Finite Group Databasebeta

← C7×#12262 #13189478

Subgroups — C7×#12262 #13189478

Order 1792.

596 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2668C211yesmaximal, characteristic
2C7×#2029C211yesmaximal, characteristic, complemented
3C7×#1614C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic, complemented
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic, complemented
7C7×#1316C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C2×C7×#121—21no—
10C2×C7×#120—21no—
11C7×#484—21no—
12C7×#450—21no—
13C7×#399C2211yescharacteristic
14C7×#397—21no—
15C7×#394—21no—
16C7×#392C2211yescharacteristic
17C7×#386—21no—
18C7×#386—21no—
19C7×#379—21no—
20C7×#379C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12262C711yesmaximal, characteristic, direct product factor, Sylow
28C25×C7D411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C22×C7×D4—21no—
32C22×C7×D4D411yescharacteristic
33C23×C4×C7—21no—
34C2×C7×SD16—41no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—41no—
38C7×C22⋊Q8—41no—
39C7×#121—44no—
40C7×#120—44no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4D411yescharacteristic
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×#107—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#2668C2×C711yescharacteristic
66#2029C2×C711yescharacteristic, complemented
67#1614C2×C711yescharacteristic, complemented
68#1596C2×C711yescharacteristic, complemented
69#1352C2×C711yescharacteristic, complemented
70#1343C2×C711yescharacteristic, complemented
71#1316C2×C711yescharacteristic, complemented
72C24×C7—21no—
73C24×C7—21no—
74C24×C7—21no—
75C24×C7—21no—
76C24×C7—41no—
77C24×C7—42no—
78C24×C7—41no—
79C24×C7—81no—
80C24×C7C2×D411yescharacteristic
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×D4—24no—
84C2×C7×D4—41no—
85C2×C7×D4—42no—
86C2×C7×D4—42no—
87C2×C7×D4—42no—
88C2×C7×D4—42no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
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—42no—
102C22×C4×C7—41no—
103C22×C4×C7—81no—
104C22×C4×C7—81no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C7×SD16—82no—
108C2×C7×C8—41no—
109C2×C7×C8—41no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—42no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—42no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—41no—
125C7×C22⋊C4—41no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—81no—
128C7×C22⋊C4—81no—
129C7×C22⋊C4—82no—
130C7×C22⋊C4—82no—
131#543C22×C711yescharacteristic
132C2×#121—21no—
133C2×#120—21no—
134#484—21no—
135#450—21no—
136#399C22×C711yescharacteristic
137#397—21no—
138#394—21no—
139#392C22×C711yescharacteristic
140#386—21no—
141#386—21no—
142#379—21no—
143#379C22×C711yescharacteristic
144#342—21no—
145#342C22×C711yescharacteristic
146#328—21no—
147#328—21no—
148#323C22×C711yescharacteristic
149#323C22×C711yescharacteristic
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—22no—
153C23×C7—22no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—42no—
163C23×C7—42no—
164C23×C7—42no—
165C23×C7—41no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—81no—
172C23×C7—82no—
173C23×C7—81no—
174C23×C7—82no—
175C23×C7—84no—
176C23×C7—81no—
177C23×C7#12011yescharacteristic
178C23×C7#13511yescharacteristic
179C23×C7#13711yescharacteristic
180C7×Q8—81no—
181C7×Q8—81no—
182C7×D4—42no—
183C7×D4—84no—
184C7×D4—81no—
185C7×D4—82no—
186C7×D4—82no—
187C7×D4—84no—
188C2×C4×C7—22no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
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—41no—
206C2×C4×C7—84no—
207C2×C4×C7—81no—
208C2×C4×C7—81no—
209C2×C4×C7—81no—
210C2×C4×C7—81no—
211C2×C4×C7—81no—
212C7×C8—81no—
213C7×C8—81no—
214C25C7×D411yescharacteristic
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C7×D411yescharacteristic
219C23×C4—21no—
220C2×SD16—41no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225#121—44no—
226#120—44no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C7×D411yescharacteristic
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—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4C7×D411yescharacteristic
241C2×C22⋊C4C23×C711yescharacteristic
242#107—41no—
243#103—41no—
244#102—41no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247#95—21no—
248#95—21no—
249#95C7×D411yescharacteristic
250#95C7×D411yescharacteristic
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—42no—
258C22×C7—42no—
259C22×C7—42no—
260C22×C7—42no—
261C22×C7—41no—
262C22×C7—41no—
263C22×C7—44no—
264C22×C7—42no—
265C22×C7—42no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7—82no—
272C22×C7—82no—
273C22×C7—82no—
274C22×C7—81no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7—81no—
278C22×C7#45211yescharacteristic
279C22×C7#45511yescharacteristic
280C22×C7B(4,𝔽2)11yescharacteristic
281C4×C7—42no—
282C4×C7—41no—
283C4×C7—41no—
284C4×C7—81no—
285C4×C7—81no—
286C4×C7—81no—
287C4×C7—81no—
288C4×C7—81no—
289C4×C7—161no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—41no—
295C24—42no—
296C24—41no—
297C24—81no—
298C24C2×C7×D411yescharacteristic
299C2×Q8—21no—
300C2×Q8—21no—
301C2×D4—24no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4—81no—
312C2×D4—81no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4C2×C7×D411yescharacteristic
324C22×C4C2×C7×D411yescharacteristic
325SD16—82no—
326C2×C8—41no—
327C2×C8—41no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C2×C7—21no—
350C2×C7—21no—
351C2×C7—41no—
352C2×C7—41no—
353C2×C7—82no—
354C2×C7—82no—
355C2×C7#160611yescharacteristic
356C2×C7#219611yescharacteristic
357C2×C7#220011yescharacteristic
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—81no—
380C23—82no—
381C23—81no—
382C23—82no—
383C23—84no—
384C23—81no—
385C23C7×#12011yescharacteristic
386C23C7×#13511yescharacteristic
387C23C7×#13711yescharacteristic
388Q8—81no—
389Q8—81no—
390D4—42no—
391D4—84no—
392D4—81no—
393D4—82no—
394D4—82no—
395D4—84no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
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—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—84no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C8—81no—
421C8—81no—
422C7#1226211yescharacteristic, direct product factor, Sylow
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22C7×#45211yescharacteristic
451C22C7×#45511yescharacteristic
452C22C7×B(4,𝔽2)11yescharacteristic
453C4—42no—
454C4—41no—
455C4—41no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—161no—
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—82no—
467C2—82no—
468C2C7×#160611yescharacteristic
469C2C7×#219611yescharacteristic
470C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2668C2
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×#1614C2semidirect
4. C7×#1596C2semidirect
5. C7×#1352C2semidirect
6. C7×#1343C2semidirect
7. C7×#1316C2semidirect
27. #12262C7direct (N × H)
66. #2029C2×C7semidirect
67. #1614C2×C7semidirect
68. #1596C2×C7semidirect
69. #1352C2×C7semidirect
70. #1343C2×C7semidirect
71. #1316C2×C7semidirect
422. C7#12262direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26682
C7×#20292
C7×#16142
C7×#15962
C7×#13522
C7×#13432
C7×#13162
#122627