Finite Group Databasebeta

← C7×#36578 #13213794

Subgroups — C7×#36578 #13213794

Order 1792.

2480 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#548C211yesmaximal, characteristic
2C22×C4×C7×Q8C211yesmaximal, characteristic, complemented
3C7×#2854C213yesmaximal, complemented
4C7×#2504C213yesmaximal, complemented
5C7×#2487C213yesmaximal, complemented
6C7×#2448C216yesmaximal, complemented
7C7×#2421C216yesmaximal, complemented
8C7×#2351C212yesmaximal, complemented
9C7×#2341C212yesmaximal, complemented
10C7×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
11C4×C7×#123C213yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C7×#548—28no—
15C2×C7×#124C2213yes—
16C2×C7×#123C2213yes—
17C2×C7×C22⋊Q8C2213yes—
18C2×C7×#121C2213yes—
19C2×C4×C7×Q8—28no—
20C2×C4×C7×Q8C2216yes—
21C2×C4×C7×Q8C2212yes—
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×Q8C2212yes—
24C2×C4×C7×D4C2213yes—
25C22×C7×C4⋊C4C2213yes—
26C22×C42×C7C2213yes—
27C7×#400C2216yes—
28C7×#399C2216yes—
29C7×#397C22112yes—
30C7×#391C2216yes—
31C7×#390C2216yes—
32C7×#390C2216yes—
33C7×#389C2216yes—
34C7×#389C2216yes—
35C7×#388C22112yes—
36C7×#388C2216yes—
37C7×#387C2216yes—
38C7×#386C2216yes—
39C7×#386C2213yes—
40C7×#385C2213yes—
41C7×#382C2216yes—
42C7×#381C2211yescharacteristic
43C7×#380C2216yes—
44C7×#380C2213yes—
45C4×C7×C4⋊C4C2216yes—
46C4×C7×C22⋊C4C2216yes—
47C4×C7×C22⋊C4C2213yes—
48#36578C711yesmaximal, characteristic, direct product factor, Sylow
49C2×C7×#41—24no—
50C2×C7×#41—28no—
51C22×C7×Q8—28no—
52C22×C7×Q8D418yes—
53C22×C7×Q8C2313yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4C2313yes—
56C22×C7×D4C2313yes—
57C23×C4×C7C2313yes—
58C23×C4×C7C2313yes—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2313yes—
61C7×#124—212no—
62C7×#123—212no—
63C7×C22⋊Q8—212no—
64C7×#121—212no—
65C4×C7×Q8—224no—
66C4×C7×Q8—28no—
67C4×C7×Q8—24no—
68C4×C7×Q8—28no—
69C4×C7×D4—212no—
70C2×C7×C4⋊C4—212no—
71C2×C7×C4⋊C4C2316yes—
72C2×C7×C4⋊C4C2316yes—
73C2×C7×C4⋊C4C2316yes—
74C2×C7×C4⋊C4C2313yes—
75C2×C7×C4⋊C4C2316yes—
76C2×C7×C4⋊C4C2313yes—
77C2×C7×C4⋊C4C2316yes—
78C2×C7×C22⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4C2313yes—
80C2×C7×C22⋊C4C2316yes—
81C2×C7×C22⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4C2313yes—
83C2×C7×C22⋊C4C2316yes—
84C2×C7×C22⋊C4C2316yes—
85C2×C7×C22⋊C4C23112yes—
86C2×C42×C7—212no—
87C2×C42×C7C2316yes—
88C2×C42×C7C2316yes—
89C2×C42×C7C2316yes—
90C2×C42×C7C2313yes—
91C7×#95C2312yes—
92C7×#95C2316yes—
93C7×#95C2316yes—
94C7×#95C2312yes—
95C7×#95C2316yes—
96C7×#95C2316yes—
97C7×#95C23112yes—
98C2×#548C2×C711yescharacteristic
99C22×C4×Q8C2×C711yescharacteristic, complemented
100#2854C2×C713yescomplemented
