Finite Group Databasebeta

← C7×Q8×#137 #13238584

Subgroups — C7×Q8×#137 #13238584

Order 1792.

1782 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×#41C211yesmaximal, characteristic
2C2×C7×Q82C211yesmaximal, characteristic, complemented
3C7×#3387C216yesmaximal, complemented
4C7×#3385C216yesmaximal, complemented
5C7×Q8×Q16C212yesmaximal, complemented
6C7×Q8×SD16C212yesmaximal, complemented
7C7×#3165C213yesmaximal, complemented
8C7×#3079C213yesmaximal
9C7×#3078C213yesmaximal, complemented
10C7×Q8×M16C211yesmaximal, characteristic, complemented
11C4×C7×#137C213yesmaximal
12C2×C7×#137C2211yescharacteristic
13C7×Q82—21no—
14C7×Q82—23no—
15C7×Q82C2212yes—
16C7×Q82C2216yes—
17C7×Q82C2211yescharacteristic
18C7×#557C2213yes—
19C7×#554—23no—
20C7×#554C2213yes—
21C7×D4×Q8—21no—
22C7×D4×Q8C2211yescharacteristic
23C7×#533—23no—
24C7×#533C2213yes—
25C2×C7×C4⋊Q8—23no—
26C2×C7×C4⋊Q8C2213yes—
27C4×C7×#41C2213yes—
28C2×C4×C7×Q8—21no—
29C2×C4×C7×Q8—21no—
30C2×C4×C7×Q8C2213yes—
31C2×C4×C7×Q8C2211yescharacteristic
32C7×#501C22112yes—
33C7×#500C2216yes—
34C7×#498C2216yes—
35C7×#479C22112yes—
36C7×#477C2216yes—
37C7×#477C2216yes—
38C7×#476C2216yes—
39C7×#475C2216yes—
40C7×#475C2216yes—
41C7×#474C2216yes—
42C7×#446C2216yes—
43C7×C8×Q8C2212yes—
44C7×#441C2216yes—
45C7×#440C2216yes—
46C4×C7×Q16C2216yes—
47C4×C7×SD16C2216yes—
48C7×#428C2213yes—
49C7×#428C2216yes—
50C7×#423C2213yes—
51C7×#419C2213yes—
52C7×#417C2213yes—
53C4×C7×M16C2213yes—
54Q8×#137C711yesmaximal, characteristic, direct product factor, Sylow
55C2×C7×#41C2311yescharacteristic
56C22×C7×Q8—21no—
57C22×C7×Q8—23no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8C2311yescharacteristic
60C7×#137Q818yesdirect product factor
61C2×C7×Q16C2312yes—
62C2×C7×SD16C2312yes—
63C2×C7×M16C2313yes—
64C2×C7×M16C2311yescharacteristic
65C7×C4⋊Q8—23no—
66C7×C4⋊Q8—26no—
67C7×C4⋊Q8—212no—
68C7×C4⋊Q8—26no—
69C7×C4⋊Q8—23no—
70C7×C4⋊Q8—23no—
71C7×C4⋊Q8—23no—
72C7×C4⋊Q8C2316yes—
73C7×C4⋊Q8C2313yes—
74C7×C4⋊Q8C2316yes—
75C7×C4⋊Q8C2313yes—
76C7×#125—26no—
77C7×#125C2316yes—
78C7×C22⋊Q8—23no—
79C7×C22⋊Q8—23no—
80C7×C22⋊Q8—23no—
81C7×C22⋊Q8—23no—
82C7×C22⋊Q8—26no—
83C4×C7×Q8—21no—
84C4×C7×Q8—26no—
85C4×C7×Q8—22no—
86C4×C7×Q8—23no—
87C4×C7×Q8—23no—
88C4×C7×Q8—23no—
89C4×C7×Q8—23no—
90C4×C7×Q8—23no—
91C4×C7×Q8D411yescharacteristic
92C4×C7×Q8D413yes—
