Finite Group Databasebeta

← C2×#11729678 #13329421

Subgroups — C2×#11729678 #13329421

Order 1792.

1054 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×#89864C211yesmaximal, characteristic, complemented
3C2×#89793C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic
5#11729678C218yesmaximal, direct product factor
6C2×#89341C211yesmaximal, characteristic, complemented
7C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic
9C7×#554—22no—
10C7×#554—22no—
11C7×#554—22no—
12C7×#554C2214yes—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×#125—21no—
15C2×C7×C22⋊Q8—21no—
16C2×C7×C22⋊Q8—22no—
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C22×C7×C4⋊C4—21no—
21C22×#1144—22no—
22C2×#6432C2211yescharacteristic
23C2×#6409—21no—
24C2×#6406—21no—
25#89864—22no—
26#89864C2214yescomplemented
27#89793—22no—
28#89793C2214yescomplemented
29#89780—22no—
30#89780C2214yes—
31C2×#6382—21no—
32C2×#6381—21no—
33C2×C4⋊(C7⋊C8)C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4×C7⋊C8C2211yescharacteristic
36#89341C2214yescomplemented
37Q8⋊(C7⋊C8)C2214yescomplemented
38D4⋊(C7⋊C8)C2214yes—
39C2×#1632—71nomaximal, Sylow
40C22×C7×Q8D411yescharacteristic
41C22×C7×Q8D412yes—
42C22×C7×D4D411yescharacteristic
43C23×C4×C7—22no—
44C23×C4×C7—21no—
45C22×Dic14—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8D412yes—
49C7×C4⋊Q8C2312yes—
50C7×#125—24no—
51C7×#125—24no—
52C7×C22⋊Q8—24no—
53C7×C22⋊Q8—28no—
54C7×C22⋊Q8—24no—
55C7×C22⋊Q8—28no—
56C4×C7×Q8—21no—
57C4×C7×Q8—21no—
58C4×C7×Q8D412yes—
59C4×C7×Q8C2312yes—
60C4×C7×D4—24no—
61C4×C7×D4—24no—
62C4×C7×D4—21no—
63C4×C7×D4—21no—
64C4×C7×D4D412yes—
65C4×C7×D4C2312yes—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—24no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×#1144—28no—
80C2×#1144—48no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C22×C7⋊C8—21no—
84C22×C7⋊C8—21no—
85C22×C7⋊C8—21no—
86#6432—21no—
87#6432—21no—
88#6432—21no—
89#6432C2312yes—
90#6409—24no—
91#6406—24no—
92#6382—24no—
93#6381—24no—
94C4⋊(C7⋊C8)—21no—
95C4⋊(C7⋊C8)—21no—
96C4⋊(C7⋊C8)C2312yes—
97C4⋊(C7⋊C8)C2312yes—
98C4×C7⋊C8—21no—
99C4×C7⋊C8C2312yes—
100C2×#554D711yescharacteristic
101C2×#494—71no—
102C2×#477—71no—
103C2×#476—71no—
104#1632—78no—
105C2×#333—71no—
106C2×Q8⋊C8—71no—
107C2×D4⋊C8—71no—
108C24×C7—21no—
109C2×C7×Q8—21no—
110C2×C7×Q8—22no—
111C2×C7×Q8—22no—
112C2×C7×Q8—22no—
113C2×C7×Q8—44no—
114C2×C7×Q8C2×D412yes—
115C2×C7×Q8C2×D414yes—
116C2×C7×D4—21no—
117C2×C7×D4—22no—
118C2×C7×D4Q1614yescomplemented
119C2×C7×D4C2×D412yes—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—28no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—42no—
128C22×C4×C7—42no—
129C22×C4×C7—84no—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×Dic7—41no—
134C2×Dic14—22no—
135C2×Dic14—44no—
136C2×Dic14—41no—
137C7×C4⋊C4—28no—
138C7×C4⋊C4—28no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4C2×D412yes—
148C7×C4⋊C4C2×D412yes—
149C7×C22⋊C4—28no—
150C7×C22⋊C4—22no—
151C7×C22⋊C4—22no—
152C42×C7—22no—
153C42×C7—22no—
154C42×C7C2×D411yescharacteristic
155C42×C7C2×D411yescharacteristic
156C42×C7C2×D411yescharacteristic
157C42×C7C2411yescharacteristic
