C7direct (N × H)
83. C2×#544C2×C7semidirect
86. #1611C2×C7semidirect
87. #1599C2×C7semidirect
88. #1520C2×C7semidirect
89. C2×#324C2×C7semidirect
114. C22×C4×C7SD16semidirect
312. C22×C4C7×SD16semidirect
394. C7#18294direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5442
C7×#31872
C7×#26182
C7×#16112
C7×#15992
C7×#15202
C2×C7×#3242
#182947