101#2504C2×C713yescomplemented
102#2487C2×C713yescomplemented
103#2448C2×C716yescomplemented
104#2421C2×C716yescomplemented
105#2351C2×C712yescomplemented
106#2341C2×C712yescomplemented
107Q8×C22⋊C4C2×C711yescharacteristic, complemented
108C4×#123C2×C713yescomplemented
109C24×C7C2413yes—
110C24×C7C2411yescharacteristic
111C7×#41—416no—
112C2×C7×Q8—232no—
113C2×C7×Q8—212no—
114C2×C7×Q8C2×D4112yes—
115C2×C7×Q8C2×D414yes—
116C2×C7×Q8#4118yes—
117C2×C7×D4—26no—
118C2×C7×D4—26no—
119C2×C7×D4—26no—
120C2×C7×D4—26no—
121C2×C7×D4—212no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—212no—
125C22×C4×C7—26no—
126C22×C4×C7—26no—
127C22×C4×C7—212no—
128C22×C4×C7—46no—
129C22×C4×C7C2×D4112yes—
130C22×C4×C7#4114yescomplemented
131C22×C4×C7C2412yes—
132C22×C4×C7C2411yescharacteristic
133C22×C4×C7C2416yes—
134C22×C4×C7C2413yes—
135C22×C4×C7C2416yes—
136C22×C4×C7C2411yescharacteristic
137C22×C4×C7C2413yes—
138C22×C4×C7C2412yes—
139C22×C4×C7C2413yes—
140C7×C4⋊C4—212no—
141C7×C4⋊C4—212no—
142C7×C4⋊C4—26no—
143C7×C4⋊C4—26no—
144C7×C4⋊C4—212no—
145C7×C4⋊C4—212no—
146C7×C4⋊C4—46no—
147C7×C22⋊C4—22no—
148C7×C22⋊C4—26no—
149C7×C22⋊C4—22no—
150C7×C22⋊C4—26no—
151C7×C22⋊C4—212no—
152C7×C22⋊C4—212no—
153C7×C22⋊C4—46no—
154C7×C22⋊C4—412no—
155C42×C7—212no—
156C42×C7—212no—
157C42×C7—26no—
158C42×C7—212no—
159C22×#41C22×C711yescharacteristic
160C23×Q8C22×C711yescharacteristic
161#548—28no—
162C2×#124C22×C713yes—
163C2×#123C22×C713yes—
164C2×C22⋊Q8C22×C713yes—
165C2×#121C22×C713yes—
166C2×C4×Q8—28no—
167C2×C4×Q8C22×C712yes—
168C2×C4×Q8C22×C712yes—
169C2×C4×Q8C22×C711yescharacteristic
170C2×C4×Q8C22×C716yes—
171C2×C4×D4C22×C713yes—
172C22×C4⋊C4C22×C713yes—
173C22×C42C22×C713yes—
174#400C22×C716yes—
175#399C22×C716yes—
176#397C22×C7112yes—
177#391C22×C716yes—
178#390C22×C716yes—
179#390C22×C716yes—
180#389C22×C716yes—
181#389C22×C716yes—
182#388C22×C7112yes—
183#388C22×C716yes—
184#387C22×C716yes—
185#386C22×C716yes—
186#386C22×C713yes—
187#385C22×C713yes—
188#382C22×C716yes—
189#381C22×C711yescharacteristic
190#380C22×C713yes—
191#380C22×C716yes—
192C4×C4⋊C4C22×C716yes—
193C4×C22⋊C4C22×C713yes—
194C4×C22⋊C4C22×C716yes—
195C23×C7—22no—
196C23×C7—22no—
197C23×C7—23no—
198C23×C7—23no—
199C23×C7—26no—
200C23×C7—23no—
201C23×C7—26no—
202C23×C7C22×D412yes—
203C23×C7C2×#4112yes—
204C23×C72−1+412yes—
205C23×C7C2511yescharacteristic
206C7×Q8—224no—
207C7×Q8#123116yescomplemented
208C7×D4—412no—
209C7×D4—412no—
210C2×C4×C7—28no—
211C2×C4×C7—22no—
212C2×C4×C7—26no—
213C2×C4×C7—22no—
214C2×C4×C7—224no—
215C2×C4×C7—21no—
216C2×C4×C7—26no—
217C2×C4×C7—23no—
218C2×C4×C7—21no—
219C2×C4×C7—22no—
220C2×C4×C7—26no—
221C2×C4×C7—26no—
