Finite Group Databasebeta

← C7×#11796 #13189012

Subgroups — C7×#11796 #13189012

Order 1792.

1068 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×C7×#392C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic, complemented
3C7×#1462C211yesmaximal, characteristic, complemented
4C7×#1343C214yesmaximal, complemented
5C23×C7×D4—21no—
6C23×C7×D4C411yescharacteristic, complemented
7C22×C7×C22⋊C4—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C7×#407—21no—
10C7×#392—21no—
11C7×#392—21no—
12C7×#392C412yescomplemented
13C7×#392C2212yes—
14C7×#379—21no—
15C2×C7×#95C411yescharacteristic, complemented
16C7×#342—21no—
17C7×#342C2212yescomplemented
18C7×#323—21no—
19C7×#323C2212yescomplemented
20#11796C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7—21no—
22C25×C7D411yescharacteristic
23C22×C7×D4—22no—
24C22×C7×D4—22no—
25C22×C7×D4—21no—
26C22×C7×D4—41no—
27C22×C7×D4—41no—
28C22×C7×D4—44no—
29C22×C7×D4—44no—
30C22×C7×D4—44no—
31C22×C7×D4—44no—
32C22×C7×D4C2×C412yes—
33C23×C4×C7—21no—
34C23×C4×C7C2×C411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C2×C7×M16—41no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—41no—
41C2×C7×C22⋊C4—41no—
42C2×C7×C22⋊C4—44no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4C2311yescharacteristic
49C7×C22⋊C8—22no—
50C7×C22⋊C8—41no—
51C7×#95—21no—
52C7×#95C2×C412yes—
53C2×#392C2×C711yescharacteristic
54#1819C2×C711yescharacteristic, complemented
55#1462C2×C711yescharacteristic, complemented
56#1343C2×C714yescomplemented
57C24×C7—24no—
58C24×C7—22no—
59C24×C7—22no—
60C24×C7—21no—
61C24×C7—21no—
62C24×C7—41no—
63C24×C7—42no—
64C24×C7—42no—
65C24×C7—42no—
66C24×C7—42no—
67C24×C7—42no—
68C24×C7—42no—
69C24×C7—42no—
70C24×C7—41no—
71C24×C7—41no—
72C24×C7C22⋊C411yescharacteristic
73C24×C7C22⋊C411yescharacteristic
74C24×C7C2×D411yescharacteristic
75C2×C7×D4—44no—
76C2×C7×D4—44no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—44no—
80C2×C7×D4—44no—
81C2×C7×D4—44no—
82C2×C7×D4—44no—
83C2×C7×D4—48no—
84C2×C7×D4—432no—
85C2×C7×D4—44no—
86C2×C7×D4—44no—
87C2×C7×D4—44no—
88C2×C7×D4—44no—
89C22×C4×C7—24no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—21no—
95C22×C4×C7—42no—
96C22×C4×C7—42no—
97C22×C4×C7—42no—
98C22×C4×C7—42no—
99C22×C4×C7—42no—
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22×C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C7×M16—82no—
105C2×C7×C8—42no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—44no—
111C7×C22⋊C4—44no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—82no—
115C7×C22⋊C4—82no—
116C23×D4—21no—
117C23×D4C4×C711yescharacteristic, complemented
118C22×C22⋊C4—21no—
119C22×C22⋊C4C22×C711yescharacteristic
120#407—21no—
121#392—21no—
122#392—21no—
123#392C4×C712yescomplemented
124#392C22×C712yes—
125#379—21no—
126C2×#95C4×C711yescharacteristic, complemented
127#342—21no—
128#342C22×C712yescomplemented
129#323—21no—
130#323C22×C712yescomplemented
131C23×C7—22no—
132C23×C7—22no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—24no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—42no—
148C23×C7—48no—
149C23×C7—48no—
150C23×C7—42no—
151C23×C7—44no—
152C23×C7—42no—
153C23×C7—48no—
154C23×C7—42no—
155C23×C7—44no—
156C23×C7—42no—
157C23×C7—42no—
158C23×C7—42no—
159C23×C7—42no—
160C23×C7—81no—
161C23×C7—81no—
162C23×C7—84no—
163C23×C7—84no—
164C23×C7—84no—
165C23×C7—82no—
166C23×C7—82no—
167C23×C7—82no—
168C23×C7—82no—
169C23×C7#9911yescharacteristic
170C23×C7#9911yescharacteristic
171C23×C7C2×C22⋊C411yescharacteristic
172C23×C7AGL(1,ℤ/8)12yes—
173C7×D4—44no—
174C7×D4—416no—
175C7×D4—416no—
176C7×D4—48no—
177C7×D4—44no—
178C2×C4×C7—24no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—48no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—81no—
194C2×C4×C7—82no—
195C2×C4×C7—82no—
196C2×C4×C7—82no—
197C7×C8—82no—
198C25—21no—
199C25C7×D411yescharacteristic
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4C2×C4×C712yes—
210C23×C4—21no—
