Finite Group Databasebeta

← C2×#11729686 #13329549

Subgroups — C2×#11729686 #13329549

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2C2×#89865C211yesmaximal, characteristic, complemented
3C2×#89793C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic
5#11729686C218yesmaximal, direct product factor
6C2×#89342C211yesmaximal, characteristic, complemented
7C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic
9C7×#556—22no—
10C7×#556—22no—
11C7×#556—22no—
12C7×#556C2214yes—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×#126—22no—
15C2×C7×#124—21no—
16C2×C7×C22⋊Q8—21no—
17C2×C4×C7×Q8—21no—
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C2×C7×#117—21no—
21C2×#6432C2211yescharacteristic
22C2×#6409—21no—
23C2×#6406—21no—
24#89865—22no—
25#89865C2214yescomplemented
26#89793—22no—
27#89793C2214yescomplemented
28#89780—22no—
29#89780C2214yes—
30C2×#6384—22no—
31C2×#6382—21no—
32C2×#6381—21no—
33C2×C4⋊(C7⋊C8)C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4×C7⋊C8C2211yescharacteristic
36#89342C2214yescomplemented
37Q8⋊(C7⋊C8)C2214yescomplemented
38D4⋊(C7⋊C8)C2214yes—
39C2×#1638—71nomaximal, Sylow
40C22×C7×Q8—21no—
41C22×C7×Q8D411yescharacteristic
42C22×C7×D4D411yescharacteristic
43C23×C4×C7—21no—
44C22×Dic14—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8D412yes—
48C7×C4⋊Q8C2312yes—
49C7×#126—28no—
50C7×#126—28no—
51C7×#124—24no—
52C7×#124—24no—
53C7×C22⋊Q8—24no—
54C7×C22⋊Q8—24no—
55C4×C7×Q8—24no—
56C4×C7×Q8—24no—
57C4×C7×Q8—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8D412yes—
60C4×C7×Q8C2312yes—
61C4×C7×D4—21no—
62C4×C7×D4—21no—
63C4×C7×D4D412yes—
64C4×C7×D4C2312yes—
65C7×#117—24no—
66C7×#117—24no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D412yes—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7—22no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C22×C7⋊C8—21no—
82C22×C7⋊C8—21no—
83C22×C7⋊C8—21no—
84#6432—21no—
85#6432—21no—
86#6432—21no—
87#6432C2312yes—
88#6409—24no—
89#6406—24no—
90#6384—28no—
91#6382—24no—
92#6381—24no—
93C4⋊(C7⋊C8)—21no—
94C4⋊(C7⋊C8)—21no—
95C4⋊(C7⋊C8)C2312yes—
96C4⋊(C7⋊C8)C2312yes—
97C4×C7⋊C8—21no—
98C4×C7⋊C8C2312yes—
99C2×#556D711yescharacteristic
100C2×#487—71no—
101C2×#477—71no—
102C2×#476—71no—
103#1638—78no—
104C2×#332—71no—
105C2×Q8⋊C8—71no—
106C2×D4⋊C8—71no—
107C24×C7—21no—
108C2×C7×Q8—21no—
109C2×C7×Q8—22no—
110C2×C7×Q8—22no—
111C2×C7×Q8—22no—
112C2×C7×Q8—22no—
113C2×C7×Q8—42no—
114C2×C7×Q8—42no—
115C2×C7×Q8C2×D412yes—
116C2×C7×D4—21no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4C2×D412yes—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—42no—
127C22×C4×C7—42no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×Dic7—41no—
132C2×Dic14—22no—
133C2×Dic14—44no—
134C2×Dic14—41no—
135C7×C4⋊C4—28no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4C2×D412yes—
144C7×C4⋊C4C2×D412yes—
145C7×C4⋊C4C2×D414yes—
146C7×C22⋊C4—28no—
147C7×C22⋊C4—28no—
148C7×C22⋊C4—22no—
149C7×C22⋊C4—22no—
150C42×C7—28no—
151C42×C7—22no—
152C42×C7—22no—
153C42×C7C2×D411yescharacteristic
