Finite Group Databasebeta

← C7×#11541 #13188757

Subgroups — C7×#11541 #13188757

Order 1792.

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

normal subgroups only

This group has 700 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×#2405C211yesmaximal, characteristic
2C7×#1462C211yesmaximal, characteristic, complemented
3C2×C7×#323C211yesmaximal, characteristic, complemented
4C7×#1326C212yesmaximal, complemented
5C7×#1316C212yesmaximal, complemented
6C23×C7×D4C411yescharacteristic
7C2×C7×#123—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C7×#407—21no—
10C2×C7×C22⋊C8—21no—
11C7×#394—21no—
12C7×#394—21no—
13C7×#386—21no—
14C7×#385C411yescharacteristic
15C7×#385C2211yescharacteristic
16C7×#379C411yescharacteristic
17C7×#379C411yescharacteristic
18C7×#379C2211yescharacteristic
19C7×#323—21no—
20C7×#323—21no—
21C7×#323C2212yes—
22C7×#323C2212yes—
23#11541C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7C2×C411yescharacteristic
25C25×C7C2×C411yescharacteristic
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4—44no—
29C22×C7×D4—41no—
30C22×C7×D4—41no—
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7D411yescharacteristic
34C2×C7×M16—41no—
35C22×C7×C8—41no—
36C7×#123—44no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4C2×C411yescharacteristic
50C2×C7×C22⋊C4D411yescharacteristic
51C2×C7×C22⋊C4D411yescharacteristic
52C2×C7×C22⋊C4C2311yescharacteristic
53C7×C22⋊C8—22no—
54C7×C22⋊C8—22no—
55C7×C22⋊C8—41no—
56C7×C22⋊C8—41no—
57C7×#95—21no—
58C7×#95—21no—
59#2405C2×C711yescharacteristic
60#1462C2×C711yescharacteristic, complemented
61C2×#323C2×C711yescharacteristic, complemented
62#1326C2×C712yescomplemented
63#1316C2×C712yescomplemented
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—21no—
67C24×C7—22no—
68C24×C7—22no—
69C24×C7—41no—
70C24×C7—42no—
71C24×C7—41no—
72C24×C7—42no—
73C24×C7—41no—
74C24×C7—81no—
75C24×C7—81no—
76C24×C7C22⋊C411yescharacteristic
77C24×C7C22⋊C411yescharacteristic
78C24×C7C22×C411yescharacteristic
79C2×C7×D4—24no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—44no—
85C2×C7×D4—44no—
86C2×C7×D4—44no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—44no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—42no—
98C22×C4×C7—41no—
99C22×C4×C7—42no—
100C22×C4×C7—81no—
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C7×M16—82no—
106C2×C7×C8—42no—
107C2×C7×C8—42no—
108C2×C7×C8—82no—
109C7×C4⋊C4—42no—
110C7×C4⋊C4—42no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—81no—
121C7×C22⋊C4—81no—
122C7×C22⋊C4—81no—
123C23×D4C4×C711yescharacteristic
124C2×#123—21no—
125C22×C22⋊C4C22×C711yescharacteristic
126#407—21no—
127C2×C22⋊C8—21no—
128#394—21no—
129#394—21no—
130#386—21no—
131#385C4×C711yescharacteristic
132#385C22×C711yescharacteristic
133#379C4×C711yescharacteristic
134#379C4×C711yescharacteristic
135#379C22×C711yescharacteristic
136#323—21no—
137#323—21no—
138#323C22×C712yes—
139#323C22×C712yes—
140C23×C7—22no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—22no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—41no—
150C23×C7—42no—
151C23×C7—42no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7—42no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—81no—
167C23×C7—81no—
168C23×C7—82no—
169C23×C7—84no—
170C23×C7—84no—
171C23×C7—84no—
172C23×C7—84no—
173C23×C7—82no—
174C23×C7#9911yescharacteristic
175C23×C7#9911yescharacteristic
176C23×C7C2×C22⋊C411yescharacteristic
177C23×C7#13112yes—
178C7×D4—44no—
179C7×D4—48no—
180C7×D4—44no—
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—42no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
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—81no—
