Finite Group Databasebeta

← C7×#61322 #13238538

Subgroups — C7×#61322 #13238538

Order 1792.

1782 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×#41C211yesmaximal, characteristic
2C2×C7×#557C211yesmaximal, characteristic, complemented
3C7×#3374C216yesmaximal, complemented
4C7×#3372C216yesmaximal, complemented
5C7×#3363C212yesmaximal, complemented
6C7×#3359C212yesmaximal, complemented
7C7×#3139C213yesmaximal, complemented
8C7×#3078C213yesmaximal, complemented
9C7×#3039C213yesmaximal
10C7×Q8×M16C211yesmaximal, characteristic, complemented
11C4×C7×#137C213yesmaximal
12C2×C7×#137C2213yes—
13C7×Q82C2211yescharacteristic
14C7×#557—21no—
15C7×#557—23no—
16C7×#557C2212yes—
17C7×#557C2216yes—
18C7×#557C2213yes—
19C7×#554—23no—
20C7×#554C2213yes—
21C7×D4×Q8—21no—
22C7×D4×Q8C2211yescharacteristic
23C7×#533—23no—
24C7×#533C2213yes—
25C2×C7×C4⋊Q8C2213yes—
26C2×C7×#125—23no—
27C4×C7×#41C2213yes—
28C2×C4×C7×Q8—21no—
29C2×C4×C7×Q8—21no—
30C2×C4×C7×Q8C2213yes—
31C2×C4×C7×Q8C2211yescharacteristic
32C7×#491C22112yes—
33C7×#487C2216yes—
34C7×#487C2216yes—
35C7×#479C22112yes—
36C7×#477C2216yes—
37C7×#475C2216yes—
38C7×#462C2216yes—
39C7×#462C2216yes—
40C7×#461C2216yes—
41C7×#461C2216yes—
42C7×#446C2216yes—
43C7×C8×Q8C2212yes—
44C7×#441C2216yes—
45C7×#440C2216yes—
46C4×C7×Q16C2216yes—
47C4×C7×SD16C2216yes—
48C7×#428C2213yes—
49C7×#423C2213yes—
50C7×#419C2213yes—
51C7×#419C2216yes—
52C7×#417C2211yescharacteristic
53C4×C7×M16C2213yes—
54#61322C711yesmaximal, characteristic, direct product factor, Sylow
55C2×C7×#41C2311yescharacteristic
56C22×C7×Q8—21no—
57C22×C7×Q8D411yescharacteristic
58C22×C7×Q8C2313yes—
59C7×#137—212no—
60C2×C7×Q16C2316yes—
61C2×C7×SD16C2316yes—
62C2×C7×M16C2311yescharacteristic
63C2×C7×M16C2313yes—
64C7×C4⋊Q8—23no—
65C7×C4⋊Q8—23no—
66C7×C4⋊Q8—23no—
67C7×C4⋊Q8—23no—
68C7×C4⋊Q8C2316yes—
69C7×C4⋊Q8C2316yes—
70C7×C4⋊Q8C2313yes—
71C7×C4⋊Q8C2313yes—
72C7×#125—26no—
73C7×#125—212no—
74C7×#125—26no—
75C7×#125—26no—
76C7×#125C2316yes—
77C7×C22⋊Q8—23no—
78C7×C22⋊Q8—23no—
79C7×C22⋊Q8—23no—
80C7×C22⋊Q8—23no—
81C7×C22⋊Q8—26no—
82C4×C7×Q8—21no—
83C4×C7×Q8—26no—
84C4×C7×Q8—22no—
85C4×C7×Q8—23no—
86C4×C7×Q8—23no—
87C4×C7×Q8—23no—
88C4×C7×Q8—23no—
89C4×C7×Q8—23no—
90C4×C7×Q8D411yescharacteristic
91C4×C7×Q8D413yes—
92C4×C7×Q8C2316yes—
93C4×C7×Q8C2316yes—
94C4×C7×Q8C2313yes—
95C4×C7×Q8C2313yes—
96C4×C7×Q8C2311yescharacteristic
97C4×C7×Q8C2313yes—
98C4×C7×D4—23no—
99C4×C7×D4—23no—
100C4×C7×D4C2313yes—
101C4×C7×D4C2313yes—
102C7×#117—23no—
103C7×#117C2313yes—
104C2×C7×C4⋊C4—23no—
105C2×C7×C4⋊C4—23no—
106C2×C7×C4⋊C4—23no—
107C2×C7×C4⋊C4—23no—
108C2×C7×C4⋊C4D413yes—
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C42×C7—23no—
111C2×C42×C7—23no—
112C2×C42×C7C2313yes—
113C7×#107C2316yes—
114C7×#106C2316yes—
115C7×C4⋊C8C2316yes—