154C42×C7C2×D411yescharacteristic
155C42×C7C2×D411yescharacteristic
156C42×C7C2411yescharacteristic
157C4⋊Dic7—22no—
158C4⋊Dic7—22no—
159C4⋊Dic7—41no—
160C4⋊Dic7—41no—
161C2×C7⋊C8—22no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—22no—
164C2×C7⋊C8—42no—
165C2×C7⋊C8—42no—
166C2×C7⋊C8—42no—
167#556—22no—
168#556—22no—
169#556—22no—
170#556C2×D714yes—
171C2×C4⋊Q8—71no—
172C2×C4⋊Q8C2×D711yescharacteristic
173C2×#126—22no—
174C2×#124—21no—
175C2×C22⋊Q8—21no—
176C2×C4×Q8—21no—
177C2×C4×Q8C2×D711yescharacteristic
178C2×C4×D4C2×D711yescharacteristic
179C2×#117—21no—
180#487—74no—
181#487—142no—
182#477—74no—
183#477—142no—
184#476—74no—
185#476—142no—
186C2×#107—141no—
187C2×#106—141no—
188C2×C4⋊C8—71no—
189C2×C4⋊C8—71no—
190C2×#103—141no—
191C2×#103—142no—
192C2×#102—141no—
193C2×C4×C8—71no—
194#332—74no—
195Q8⋊C8—74no—
196D4⋊C8—74no—
197C23×C7—22no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7#12011yescharacteristic
202C7×Q8—22no—
203C7×Q8—22no—
204C7×Q8—42no—
205C7×Q8—42no—
206C7×Q8—42no—
207C7×Q8—44no—
208C7×D4—22no—
209C7×D4—22no—
210C7×D4—42no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—22no—
215C2×C4×C7—24no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—44no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—44no—
223C2×C4×C7#12011yescharacteristic
224C2×C4×C7#12011yescharacteristic
225C2×C4×C7#12011yescharacteristic
226C2×C4×C7#13512yes—
227C2×C4×C7#13512yes—
228C2×C4×C7#13712yes—
229C2×C4×C7#13712yes—
230C2×C4×C7C22×D411yescharacteristic
231C2×C4×C7C22×D411yescharacteristic
232C2×C4×C7C22×D411yescharacteristic
233C2×Dic7—42no—
234C2×Dic7—82no—
235Dic14—44no—
236Dic14—82no—
237C7⋊C8—42no—
238C7⋊C8—42no—
239C7⋊C8—42no—
240C22×Q8—21no—
241C22×Q8—141no—
242C22×Q8C22⋊D711yescharacteristic
243C22×D4C22⋊D711yescharacteristic
244C23×C4—21no—
245C22×C8—141no—
246C22×C8—141no—
247C22×C8—141no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—72no—
251C4⋊Q8—141no—
252C4⋊Q8—141no—
253C4⋊Q8—141no—
254C4⋊Q8C22⋊D712yes—
255C4⋊Q8C22×D712yes—
256#126—28no—
257#126—28no—
258#124—24no—
259#124—24no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8C22⋊D712yes—
267C4×Q8C22×D712yes—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4C22⋊D712yes—
271C4×D4C22×D712yes—
272#117—24no—
273#117—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—141no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4C22⋊D711yescharacteristic
280C2×C4⋊C4C22⋊D711yescharacteristic
281C2×C4⋊C4C22⋊D712yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42C22×D711yescharacteristic
288#107—144no—
289#106—144no—
290C4⋊C8—72no—
291C4⋊C8—72no—
292C4⋊C8—141no—
293C4⋊C8—141no—
294#103—144no—
295#103—148no—
296#102—144no—
297C4×C8—72no—
298C4×C8—141no—
299C22×C7—42no—
300C22×C7—42no—
301C22×C7—42no—
302C22×C7—84no—
303C22×C7#45212yes—
304C22×C7#45212yes—
305C22×C7#45412yes—
306C22×C7C2×#12011yescharacteristic
307C4×C7—21no—
308C4×C7—21no—
309C4×C7—42no—
310C4×C7—42no—
311C4×C7—44no—
312C4×C7—42no—
313C4×C7#45212yes—
314C4×C7#45212yes—
315C4×C7C2×#13511yescharacteristic