211C23×C4C2×C4×C711yescharacteristic
212C23×C4C7×D411yescharacteristic
213C2×M16—41no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4C2×C4×C711yescharacteristic
223C2×C22⋊C4C7×D411yescharacteristic
224C2×C22⋊C4C7×D411yescharacteristic
225C2×C22⋊C4C23×C711yescharacteristic
226C22⋊C8—22no—
227C22⋊C8—41no—
228#95—21no—
229#95C2×C4×C712yes—
230C22×C7—24no—
231C22×C7—24no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—22no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—44no—
242C22×C7—44no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—44no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—81no—
250C22×C7—84no—
251C22×C7—84no—
252C22×C7—81no—
253C22×C7—82no—
254C22×C7—82no—
255C22×C7—88no—
256C22×C7—88no—
257C22×C7—82no—
258C22×C7—82no—
259C22×C7AΣL(1,𝔽16)12yes—
260C22×C7#35312yes—
261C22×C7C2×#9911yescharacteristic
262C22×C7#41811yescharacteristic
263C22×C7#42111yescharacteristic
264C4×C7—44no—
265C4×C7—41no—
266C4×C7—41no—
267C4×C7—41no—
268C4×C7—41no—
269C4×C7—162no—
270C24—24no—
271C24—22no—
272C24—22no—
273C24—21no—
274C24—21no—
275C24—41no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—41no—
284C24—41no—
285C24C7×C22⋊C411yescharacteristic
286C24C7×C22⋊C411yescharacteristic
287C24C2×C7×D411yescharacteristic
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4—432no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4C7×C22⋊C411yescharacteristic
314C22×C4C7×C22⋊C411yescharacteristic
315C22×C4C22×C4×C711yescharacteristic
316C22×C4C2×C7×D411yescharacteristic
317M16—82no—
318C2×C8—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C2×C7—21no—
330C2×C7—21no—
331C2×C7—22no—
332C2×C7—22no—
333C2×C7—42no—
334C2×C7—81no—
335C2×C7—84no—
336C2×C7#151011yescharacteristic
337C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
338C2×C7C2×#35311yescharacteristic
339C23—22no—
340C23—22no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—24no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—48no—
357C23—48no—
358C23—42no—
359C23—44no—
360C23—42no—
361C23—48no—
362C23—42no—
363C23—44no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—84no—
371C23—84no—
372C23—84no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23C7×#9911yescharacteristic
378C23C7×#9911yescharacteristic
379C23C2×C7×C22⋊C411yescharacteristic
380C23C7×AGL(1,ℤ/8)12yes—
381D4—44no—
382D4—416no—
383D4—416no—
384D4—44no—
385D4—48no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—48no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C8—82no—
406C7#1179611yescharacteristic, direct product factor, Sylow
407C22—24no—
408C22—24no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—44no—
419C22—44no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—44no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—84no—
428C22—84no—
429C22—81no—
430C22—82no—
431C22—82no—
432C22—88no—
433C22—88no—
434C22—82no—
435C22—82no—
436C22C7×AΣL(1,𝔽16)12yes—
437C22C7×#35312yes—
438C22C2×C7×#9911yescharacteristic
439C22C7×#41811yescharacteristic
440C22C7×#42111yescharacteristic
441C4—44no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—162no—
447C2—21no—
448C2—21no—
449C2—22no—
450C2—22no—
451C2—42no—
452C2—81no—
453C2—84no—
454C2C7×#151011yescharacteristic
455C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
456C2C2×C7×#35311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC2×C7×#392C2
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. C7×#1462C2semidirect
4. C7×#1343C2semidirect
6. C23×C7×D4C4semidirect
12. C7×#392C4semidirect
15. C2×C7×#95C4semidirect
17. C7×#342C22semidirect
19. C7×#323C22semidirect
20. #11796C7direct (N × H)
54. #1819C2×C7semidirect
55. #1462C2×C7semidirect
56. #1343C2×C7semidirect
117. C23×D4C4×C7semidirect
123. #392C4×C7semidirect
126. C2×#95C4×C7semidirect
128. #342C22×C7semidirect
130. #323C22×C7semidirect
406. C7#11796direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3922
C7×#18192
C7×#14622
C7×#13432
#117967