Finite Group Databasebeta

← C7×#9361 #13186577

Subgroups — C7×#9361 #13186577

Order 1792.

760 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3007C211yesmaximal, characteristic
2C7×#2339C211yesmaximal, characteristic
3C7×#1996C211yesmaximal, characteristic, complemented
4C7×#1789C211yesmaximal, characteristic
5C7×#1508C211yesmaximal, characteristic, complemented
6C7×#1305C211yesmaximal, characteristic, complemented
7C7×#1297C211yesmaximal, characteristic, complemented
8C22×C7×#41C2211yescharacteristic
9C2×C7×#137C412yescomplemented
10C2×C7×C22⋊Q8C2211yescharacteristic
11C2×C4×C7×D4—21no—
12C7×#452—22no—
13C7×#451—21no—
14C7×#450—21no—
15C2×C7×#104—22no—
16C7×#417C412yescomplemented
17C7×#407C2211yescharacteristic
18C7×#407C2211yescharacteristic
19C7×#391—21no—
20C7×#386—22no—
21C7×#381—21no—
22C7×#380C2211yescharacteristic
23C4×C7×C22⋊C4C2211yescharacteristic
24C4×C7×C22⋊C4C2211yescharacteristic
25C2×C7×#95—21no—
26C7×#342—22no—
27C7×#328—21no—
28C7×#328—21no—
29#9361C711yesmaximal, characteristic, direct product factor, Sylow
30C2×C7×#41—21no—
31C2×C7×#41—21no—
32C2×C7×#41—44no—
33C2×C7×#41C2×C412yes—
34C2×C7×#41D412yes—
35C22×C7×Q8C2×C411yescharacteristic, complemented
36C22×C7×Q8D411yescharacteristic
37C22×C7×D4—21no—
38C22×C7×D4D411yescharacteristic
39C23×C4×C7—21no—
40C23×C4×C7—22no—
41C23×C4×C7C2311yescharacteristic
42C7×#137—44no—
43C2×C7×Q16—22no—
44C2×C7×SD16—22no—
45C2×C7×M16—21no—
46C2×C7×M16C2×C412yescomplemented
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C4×C7×D4—44no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—22no—
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4D412yes—
60C2×C42×C7—21no—
61C2×C42×C7—21no—
62C7×#104—44no—
63C7×#103—22no—
64C7×#102—22no—
65C7×C22⋊C8—21no—
66C7×C22⋊C8—21no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8—21no—
69C7×#95—22no—
70C7×#95—22no—
71C7×#95—21no—
72C7×#95—21no—
73C7×#95D411yescharacteristic
74C7×#95D411yescharacteristic
75#3007C2×C711yescharacteristic
76#2339C2×C711yescharacteristic
77#1996C2×C711yescharacteristic, complemented
78#1789C2×C711yescharacteristic
79#1508C2×C711yescharacteristic, complemented
80#1305C2×C711yescharacteristic, complemented
81#1297C2×C711yescharacteristic, complemented
82C24×C7—21no—
83C24×C7C2×D411yescharacteristic
84C7×#41—24no—
85C7×#41—24no—
86C7×#41—42no—
87C7×#41—42no—
88C7×#41—44no—
89C7×#41—44no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—42no—
94C2×C7×Q8—42no—
95C2×C7×Q8C22⋊C412yes—
96C2×C7×D4—24no—
97C2×C7×D4—22no—
98C2×C7×D4—22no—
99C2×C7×D4—21no—
100C2×C7×D4—44no—
101C2×C7×D4C22⋊C412yes—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—42no—
111C22×C4×C7—44no—
112C22×C4×C7—44no—
113C22×C4×C7—44no—
114C22×C4×C7—41no—
115C22×C4×C7—82no—
116C22×C4×C7C22×C411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7#4111yescharacteristic
121C22×C4×C7#4111yescharacteristic
122C7×Q16—44no—
123C7×SD16—44no—
124C7×M16—42no—
125C7×M16—42no—
126C2×C7×C8—22no—
127C2×C7×C8—41no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—42no—
131C7×C4⋊C4—41no—
132C7×C4⋊C4—41no—
133C7×C22⋊C4—22no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—44no—
137C7×C22⋊C4—41no—
138C7×C22⋊C4—41no—
139C7×C22⋊C4—42no—
140C42×C7—42no—
141C42×C7—42no—
142C22×#41C22×C711yescharacteristic
143C2×#137C4×C712yescomplemented
144C2×C22⋊Q8C22×C711yescharacteristic
145C2×C4×D4—21no—
146#452—22no—
147#451—21no—
148#450—21no—
149C2×#104—22no—
150#417C4×C712yescomplemented
151#407C22×C711yescharacteristic
152#407C22×C711yescharacteristic
153#391—21no—
154#386—22no—
155#381—21no—
156#380C22×C711yescharacteristic
157C4×C22⋊C4C22×C711yescharacteristic
158C4×C22⋊C4C22×C711yescharacteristic
159C2×#95—21no—
160#342—22no—
161#328—21no—
162#328—21no—
163C23×C7—24no—
164C23×C7—22no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7—42no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7C4×D411yescharacteristic
173C23×C7#12011yescharacteristic
174C23×C7#12111yescharacteristic
175C7×Q8—22no—
176C7×Q8—22no—
177C7×Q8—42no—
178C7×Q8—42no—
179C7×Q8—81no—
180C7×D4—24no—
181C7×D4—22no—
182C7×D4—44no—
183C7×D4—44no—
184C7×D4#9914yes—
185C2×C4×C7—22no—
186C2×C4×C7—22no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—44no—
194C2×C4×C7—44no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—41no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—42no—
206C2×C4×C7—41no—
207C2×C4×C7—88no—
208C2×C4×C7—81no—