116C7×C4⋊C8C2316yes—
117C7×#103C23112yes—
118C7×#103C23112yes—
119C7×#103C2316yes—
120C7×#103C2316yes—
121C7×#103C2312yes—
122C7×#102C2312yes—
123C7×#97C2316yes—
124C4×C7×C8C2316yes—
125Q8×#41C2×C711yescharacteristic
126C2×#557C2×C711yescharacteristic, complemented
127#3374C2×C716yescomplemented
128#3372C2×C716yescomplemented
129#3363C2×C712yescomplemented
130#3359C2×C712yescomplemented
131#3139C2×C713yescomplemented
132#3078C2×C713yescomplemented
133#3039C2×C713yes—
134Q8×M16C2×C711yescharacteristic, complemented
135C4×#137C2×C713yes—
136C7×#41—22no—
137C7×#41#4114yes—
138C2×C7×Q8—24no—
139C2×C7×Q8—23no—
140C2×C7×Q8—23no—
141C2×C7×Q8—23no—
142C2×C7×Q8—26no—
143C2×C7×Q8—26no—
144C2×C7×Q8—44no—
145C2×C7×Q8C2×D413yes—
146C2×C7×Q8C2×D411yescharacteristic
147C2×C7×Q8C2416yes—
148C2×C7×Q8C2411yescharacteristic
149C2×C7×D4—21no—
150C2×C7×D4C2411yescharacteristic
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—23no—
154C22×C4×C7—23no—
155C22×C4×C7—23no—
156C22×C4×C7C2×D413yes—
157C22×C4×C7C2×D413yes—
158C22×C4×C7C2411yescharacteristic
159C7×Q16—212no—
160C7×SD16—212no—
161C7×M16—26no—
162C7×M16—22no—
163C2×C7×C8C2412yes—
164C2×C7×C8C2416yes—
165C7×C4⋊C4—26no—
166C7×C4⋊C4—23no—
167C7×C4⋊C4—26no—
168C7×C4⋊C4—26no—
169C7×C4⋊C4—26no—
170C7×C4⋊C4—26no—
171C7×C4⋊C4—26no—
172C7×C4⋊C4—26no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—23no—
175C7×C4⋊C4—23no—
176C7×C4⋊C4—23no—
177C7×C4⋊C4—26no—
178C7×C4⋊C4C2×D413yes—
179C7×C4⋊C4C2×D416yes—
180C7×C4⋊C4C2×D413yes—
181C7×C4⋊C4C2412yes—
182C7×C4⋊C4C2413yes—
183C7×C4⋊C4C2413yes—
184C7×C22⋊C4—23no—
185C7×C22⋊C4—23no—
186C7×C22⋊C4—23no—
187C42×C7—23no—
188C42×C7—23no—
189C42×C7—23no—
190C42×C7—26no—
191C42×C7—26no—
192C42×C7C2×D413yes—
193C42×C7C2×D413yes—
194C42×C7C2413yes—
195C42×C7C2413yes—
196C2×#137C22×C713yes—
197Q82C22×C711yescharacteristic
198#557—21no—
199#557—23no—
200#557C22×C712yes—
201#557C22×C716yes—
202#557C22×C713yes—
203#554—23no—
204#554C22×C713yes—
205D4×Q8—21no—
206D4×Q8C22×C711yescharacteristic
207#533—23no—
208#533C22×C713yes—
209C2×C4⋊Q8C22×C713yes—
210C2×#125—23no—
211C4×#41C22×C713yes—
212C2×C4×Q8—21no—
213C2×C4×Q8—21no—
214C2×C4×Q8C22×C713yes—
215C2×C4×Q8C22×C711yescharacteristic
216#491C22×C7112yes—
217#487C22×C716yes—
218#487C22×C716yes—
219#479C22×C7112yes—
220#477C22×C716yes—
221#475C22×C716yes—
222#462C22×C716yes—
223#462C22×C716yes—
224#461C22×C716yes—
225#461C22×C716yes—
226#446C22×C716yes—
227C8×Q8C22×C712yes—
228#441C22×C716yes—
229#440C22×C716yes—
230C4×Q16C22×C716yes—
231C4×SD16C22×C716yes—
232#428C22×C713yes—
233#423C22×C713yes—
234#419C22×C716yes—
235#419C22×C713yes—
236#417C22×C711yescharacteristic
237C4×M16C22×C713yes—
238C23×C7—21no—
239C23×C7C22×D411yescharacteristic
240C7×Q8—26no—
241C7×Q8—21no—
242C7×Q8—26no—
243C7×Q8—26no—