93C4×C7×Q8C2313yes—
94C4×C7×Q8C2316yes—
95C4×C7×Q8C2313yes—
96C4×C7×Q8C2316yes—
97C4×C7×Q8C2311yescharacteristic
98C4×C7×Q8C2313yes—
99C4×C7×D4—23no—
100C4×C7×D4—23no—
101C4×C7×D4C2313yes—
102C4×C7×D4C2313yes—
103C7×#117—23no—
104C7×#117C2313yes—
105C2×C7×C4⋊C4—23no—
106C2×C7×C4⋊C4—23no—
107C2×C7×C4⋊C4—23no—
108C2×C7×C4⋊C4D413yes—
109C2×C7×C4⋊C4C2313yes—
110C2×C42×C7—23no—
111C2×C42×C7—23no—
112C2×C42×C7C2313yes—
113C7×#107C23112yes—
114C7×#107C2316yes—
115C7×#106C23112yes—
116C7×#106C2316yes—
117C7×C4⋊C8C2316yes—
118C7×C4⋊C8C2316yes—
119C7×#103C2316yes—
120C7×#103C2316yes—
121C7×#103C2316yes—
122C7×#102C2316yes—
123C7×#97C2316yes—
124C4×C7×C8C2316yes—
125Q8×#41C2×C711yescharacteristic
126C2×Q82C2×C711yescharacteristic, complemented
127#3387C2×C716yescomplemented
128#3385C2×C716yescomplemented
129Q8×Q16C2×C712yescomplemented
130Q8×SD16C2×C712yescomplemented
131#3165C2×C713yescomplemented
132#3079C2×C713yes—
133#3078C2×C713yescomplemented
134Q8×M16C2×C711yescharacteristic, complemented
135C4×#137C2×C713yes—
136C7×#41—22no—
137C7×#41C2×Q814yes—
138C2×C7×Q8—24no—
139C2×C7×Q8—21no—
140C2×C7×Q8—23no—
141C2×C7×Q8—23no—
142C2×C7×Q8—26no—
143C2×C7×Q8—26no—
144C2×C7×Q8—22no—
145C2×C7×Q8—44no—
146C2×C7×Q8—86no—
147C2×C7×Q8C2×D413yes—
148C2×C7×Q8C2×D411yescharacteristic
149C2×C7×Q8C2×Q814yescomplemented
150C2×C7×Q8C2412yes—
151C2×C7×Q8C2411yescharacteristic
152C2×C7×D4—21no—
153C2×C7×D4C2411yescharacteristic
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—23no—
157C22×C4×C7—23no—
158C22×C4×C7—23no—
159C22×C4×C7C2×D413yes—
160C22×C4×C7C2×D413yes—
161C22×C4×C7C2411yescharacteristic
162C7×Q16C2×Q818yescomplemented
163C7×SD16C2×Q818yescomplemented
164C7×M16—26no—
165C7×M16C2×Q814yescomplemented
166C2×C7×C8C2416yes—
167C2×C7×C8C2412yes—
168C7×C4⋊C4—26no—
169C7×C4⋊C4—23no—
170C7×C4⋊C4—26no—
171C7×C4⋊C4—26no—
172C7×C4⋊C4—26no—
173C7×C4⋊C4—26no—
174C7×C4⋊C4—23no—
175C7×C4⋊C4—23no—
176C7×C4⋊C4—23no—
177C7×C4⋊C4—23no—
178C7×C4⋊C4—26no—
179C7×C4⋊C4C2×D413yes—
180C7×C4⋊C4C2×D416yes—
181C7×C4⋊C4C2×D413yes—
182C7×C4⋊C4C2416yes—
183C7×C4⋊C4C2413yes—
184C7×C4⋊C4C2413yes—
185C7×C22⋊C4—23no—
186C7×C22⋊C4—23no—
187C7×C22⋊C4—23no—
188C42×C7—23no—
189C42×C7—23no—
190C42×C7—23no—
191C42×C7—26no—
192C42×C7—26no—
193C42×C7C2×D413yes—
194C42×C7C2×D413yes—
195C42×C7C2413yes—
196C42×C7C2413yes—
197C2×#137C22×C711yescharacteristic
