Finite Group Databasebeta

← C7×#11839 #13189055

Subgroups — C7×#11839 #13189055

Order 1792.

698 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2404C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic, complemented
3C2×C7×#323C211yesmaximal, characteristic, complemented
4C7×#1351C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C2×C7×#120—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C2×C7×C22⋊C8—21no—
10C7×#393C411yescharacteristic, complemented
11C7×#393C2211yescharacteristic
12C7×#392C411yescharacteristic, complemented
13C7×#392C2211yescharacteristic
14C7×#386—21no—
15C7×#386—21no—
16C7×#386—21no—
17C7×#381C411yescharacteristic, complemented
18C7×#379—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11839C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7D411yescharacteristic
25C2×C7×#41—41no—
26C2×C7×#41—41no—
27C2×C7×#41—44no—
28C22×C7×Q8C2×C411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C22×C7×D4—21no—
32C22×C7×D4C2×C411yescharacteristic
33C23×C4×C7—21no—
34C23×C4×C7C2×C411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C22×C7×C8—41no—
37C7×#120—44no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4D411yescharacteristic
51C2×C7×C22⋊C4C2311yescharacteristic
52C7×C22⋊C8—22no—
53C7×C22⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#2404C2×C711yescharacteristic
60#1819C2×C711yescharacteristic, complemented
61C2×#323C2×C711yescharacteristic, complemented
62#1351C2×C712yescomplemented
63#1343C2×C712yescomplemented
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—21no—
67C24×C7—22no—
68C24×C7—41no—
69C24×C7—42no—
70C24×C7—81no—
71C24×C7C22⋊C411yescharacteristic
72C24×C7C22⋊C411yescharacteristic
73C24×C7C2×D411yescharacteristic
74C7×#41—44no—
75C7×#41—44no—
76C7×#41—44no—
77C7×#41—44no—
78C2×C7×Q8—42no—
79C2×C7×Q8—41no—
80C2×C7×D4—24no—
81C2×C7×D4—41no—
82C2×C7×D4—42no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—42no—
86C2×C7×D4—42no—
87C2×C7×D4—42no—
88C2×C7×D4—42no—
89C2×C7×D4—42no—
90C2×C7×D4—81no—
91C2×C7×D4—81no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—42no—
100C22×C4×C7—42no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7—42no—
104C22×C4×C7—81no—
105C22×C4×C7C22⋊C411yescharacteristic
106C22×C4×C7C22⋊C411yescharacteristic
107C22×C4×C7C22×C411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C2×C7×C8—42no—
110C2×C7×C8—82no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—81no—
121C7×C22⋊C4—81no—
122C7×C22⋊C4—82no—
123C7×C22⋊C4—82no—
124C22×#41C4×C711yescharacteristic, complemented
125C2×#120—21no—
126C22×C22⋊C4C22×C711yescharacteristic
127C2×C22⋊C8—21no—
128#393C4×C711yescharacteristic, complemented
129#393C22×C711yescharacteristic
130#392C4×C711yescharacteristic, complemented
131#392C22×C711yescharacteristic
132#386—21no—
133#386—21no—
134#386—21no—
135#381C4×C711yescharacteristic, complemented
136#379—21no—
137#342—21no—
138#342C22×C712yes—
139#323—21no—
140#323C22×C712yes—
141C23×C7—22no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—22no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—41no—
151C23×C7—42no—
152C23×C7—42no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7—41no—
161C23×C7—42no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—81no—
166C23×C7—81no—
167C23×C7—84no—
168C23×C7—84no—
169C23×C7#9911yescharacteristic
170C23×C7#9911yescharacteristic
171C23×C7C2×C22⋊C411yescharacteristic
172C23×C7#13512yes—
173C7×Q8—42no—
174C7×Q8—42no—
175C7×D4—44no—
176C7×D4—42no—
177C7×D4—42no—
178C7×D4—44no—
179C7×D4—84no—
180C7×D4—84no—
181C2×C4×C7—22no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—42no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—48no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—81no—
203C2×C4×C7—82no—
204C2×C4×C7—82no—
205C2×C4×C7—82no—
206C7×C8—82no—
207C25C7×D411yescharacteristic
208C2×#41—41no—
209C2×#41—41no—