244C7×Q8—46no—
245C7×Q8—46no—
246C7×Q8—83no—
247C7×Q8#13714yescomplemented
248C7×Q8C2×#4112yes—
249C7×D4—24no—
250C7×D4—21no—
251C7×D4C2×#4112yes—
252C2×C4×C7—21no—
253C2×C4×C7—24no—
254C2×C4×C7—26no—
255C2×C4×C7—22no—
256C2×C4×C7—23no—
257C2×C4×C7—23no—
258C2×C4×C7—26no—
259C2×C4×C7—43no—
260C2×C4×C7—42no—
261C2×C4×C7—46no—
262C2×C4×C7—46no—
263C2×C4×C7—83no—
264C2×C4×C7C22×D413yes—
265C2×C4×C7C22×D413yes—
266C2×C4×C7C22×D413yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D413yes—
269C2×C4×C7C2×#4112yes—
270C2×C4×C72−1+412yes—
271C2×C4×C7C2511yescharacteristic
272C7×C8—26no—
273C7×C8—22no—
274C2×#41C23×C711yescharacteristic
275C22×Q8—21no—
276C22×Q8C7×D411yescharacteristic
277C22×Q8C23×C713yes—
278#137—212no—
279C2×Q16C23×C716yes—
280C2×SD16C23×C716yes—
281C2×M16C23×C713yes—
282C2×M16C23×C711yescharacteristic
283C4⋊Q8—23no—
284C4⋊Q8—23no—
285C4⋊Q8—23no—
286C4⋊Q8—23no—
287C4⋊Q8C23×C716yes—
288C4⋊Q8C23×C716yes—
289C4⋊Q8C23×C713yes—
290C4⋊Q8C23×C713yes—
291#125—26no—
292#125—212no—
293#125—26no—
294#125—26no—
295#125C23×C716yes—
296C22⋊Q8—23no—
297C22⋊Q8—23no—
298C22⋊Q8—23no—
299C22⋊Q8—23no—
300C22⋊Q8—26no—
301C4×Q8—21no—
302C4×Q8—26no—
303C4×Q8—22no—
304C4×Q8—23no—
305C4×Q8—23no—
306C4×Q8—23no—
307C4×Q8—23no—
308C4×Q8—23no—
309C4×Q8C7×D411yescharacteristic
310C4×Q8C7×D413yes—
311C4×Q8C23×C716yes—
312C4×Q8C23×C716yes—
313C4×Q8C23×C713yes—
314C4×Q8C23×C713yes—
315C4×Q8C23×C711yescharacteristic
316C4×Q8C23×C713yes—
317C4×D4—23no—
318C4×D4—23no—
319C4×D4C23×C713yes—
320C4×D4C23×C713yes—
321#117—23no—
322#117C23×C713yes—
323C2×C4⋊C4—23no—
324C2×C4⋊C4—23no—
325C2×C4⋊C4—23no—
326C2×C4⋊C4—23no—
327C2×C4⋊C4C7×D413yes—
328C2×C4⋊C4C23×C711yescharacteristic
329C2×C42—23no—
330C2×C42—23no—
331C2×C42C23×C713yes—
332#107C23×C716yes—
333#106C23×C716yes—
334C4⋊C8C23×C716yes—
335C4⋊C8C23×C716yes—
336#103C23×C712yes—
337#103C23×C716yes—
338#103C23×C716yes—
339#103C23×C7112yes—
340#103C23×C7112yes—
341#102C23×C712yes—
342#97C23×C716yes—
343C4×C8C23×C716yes—
344C22×C7—21no—
345C22×C7—21no—
346C22×C7—22no—
347C22×C7—41no—
348C22×C7—41no—
349C22×C7#54812yes—
350C22×C7C23×D411yescharacteristic
351C4×C7—23no—
352C4×C7—23no—
353C4×C7—22no—
354C4×C7—23no—
355C4×C7—42no—
356C4×C7—43no—
357C4×C7—46no—
358C4×C7—43no—
359C4×C7#54812yes—
360C4×C7C2×#13716yes—
361C4×C7C22×#4111yescharacteristic
362C4×C7C2×2−1+411yescharacteristic
363#41—22no—
364#41C7×#4114yes—
365C2×Q8—24no—
366C2×Q8—23no—
367C2×Q8—23no—
368C2×Q8—23no—
369C2×Q8—26no—
370C2×Q8—26no—
371C2×Q8—44no—
372C2×Q8C2×C7×D413yes—
373C2×Q8C2×C7×D411yescharacteristic
374C2×Q8C24×C716yes—
375C2×Q8C24×C711yescharacteristic
376C2×D4—21no—
377C2×D4C24×C711yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—23no—
383C22×C4C2×C7×D413yes—
384C22×C4C2×C7×D413yes—
385C22×C4C24×C711yescharacteristic
