Finite Group Databasebeta

← C2×#11741122 #14233515

Subgroups — C2×#11741122 #14233515

Order 1792.

1842 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2C2×Q8×Dic14C211yesmaximal, characteristic, complemented
3#11741122C2116yesmaximal, direct product factor
4C2×#89862C211yesmaximal, characteristic, complemented
5C2×#89853C211yesmaximal, characteristic, complemented
6C2×#89826C212yesmaximal
7C2×#89827C212yesmaximal
8C2×#89796C211yesmaximal, characteristic, complemented
9C2×#89792C212yesmaximal, complemented
10C2×#89802C212yesmaximal, complemented
11C2×C4×#1142C211yesmaximal, characteristic
12C2×#89780C211yesmaximal, characteristic
13C7×#556—24no—
14C7×#556C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#126—22no—
17C2×C7×#124C2212yes—
18C2×C7×C22⋊Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C7×#117—21no—
23C2×Q8×Dic7—21no—
24C22×#1144C2212yes—
25C22×#1142C2211yescharacteristic
26Q8×Dic14—24no—
27Q8×Dic14C2218yescomplemented
28C2×#6451—21no—
29C2×#6451—22no—
30C2×#6432C2211yescharacteristic
31C2×#6432C2212yes—
32C2×C4×Dic14C2211yescharacteristic
33C2×C4×Dic14C2212yes—
34C2×#6409C2212yes—
35C2×#6409C2212yes—
36C2×#6409C2212yes—
37C2×#6406C2212yes—
38C2×#6406C2211yescharacteristic
39#89862C2218yescomplemented
40#89853C2218yescomplemented
41#89827C22116yes—
42#89826C22116yes—
43#89802C22116yescomplemented
44#89796C2218yescomplemented
45#89792C22116yescomplemented
46C4×#1142C2218yes—
47#89780C2218yes—
48C2×#6384C2211yescharacteristic
49C2×#6382C2212yes—
50C2×#6382C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52C2×C4⋊(C7⋊C8)C2212yes—
53C2×#6377C2212yes—
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3369—71nomaximal, Sylow
56C22×C7×Q8D411yescharacteristic
57C22×C7×Q8C2312yes—
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C22×Dic14—22no—
61C22×Dic14—21no—
62C22×Dic14C2311yescharacteristic
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8C2314yes—
65C7×#126—216no—
66C7×#124—24no—
67C7×#124C2318yes—
68C7×C22⋊Q8—28no—
69C4×C7×Q8—24no—
70C4×C7×Q8D414yes—
71C4×C7×Q8C2318yes—
72C4×C7×Q8C2314yes—
73C4×C7×D4—22no—
74C4×C7×D4C2314yes—
75C7×#117—28no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4D412yes—
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—22no—
85C2×C42×C7—22no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2312yes—
88Q8×Dic7—28no—
89C2×#1144—28no—
90C2×#1144C2318yes—
91C2×#1142—24no—
92C2×#1142C2314yes—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2312yes—
97C2×Dic7⋊C4—22no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—22no—
100C2×C4×Dic7—21no—
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2312yes—
104#6451—28no—
105#6451—216no—
106#6432—22no—
107#6432—24no—
108#6432C2314yes—
109#6432C2318yes—
110C4×Dic14—22no—
111C4×Dic14—24no—
112C4×Dic14C2314yes—
113C4×Dic14C2318yes—
114#6409C2318yes—
115#6409C2318yes—
116#6409C2318yes—
117#6406C2318yes—
118#6406C2314yes—
119#6384C2314yes—
120#6382C2318yes—
121#6382C2314yes—
122C4⋊(C7⋊C8)C2314yes—
123C4⋊(C7⋊C8)C2318yes—
124#6377C2318yes—
125C4×C7⋊C8C2314yes—
126C2×Q82—71no—
127C2×#556D711yescharacteristic
128#3369—716no—
129C2×#489—72no—
130C2×#487—71no—
131C2×#475—72no—
132C2×#476—71no—
133C2×#464—72no—
134C2×#462—71no—
135C2×#446—71no—
136C2×#441—72no—
137C2×C4×SD16—71no—
138C24×C7—21no—
139C2×C7×Q8—24no—
140C2×C7×Q8—22no—
141C2×C7×Q8—24no—
142C2×C7×Q8—24no—
143C2×C7×Q8C2×D414yes—
144C2×C7×Q8C2414yes—
145C2×C7×D4—21no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4C2412yes—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—44no—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C2×Dic14—21no—