198Q82—21no—
199Q82—23no—
200Q82C22×C712yes—
201Q82C22×C711yescharacteristic
202Q82C22×C716yes—
203#557C22×C713yes—
204#554—23no—
205#554C22×C713yes—
206D4×Q8—21no—
207D4×Q8C22×C711yescharacteristic
208#533—23no—
209#533C22×C713yes—
210C2×C4⋊Q8—23no—
211C2×C4⋊Q8C22×C713yes—
212C4×#41C22×C713yes—
213C2×C4×Q8—21no—
214C2×C4×Q8—21no—
215C2×C4×Q8C22×C713yes—
216C2×C4×Q8C22×C711yescharacteristic
217#501C22×C7112yes—
218#500C22×C716yes—
219#498C22×C716yes—
220#479C22×C7112yes—
221#477C22×C716yes—
222#477C22×C716yes—
223#476C22×C716yes—
224#475C22×C716yes—
225#475C22×C716yes—
226#474C22×C716yes—
227#446C22×C716yes—
228C8×Q8C22×C712yes—
229#441C22×C716yes—
230#440C22×C716yes—
231C4×Q16C22×C716yes—
232C4×SD16C22×C716yes—
233#428C22×C716yes—
234#428C22×C713yes—
235#423C22×C713yes—
236#419C22×C713yes—
237#417C22×C713yes—
238C4×M16C22×C713yes—
239C23×C7—21no—
240C23×C7C22×D411yescharacteristic
241C7×Q8—24no—
242C7×Q8—26no—
243C7×Q8—22no—
244C7×Q8—21no—
245C7×Q8—46no—
246C7×Q8—83no—
247C7×Q8—86no—
248C7×Q8—86no—
249C7×Q8#13714yesdirect product factor
250C7×Q8C22×Q814yes—
251C7×Q8C22×Q812yes—
252C7×D4—24no—
253C7×D4—21no—
254C7×D4C22×Q812yes—
255C2×C4×C7—24no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—26no—
260C2×C4×C7—23no—
261C2×C4×C7—23no—
262C2×C4×C7—26no—
263C2×C4×C7—43no—
264C2×C4×C7—46no—
265C2×C4×C7—46no—
266C2×C4×C7—83no—
267C2×C4×C7C22×D413yes—
268C2×C4×C7C22×D413yes—
269C2×C4×C7C22×D413yes—
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D413yes—
272C2×C4×C7C22×Q812yes—
273C2×C4×C72−1+412yes—
274C2×C4×C7C2511yescharacteristic
275C7×C8—26no—
276C7×C8C22×Q814yes—
277C2×#41C23×C711yescharacteristic
278C22×Q8—21no—
279C22×Q8—23no—
280C22×Q8C7×D411yescharacteristic
281C22×Q8C23×C711yescharacteristic
282#137C7×Q818yesdirect product factor
283C2×Q16C23×C712yes—
284C2×SD16C23×C712yes—
285C2×M16C23×C713yes—
286C2×M16C23×C711yescharacteristic
287C4⋊Q8—23no—
288C4⋊Q8—26no—
289C4⋊Q8—212no—
290C4⋊Q8—26no—
291C4⋊Q8—23no—
292C4⋊Q8—23no—
293C4⋊Q8—23no—
294C4⋊Q8C23×C716yes—
295C4⋊Q8C23×C716yes—
296C4⋊Q8C23×C713yes—
297C4⋊Q8C23×C713yes—
298#125—26no—
299#125C23×C716yes—
300C22⋊Q8—23no—
301C22⋊Q8—23no—
302C22⋊Q8—23no—
303C22⋊Q8—23no—
304C22⋊Q8—26no—
305C4×Q8—21no—
306C4×Q8—26no—
307C4×Q8—22no—
308C4×Q8—23no—
309C4×Q8—23no—
310C4×Q8—23no—
311C4×Q8—23no—
312C4×Q8—23no—
