Finite Group Databasebeta

← C7×#12242 #13189458

Subgroups — C7×#12242 #13189458

Order 1792.

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

normal subgroups only

This group has 588 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×#2675C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic, complemented
3C7×#1596C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic, complemented
5C7×#1343C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic, complemented
7C7×#1326C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C2×C7×#121—21no—
10C2×C7×#121—21no—
11C2×C7×#121—21no—
12C7×#450—21no—
13C7×#450—21no—
14C7×#396—21no—
15C7×#392—21no—
16C7×#392C2211yescharacteristic
17C7×#392C2211yescharacteristic
18C7×#385—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12242C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C22×C7×D4—21no—
32C22×C7×D4D411yescharacteristic
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C23×C4×C7D411yescharacteristic
36C2×C7×SD16—41no—
37C2×C7×SD16—41no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—41no—
41C7×C22⋊Q8—41no—
42C7×#121—44no—
43C7×#121—44no—
44C7×#121—42no—
45C7×#121—42no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—41no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4—21no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4D411yescharacteristic
58C2×C7×C22⋊C4C2311yescharacteristic
59C7×#102—41no—
60C7×#102—41no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×#95—21no—
64C7×#95—41no—
65C7×#95D411yescharacteristic
66C7×#95D411yescharacteristic
67#2675C2×C711yescharacteristic
68#2062C2×C711yescharacteristic, complemented
69#1596C2×C711yescharacteristic, complemented
70#1596C2×C711yescharacteristic, complemented
71#1343C2×C711yescharacteristic, complemented
72#1343C2×C711yescharacteristic, complemented
73#1326C2×C711yescharacteristic, complemented
74C24×C7—21no—
75C24×C7—21no—
76C24×C7C2×D411yescharacteristic
77C2×C7×Q8—21no—
78C2×C7×Q8—21no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—42no—
85C2×C7×D4—42no—
86C2×C7×D4—42no—
87C2×C7×D4—42no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—41no—
92C2×C7×D4—81no—
93C2×C7×D4—81no—
94C2×C7×D4—81no—
95C2×C7×D4—81no—
96C2×C7×D4—81no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—42no—
104C22×C4×C7—42no—
105C22×C4×C7—42no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—81no—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C7×SD16—82no—
112C7×SD16—82no—
113C2×C7×C8—41no—
114C2×C7×C8—41no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—42no—
118C7×C4⋊C4—42no—
119C7×C4⋊C4—42no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—82no—
123C7×C22⋊C4—24no—
124C7×C22⋊C4—21no—
125C7×C22⋊C4—22no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—42no—
128C7×C22⋊C4—42no—
129C7×C22⋊C4—42no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—81no—
134C7×C22⋊C4SD1612yescomplemented
135#543C22×C711yescharacteristic
136C2×#121—21no—
137C2×#121—21no—
138C2×#121—21no—
139#450—21no—
140#450—21no—
141#396—21no—
142#392—21no—
143#392C22×C711yescharacteristic
144#392C22×C711yescharacteristic
145#385—21no—
146#385—21no—
147#385C22×C711yescharacteristic
148#342—21no—
149#342C22×C711yescharacteristic
150#328—21no—
151#328—21no—
152#323C22×C711yescharacteristic
153#323C22×C711yescharacteristic
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—22no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7#12011yescharacteristic
173C23×C7C2×SD1611yescharacteristic
174C23×C7AGL(1,ℤ/8)11yescharacteristic
175C7×Q8—81no—
176C7×Q8—81no—
177C7×D4—42no—
178C7×D4—42no—
179C7×D4—81no—
180C7×D4—82no—
181C7×D4—82no—
182C7×D4—84no—
183C7×D4—84no—
184C7×D4—81no—
185C7×D4—82no—
186C7×D4—82no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—22no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
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—42no—
201C2×C4×C7—42no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—81no—
208C2×C4×C7—84no—
209C2×C4×C7—81no—
210C2×C4×C7—81no—
211C2×C4×C7—81no—
212C2×C4×C7—81no—
213C7×C8—81no—
214C7×C8—81no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4C7×D411yescharacteristic
220C22×D4C7×D411yescharacteristic
221C23×C4—21no—
222C23×C4C7×D411yescharacteristic
223C2×SD16—41no—
224C2×SD16—41no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229#121—44no—
230#121—44no—
231#121—42no—
232#121—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—41no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4C7×D411yescharacteristic
245C2×C22⋊C4C23×C711yescharacteristic
246#102—41no—
247#102—41no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250#95—21no—
251#95—41no—
252#95C7×D411yescharacteristic
253#95C7×D411yescharacteristic
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—42no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7—81no—
273C22×C7—81no—
274C22×C7—81no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7#45011yescharacteristic
278C22×C7#45611yescharacteristic
279C22×C7B(4,𝔽2)11yescharacteristic
280C4×C7—42no—
281C4×C7—42no—
282C4×C7—41no—
283C4×C7—42no—
284C4×C7—41no—
285C4×C7—81no—
286C4×C7—82no—
287C4×C7—81no—
288C4×C7—81no—
289C24—21no—
290C24—21no—
291C24C2×C7×D411yescharacteristic
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4—81no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—81no—
324C22×C4C2×C7×D411yescharacteristic
325C22×C4C2×C7×D411yescharacteristic
326SD16—82no—
327SD16—82no—
328C2×C8—41no—
329C2×C8—41no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—82no—
338C22⋊C4—24no—
339C22⋊C4—21no—
340C22⋊C4—22no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—81no—
349C22⋊C4C7×SD1612yescomplemented
350C2×C7—21no—
351C2×C7—21no—
352C2×C7—41no—
353C2×C7—41no—
354C2×C7—82no—
355C2×C7—161no—
356C2×C7#160411yescharacteristic
357C2×C7#219611yescharacteristic
358C2×C7#220011yescharacteristic
359C23—21no—
360C23—21no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23C7×#12011yescharacteristic
378C23C2×C7×SD1611yescharacteristic
379C23C7×AGL(1,ℤ/8)11yescharacteristic
380Q8—81no—
381Q8—81no—
382D4—42no—
383D4—42no—
384D4—81no—
385D4—82no—
386D4—82no—
387D4—84no—
388D4—84no—
389D4—81no—
390D4—82no—
391D4—82no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
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—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—84no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C8—81no—
419C8—81no—
420C7#1224211yescharacteristic, direct product factor, Sylow
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—42no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22C7×#45011yescharacteristic
445C22C7×#45611yescharacteristic
446C22C7×B(4,𝔽2)11yescharacteristic
447C4—42no—
448C4—42no—
449C4—41no—
450C4—42no—
451C4—41no—
452C4—81no—
453C4—82no—
454C4—81no—
455C4—81no—
456C2—21no—
457C2—21no—
458C2—41no—
459C2—41no—
460C2—82no—
461C2—161no—
462C2C7×#160411yescharacteristic
463C2C7×#219611yescharacteristic
464C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2675C2
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×#2062C2semidirect
3. C7×#1596C2semidirect
4. C7×#1596C2semidirect
5. C7×#1343C2semidirect
6. C7×#1343C2semidirect
7. C7×#1326C2semidirect
27. #12242C7direct (N × H)
68. #2062C2×C7semidirect
69. #1596C2×C7semidirect
70. #1596C2×C7semidirect
71. #1343C2×C7semidirect
72. #1343C2×C7semidirect
73. #1326C2×C7semidirect
134. C7×C22⋊C4SD16semidirect
349. C22⋊C4C7×SD16semidirect
420. C7#12242direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26752
C7×#20622
C7×#15962
C7×#13432
C7×#13262
#122427