316C4×C7C2×#13511yescharacteristic
317C4×C7C2×#13711yescharacteristic
318C4×C7C2×#13711yescharacteristic
319Dic7—82no—
320C24—21no—
321C2×Q8—21no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—142no—
329C2×Q8—284no—
330C2×Q8—281no—
331C2×Q8C2×C22⋊D712yes—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4C2×C22⋊D712yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—281no—
345C22×C4C2×C22⋊D711yescharacteristic
346C22×C4C2×C22⋊D711yescharacteristic
347C22×C4C2×C22⋊D711yescharacteristic
348C2×C8—142no—
349C2×C8—142no—
350C2×C8—142no—
351C2×C8—282no—
352C2×C8—282no—
353C2×C8—282no—
354C4⋊C4—28no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—142no—
363C4⋊C4—142no—
364C4⋊C4—281no—
365C4⋊C4—281no—
366C4⋊C4C2×C22⋊D714yes—
367C4⋊C4C2×C22⋊D712yes—
368C4⋊C4C2×C22⋊D712yes—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C42—28no—
374C42—22no—
375C42—22no—
376C42C2×C22⋊D711yescharacteristic
377C42C2×C22⋊D711yescharacteristic
378C42C2×C22⋊D711yescharacteristic
379C42C23×D711yescharacteristic
380C2×C7—82no—
381C2×C7#163814yescomplemented
382C2×C7C2×#45211yescharacteristic
383C2×C7C2×#45211yescharacteristic
384C2×C7C2×#45411yescharacteristic
385C23—22no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23#651611yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—42no—
393Q8—42no—
394Q8—42no—
395Q8—44no—
396Q8—284no—
397Q8—562no—
398D4—22no—
399D4—22no—
400D4—42no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—282no—
414C2×C4—562no—
415C2×C4#651312yes—
416C2×C4#651312yes—
417C2×C4#651412yes—
418C2×C4#651412yes—
419C2×C4#651611yescharacteristic
420C2×C4#651611yescharacteristic
421C2×C4#651611yescharacteristic
422C2×C4C22×C22⋊D711yescharacteristic
423C2×C4C22×C22⋊D711yescharacteristic
424C2×C4C22×C22⋊D711yescharacteristic
425C8—282no—
426C8—282no—
427C8—282no—
428C7C2×#163811yescharacteristic, complemented, Sylow
429C22—42no—
430C22—42no—
431C22—42no—
432C22—84no—
433C22#9001112yes—
434C22#9001112yes—
435C22#9001712yes—
436C22C2×#651611yescharacteristic
437C4—21no—
438C4—21no—
439C4—42no—
440C4—42no—
441C4—44no—
442C4—42no—
443C4—562no—
444C4#9001112yes—
445C4#9001112yes—
446C4C2×#651311yescharacteristic
447C4C2×#651311yescharacteristic
448C4C2×#651411yescharacteristic
449C4C2×#651411yescharacteristic
450C2—82no—
451C2#1172968614yesdirect product factor
452C2C2×#9001111yescharacteristic
453C2C2×#9001111yescharacteristic
454C2C2×#9001711yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#556C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenterC2×#556D7
hypocenterC7C2×#1638
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#89865C2semidirect
3. C2×#89793C2semidirect
5. #11729686C2direct (N × H)
6. C2×#89342C2semidirect
7. C2×Q8⋊(C7⋊C8)C2semidirect
25. #89865C22semidirect
27. #89793C22semidirect
36. #89342C22semidirect
37. Q8⋊(C7⋊C8)C22semidirect
381. C2×C7#1638semidirect
428. C7C2×#1638semidirect
451. C2#11729686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#898652
C2×#897932
C2×#897802
#117296862
C2×#893422
C2×Q8⋊(C7⋊C8)2
C2×D4⋊(C7⋊C8)2
C2×#16387