Finite Group Databasebeta

← C2×#11741076 #14234285

Subgroups — C2×#11741076 #14234285

Order 1792.

1854 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×Q8×Dic14C211yesmaximal, characteristic, complemented
3#11741076C2116yesmaximal, direct product factor
4C2×#89863C211yesmaximal, characteristic, complemented
5C2×#89850C211yesmaximal, characteristic, complemented
6C2×#89835C212yesmaximal, complemented
7C2×#89837C212yesmaximal, complemented
8C2×#89796C211yesmaximal, characteristic, complemented
9C2×C4×#1142C211yesmaximal, characteristic
10C2×#89780C211yesmaximal, characteristic
11C2×#89779C212yesmaximal
12C2×#89790C212yesmaximal
13C7×#554—24no—
14C7×#554C2218yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C22×C7×C4⋊C4—21no—
23C2×Q8×Dic7—21no—
24C22×#1142C2211yescharacteristic
25Q8×Dic14—24no—
26Q8×Dic14C2218yescomplemented
27C2×#6451—21no—
28C2×#6451—22no—
29C2×#6432C2212yes—
30C2×#6432C2211yescharacteristic
31C2×C4×Dic14C2212yes—
32C2×C4×Dic14C2211yescharacteristic
33C2×#6406C2212yes—
34C2×#6406C2211yescharacteristic
35#89863C2218yescomplemented
36#89850C2218yescomplemented
37#89837C22116yescomplemented
38#89835C22116yescomplemented
39#89796C2218yescomplemented
40#89790C22116yes—
41C4×#1142C2218yes—
42#89780C2218yes—
43#89779C22116yes—
44C2×#6384C2211yescharacteristic
45C2×#6384C2212yes—
46C2×#6382C2211yescharacteristic
47C2×#6382C2212yes—
48C2×#6381C2212yes—
49C2×#6381C2212yes—
50C2×#6381C2212yes—
51C2×C4⋊(C7⋊C8)C2212yes—
52C2×C4⋊(C7⋊C8)C2211yescharacteristic
53C2×#6377C2212yes—
54C2×C4×C7⋊C8C2211yescharacteristic
55C2×#3385—71nomaximal, Sylow
56C22×C7×Q8—21no—
57C22×C7×Q8D411yescharacteristic
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7—22no—
61C22×Dic14—21no—
62C22×Dic14—22no—
63C22×Dic14C2311yescharacteristic
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8C2314yes—
66C7×#125—24no—
67C7×#125C2318yes—
68C7×C22⋊Q8—28no—
69C7×C22⋊Q8—216no—
70C4×C7×Q8D414yes—
71C4×C7×Q8C2314yes—
72C4×C7×D4—24no—
73C4×C7×D4—22no—
74C4×C7×D4C2318yes—
75C4×C7×D4C2314yes—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—28no—
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4D412yes—
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—22no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2312yes—
88Q8×Dic7—28no—
89C2×#1142Q818yescomplemented
90C2×#1142C2314yes—
91C2×C4⋊Dic7—22no—
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7C2312yes—
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—22no—
97C2×C4×Dic7—22no—
98C2×C4×Dic7—21no—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2312yes—
101C22×C7⋊C8C2311yescharacteristic
102#6451—28no—
103#6451—216no—
104#6432—24no—
105#6432—22no—
106#6432C2318yes—
107#6432C2314yes—