313C4×Q8C7×D411yescharacteristic
314C4×Q8C7×D413yes—
315C4×Q8C23×C716yes—
316C4×Q8C23×C716yes—
317C4×Q8C23×C713yes—
318C4×Q8C23×C713yes—
319C4×Q8C23×C711yescharacteristic
320C4×Q8C23×C713yes—
321C4×D4—23no—
322C4×D4—23no—
323C4×D4C23×C713yes—
324C4×D4C23×C713yes—
325#117—23no—
326#117C23×C713yes—
327C2×C4⋊C4—23no—
328C2×C4⋊C4—23no—
329C2×C4⋊C4—23no—
330C2×C4⋊C4C7×D413yes—
331C2×C4⋊C4C23×C713yes—
332C2×C42—23no—
333C2×C42—23no—
334C2×C42C23×C713yes—
335#107C23×C716yes—
336#107C23×C7112yes—
337#106C23×C716yes—
338#106C23×C7112yes—
339C4⋊C8C23×C716yes—
340C4⋊C8C23×C716yes—
341#103C23×C716yes—
342#103C23×C716yes—
343#103C23×C716yes—
344#102C23×C716yes—
345#97C23×C716yes—
346C4×C8C23×C716yes—
347C22×C7—21no—
348C22×C7—21no—
349C22×C7—22no—
350C22×C7—41no—
351C22×C7—41no—
352C22×C7D4×Q812yes—
353C22×C7C23×D411yescharacteristic
354C4×C7—23no—
355C4×C7—23no—
356C4×C7—22no—
357C4×C7—24no—
358C4×C7—23no—
359C4×C7—46no—
360C4×C7—43no—
361C4×C7—43no—
362C4×C7D4×Q812yes—
363C4×C7C2×#13716yes—
364C4×C7C23×Q811yescharacteristic
365C4×C7C2×2−1+411yescharacteristic
366#41—22no—
367#41C2×C7×Q814yes—
368C2×Q8—24no—
369C2×Q8—21no—
370C2×Q8—23no—
371C2×Q8—23no—
372C2×Q8—26no—
373C2×Q8—26no—
374C2×Q8—22no—
375C2×Q8—44no—
376C2×Q8—86no—
377C2×Q8C2×C7×D413yes—
378C2×Q8C2×C7×D411yescharacteristic
379C2×Q8C2×C7×Q814yescomplemented
380C2×Q8C24×C712yes—
381C2×Q8C24×C711yescharacteristic
382C2×D4—21no—
383C2×D4C24×C711yescharacteristic
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—23no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4C2×C7×D413yes—
390C22×C4C2×C7×D413yes—
391C22×C4C24×C711yescharacteristic
392Q16C2×C7×Q818yescomplemented
393SD16C2×C7×Q818yescomplemented
394M16—26no—
395M16C2×C7×Q814yescomplemented
396C2×C8C24×C716yes—
397C2×C8C24×C712yes—
398C4⋊C4—26no—
399C4⋊C4—23no—
400C4⋊C4—26no—
401C4⋊C4—26no—
402C4⋊C4—26no—
403C4⋊C4—26no—
404C4⋊C4—23no—
405C4⋊C4—23no—
406C4⋊C4—23no—
407C4⋊C4—23no—
408C4⋊C4—26no—
409C4⋊C4C2×C7×D413yes—
410C4⋊C4C2×C7×D416yes—
411C4⋊C4C2×C7×D413yes—
412C4⋊C4C24×C716yes—
413C4⋊C4C24×C713yes—
414C4⋊C4C24×C713yes—
415C22⋊C4—23no—
416C22⋊C4—23no—
417C22⋊C4—23no—
418C42—23no—
419C42—23no—
420C42—23no—
421C42—26no—
422C42—26no—
423C42C2×C7×D413yes—
424C42C2×C7×D413yes—
425C42C24×C713yes—
426C42C24×C713yes—
427C2×C7—22no—
428C2×C7—42no—
429C2×C7C2×D4×Q811yescharacteristic