204C7×C8—82no—
205C7×C8—82no—
206C25C2×C4×C711yescharacteristic
207C25C2×C4×C711yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—44no—
211C22×D4—41no—
212C22×D4—41no—
213C23×C4C2×C4×C711yescharacteristic
214C23×C4C7×D411yescharacteristic
215C23×C4C7×D411yescharacteristic
216C2×M16—41no—
217C22×C8—41no—
218#123—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4C2×C4×C711yescharacteristic
230C2×C22⋊C4C2×C4×C711yescharacteristic
231C2×C22⋊C4C2×C4×C711yescharacteristic
232C2×C22⋊C4C7×D411yescharacteristic
233C2×C22⋊C4C7×D411yescharacteristic
234C2×C22⋊C4C23×C711yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239#95—21no—
240#95—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—22no—
248C22×C7—21no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—42no—
254C22×C7—42no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—42no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—84no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—84no—
267C22×C7—84no—
268C22×C7—84no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7AΣL(1,𝔽16)12yes—
272C22×C7#40811yescharacteristic
273C22×C7C2×#9911yescharacteristic
274C22×C7#41311yescharacteristic
275C4×C7—41no—
276C4×C7—42no—
277C4×C7—41no—
278C4×C7—41no—
279C4×C7—41no—
280C4×C7—81no—
281C4×C7—81no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—41no—
288C24—42no—
289C24—41no—
290C24—42no—
291C24—41no—
292C24—81no—
293C24—81no—
294C24C7×C22⋊C411yescharacteristic
295C24C7×C22⋊C411yescharacteristic
296C24C22×C4×C711yescharacteristic
297C2×D4—24no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—44no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—81no—
319C22×C4C7×C22⋊C411yescharacteristic
320C22×C4C7×C22⋊C411yescharacteristic
321C22×C4C2×C7×D411yescharacteristic
322C22×C4C2×C7×D411yescharacteristic
323M16—82no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—82no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C2×C7—21no—
342C2×C7—21no—
343C2×C7—22no—
344C2×C7—41no—
345C2×C7—42no—
346C2×C7—82no—
347C2×C7—81no—
348C2×C7#146811yescharacteristic
349C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
350C2×C7#212111yescharacteristic
351C23—22no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—81no—
378C23—81no—
379C23—82no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—82no—
385C23C7×#9911yescharacteristic
386C23C7×#9911yescharacteristic
387C23C2×C7×C22⋊C411yescharacteristic
388C23C7×#13112yes—
389D4—44no—
390D4—48no—
391D4—44no—
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—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C8—82no—
416C8—82no—
417C7#1154111yescharacteristic, direct product factor, Sylow
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—81no—
447C22—81no—
448C22C7×AΣL(1,𝔽16)12yes—
449C22C7×#40811yescharacteristic
450C22C2×C7×#9911yescharacteristic
451C22C7×#41311yescharacteristic
452C4—41no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—81no—
458C4—81no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—41no—
463C2—42no—
464C2—82no—
465C2—81no—
466C2C7×#146811yescharacteristic
467C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
468C2C7×#212111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC23×C7×D4C4
LCSG ▷ C24 ▷ C22 ▷ 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×#1462C2semidirect
3. C2×C7×#323C2semidirect
4. C7×#1326C2semidirect
5. C7×#1316C2semidirect
23. #11541C7direct (N × H)
60. #1462C2×C7semidirect
61. C2×#323C2×C7semidirect
62. #1326C2×C7semidirect
63. #1316C2×C7semidirect
417. C7#11541direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24052
C7×#14622
C2×C7×#3232
C7×#13262
C7×#13162
#115417