108C4×Dic14—24no—
109C4×Dic14—22no—
110C4×Dic14C2314yes—
111C4×Dic14C2318yes—
112#6406C2314yes—
113#6406C2318yes—
114#6384C2318yes—
115#6384C2314yes—
116#6382C2318yes—
117#6382C2314yes—
118#6381C2318yes—
119#6381C2318yes—
120#6381C2318yes—
121C4⋊(C7⋊C8)C2314yes—
122C4⋊(C7⋊C8)C2318yes—
123#6377C2318yes—
124C4×C7⋊C8C2314yes—
125C2×Q82—71no—
126C2×#554D711yescharacteristic
127#3385—716no—
128C2×#501—72no—
129C2×#500—71no—
130C2×#479—72no—
131C2×#477—71no—
132C2×#476—71no—
133C2×#474—72no—
134C2×#446—71no—
135C2×#440—72no—
136C2×C4×SD16—71no—
137C24×C7—21no—
138C2×C7×Q8—24no—
139C2×C7×Q8—24no—
140C2×C7×Q8—82no—
141C2×C7×Q8C2×D414yes—
142C2×C7×D4—21no—
143C2×C7×D4—22no—
144C2×C7×D4C2×Q814yes—
145C2×C7×D4C2412yes—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—24no—
149C22×C4×C7—28no—
150C22×C4×C7—44no—
151C22×C4×C7—84no—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—22no—
159C22×Dic7—21no—
160C2×Dic14—21no—
161C2×Dic14—22no—
162C2×Dic14—24no—
163C2×Dic14—28no—
164C2×Dic14—84no—
165C2×Dic14—82no—
166C2×Dic14C2×Q814yes—
167C2×Dic14C2412yes—
168C7×C4⋊C4—28no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4C2×D414yes—
174C7×C4⋊C4C2×D418yes—
175C7×C4⋊C4C2412yes—
176C7×C4⋊C4C2414yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—28no—
179C42×C7C2×D414yes—
180C42×C7C2×D412yes—
181C42×C7C2414yes—
182C42×C7C2412yes—
183#1142C2×Q8116yescomplemented
184C4⋊Dic7—22no—
185C4⋊Dic7—21no—
186C4⋊Dic7—28no—
187C4⋊Dic7—24no—
188C4⋊Dic7C2412yes—
189C4⋊Dic7C2414yes—
190Dic7⋊C4—24no—
191Dic7⋊C4—28no—
192C4×Dic7—28no—
193C4×Dic7—24no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—24no—
196C2×C7⋊C8C2×Q814yes—
197C2×C7⋊C8C2412yes—
198C2×C7⋊C8C2414yes—
199C2×C7⋊C8C2412yes—
200C22×SD16—71no—
201Q82—78no—
202Q82—144no—
203#554—24no—
204#554C2×D718yes—
205C2×C4⋊Q8—71no—
206C2×C4⋊Q8—72no—
207C2×C4⋊Q8—141no—
208C2×C4⋊Q8—142no—
209C2×C4⋊Q8C2×D711yescharacteristic
210C2×#125C2×D712yes—
211C2×C22⋊Q8—21no—
212C2×C22⋊Q8—22no—
213C2×C4×Q8—71no—
214C2×C4×Q8—72no—
215C2×C4×Q8—141no—
216C2×C4×Q8C2×D711yescharacteristic
217C2×C4×D4C2×D711yescharacteristic
218C2×C4×D4C2×D712yes—
219C22×C4⋊C4—21no—
220#501—716no—
221#500—78no—
222#479—716no—
223#477—78no—
224#476—78no—
225#474—716no—
226#446—78no—
227#440—716no—
228C4×SD16—78no—
229C2×#107—72no—
230C2×#107—72no—
231C2×#107—72no—
232C2×#106—72no—
233C2×#106—71no—
234C2×C4⋊C8—71no—
235C2×C4⋊C8—72no—
236C2×#103—72no—
237C2×#103—71no—
238C2×#102—71no—
239C2×#102—72no—
240C2×#97—72no—
241C2×C4×C8—71no—
242C23×C7—24no—
243C23×C7—22no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7C22×D411yescharacteristic
247C7×Q8—28no—
248C7×Q8—84no—
249C7×D4—24no—
250C7×D4—22no—
251C7×D4C22×Q814yes—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—42no—
