Finite Group Databasebeta

← C7×#12205 #13189421

Subgroups — C7×#12205 #13189421

Order 1792.

544 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2812C211yesmaximal, characteristic
2C7×#2029C211yesmaximal, characteristic, complemented
3C7×#1608C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic, complemented
5C7×#1344C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic, complemented
7C7×#1316C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C2×C7×#120—21no—
10C7×#483—21no—
11C7×#450—21no—
12C7×#397—21no—
13C7×#397C2211yescharacteristic
14C7×#394—21no—
15C7×#394—21no—
16C7×#394—21no—
17C7×#392—21no—
18C7×#392C2211yescharacteristic
19C7×#379—21no—
20C7×#379C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12205C711yesmaximal, characteristic, direct product factor, Sylow
28C25×C7D411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C22×C7×D4—21no—
32C22×C7×D4D411yescharacteristic
33C2×C7×SD16—41no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—41no—
37C7×C22⋊Q8—41no—
38C7×#120—42no—
39C7×#120—42no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—41no—
53C2×C7×C22⋊C4—41no—
54C2×C7×C22⋊C4D411yescharacteristic
55C2×C7×C22⋊C4C2311yescharacteristic
56C7×#106—41no—
57C7×#103—41no—
58C7×#102—41no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95D411yescharacteristic
64C7×#95D411yescharacteristic
65#2812C2×C711yescharacteristic
66#2029C2×C711yescharacteristic, complemented
67#1608C2×C711yescharacteristic, complemented
68#1596C2×C711yescharacteristic, complemented
69#1344C2×C711yescharacteristic, complemented
70#1343C2×C711yescharacteristic, complemented
71#1316C2×C711yescharacteristic, complemented
72C24×C7—21no—
73C24×C7—21no—
74C24×C7—21no—
75C24×C7—41no—
76C24×C7—41no—
77C24×C7—81no—
78C24×C7—81no—
79C24×C7SD1612yescomplemented
80C24×C7C2×D411yescharacteristic
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×D4—41no—
84C2×C7×D4—42no—
85C2×C7×D4—42no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—81no—
89C2×C7×D4—81no—
90C2×C7×D4—81no—
91C2×C7×D4—81no—
92C2×C7×D4—81no—
93C2×C7×D4—81no—
94C2×C7×D4—81no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—41no—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C7×SD16—82no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4—81no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—81no—
124C7×C22⋊C4—81no—
125C7×C22⋊C4—81no—
126C7×C22⋊C4—81no—
127C7×C22⋊C4—81no—
128C7×C22⋊C4—82no—
129C7×C22⋊C4—82no—
130#543C22×C711yescharacteristic
131C2×#120—21no—
132#483—21no—
133#450—21no—
134#397—21no—
135#397C22×C711yescharacteristic
136#394—21no—
137#394—21no—
138#394—21no—
139#392—21no—
140#392C22×C711yescharacteristic
141#379—21no—
142#379C22×C711yescharacteristic
143#342—21no—
144#342C22×C711yescharacteristic
145#328—21no—
146#328—21no—
147#323C22×C711yescharacteristic
148#323C22×C711yescharacteristic
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—22no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—42no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—42no—
164C23×C7—42no—
165C23×C7—41no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—81no—
171C23×C7—82no—
172C23×C7—82no—
173C23×C7—81no—
174C23×C7—81no—
175C23×C7—84no—
176C23×C7#12011yescharacteristic
177C23×C7C2×SD1611yescharacteristic
178C23×C7AGL(1,ℤ/8)11yescharacteristic
179C7×Q8—81no—
180C7×Q8—81no—
181C7×D4—42no—
182C7×D4—81no—
183C7×D4—82no—
184C7×D4—82no—
185C7×D4—84no—
186C7×D4—84no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
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—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—81no—
206C2×C4×C7—81no—
207C2×C4×C7—81no—
208C2×C4×C7—81no—
209C2×C4×C7—81no—
210C7×C8—81no—
211C7×C8—81no—
212C25C7×D411yescharacteristic
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4C7×D411yescharacteristic
217C2×SD16—41no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222#120—42no—
223#120—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4C7×D411yescharacteristic
239C2×C22⋊C4C23×C711yescharacteristic
240#106—41no—
241#103—41no—
242#102—41no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245#95—21no—
246#95—21no—
247#95C7×D411yescharacteristic
248#95C7×D411yescharacteristic
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—42no—
256C22×C7—42no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—44no—
260C22×C7—42no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—82no—
269C22×C7—82no—
270C22×C7—82no—
271C22×C7—81no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7—81no—
275C22×C7—81no—
276C22×C7#45011yescharacteristic
277C22×C7#45511yescharacteristic
278C22×C7B(4,𝔽2)11yescharacteristic
279C4×C7—41no—
280C4×C7—41no—
281C4×C7—81no—
282C4×C7—81no—
283C4×C7—81no—
284C4×C7—81no—
285C4×C7—81no—
286C4×C7—81no—
287C4×C7—161no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—41no—
292C24—41no—
293C24—81no—
294C24—81no—
295C24C7×SD1612yescomplemented
296C24C2×C7×D411yescharacteristic
297C2×Q8—21no—
298C2×Q8—21no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—81no—
305C2×D4—81no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—81no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4C2×C7×D411yescharacteristic
319C22×C4C2×C7×D411yescharacteristic
320SD16—82no—
321C2×C8—41no—
322C2×C8—41no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—81no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C22⋊C4—81no—
343C22⋊C4—81no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C2×C7—21no—
347C2×C7—21no—
348C2×C7—41no—
349C2×C7—41no—
350C2×C7—82no—
351C2×C7—161no—
352C2×C7#160111yescharacteristic
353C2×C7#219911yescharacteristic
354C2×C7#220011yescharacteristic
355C23—21no—
356C23—21no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—81no—
377C23—82no—
378C23—82no—
379C23—81no—
380C23—81no—
381C23—84no—
382C23C7×#12011yescharacteristic
383C23C2×C7×SD1611yescharacteristic
384C23C7×AGL(1,ℤ/8)11yescharacteristic
385Q8—81no—
386Q8—81no—
387D4—42no—
388D4—81no—
389D4—82no—
390D4—82no—
391D4—84no—
392D4—84no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
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—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C8—81no—
417C8—81no—
418C7#1220511yescharacteristic, direct product factor, Sylow
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—44no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22C7×#45011yescharacteristic
447C22C7×#45511yescharacteristic
448C22C7×B(4,𝔽2)11yescharacteristic
449C4—41no—
450C4—41no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—161no—
458C2—21no—
459C2—21no—
460C2—41no—
461C2—41no—
462C2—82no—
463C2—161no—
464C2C7×#160111yescharacteristic
465C2C7×#219911yescharacteristic
466C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2812C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2029C2semidirect
3. C7×#1608C2semidirect
4. C7×#1596C2semidirect
5. C7×#1344C2semidirect
6. C7×#1343C2semidirect
7. C7×#1316C2semidirect
27. #12205C7direct (N × H)
66. #2029C2×C7semidirect
67. #1608C2×C7semidirect
68. #1596C2×C7semidirect
69. #1344C2×C7semidirect
70. #1343C2×C7semidirect
71. #1316C2×C7semidirect
79. C24×C7SD16semidirect
295. C24C7×SD16semidirect
418. C7#12205direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28122
C7×#20292
C7×#16082
C7×#15962
C7×#13442
C7×#13432
C7×#13162
#122057