430C2×C7C22×#13711yescharacteristic
431C2×C7#358511yescharacteristic
432C23—21no—
433C23C22×C7×D411yescharacteristic
434Q8—22no—
435Q8—21no—
436Q8—24no—
437Q8—26no—
438Q8—46no—
439Q8—83no—
440Q8—86no—
441Q8—86no—
442Q8C7×#13714yesdirect product factor
443Q8C22×C7×Q814yes—
444Q8C22×C7×Q812yes—
445D4—24no—
446D4—21no—
447D4C22×C7×Q812yes—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—26no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—26no—
456C2×C4—43no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—83no—
460C2×C4C22×C7×D413yes—
461C2×C4C22×C7×D413yes—
462C2×C4C22×C7×D413yes—
463C2×C4C22×C7×D411yescharacteristic
464C2×C4C22×C7×D413yes—
465C2×C4C22×C7×Q812yes—
466C2×C4C7×2−1+412yes—
467C2×C4C25×C711yescharacteristic
468C8—26no—
469C8C22×C7×Q814yes—
470C7Q8×#13711yescharacteristic, direct product factor, Sylow
471C22—21no—
472C22—21no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22C7×D4×Q812yes—
477C22C23×C7×D411yescharacteristic
478C4—23no—
479C4—23no—
480C4—23no—
481C4—22no—
482C4—24no—
483C4—43no—
484C4—46no—
485C4—43no—
486C4C7×D4×Q812yes—
487C4C2×C7×#13716yes—
488C4C23×C7×Q811yescharacteristic
489C4C2×C7×2−1+411yescharacteristic
490C2—22no—
491C2—42no—
492C2C2×C7×D4×Q811yescharacteristic
493C2C22×C7×#13711yescharacteristic
494C2C7×#358511yescharacteristic
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×Q82C2semidirect
3. C7×#3387C2semidirect
4. C7×#3385C2semidirect
5. C7×Q8×Q16C2semidirect
6. C7×Q8×SD16C2semidirect
7. C7×#3165C2semidirect
9. C7×#3078C2semidirect
10. C7×Q8×M16C2semidirect
54. Q8×#137C7direct (N × H)
60. C7×#137Q8direct (N × H)
126. C2×Q82C2×C7semidirect
127. #3387C2×C7semidirect
128. #3385C2×C7semidirect
129. Q8×Q16C2×C7semidirect
130. Q8×SD16C2×C7semidirect
131. #3165C2×C7semidirect
133. #3078C2×C7semidirect
134. Q8×M16C2×C7semidirect
149. C2×C7×Q8C2×Q8semidirect
162. C7×Q16C2×Q8semidirect
163. C7×SD16C2×Q8semidirect
165. C7×M16C2×Q8semidirect
249. C7×Q8#137direct (N × H)
282. #137C7×Q8direct (N × H)
379. C2×Q8C2×C7×Q8semidirect
392. Q16C2×C7×Q8semidirect
393. SD16C2×C7×Q8semidirect
395. M16C2×C7×Q8semidirect
442. Q8C7×#137direct (N × H)
470. C7Q8×#137direct (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×Q822
C7×#33872
C7×#33852
C7×Q8×Q162
C7×Q8×SD162
C7×#31652
C7×#30792
C7×#30782
C7×Q8×M162
C4×C7×#1372
Q8×#1377