158#1144—416no—
159C4⋊Dic7—22no—
160C4⋊Dic7—22no—
161C4⋊Dic7—41no—
162C4⋊Dic7—41no—
163C2×C7⋊C8—24no—
164C2×C7⋊C8—22no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—42no—
168C2×C7⋊C8—42no—
169C22×Q16—142no—
170#554—22no—
171#554—22no—
172#554—22no—
173#554C2×D714yes—
174C2×C4⋊Q8—71no—
175C2×C4⋊Q8C2×D711yescharacteristic
176C2×#125—21no—
177C2×C22⋊Q8—21no—
178C2×C22⋊Q8—22no—
179C2×C4×Q8C2×D711yescharacteristic
180C2×C4×D4—21no—
181C2×C4×D4C2×D711yescharacteristic
182C22×C4⋊C4—21no—
183#494—74no—
184#494—142no—
185#477—74no—
186#477—142no—
187#476—74no—
188#476—142no—
189C2×#107—141no—
190C2×#106—141no—
191C2×C4⋊C8—71no—
192C2×C4⋊C8—71no—
193C2×#103—141no—
194C2×#102—141no—
195C2×C4×C8—71no—
196#333—74no—
197Q8⋊C8—74no—
198D4⋊C8—74no—
199C23×C7—22no—
200C23×C7—24no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7#12011yescharacteristic
204C7×Q8—22no—
205C7×Q8—22no—
206C7×Q8—44no—
207C7×Q8—42no—
208C7×Q8—82no—
209C7×D4—22no—
210C7×D4—24no—
211C7×D4C2×Q1614yescomplemented
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—22no—
215C2×C4×C7—22no—
216C2×C4×C7—24no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—44no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—44no—
224C2×C4×C7—88no—
225C2×C4×C7#12011yescharacteristic
226C2×C4×C7#12011yescharacteristic
227C2×C4×C7#12011yescharacteristic
228C2×C4×C7C2×Q1612yes—
229C2×C4×C7#13512yes—
230C2×C4×C7#13712yes—
231C2×C4×C7#13712yes—
232C2×C4×C7C22×D411yescharacteristic
233C2×C4×C7C22×D411yescharacteristic
234C2×C4×C7C22×D411yescharacteristic
235C2×Dic7—42no—
236C2×Dic7—82no—
237Dic14—44no—
238Dic14—82no—
239C7⋊C8—24no—
240C7⋊C8—42no—
241C7⋊C8—42no—
242C22×Q8—141no—
243C22×Q8C22⋊D711yescharacteristic
244C22×Q8C22⋊D712yes—
245C22×D4C22⋊D711yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C2×Q16—148no—
249C2×Q16—288no—
250C22×C8—141no—
251C22×C8—141no—
252C22×C8—141no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—72no—
256C4⋊Q8—141no—
257C4⋊Q8—141no—
258C4⋊Q8—141no—
259C4⋊Q8C22⋊D712yes—
260C4⋊Q8C22×D712yes—
261#125—24no—
262#125—24no—
263C22⋊Q8—24no—
264C22⋊Q8—28no—
265C22⋊Q8—24no—
266C22⋊Q8—28no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C22⋊D712yes—
270C4×Q8C22×D712yes—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4C22⋊D712yes—
276C4×D4C22×D712yes—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—141no—
285C2×C4⋊C4—141no—
286C2×C4⋊C4C22⋊D711yescharacteristic
287C2×C4⋊C4C22⋊D711yescharacteristic
288C2×C22⋊C4—22no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42C22×D711yescharacteristic
292#107—144no—
293#106—144no—
294C4⋊C8—72no—
295C4⋊C8—72no—
296C4⋊C8—141no—
297C4⋊C8—141no—
298#103—144no—
299#102—144no—
300C4×C8—72no—
301C4×C8—141no—
302C22×C7—24no—
303C22×C7—48no—
304C22×C7—42no—
305C22×C7—42no—
306C22×C7#45112yes—
307C22×C7#45212yes—
308C22×C7#45312yes—
309C22×C7C2×#12011yescharacteristic
310C4×C7—21no—
311C4×C7—21no—
312C4×C7—42no—
313C4×C7—42no—
314C4×C7—42no—
315C4×C7—44no—
316C4×C7#45112yes—
317C4×C7#45212yes—
318C4×C7C22×Q1611yescharacteristic
319C4×C7C2×#13511yescharacteristic
320C4×C7C2×#13711yescharacteristic
321C4×C7C2×#13711yescharacteristic