258C2×C4×C7—44no—
259C2×C4×C7—48no—
260C2×C4×C7—88no—
261C2×C4×C7#13714yes—
262C2×C4×C7C22×D414yes—
263C2×C4×C7C22×D412yes—
264C2×C4×C7C22×D414yes—
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D412yes—
267C2×C4×C7C22×Q812yes—
268C2×C4×C72−1+412yes—
269C2×C4×C7C2511yescharacteristic
270C2×Dic7—24no—
271C2×Dic7—22no—
272C2×Dic7—24no—
273C2×Dic7—22no—
274C2×Dic7—42no—
275C2×Dic7—44no—
276Dic14—24no—
277Dic14—22no—
278Dic14—88no—
279Dic14—84no—
280Dic14C22×Q814yes—
281C7⋊C8—22no—
282C7⋊C8—24no—
283C7⋊C8C22×Q814yes—
284C22×Q8—21no—
285C22×Q8—71no—
286C22×Q8—141no—
287C22×Q8—142no—
288C22×Q8C22⋊D711yescharacteristic
289C22×D4C22×D711yescharacteristic
290C23×C4—21no—
291C23×C4—22no—
292C2×SD16—78no—
293C2×SD16—74no—
294C22×C8—71no—
295C22×C8—71no—
296C22×C8—72no—
297C4⋊Q8—22no—
298C4⋊Q8—78no—
299C4⋊Q8—74no—
300C4⋊Q8—148no—
301C4⋊Q8—1416no—
302C4⋊Q8—144no—
303C4⋊Q8—142no—
304C4⋊Q8C22×D714yes—
305#125—24no—
306#125C22×D718yes—
307C22⋊Q8—28no—
308C22⋊Q8—216no—
309C4×Q8—74no—
310C4×Q8—78no—
311C4×Q8—148no—
312C4×Q8—144no—
313C4×Q8—142no—
314C4×Q8C22⋊D714yes—
315C4×Q8C22×D714yes—
316C4×D4—24no—
317C4×D4—22no—
318C4×D4C22×D714yes—
319C4×D4C22×D718yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—72no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—142no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4C22⋊D711yescharacteristic
331C2×C4⋊C4C22⋊D712yes—
332C2×C4⋊C4C22×D711yescharacteristic
333C2×C4⋊C4C22×D712yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C42—142no—
337C2×C42—141no—
338C2×C42C22×D712yes—
339C2×C42C22×D711yescharacteristic
340#107—78no—
341#107—78no—
342#107—78no—
343#106—74no—
344#106—78no—
345C4⋊C8—78no—
346C4⋊C8—74no—
347#103—74no—
348#103—78no—
349#102—74no—
350#102—78no—
351#97—78no—
352C4×C8—74no—
353C22×C7—24no—
354C22×C7—48no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7D4×Q812yes—
358C22×C7C2×#13712yes—
359C22×C7#57612yes—
360C22×C7C23×D411yescharacteristic
361C4×C7—24no—
362C4×C7—24no—
363C4×C7—22no—
364C4×C7—42no—
365C4×C7—44no—
366C4×C7D4×Q812yes—
367C4×C7C2×#13714yes—
368C4×C7C23×Q811yescharacteristic
369C4×C7C2×2−1+411yescharacteristic
370Dic7—24no—
371Dic7—42no—
372Dic7—44no—
373C24—21no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—74no—
377C2×Q8—72no—
378C2×Q8—82no—
379C2×Q8—142no—
380C2×Q8—141no—
381C2×Q8—144no—
382C2×Q8—148no—
383C2×Q8—564no—
384C2×Q8—562no—
385C2×Q8C2×C22⋊D714yes—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4Q8×D714yes—
389C2×D4C23×D712yes—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—44no—
395C22×C4—84no—
396C22×C4—141no—
397C22×C4—142no—
398C22×C4—141no—
399C22×C4C2×C22⋊D712yes—
400C22×C4C2×C22⋊D711yescharacteristic
401C22×C4C2×C22⋊D712yes—