222C2×C4×C7—23no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—23no—
226C2×C4×C7—22no—
227C2×C4×C7—26no—
228C2×C4×C7—26no—
229C2×C4×C7—26no—
230C2×C4×C7—22no—
231C2×C4×C7—412no—
232C2×C4×C7—412no—
233C2×C4×C7C22×D416yes—
234C2×C4×C7C22×D416yes—
235C2×C4×C7C2×#4114yes—
236C2×C4×C7C2×#41112yes—
237C2×#41—24no—
238C2×#41—28no—
239C22×Q8—28no—
240C22×Q8C7×D418yes—
241C22×Q8C23×C713yes—
242C22×Q8C23×C711yescharacteristic
243C22×D4C23×C713yes—
244C22×D4C23×C713yes—
245C23×C4C23×C713yes—
246C23×C4C23×C713yes—
247C23×C4C23×C711yescharacteristic
248C23×C4C23×C713yes—
249#124—212no—
250#123—212no—
251C22⋊Q8—212no—
252#121—212no—
253C4×Q8—224no—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—28no—
257C4×D4—212no—
258C2×C4⋊C4—212no—
259C2×C4⋊C4C23×C716yes—
260C2×C4⋊C4C23×C716yes—
261C2×C4⋊C4C23×C713yes—
262C2×C4⋊C4C23×C713yes—
263C2×C4⋊C4C23×C716yes—
264C2×C4⋊C4C23×C716yes—
265C2×C4⋊C4C23×C716yes—
266C2×C22⋊C4C23×C711yescharacteristic
267C2×C22⋊C4C23×C713yes—
268C2×C22⋊C4C23×C716yes—
269C2×C22⋊C4C23×C713yes—
270C2×C22⋊C4C23×C716yes—
271C2×C22⋊C4C23×C711yescharacteristic
272C2×C22⋊C4C23×C716yes—
273C2×C22⋊C4C23×C7112yes—
274C2×C42—212no—
275C2×C42C23×C713yes—
276C2×C42C23×C716yes—
277C2×C42C23×C716yes—
278C2×C42C23×C716yes—
279#95C23×C712yes—
280#95C23×C716yes—
281#95C23×C716yes—
282#95C23×C712yes—
283#95C23×C716yes—
284#95C23×C716yes—
285#95C23×C7112yes—
286C22×C7—22no—
287C22×C7—24no—
288C22×C7—22no—
289C22×C7—24no—
290C22×C7—43no—
291C22×C7—43no—
292C22×C7—46no—
293C22×C7—43no—
294C22×C7—46no—
295C22×C7#54814yes—
296C22×C7C23×D411yescharacteristic
297C22×C7C22×#4112yes—
298C22×C7C22×#4111yescharacteristic
299C22×C7C2×2−1+411yescharacteristic
300C22×C7#58512yes—
301C4×C7—26no—
302C4×C7—24no—
303C4×C7—41no—
304C4×C7—43no—
305C4×C7—46no—
306C4×C7—41no—
307C4×C7—46no—
308C4×C7—42no—
309C4×C7C2×#123112yes—
310C24C24×C713yes—
311C24C24×C711yescharacteristic
312#41—416no—
313C2×Q8—232no—
314C2×Q8—212no—
315C2×Q8C2×C7×D4112yes—
316C2×Q8C2×C7×D414yes—
317C2×Q8C7×#4118yes—
318C2×D4—26no—
319C2×D4—26no—
320C2×D4—26no—
321C2×D4—26no—
322C2×D4—212no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—212no—
326C22×C4—26no—
327C22×C4—26no—
328C22×C4—212no—
329C22×C4—46no—
330C22×C4C2×C7×D4112yes—
331C22×C4C7×#4114yescomplemented
332C22×C4C24×C712yes—
333C22×C4C24×C711yescharacteristic
334C22×C4C24×C716yes—
335C22×C4C24×C713yes—
336C22×C4C24×C716yes—
337C22×C4C24×C711yescharacteristic
338C22×C4C24×C713yes—
339C22×C4C24×C712yes—
340C22×C4C24×C713yes—
341C4⋊C4—212no—
342C4⋊C4—212no—
343C4⋊C4—26no—
344C4⋊C4—26no—
345C4⋊C4—212no—
346C4⋊C4—212no—
347C4⋊C4—46no—
348C22⋊C4—22no—
349C22⋊C4—26no—
350C22⋊C4—22no—
351C22⋊C4—26no—