210C2×#41—44no—
211C22×Q8C2×C4×C711yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C2×C4×C711yescharacteristic
216C23×C4—21no—
217C23×C4C2×C4×C711yescharacteristic
218C23×C4C7×D411yescharacteristic
219C22×C8—41no—
220#120—44no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4C2×C4×C711yescharacteristic
232C2×C22⋊C4C7×D411yescharacteristic
233C2×C22⋊C4C7×D411yescharacteristic
234C2×C22⋊C4C23×C711yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—41no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C2×C4×C711yescharacteristic
241#95C2×C4×C711yescharacteristic
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—22no—
249C22×C7—21no—
250C22×C7—42no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—42no—
254C22×C7—42no—
255C22×C7—42no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—42no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—81no—
263C22×C7—84no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—84no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7AΣL(1,𝔽16)12yes—
271C22×C7C2×#9911yescharacteristic
272C22×C7#41611yescharacteristic
273C22×C7#42111yescharacteristic
274C4×C7—41no—
275C4×C7—42no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—41no—
279C4×C7—81no—
280C4×C7—81no—
281C4×C7—162no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—41no—
287C24—42no—
288C24—81no—
289C24C7×C22⋊C411yescharacteristic
290C24C7×C22⋊C411yescharacteristic
291C24C2×C7×D411yescharacteristic
292#41—44no—
293#41—44no—
294#41—44no—
295#41—44no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×D4—24no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—81no—
309C2×D4—81no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—81no—
323C22×C4C7×C22⋊C411yescharacteristic
324C22×C4C7×C22⋊C411yescharacteristic
325C22×C4C22×C4×C711yescharacteristic
326C22×C4C2×C7×D411yescharacteristic
327C2×C8—42no—
328C2×C8—82no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C2×C7—21no—
343C2×C7—21no—
344C2×C7—22no—
345C2×C7—42no—
346C2×C7—41no—
347C2×C7—81no—
348C2×C7—82no—
349C2×C7#151611yescharacteristic
350C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
351C2×C7#212711yescharacteristic
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—81no—
377C23—81no—
378C23—84no—
379C23—84no—
380C23C7×#9911yescharacteristic
381C23C7×#9911yescharacteristic
382C23C2×C7×C22⋊C411yescharacteristic
383C23C7×#13512yes—
384Q8—42no—
385Q8—42no—
386D4—44no—
387D4—42no—
388D4—42no—
389D4—44no—
390D4—84no—
391D4—84no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C8—82no—
418C7#1183911yescharacteristic, direct product factor, Sylow
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—81no—
440C22—84no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—84no—
445C22—81no—
446C22—81no—
447C22C7×AΣL(1,𝔽16)12yes—
448C22C2×C7×#9911yescharacteristic
449C22C7×#41611yescharacteristic
450C22C7×#42111yescharacteristic
451C4—41no—
452C4—42no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—81no—
457C4—81no—
458C4—162no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—42no—
463C2—41no—
464C2—81no—
465C2—82no—
466C2C7×#151611yescharacteristic
467C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
468C2C7×#212711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2404C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1819C2semidirect
3. C2×C7×#323C2semidirect
4. C7×#1351C2semidirect
5. C7×#1343C2semidirect
6. C22×C7×#41C4semidirect
10. C7×#393C4semidirect
12. C7×#392C4semidirect
17. C7×#381C4semidirect
23. #11839C7direct (N × H)
60. #1819C2×C7semidirect
61. C2×#323C2×C7semidirect
62. #1351C2×C7semidirect
63. #1343C2×C7semidirect
124. C22×#41C4×C7semidirect
128. #393C4×C7semidirect
130. #392C4×C7semidirect
135. #381C4×C7semidirect
418. C7#11839direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24042
C7×#18192
C2×C7×#3232
C7×#13512
C7×#13432
#118397