162C2×Dic14—22no—
163C2×Dic14—28no—
164C2×Dic14—24no—
165C2×Dic14—84no—
166C2×Dic14—82no—
167C2×Dic14#4114yes—
168C2×Dic14C2412yes—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4C2×D414yes—
174C7×C4⋊C4C2×D418yes—
175C7×C4⋊C4C2412yes—
176C7×C22⋊C4—28no—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—28no—
179C42×C7—28no—
180C42×C7C2×D412yes—
181C42×C7C2×D414yes—
182C42×C7C2412yes—
183C42×C7C2414yes—
184#1144—216no—
185#1142—28no—
186C4⋊Dic7—21no—
187C4⋊Dic7—22no—
188C4⋊Dic7—28no—
189C4⋊Dic7—24no—
190C4⋊Dic7C2414yes—
191C4⋊Dic7C2412yes—
192Dic7⋊C4—28no—
193Dic7⋊C4—24no—
194C4×Dic7—28no—
195C4×Dic7—24no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—24no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8C2412yes—
200C2×C7⋊C8C2414yes—
201C2×C7⋊C8C2412yes—
202C22×Q16—72no—
203C22×SD16—71no—
204Q82—78no—
205Q82—144no—
206#556—24no—
207#556C2×D718yes—
208C2×C4⋊Q8—72no—
209C2×C4⋊Q8—71no—
210C2×C4⋊Q8—141no—
211C2×C4⋊Q8—142no—
212C2×C4⋊Q8C2×D711yescharacteristic
213C2×#126—22no—
214C2×#124C2×D712yes—
215C2×C22⋊Q8—21no—
216C2×C4×Q8—71no—
217C2×C4×Q8—72no—
218C2×C4×Q8—141no—
219C2×C4×Q8C2×D712yes—
220C2×C4×Q8C2×D711yescharacteristic
221C2×C4×D4C2×D711yescharacteristic
222C2×#117—21no—
223#489—716no—
224#487—78no—
225#476—78no—
226#475—716no—
227#464—716no—
228#462—78no—
229#446—78no—
230#441—716no—
231C4×SD16—78no—
232C2×#106—71no—
233C2×#106—72no—
234C2×C4⋊C8—72no—
235C2×C4⋊C8—71no—
236C2×#103—72no—
237C2×#103—72no—
238C2×#103—72no—
239C2×#103—71no—
240C2×#102—71no—
241C2×#102—72no—
242C2×#97—72no—
243C2×C4×C8—71no—
244C23×C7—22no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7C22×D411yescharacteristic
249C7×Q8—28no—
250C7×Q8—24no—
251C7×Q8—24no—
252C7×Q8—44no—
253C7×D4—22no—
254C7×D4—22no—
255C7×D4—42no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—44no—
262C2×C4×C7—42no—
263C2×C4×C7—48no—
264C2×C4×C7#13714yes—
265C2×C4×C7C22×D414yes—
266C2×C4×C7C22×D414yes—
267C2×C4×C7C22×D412yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×D412yes—
270C2×C4×C7C2×#4112yes—
271C2×C4×C72−1+412yes—
272C2×C4×C7C2511yescharacteristic
273C2×Dic7—24no—
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—22no—
277C2×Dic7—42no—
278C2×Dic7—44no—
279Dic14—24no—
280Dic14—22no—
281Dic14—88no—
282Dic14—84no—
283Dic14C2×#4114yes—
284C7⋊C8—22no—
285C7⋊C8—24no—
286C7⋊C8—22no—
287C22×Q8—71no—
288C22×Q8—142no—
289C22×Q8—141no—
290C22×Q8C22⋊D711yescharacteristic
291C22×Q8C22×D712yes—
292C22×D4C22×D711yescharacteristic
293C23×C4—21no—
294C2×Q16—78no—
295C2×Q16—148no—
296C2×SD16—74no—
297C2×SD16—144no—
298C22×C8—71no—
299C22×C8—72no—
300C22×C8—71no—
301C4⋊Q8—22no—
302C4⋊Q8—78no—
303C4⋊Q8—74no—
304C4⋊Q8—148no—
305C4⋊Q8—1416no—
306C4⋊Q8—142no—
307C4⋊Q8—144no—
308C4⋊Q8C22×D714yes—
309#126—216no—
310#124—24no—
311#124C22×D718yes—
312C22⋊Q8—28no—
313C4×Q8—24no—
314C4×Q8—74no—
315C4×Q8—78no—
316C4×Q8—148no—
317C4×Q8—142no—
318C4×Q8—144no—
319C4×Q8C22⋊D714yes—
320C4×Q8C22×D718yes—
321C4×Q8C22×D714yes—
322C4×D4—22no—
323C4×D4C22×D714yes—
324#117—28no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—72no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22⋊D711yescharacteristic
335C2×C4⋊C4C22⋊D712yes—
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C42—22no—
341C2×C42—142no—
342C2×C42—141no—
343C2×C42C22×D711yescharacteristic
344C2×C42C22×D712yes—
345#106—74no—
346#106—78no—
347C4⋊C8—74no—
348C4⋊C8—78no—
349#103—78no—
350#103—78no—
351#103—78no—
352#103—74no—
353#102—74no—
354#102—78no—
355#97—78no—
356C4×C8—74no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—84no—