386Q16—212no—
387SD16—212no—
388M16—26no—
389M16—22no—
390C2×C8C24×C712yes—
391C2×C8C24×C716yes—
392C4⋊C4—26no—
393C4⋊C4—23no—
394C4⋊C4—26no—
395C4⋊C4—26no—
396C4⋊C4—26no—
397C4⋊C4—26no—
398C4⋊C4—26no—
399C4⋊C4—26no—
400C4⋊C4—21no—
401C4⋊C4—23no—
402C4⋊C4—23no—
403C4⋊C4—23no—
404C4⋊C4—26no—
405C4⋊C4C2×C7×D413yes—
406C4⋊C4C2×C7×D416yes—
407C4⋊C4C2×C7×D413yes—
408C4⋊C4C24×C712yes—
409C4⋊C4C24×C713yes—
410C4⋊C4C24×C713yes—
411C22⋊C4—23no—
412C22⋊C4—23no—
413C22⋊C4—23no—
414C42—23no—
415C42—23no—
416C42—23no—
417C42—26no—
418C42—26no—
419C42C2×C7×D413yes—
420C42C2×C7×D413yes—
421C42C24×C713yes—
422C42C24×C713yes—
423C2×C7—22no—
424C2×C7—42no—
425C2×C7C2×#54811yescharacteristic
426C2×C7C22×#13711yescharacteristic
427C2×C7#358611yescharacteristic
428C23—21no—
429C23C22×C7×D411yescharacteristic
430Q8—21no—
431Q8—26no—
432Q8—26no—
433Q8—26no—
434Q8—46no—
435Q8—46no—
436Q8—83no—
437Q8C7×#13714yescomplemented
438Q8C2×C7×#4112yes—
439D4—24no—
440D4—21no—
441D4C2×C7×#4112yes—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—26no—
445C2×C4—22no—
446C2×C4—23no—
447C2×C4—23no—
448C2×C4—26no—
449C2×C4—43no—
450C2×C4—42no—
451C2×C4—46no—
452C2×C4—46no—
453C2×C4—83no—
454C2×C4C22×C7×D413yes—
455C2×C4C22×C7×D413yes—
456C2×C4C22×C7×D413yes—
457C2×C4C22×C7×D411yescharacteristic
458C2×C4C22×C7×D413yes—
459C2×C4C2×C7×#4112yes—
460C2×C4C7×2−1+412yes—
461C2×C4C25×C711yescharacteristic
462C8—26no—
463C8—22no—
464C7#6132211yescharacteristic, direct product factor, Sylow
465C22—21no—
466C22—21no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22C7×#54812yes—
471C22C23×C7×D411yescharacteristic
472C4—23no—
473C4—23no—
474C4—23no—
475C4—22no—
476C4—43no—
477C4—42no—
478C4—43no—
479C4—46no—
480C4C7×#54812yes—
481C4C2×C7×#13716yes—
482C4C22×C7×#4111yescharacteristic
483C4C2×C7×2−1+411yescharacteristic
484C2—22no—
485C2—42no—
486C2C2×C7×#54811yescharacteristic
487C2C22×C7×#13711yescharacteristic
488C2C7×#358611yescharacteristic
subgroupG/H
center, socleC22×C7C23×D4
commutator, Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#557C2semidirect
3. C7×#3374C2semidirect
4. C7×#3372C2semidirect
5. C7×#3363C2semidirect
6. C7×#3359C2semidirect
7. C7×#3139C2semidirect
8. C7×#3078C2semidirect
10. C7×Q8×M16C2semidirect
54. #61322C7direct (N × H)
126. C2×#557C2×C7semidirect
127. #3374C2×C7semidirect
128. #3372C2×C7semidirect
129. #3363C2×C7semidirect
130. #3359C2×C7semidirect
131. #3139C2×C7semidirect
132. #3078C2×C7semidirect
134. Q8×M16C2×C7semidirect
247. C7×Q8#137semidirect
437. Q8C7×#137semidirect
464. C7#61322direct (N × H)

Maximal subgroups

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

subgroupindex
C7×Q8×#412
C2×C7×#5572
C7×#33742
C7×#33722
C7×#33632
C7×#33592
C7×#31392
C7×#30782
C7×#30392
C7×Q8×M162
C4×C7×#1372
#613227