352C22⋊C4—212no—
353C22⋊C4—212no—
354C22⋊C4—46no—
355C22⋊C4—412no—
356C42—212no—
357C42—212no—
358C42—26no—
359C42—212no—
360C2×C7—24no—
361C2×C7—83no—
362C2×C7C22×#12311yescharacteristic
363C2×C7C2×#54811yescharacteristic
364C2×C7#348012yes—
365C2×C7#348711yescharacteristic
366C2×C7#351612yes—
367C23—22no—
368C23—22no—
369C23—23no—
370C23—23no—
371C23—26no—
372C23—23no—
373C23—26no—
374C23C22×C7×D412yes—
375C23C2×C7×#4112yes—
376C23C7×2−1+412yes—
377C23C25×C711yescharacteristic
378Q8—224no—
379Q8C7×#123116yescomplemented
380D4—412no—
381D4—412no—
382C2×C4—22no—
383C2×C4—26no—
384C2×C4—22no—
385C2×C4—28no—
386C2×C4—224no—
387C2×C4—21no—
388C2×C4—26no—
389C2×C4—23no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—26no—
393C2×C4—26no—
394C2×C4—23no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—23no—
398C2×C4—22no—
399C2×C4—26no—
400C2×C4—26no—
401C2×C4—26no—
402C2×C4—22no—
403C2×C4—412no—
404C2×C4—412no—
405C2×C4C22×C7×D416yes—
406C2×C4C22×C7×D416yes—
407C2×C4C2×C7×#4114yes—
408C2×C4C2×C7×#41112yes—
409C7#3657811yescharacteristic, direct product factor, Sylow
410C22—22no—
411C22—24no—
412C22—22no—
413C22—24no—
414C22—43no—
415C22—43no—
416C22—46no—
417C22—43no—
418C22—46no—
419C22C7×#54814yes—
420C22C23×C7×D411yescharacteristic
421C22C22×C7×#4112yes—
422C22C22×C7×#4111yescharacteristic
423C22C2×C7×2−1+411yescharacteristic
424C22C7×#58512yes—
425C4—26no—
426C4—24no—
427C4—41no—
428C4—43no—
429C4—46no—
430C4—46no—
431C4—41no—
432C4—42no—
433C4C2×C7×#123112yes—
434C2—24no—
435C2—83no—
436C2C22×C7×#12311yescharacteristic
437C2C2×C7×#54811yescharacteristic
438C2C7×#348012yes—
439C2C7×#348711yescharacteristic
440C2C7×#351612yes—
subgroupG/H
center, socleC23×C7C25
commutator, Frattini, cosocleC23C25×C7
prime radicalC2×C7×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×C7×Q8C2semidirect
3. C7×#2854C2semidirect
4. C7×#2504C2semidirect
5. C7×#2487C2semidirect
6. C7×#2448C2semidirect
7. C7×#2421C2semidirect
8. C7×#2351C2semidirect
9. C7×#2341C2semidirect
10. C7×Q8×C22⋊C4C2semidirect
11. C4×C7×#123C2semidirect
48. #36578C7direct (N × H)
99. C22×C4×Q8C2×C7semidirect
100. #2854C2×C7semidirect
101. #2504C2×C7semidirect
102. #2487C2×C7semidirect
103. #2448C2×C7semidirect
104. #2421C2×C7semidirect
105. #2351C2×C7semidirect
106. #2341C2×C7semidirect
107. Q8×C22⋊C4C2×C7semidirect
108. C4×#123C2×C7semidirect
130. C22×C4×C7#41semidirect
207. C7×Q8#123semidirect
331. C22×C4C7×#41semidirect
379. Q8C7×#123semidirect
409. C7#36578direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5482
C22×C4×C7×Q82
C7×#28542
C7×#25042
C7×#24872
C7×#24482
C7×#24212
C7×#23512
C7×#23412
C7×Q8×C22⋊C42
C4×C7×#1232
#365787