361C22×C7#54812yes—
362C22×C7C2×#13712yes—
363C22×C7#57712yes—
364C22×C7C23×D411yescharacteristic
365C4×C7—24no—
366C4×C7—24no—
367C4×C7—22no—
368C4×C7—44no—
369C4×C7—42no—
370C4×C7#54812yes—
371C4×C7C2×#13714yes—
372C4×C7C22×#4111yescharacteristic
373C4×C7C2×2−1+411yescharacteristic
374Dic7—24no—
375Dic7—42no—
376Dic7—44no—
377C24—21no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—74no—
383C2×Q8—72no—
384C2×Q8—142no—
385C2×Q8—141no—
386C2×Q8—148no—
387C2×Q8—144no—
388C2×Q8—564no—
389C2×Q8—562no—
390C2×Q8C2×C22⋊D714yes—
391C2×Q8C23×D714yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4C23×D712yes—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—44no—
400C22×C4—141no—
401C22×C4—142no—
402C22×C4—141no—
403C22×C4C2×C22⋊D712yes—
404C22×C4C2×C22⋊D712yes—
405C22×C4C2×C22⋊D711yescharacteristic
406C22×C4C2×C22⋊D711yescharacteristic
407C22×C4C23×D711yescharacteristic
408Q16—1416no—
409SD16—148no—
410C2×C8—72no—
411C2×C8—72no—
412C2×C8—74no—
413C2×C8—142no—
414C2×C8—144no—
415C2×C8—142no—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—21no—
420C4⋊C4—72no—
421C4⋊C4—74no—
422C4⋊C4—148no—
423C4⋊C4—144no—
424C4⋊C4—141no—
425C4⋊C4—142no—
426C4⋊C4—148no—
427C4⋊C4—144no—
428C4⋊C4C2×C22⋊D714yes—
429C4⋊C4C2×C22⋊D718yes—
430C4⋊C4C23×D712yes—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—28no—
434C42—28no—
435C42—148no—
436C42—144no—
437C42C2×C22⋊D712yes—
438C42C2×C22⋊D714yes—
439C42C23×D712yes—
440C42C23×D714yes—
441C2×C7—82no—
442C2×C7#336914yescomplemented
443C2×C7C2×#54811yescharacteristic
444C2×C7C22×#13711yescharacteristic
445C2×C7C2×#57711yescharacteristic
446C23—22no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23C22×C22⋊D711yescharacteristic
451Q8—28no—
452Q8—24no—
453Q8—24no—
454Q8—44no—
455Q8—74no—
456Q8—144no—
457Q8—142no—
458Q8—568no—
459Q8—564no—
460D4—22no—
461D4—22no—
462D4—42no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—144no—
472C2×C4—142no—
473C2×C4—144no—
474C2×C4—142no—
475C2×C4—284no—
476C2×C4—282no—
477C2×C4#651414yes—
478C2×C4D7×#4112yes—
479C2×C4#655412yes—
480C2×C4C22×C22⋊D714yes—
481C2×C4C22×C22⋊D714yes—
482C2×C4C22×C22⋊D712yes—
483C2×C4C22×C22⋊D712yes—
484C2×C4C22×C22⋊D711yescharacteristic
485C2×C4C24×D711yescharacteristic
486C8—142no—
487C8—144no—
488C8—142no—
489C7C2×#336911yescharacteristic, complemented, Sylow
490C22—42no—
491C22—42no—
492C22—42no—
493C22—84no—
494C22C2×#651412yes—
495C22#9052212yes—
496C22#9052812yes—
497C22C23×C22⋊D711yescharacteristic
498C4—24no—
499C4—24no—
500C4—22no—
501C4—42no—
502C4—44no—
503C4—144no—
504C4—284no—
505C4—282no—
506C4C2×#651414yes—
507C4#9052212yes—
508C4C2×D7×#4111yescharacteristic
509C4C2×#655411yescharacteristic
510C2—82no—
511C2#1174112214yesdirect product factor
512C2C22×#651411yescharacteristic
513C2C2×#9052211yescharacteristic
514C2C2×#9052811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×#556D7
hypocenterC7C2×#3369
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic14C2semidirect
3. #11741122C2direct (N × H)
4. C2×#89862C2semidirect
5. C2×#89853C2semidirect
8. C2×#89796C2semidirect
9. C2×#89792C2semidirect
10. C2×#89802C2semidirect
27. Q8×Dic14C22semidirect
39. #89862C22semidirect
40. #89853C22semidirect
43. #89802C22semidirect
44. #89796C22semidirect
45. #89792C22semidirect
442. C2×C7#3369semidirect
489. C7C2×#3369semidirect
511. C2#11741122direct (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×Q8×Dic142
#117411222
C2×#898622
C2×#898532
C2×#898262
C2×#898272
C2×#897962
C2×#897922
C2×#898022
C2×C4×#11422
C2×#897802
C2×#33697