322Dic7—82no—
323C24—21no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—44no—
329C2×Q8—142no—
330C2×Q8—284no—
331C2×Q8—281no—
332C2×Q8C2×C22⋊D712yes—
333C2×Q8C2×C22⋊D714yes—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4#114414yescomplemented
337C2×D4C2×C22⋊D712yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—84no—
348C22×C4—281no—
349C22×C4C2×C22⋊D711yescharacteristic
350C22×C4C2×C22⋊D711yescharacteristic
351C22×C4C2×C22⋊D711yescharacteristic
352Q16—2816no—
353C2×C8—144no—
354C2×C8—142no—
355C2×C8—142no—
356C2×C8—142no—
357C2×C8—282no—
358C2×C8—282no—
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—142no—
370C4⋊C4—142no—
371C4⋊C4—281no—
372C4⋊C4—281no—
373C4⋊C4C2×C22⋊D712yes—
374C4⋊C4C2×C22⋊D712yes—
375C22⋊C4—28no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C42—22no—
379C42—22no—
380C42C2×C22⋊D711yescharacteristic
381C42C2×C22⋊D711yescharacteristic
382C42C2×C22⋊D711yescharacteristic
383C42C23×D711yescharacteristic
384C2×C7—44no—
385C2×C7#163214yescomplemented
386C2×C7C2×#45111yescharacteristic
387C2×C7C2×#45211yescharacteristic
388C2×C7C2×#45311yescharacteristic
389C23—24no—
390C23—22no—
391C23—42no—
392C23—42no—
393C23#651611yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—44no—
397Q8—42no—
398Q8—82no—
399Q8—284no—
400Q8—562no—
401D4—22no—
402D4—24no—
403D4C2×#114414yescomplemented
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—88no—
417C2×C4—282no—
418C2×C4—562no—
419C2×C4#650512yes—
420C2×C4C2×#114412yes—
421C2×C4#651312yes—
422C2×C4#651412yes—
423C2×C4#651611yescharacteristic
424C2×C4#651611yescharacteristic
425C2×C4#651611yescharacteristic
426C2×C4C22×C22⋊D711yescharacteristic
427C2×C4C22×C22⋊D711yescharacteristic
428C2×C4C22×C22⋊D711yescharacteristic
429C8—144no—
430C8—282no—
431C8—282no—
432C7C2×#163211yescharacteristic, complemented, Sylow
433C22—24no—
434C22—48no—
435C22—42no—
436C22—42no—
437C22#9000012yes—
438C22#9001112yes—
439C22#9001312yes—
440C22C2×#651611yescharacteristic
441C4—21no—
442C4—21no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—44no—
447C4—562no—
448C4#9000012yes—
449C4#9001112yes—
450C4C2×#650511yescharacteristic
451C4C22×#114411yescharacteristic
452C4C2×#651311yescharacteristic
453C4C2×#651411yescharacteristic
454C2—44no—
455C2#1172967814yesdirect product factor
456C2C2×#9000011yescharacteristic
457C2C2×#9001111yescharacteristic
458C2C2×#9001311yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#554C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenterC2×#554D7
hypocenterC7C2×#1632
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#89864C2semidirect
3. C2×#89793C2semidirect
5. #11729678C2direct (N × H)
6. C2×#89341C2semidirect
7. C2×Q8⋊(C7⋊C8)C2semidirect
26. #89864C22semidirect
28. #89793C22semidirect
36. #89341C22semidirect
37. Q8⋊(C7⋊C8)C22semidirect
118. C2×C7×D4Q16semidirect
211. C7×D4C2×Q16semidirect
336. C2×D4#1144semidirect
385. C2×C7#1632semidirect
403. D4C2×#1144semidirect
432. C7C2×#1632semidirect
455. C2#11729678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#898642
C2×#897932
C2×#897802
#117296782
C2×#893412
C2×Q8⋊(C7⋊C8)2
C2×D4⋊(C7⋊C8)2
C2×#16327