402C22×C4C2×C22⋊D711yescharacteristic
403C22×C4C23×D711yescharacteristic
404SD16—716no—
405C2×C8—74no—
406C2×C8—72no—
407C2×C8—72no—
408C2×C8—74no—
409C2×C8—142no—
410C2×C8—144no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—72no—
417C4⋊C4—74no—
418C4⋊C4—144no—
419C4⋊C4—148no—
420C4⋊C4—141no—
421C4⋊C4—142no—
422C4⋊C4—148no—
423C4⋊C4—144no—
424C4⋊C4C2×C22⋊D714yes—
425C4⋊C4C2×C22⋊D718yes—
426C4⋊C4C23×D712yes—
427C4⋊C4C23×D714yes—
428C22⋊C4—24no—
429C22⋊C4—28no—
430C42—148no—
431C42—144no—
432C42C2×C22⋊D714yes—
433C42C2×C22⋊D712yes—
434C42C23×D712yes—
435C42C23×D714yes—
436C2×C7—44no—
437C2×C7#338514yescomplemented
438C2×C7C2×D4×Q811yescharacteristic
439C2×C7C22×#13711yescharacteristic
440C2×C7C2×#57611yescharacteristic
441C23—24no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23C22×C22⋊D711yescharacteristic
446Q8—28no—
447Q8—74no—
448Q8—84no—
449Q8—144no—
450Q8—142no—
451Q8—568no—
452Q8—564no—
453D4—24no—
454D4—22no—
455D4C2×Q8×D714yes—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—88no—
465C2×C4—144no—
466C2×C4—142no—
467C2×C4—144no—
468C2×C4—142no—
469C2×C4—282no—
470C2×C4—284no—
471C2×C4#651414yes—
472C2×C4C2×Q8×D712yes—
473C2×C4#655112yes—
474C2×C4C22×C22⋊D714yes—
475C2×C4C22×C22⋊D712yes—
476C2×C4C22×C22⋊D714yes—
477C2×C4C22×C22⋊D711yescharacteristic
478C2×C4C22×C22⋊D712yes—
479C2×C4C24×D711yescharacteristic
480C8—74no—
481C8—142no—
482C8—144no—
483C7C2×#338511yescharacteristic, complemented, Sylow
484C22—24no—
485C22—42no—
486C22—42no—
487C22—48no—
488C22Q8×C22⋊D712yes—
489C22C2×#651412yes—
490C22#9052612yes—
491C22C23×C22⋊D711yescharacteristic
492C4—24no—
493C4—24no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—144no—
498C4—282no—
499C4—284no—
500C4Q8×C22⋊D712yes—
501C4C2×#651414yes—
502C4C22×Q8×D711yescharacteristic
503C4C2×#655111yescharacteristic
504C2—44no—
505C2#1174107614yesdirect product factor
506C2C2×Q8×C22⋊D711yescharacteristic
507C2C22×#651411yescharacteristic
508C2C2×#9052611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×#554D7
hypocenterC7C2×#3385
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic14C2semidirect
3. #11741076C2direct (N × H)
4. C2×#89863C2semidirect
5. C2×#89850C2semidirect
6. C2×#89835C2semidirect
7. C2×#89837C2semidirect
8. C2×#89796C2semidirect
26. Q8×Dic14C22semidirect
35. #89863C22semidirect
36. #89850C22semidirect
37. #89837C22semidirect
38. #89835C22semidirect
39. #89796C22semidirect
89. C2×#1142Q8semidirect
183. #1142C2×Q8semidirect
437. C2×C7#3385semidirect
483. C7C2×#3385semidirect
505. C2#11741076direct (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×Q8×Dic142
#117410762
C2×#898632
C2×#898502
C2×#898352
C2×#898372
C2×#897962
C2×C4×#11422
C2×#897802
C2×#897792
C2×#897902
C2×#33857