209C2×C4×C7—81no—
210C2×C4×C7—81no—
211C2×C4×C7—81no—
212C2×C4×C7C2×C22⋊C411yescharacteristic
213C2×C4×C7C4×D411yescharacteristic
214C2×C4×C7#12111yescharacteristic
215C2×C4×C7#12311yescharacteristic
216C7×C8—42no—
217C7×C8—81no—
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—44no—
221C2×#41C2×C4×C712yes—
222C2×#41C7×D412yes—
223C22×Q8C2×C4×C711yescharacteristic, complemented
224C22×Q8C7×D411yescharacteristic
225C22×D4—21no—
226C22×D4C7×D411yescharacteristic
227C23×C4—21no—
228C23×C4—22no—
229C23×C4C23×C711yescharacteristic
230#137—44no—
231C2×Q16—22no—
232C2×SD16—22no—
233C2×M16—21no—
234C2×M16C2×C4×C712yescomplemented
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×D4—44no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×C4×C711yescharacteristic, complemented
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4C7×D412yes—
248C2×C42—21no—
249C2×C42—21no—
250#104—44no—
251#103—22no—
252#102—22no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95C7×D411yescharacteristic
262#95C7×D411yescharacteristic
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—24no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—44no—
273C22×C7—44no—
274C22×C7—42no—
275C22×C7—42no—
276C22×C7—81no—
277C22×C7—81no—
278C22×C7#38611yescharacteristic
279C22×C7#45511yescharacteristic
280C22×C7#45611yescharacteristic
281C4×C7—21no—
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—42no—
285C4×C7—42no—
286C4×C7—81no—
287C4×C7—82no—
288C4×C7—82no—
289C4×C7—81no—
290C4×C7—81no—
291C4×C7C2×#9912yes—
292C24—21no—
293C24C2×C7×D411yescharacteristic
294#41—24no—
295#41—24no—
296#41—42no—
297#41—42no—
298#41—44no—
299#41—44no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8C7×C22⋊C412yes—
306C2×D4—24no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—21no—
310C2×D4—44no—
311C2×D4C7×C22⋊C412yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4C22×C4×C711yescharacteristic
327C22×C4C2×C7×D411yescharacteristic
328C22×C4C2×C7×D411yescharacteristic
329C22×C4C2×C7×D411yescharacteristic
330C22×C4C7×#4111yescharacteristic
331C22×C4C7×#4111yescharacteristic
332Q16—44no—
333SD16—44no—
334M16—42no—
335M16—42no—
336C2×C8—22no—
337C2×C8—41no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C22⋊C4—22no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—42no—
350C42—42no—
351C42—42no—
352C2×C7—21no—
353C2×C7—21no—
354C2×C7—41no—
355C2×C7—41no—
356C2×C7—44no—
357C2×C7#188211yescharacteristic
358C2×C7#189811yescharacteristic
359C2×C7#190611yescharacteristic
360C23—24no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23C4×C7×D411yescharacteristic
370C23C7×#12011yescharacteristic
371C23C7×#12111yescharacteristic
372Q8—22no—
373Q8—22no—
374Q8—42no—
375Q8—42no—
376Q8—81no—
377D4—24no—
378D4—22no—
379D4—44no—
380D4—44no—
381D4C7×#9914yes—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—88no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C2×C7×C22⋊C411yescharacteristic
410C2×C4C4×C7×D411yescharacteristic
411C2×C4C7×#12111yescharacteristic
412C2×C4C7×#12311yescharacteristic
413C8—42no—
414C8—81no—
415C7#936111yescharacteristic, direct product factor, Sylow
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—24no—
423C22—21no—
424C22—21no—
425C22—44no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22C7×#38611yescharacteristic
432C22C7×#45511yescharacteristic
433C22C7×#45611yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—42no—
439C4—82no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4C2×C7×#9912yes—
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2—44no—
450C2C7×#188211yescharacteristic
451C2C7×#189811yescharacteristic
452C2C7×#190611yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×#41C22
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
3. C7×#1996C2semidirect
5. C7×#1508C2semidirect
6. C7×#1305C2semidirect
7. C7×#1297C2semidirect
9. C2×C7×#137C4semidirect
16. C7×#417C4semidirect
29. #9361C7direct (N × H)
35. C22×C7×Q8C2×C4semidirect
46. C2×C7×M16C2×C4semidirect
54. C2×C7×C4⋊C4C2×C4semidirect
77. #1996C2×C7semidirect
79. #1508C2×C7semidirect
80. #1305C2×C7semidirect
81. #1297C2×C7semidirect
143. C2×#137C4×C7semidirect
150. #417C4×C7semidirect
223. C22×Q8C2×C4×C7semidirect
234. C2×M16C2×C4×C7semidirect
242. C2×C4⋊C4C2×C4×C7semidirect
415. C7#9361direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30072
C7×#23392
C7×#19962
C7×#17892
C7×#15082
C7×#13052
C7×#12972
#93617